FRIEDRICH-SCHILLER-

UNIVERSITÄT JENA

JENAER SCHRIFTENZUR

MATHEMATIK UND INFORMATIKEingang : 23.11.2005 Math/Inf/11/05 Als Manuskript gedruckt

Entropy Estimation Methods in HRV Analysis of Patientswith Myocardial Infarction

S. LAU, J. HAUEISEN, E. G. SCHUKAT-TALAMAZZINI, A. VOSS, M. GOERNIG, U. LEDERAND H.-R. FIGULLA

Abstract

Heart rate variability (HRV) is a marker for autonomous activity in the heart. A key application of HRVmeasures is the stratification of mortality risk after myocardial infarction. The information entropy is a promisingmeasure of HRV. Our hypothesis is that the information entropy of HRV, a non-linear approach, is a suitablemeasure for this. As a first step, we aimed at evaluating the effect of myocardial infarction on the entropy.

Our method was to compare the entropy to standard HRV parameters. Essentially, one multivariate classifica-tion rule was generated based on existing HRV measures and one based on existing and new entropy measures.The gain in classification accuracy was then an evaluation criterion. The classification rules were expressed asdecision trees. The simplicity and parameter choice of the augmented tree was the second criterion. Additionally,five entropy estimation techniques were compared in terms of estimation accuracy and discrimination strength.

A key finding is that the entropy is reduced in patients with myocardial infarction with very high significance.Additionally, a simple threshold of the meanNN-normalised entropy outperforms the best multivariate standards-based infarct classifier by 5-10%. The statistical and compression-based entropy estimations are with a correlationof >94% highly consistent and thus reliable. The entropy based on Burrows-Wheeler compression, implementedin Bzip2, yields the best entropy estimation for infarct analysis purposes.Keywords: Heart rate variability (HRV), myocardial infarction, entropy, entropy estimation, Burrows-Wheeler compression, Bzip2, Ngram, LZ77, Gzip, adaptive linear regression, classification, decision tree

2000 Math. Subject Classification: 62P10, 92C55, 94A17

Contents

1 Introduction 41.1 Heart and Myocardial Infarction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.1 Physiology of the Heart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Regulation of Heart Activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.3 Myocardial Infarction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Heart Rate Variability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.1 HRV Signal and Influencing Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.2 HRV Analysis Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.3 HRV after Myocardial Infarction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3 Entropy and Estimation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Definition and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.2 Estimation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Recent Entropy-Approaches in HRV Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.1 Approximate Entropy (ApEn) by Pincus . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.2 Multiscale Entropy (MSE) by Costa et al. . . . . . . . . . . . . . . . . . . . . . . . . . . 151.4.3 Renormalised Entropy (RE) by Wessel et al. . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Research Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Materials and Methods 182.1 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2 Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Template Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.2 Artifact Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.3 Length Normalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Entropy Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.1 NGRAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 ALRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.3 LZ77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.4 GZIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.5 BZIP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Alternative HRV Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Data Set Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Population Subgrouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.8 Infarct Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.9 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Results 283.1 Entropy Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1.1 LZ77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.2 NGRAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 ALRM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.4 GZIP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.5 BZIP2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.1.6 Cut, Interp and Diff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.7 Convergence with Signal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.2 Population Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Statistical Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.1 Infarct Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3.2 Infarct Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.3 Risk Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.3.4 Medication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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3.4 Infarct Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.1 Statistical Classifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.2 Without Grouping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4.3 Within Age Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.4 Within Age and Gender Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4.5 Within Age Groups and Stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Investigation of Infarct Stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Discussion 704.1 Interpretation of Classification Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.1.1 Nature of Classification Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.2 Standards-based Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.1.3 Entropy-based Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.1.4 Partial Classifier for Risk Stratification . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

4.2 Entropy as HRV Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3 Final Infarct Classification Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5 Conclusion 78

Acknowledgments 78

References 81

Contacts 82

List of Figures 84

List of Tables 85

A Formal Concept Analysis Basics 86A.1 Context and Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.2 Concept Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A.3 Conceptual Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

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1 Introduction

The heart is one of the most central organs of the human body. It pumps the blood through all parts of the body andthus supplies nutrients and oxygen, removes metabolic products, transports messenger substances, etc.. Its activityis regulated by a number of control mechanisms. A fundamental measure of the heart activity is the heart rate. Thevariability of the heart rate changes with the condition of the heart.

Myocardial infarction is an acute, life-threatening condition of the heart. Reliable diagnostics are important forthe immediate treatment as well as for the stratification of the risk of a future life-threatening condition after aninfarct. The heart rate variability (HRV) is one of these diagnostic measures.

The HRV is in the focus of this study. A new approach to quantifying the HRV, the information entropy, isevaluated and compared to existing standard measures in the case of myocardial infarction.

1.1 Heart and Myocardial Infarction

1.1.1 Physiology of the Heart

Structure The purpose of the heart is to pump blood through the body and the lungs. Because the systemiccirculation and the pulmonary circulation are separated, the heart consists of two simultaneous pumps: the left andthe right heart. Figure 1 shows a diagram of the anatomical parts. Each side consists of an atrial chamber, in whichblood accumulates, and a ventricle, which pumps the blood into the respective circulation. [16]

Figure 1: Structure of the heart and course of blood flow through the heart chambers [16, p. 99]

Cardiac Muscle The heart muscle of the atrium and ventricle is very similar to the skelet muscle, but its con-traction is slower. Additionally, the cardiac muscle is a syncytium. The muscle cells are connected through gapjunctions that allow ions to pass relatively freely. The action potential is therefore transmitted from one cell to thenext. The atrial muscle is separated from the ventricular one by fibrous tissue around the valvular openings. Thus,the atria and ventricles respectively act as functional syncytia. They are only connected through the atrioventricularbundle, which is shown in Figure 2. [16]

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Figure 2: The sinus node and the Purkinje system of the heart, showing also the A-V node, the atrial internodalpathways, and the ventricular bundle branches. [16, p. 112]

Excitatory System The excitatory system of the heart consists of specialised conductive muscle fibres. They onlycontain few contractile fibrils. Instead they are able to transmit excitation faster. This allows the cardiac muscle tocontract at once. Additionally, most parts of the excitatory system are capable of rhythmic self-excitation at respec-tive rates. This causes the heart to beat, even under pathological conditions. The reason is that the cell membraneleaks ions, which depolarises the membrane continuously and rhythmically causes action potentials. [16]

The parts of the excitatory system are shown in Figure 2. The most important is the sinus node, located at thesuperior lateral wall of the right atrium. The impulse of the sinus node is conducted by the internodal pathways.This causes the atria to contract. The atrioventricular (AV) node picks up the impulse, delays it and passes it ontothe ventricles. The AV bundle conducts the impulse from atria to ventricles. The left and right Purkinje fibresdistribute the impulse to all parts of the ventricle. This causes the ventricles to contract at once and with a smalldelay to the atria. [16]

Control of Excitation The physiological pace maker of the heart is the sinus node. It generates impulses at arate of 60-80/min. This impulse is conducted through the excitatory system and causes the coordinated contractionof the cardiac muscle. The rhythm of the heart is then a ’sinus rhythm’. [10]

The AV node is also capable of self-excitation, but at a rate of 40-60/min. The impulse generated by the sinusnode normally discharges the AV node early. Thus, the AV node is only activated as pace maker if there is noimpulse from the sinus node or similar sources. The excitatory system of the ventricles self-excites at rates of20-40 beats per minute (bpm) if it is not discharged by another impulse. [10]

Electric and Magnetic Measurements The excitation of the heart is accomplished by ions flowing along themuscle fibre, thus causing an electric field and a corresponding magnetic field. The electrocardiogram (ECG) is aprojection of the electric field in one or several directions.Figure 3 shows the essential shape of an ECG and itsmapping to the conduction steps. The P wave is caused by the spread of stimulus in the atria. The QRS complexrepresents the spread across the ventricles and the T wave marks the repolarisation of the ventricles.

The magnetocardiogram (MCG) records the magnetic field, usually in an array of locations on the body surfaceabove the heart. Figure 4 shows a sample recording.

The ECG and MCG are used to assess the functional condition of the heart. In this study however only the heartrate is the focus. It can be extracted from an ECG or MCG using template matching. A PQRST or QRS template ischosen and compared to the signal. The matches are recorded as beats. This leads to highly accurate beat intervallengths.

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Figure 3: Relation between excitation and ECG with sinus rhythm [46, p. 201]

Figure 4: 31-channel MCG of 1000 ms. The sensors are arranged on the body surface above the heart. The feet arein the direction to the left.

1.1.2 Regulation of Heart Activity

There are three main aspects of heart activity that can be regulated: frequency (chronotropic), contraction strength(inotropic) and contraction speed (dromotropic). The regulation is achieved with the following four mechanisms [41]:

Frank-Starling-Mechanism The Frank-Starling-Mechanism is an intrinsic regulation of the heart. If more bloodarrives at the heart, the heart increases the cardiac output. This prevents damming of blood in the veins. Even ifthe peripheral vascular resistance changes, the cardiac output is maintained. [41] The explanation is that the bloodstretches the cardiac muscle, so that a better degree of interdigitation between the actin and myosin filaments isachieved, which increases the force. [16, p. 112]

Sympathetic and Vagal Innervation The autonomous nervous system allows the heart to adapt to various con-ditions, such as physical exercise. Figure 5 shows where the nerves are located. In general the sympathicus has apositive chronotropic, inotropic and dromotropic effect, while the parasympathicus has a negative one. The under-lying mechanism is to increase the ion conductivity of the cell membranes. [10]

The sympathetic nerves reach all parts of the cardiac muscle, particularly the ventricles. Their transmitter isNoradrenaline which binds to β-receptors on the membrane of the cardiac muscle. This causes the Ca2+ con-

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Figure 5: Cardiac nerves [16, p. 107]

ductivity to increase. This depolarises the membrane additionally. In the sinus node this causes the membrane todepolarise faster. Thus, the heart rate increases (positive chronotropic effect). The same process also causes theinotropic and dromotropic effect. [10]

At rest the sympathetic nerves discharge continuously at a slow rate causing a 30% increase in cardiac outputcompared to no sympathetic discharge. The sympathetic stimulation can therefore also be inhibited to decreasecardiac activity. [10]

A clinical use of the β-receptors, such as in case of myocardial infarction, is to block them. This reducesexcessive sympathetic stimulation. Such substances are also called β-blockers. [10]

The parasympathetic nerves reach the sinus node, AV node and atria, not the ventricles. Their transmitter isacetylcholine. It causes the K+ conductivity of the cell membranes to increase. This hyperpolarises the mem-branes. In the sinus node this means that the time to depolarise the membrane up to the activation threshold is in-creased. The heart rate decreases (negative chronotropic effect). The vagus can reduce the heart rate to 20-30/min.It also has a negative inotropic and dromotropic effect. [10]

Hormones The thyroid hormone trijodthyroxin has a positive inotropic effect. It increases the synthesis of V1-isomyosin, which has the highest myosin-ATPase activity. Thyroid hyperfunction therefore has a positive inotropicand chronotropic effect, while hypofunction causes the opposite effects. [41]

Cardiac Glycosides Cardiac glycosides, such as digoxin, digitoxin and ouabain, have a positive inotropic effect.They inhibit the Na+/K+-ATPase in the membranes. This causes Ca2+ to accumulate in the cells, which is thenavailable for the contraction process. Cardiac glycosides also have a negative chronotropic effect. Presumably, thesympathetic tone decreases because of the improved cardiac function. [35] [41]

1.1.3 Myocardial Infarction

Definition Myocardial infarction is an absolute persistent ischemia of an area of the myocardium. Its pathologic-anatomic correlative is a coagulation necrosis. It is most commonly caused by a rupture of arteriosclerotic plaqueor an occlusive thrombus. [39]

Factors that promote infarcts are hypertension, hypercholesterolemia, smoking and diabetes mellitus. Womenbetween 35 and 55 years have a hormonal protection through estrogen and thus have a much lower risk of infarc-tion. [39]

Three types of penetration of the layers of the heart wall are differentiated [54]:

1. transmural infarction extends from the endocard to the epicard,2. subepicardial infarction is limited to the inner half of the myocardium,

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3. subendocardial infarction is limited to the outer half of the myocardium.

Six locations of coronary occlusion have been identified. These are pointed out in Figure 6. Each of the oc-cluded arteries has a myocardial area that it supplies. This area is affected. Table 1 maps arteries to areas ofischemia. In 95% of all cases only the left ventricle is affected. Isolated right ventricle infarcts are rare. [39]

Figure 6: Preferred localisation of vascular obliteration for different infarct types [39, p. 90]

No Infarct Location Frequency1 large anteroapical infarction (ventroapical)

50%2 medium anterior infarction (supraapikal, anteroseptal)3 no anterior infarction4 lateral infarction (apikolateral, basolateral) 20%5 posterior infarction (posteroapikal, posteroseptal, posterobasal) 30%6 large combined anterior-posterior infarction, large septum infarction rare

Table 1: Locations of myocardial infarction and frequency for balanced coronary blood supply according toFigure 6 [39]

Etiology Myocardial infarction can also be defined as a complication of coronary heart disease (CHD). [54]CHD comprises all effects of coronary arterioscleosis, such as angina pectoris. The arteriosclerosis reduces thewidth of the coronary vessels by as much as 60%. This leads to a progressively reduced blood flow into areasof the myocardium. CHD causes arrhythmias, myocardial fibrosis, micronecrosis and eventually infarction. Therupture of arteriosclerotic plaque causes vascular occlusions and promotes the development of thrombi, which canboth cause acute infarction. [54] [15]

Other causes of myocardial infarction are spasms of the coronary arteries, a block of the oxygen diffusion fromartery to the myocardial cells, intramural placement of extramural arteries, kinking of arteries and embolisms. [54]

Pathogenesis The ischemia can develop suddenly or gradually. It is accompanied by an angina pectoris. Theischemia leads to a lack of oxygen and nutrients in the supply area of the obliterated artery. The lack of intracellularATP causes Ca2+ to accumulate in the muscle fibres. This causes a contraction which increases the rigidity of theventricle. This causes the enddiastolic pressure to increase in the left ventricle and the intramural vessels to beconstricted. [54]

The factors that determine the infarct size are the size of the supply area of the obliterated artery, the duration ofthe obliteration, the demand of oxygen at the time of infarct, the degree of blood flow and the ischemic tolerance.Due to preconditioning through CHD the ischemic tolerance can be high. [54]

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In reaction to the ischemia anastomoses and collaterals supply blood to the edge zone of the infarction area.But this is not sufficient to supply the whole infarcted tissue. In the centre of the infarct a necrosis develops. Thistissue is replaced by fibrous scar tissue in the following weeks. A short-term complication is an apposition infarct,where upstream of the occlusion a new thrombus develops which extends the necrotic focus. [54]

Diagnostics The key diagnostic tool is the ECG. The shape of the ECG shows characteristic changes in differentstages of the infarct, such as in Figure 7. However, 10% of all cases show unspecific ECG changes. Prominentsymptoms are pain that can radiate into the left arm or chest and cardiac insufficiency. [54] Additionally, in thefirst 12 hours an increase of the key enzymes creatine kinase (CK), glutamat-oxalacetat-transaminase (GOT) andlactate dehydrogenase (LDH) indicates an infarct. [23]

Figure 7: ECG characteristics of myocardial infarction [46, p. 199]

Ultrasonic cardiography (UCG) can be used to examine the heart and to determine the ejection fraction andventricle measurements. Myocardial szintigraphy and coronary angiography can be used to display the location ofthe obliteration and the extent of the ischemia. [15]

Treatment The acute treatment goal is the prevention of necrosis of the myocardium. Depending on the causeof the infarct, a vascular dilator or thrombolyticum can achieve this. [54]

To counteract the ischemia, nitrogylcerin can be used. It widens the periphery vessels and thus reduces thecardiac work load as well as the enddiastolic pressure. This reduces the oxygen consumption. Other symptoms,such as left or right ventricular insufficiency, arrhythmias, tachy- or bradycardia or hyper- or hypotension have tobe considered and treated respectively. [23]

1.2 Heart Rate Variability Analysis

The heart rate variability (HRV) is another important diagnostic tool. It is the focus of this study. Even withoutexternal demands, such as exercise, the heart rate (HR) continuously varies on a small scale. This is due to thesympathetic and vagal tone. The heart rate variability (HRV) is therefore a promising marker of the activity of theautonomous nervous system.

One of the key observations was made by Wolf et al. [51] in 1977. It was found that a reduced HRV is relatedto a higher risk of post-infarction mortality. Since then a wide range of HRV analysis methods emerged. In 1996 atask force of the European Society of Cardiology and the North American Society of Pacing and Electrophysiologydeveloped a standard of measurements and physiological correlates of HRV and clinical applications. The two keyapplications are the risk stratification after myocardial infarction and the assessment of diabetic neuropathy. [33]

1.2.1 HRV Signal and Influencing Factors

Although HRV means heart rate variability, the signal that is analysed is not the heart rate directly but the sequenceof beat-to-beat intervals, also called normal-to-normal (NN) intervals. Figure 8 shows such tachograms from ahealthy and a post-infarction recording matched in age and gender. The two other representations of the HRVsignal are the successive differences of intervals and the interpolated continuous-time interval length.

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0 100 200 300 400 500 600 700600

800

1000

Tachogram of healthy individual

beat interval

inte

rval

leng

th in

ms

0 100 200 300 400 500 600 700 800600

800

1000

Tachogram of post−infarction patient

beat interval

inte

rval

leng

th in

ms

Figure 8: Sample tachogram of healthy and post-infarct individual

Sympatho-vagal Balance Under physiological conditions the heart rate variability is the result of a constantbalance of the vagal and sympathetic tone. The intrinsic heart rate of the sinus node is 100-120 bpm. That meanswithout innervation, e.g. in transplanted hearts, the heart would beat at this rate and with virtually no variability.While vagal stimulation reduces the heart rate, sympathetic stimulation increases it. At rest the vagal stimulationdominates and thus the normal heart rate lies around to 60 bpm. [17]

Latencies The latency of vagal effects is very short with a maximum of 400 ms. The vagus nerve thereforeadapts the interval length on a beat-to-beat basis. The effect of one stimulus only lasts at most 5 seconds. The vagalstrength is modulated through the frequency of stimuli. In fact, the beat-to-beat interval length increases linearlywith the frequency of vagal stimuli, which Figure 9. [17]

Figure 9: Chronotropic responses to graded efferent vagal stimulation [17, p. 5]

The latency of the sympathetic effects is much higher. Between stimulus and the start of effect lie at most 5seconds. The effect progressively increases and reaches a steady level in 20-30 s. The relation between sympa-thetic frequency and interval length is approximately linear. The sympathetic effect is much slower than the vagalone. [17]

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Reflexes The autonomous nervous system mediates a number of reflexes which jointly moderate the sympatho-vagal balance. The most important ones are the following. Table 2 summarises the causes for tachycardia andbradycardia [17].

• Baroreceptor Reflex Baroreceptors situated in the adventitia of the carotid sinus and the aortic arch areexcited if increasing blood pressure stretches the vessels. This increases vagal and decreases sympatheticactivity to reduce the blood pressure.

• Chemoreceptor Reflex Chemoreceptors react to hypoxia, hypercapnia and acidemia in the arteries. Theyare situated in the carotid and aortic sections. An excitation results in an increased rate and depth of in-spiration. The carotid receptors cause the HR to decrease, while the aortic ones increase HR. However, theinspiration indirectly influences the HR and thus masks the direct effects.

• Atrial Receptor Reflex Receptors are excited if a high atrial volume stretches the atrial wall. This in-creases the sympathetic tone and thus the HR. Additionally, the urine flow is increased to reduce the bloodvolume.

• Coronary Chemoreflex Chemicals, such as phenyl diguanide, capsaicin and substances resulting frommyocardial ischemia, including bradykinin and prostaglandins, are detected by receptors. This reflex causesbradycardia and hypotension.

• Others Other reflexes include the lungs and abdominal viscera. For example, lung inflation stimulatesairway stretch receptors. This increases the HR. However, hyperinflation, pulmonary congestion, embolismsand chemicals cause bradycardia. The abdominal viscera can cause increased sympathetic activity in casevenous congestion.

Reflexes causing bradycardia Reflexes causing tachycardiaBaroreceptorsCarotid chemoreceptorsCoronary chemoreflex (Bezold-Jarisch)Lung hyperinflation

Atrial receptorsAortic chemoreceptorsMuscle receptorsLung inflation (moderate)

Table 2: Reflexes influencing heart rate [17, p. 7]

In addition to reflexes, the HR also follows a circadian rhythm. Therefore short-term recordings of 5-10 minand long-term recordings of 24 h have to be commonly distinguished.

1.2.2 HRV Analysis Methods

The methods of HRV analysis can be categorised into four groups: time domain measures, frequency domainmeasures, rhythm analysis and non-linear methods. [33]

Time Domain The simplest measures that can be applied to the tachogram are statistical ones, for example themean interval length (meanNN) and the standard deviation (sdNN). SdaNN is the standard deviation of intervalsaveraged over 5 min sections. It only reflects cyclical components with a length of more than 5 min. SdNN indexis the mean of 5 min sdNN values. It only reflects cyclical components of less than 5 min length. The parameterpNN50 is the proportion of interval differences of less than 50 ms. It is a marker for vagal activity. [33]

Geometrical measures are based on the density distribution of NN intervals. One approach is to use somegeometrical feature of the histogram is selected, such as the width at a certain height. Alternatively, the shape ofthe histogram can be approximated by a geometrical shape, such as a triangle, and a measure of that can be used.A third approach is to classify the shape of the histogram into different categories. [33]

Frequency Domain The spectral analysis of the tachogram was a breakthrough in HRV analysis, because thefactors influencing the HR have different frequencies. In particular, the vagal and sympathetic stimulation appearto have different frequencies. The frequency domain is divided into bands, which isolate typical peaks. For short-term recordings, the high band (HF) lies between 0.15 Hz and 0.4 Hz, the low band (LF) from 0.04 Hz to 0.15 Hz

11

0 0.1 0.2 0.3 0.4 0.50

0.005

0.01

0.015healthy individual

frequency (Hz)

PS

D (

s2 /Hz)

0 0.1 0.2 0.3 0.40

0.005

0.01

0.015post−infarct patient

frequency (Hz)

PS

D (

s2 /Hz)

VLF

LF

HF

VLF

LF HF

Figure 10: Power spectral density of the healthy and post-infarct recording from Figure 8

and the very low band (VLF) contains frequencies below 0.04 Hz.Figure 10 shows the power spectral density forthe two recordings from Figure 8. [33]

The mapping of LF and HF to regulatory influences works particularly well. The HF peak is called the respira-tory peak, because it is caused by the respiratory arrhythmia. Breathing rate and volume can modify the frequencyresponse. The power in HF is also a measure of vagal activity. Vagal stimulation takes effect almost instantly, thuswith a high frequency. Sympathetic activation has a significant latency and recovery time. It can only account forspectral power in the LF band. LF can however also be influenced by vagal activation, which experiments showed.The sympathico-vagal balance is then expressed in the parameter LF/HF. [1]

Rhythm Analysis The limitation of time and frequency parameters is that regularities in the RR series are notrecognised. For example, sdNN does not care about the sequence of values at all. If the values would be sorted,sdNN would remain the same. The rhythm analysis approach is to measure blocks of intervals with specific rhythmsand to investigate their relationship. For example the heart rate turbulence (HRT) [2] after a premature ventricularbeat is a useful parameter for post-infarct risk stratification. [33]

Non-linear Methods More recent approaches aim to model the complex interplay of influencing factors withnon-linear approaches. Some of these are the 1/f scaling of Fourier spectra, H scaling exponent or Coarse Grain-ing Spectral Analysis (CGSA). Additionally, different data representations may improve interpretability. Theseare for example low-dimension attractor plots or singular value decompositions. Another approach are entropymeasures. [33]

1.2.3 HRV after Myocardial Infarction

In the general understanding a depressed HRV indicates a pathological condition. This is also true for myocardialinfarction. [33]

Acute Myocardial Infarction In the first hours after the infarct a significant reduction in sdNN can be found.This is due to the ventricular dysfunction and the peak in creatine kinase. [33]

The vagal stimulation is reduced and thus the sympathicus dominates. The mechanism of neural response is notcompletely uncovered. It is assumed that the changes of the geometry of the beating heart mechanically cause anincrease in the firing of sympathetic fibres. This attenuates the vagal activity at the sinus node. Another explanationis a diminished responsiveness of the sinus node. [33] [27]

This shift in sympathico-vagal balance shows in a predominant LF and a reduced HF. The LF/HF ratio istherefore high. [27]

Post-acute Myocardial Infarction In the days after the infarct the depression in HRV can be observed in the timeand frequency domain parameters. The spectral power reduces significantly. The shift to increased sympatheticstimulation and reduced vagal activity remains. However, betablockers may blunt the HRV result to some degree.They reduce the sympathetic tone and thus LF. In the following weeks to months the HRV recovers partially, butnot completely. [27]

12

Risk Stratification Risk stratification aims to identify the patients with a high risk of post-infarct mortality.These patients can greatly benefit from further treatment. Apart from a depressed HRV, predictive factors are adepressed left ventricular ejection fraction (LVEF), increased ventricular ectopic activity, and presence of latepotentials. [22, 5, 27]

The HRV is independent from these other predictors. For all-cause death risk its predictive strength is similarto that of LVEF. However, the HRV is superior for predicting arrhythmic events, such as sudden cardiac death orventricular tachycardia. Figure 11 shows an example. Both the 5 min sdNN and the HRV triangular index showsignificant differences between survivors and non-survivors. However, the HRV index has a superior predictionstrength in this data set. [12] The taskforce guideline describes the thresholds sdNN (24 h) < 50 ms and HRVtriangular index < 15 for highly depressed HRV. [33]

Figure 11: Kaplan-Meier survival curves for all cause cardiac mortality during the year following acute myocardialinfarction. HRVi: HRV index over 24h; SDNN: standard deviation of all normal RR intervals over 5 min stationaryperiod [12, p. 344]

Day 11 ± 3 after the infarct is reported to have the highest predictive strength. This is also the usual time ofdischarge from hospital, which is very practical for clinical applications. The other advantages are that it is aninexpensive and non-invasive way to screen infarct patients. [27]

Side Factors A number of other factors influence the HRV. They need to be considered during analysis. Theseare for example:

• Age With increasing age the the HRV declines non-linearly. The decline is steepest for the 20’s and 30’s.However, the LF/HF ratio, indicating the sympathico-vagal balance, remains the same in a standing positionwith metronomic respiration. The predictive strength for sudden cardiac death also declines with increasingage. [32] [1]

• Breathing The lung inflation leads to an increased HR, deflation to a reduced one. The frequency ofbreathing is also reflected in the HRV. The breathing should best be normalised. [32]

• Diabetic Neuropathy A reduced HRV can also be used to identify degeneration of small nerve fibres inpatients with diabetic mellitus. This has to be considered when assessing the mortality risk. [33]

1.3 Entropy and Estimation Techniques

1.3.1 Definition and Motivation

Concept The entropy is a central concept of information theory. The entropy of a message is a measure of theamount of information contained in it. For example, the message "aaa aaa aaa" does not contain much information.In fact it can be shortened by saying "9*a". Most of the original message is redundant. However, the message"acd bgx mth" contains a lot of information. There is no redundancy or regularity. Therefore in order to transmitthis message at least 9 characters are necessary. The message "bda bda bda" lies in between. It does contain someredundancy, but in order to transmit it one would need 5 characters: "3xbda".

13

Definition A more formal definition of the entropy has been provided by Shannon [45]. Imagine a messagesource that has n possible messages with probabilities p1, .., pn. Then the entropy of one message i is:

Ei = −log2pi [bit] (1)

This is intuitive, because a message with high probability will have a low entropy and vice versa. For example themessage "hello" has a high probability and thus a low information content. But the message "tachycardia" has afairly low probability in everyday life and thus conveys more information. [4]

The average entropy E of all messages, also called the entropy of the message source, is then:

E = −

n∑

i=1

pilog2pi [bit] (2)

The entropy thus measures the amount of choice involved in choosing one message. For example, if a source canonly emit 8 equally likely messages, then the average entropy is 3 bits. Because with 3 bits 23 = 8 numbers orchoices can be encoded. In this study, each NN interval is considered a message and the sinus node is the source.A tachogram is a sample of the messages that a source emits. [4]

This also highlights that the entropy is dependent on the probabilities of the messages. These probabilitiesdepend on the context. For example, the word "tachycardia" has a low probability in everyday life, but a fairly highone in the cardiology department of any hospital. The entropy can therefore only be estimated with reference to aprobabilistic model of the source. [4]

Motivation The concept of heart rate variability is very similar to that of the entropy. A healthy tachogram, suchas in Figure 8, with a lot of variability is unordered and non-repetitive. Its entropy would be high. A pathologicaltachogram with a low variability would be ordered and highly repetitive. Its entropy would be low. It is thereforeonly natural to use the entropy of the tachogram as a measure of heart rate variability. The entropy is the counterpartof HRV in information theory.

1.3.2 Estimation Techniques

There are two principle ways of estimating the entropy of a message. The first approach is to estimate the proba-bility of the message in some statistical way and compute the entropy from it. The second one is to compress themessage. [4]

By compressing the message the redundancy contained in it is eliminated to a large degree. What remains is ashort representation of the message. The number of bits of this representation is an upper bound of the entropy ofthe message. The better the compression, the closer the entropy estimation. [4]

In the field of data compression a range of efficient algorithms have been implemented. One of the first all-purpose algorithms was introduced by Ziv and Lempel [53] in 1977 and was followed by a host of variants. A morerecent technique is Burrows-Wheeler block sorting [7], which is implemented in the bzip2 format.

The five different estimation techniques used in this study are described inSection 2.3.

1.4 Recent Entropy-Approaches in HRV Analysis

The following entropy-based approaches have been applied to cardiovascular signals, in particular the tachogram.Each of them aims to highlight a certain aspect of the complexity that the entropy measures.

1.4.1 Approximate Entropy (ApEn) by Pincus

The approximate entropy (ApEn) is a measure of system complexity that was introduced by Pincus [37]. It hasbeen widely applied in cardiovascular studies. [19] [36] [21] [26]

ApEn measures the degree to which the occurrence of a value depends on its predecessors in the input. Highvalues thus imply a low predictability and regularity. It does this for a fixed input length N , a fixed number ofpredecessors m and a fixed similarity tolerance r. Therefore there is not one ApEn value per input SN , but a familyof ApEn(SN , m, r) values. The similarity tolerance r determines how much two sequences can differ while stillbeing considered a match:

|HR(i + k) − HR(j + k)| < r for 0 ≤ k ≤ m (3)

14

Consider the set Pm of all patterns of length m within Sm. Pm has elements pm(1), pm(2), .., pm(N − m + 1).Then the fraction Cim of each unique pattern in the input is defined as:

Cim =nim(r)

N − m + 1(4)

where nim is the number of patterns in Pm that are similar to pm(i). The mean of all Cim is denoted with Cm andmeasures the prevalence of repetitive patterns of length m in SN . The approximate entropy is then:

ApEn(SN , m, r) = ln

(

Cm(r)

Cm+1(r)

)

(5)

ApEn is therefore the logarithm of the relative prevalence of repetitive patterns of length m compared to patternswith length m + 1. [28] [38]

Although this measure has been used widely is has significant short- comings. [38] Firstly, the input has tohave a certain length. Pincus [37] suggests at least 1000 values. Richman and Moorman [38] point out, the ApEnis heavily dependent on the input length. Especially short signals lead to a lower value than expected. The secondissue is that when counting the occurrences of a pattern the original pattern is counted as well, which leads to abias. [38] A third problem is that the ApEn is not consistent across different pattern lengths. If one input has ahigher ApEn than another one for pattern length m, this is not necessarily the case for m + 1. [38]

1.4.2 Multiscale Entropy (MSE) by Costa et al.

The multiscale entropy by Costa et al. [9] aims to separate short-range from long-range correlations in the inputsignal. Entropy values are computed for different scales of the signal. The rationale is that pathological states donot necessarily alter the variability of the heart rate for all ranges of correlations. [9]

The mechanism to achieve this is to coarse-grain the signal using different scale factors τ . The signal is simplydivided into blocks of size τ . For each block the average is computed. The resulting signal can thus be describedas:

yτj =

1

τ

jτ∑

i=(j−1)τ+1

xi 1 ≤ j ≤N

τ(6)

For each coarse-grained input the sample entropy (SampEn) [38] is computed. SampEn is very similar toApEn. Its advantage is mainly that self-matches are not counted, which reduces the bias observed in the ApEn.Additionally, the entropy values are more consistent across different pattern lengths. [38] [9]

This approach has been applied to the tachogram of subjects of different age and pathologies. The parametersused to compute the sample entropy were N = 20000, m = 2 and r = 0.15. For example, the discriminationstrength between young and elderly subjects was highest with a scale τ = 5. [8] [9]

The scaling approach can be compared to frequency domain analysis. The coarse-graining simply removesthe higher frequencies from the signal. The entropy is then only influenced by the remaining lower frequencies.However no frequency bands can be isolated, only all frequencies below a given threshold are examined. Someof the disadvantages of ApEn are inherited. Especially, the signal length is crucial. Costa et al. used 20000 valueswhich were then scaled down. In order to acquire 20000 heart beats the recording should be at least 5 hours inlength.

1.4.3 Renormalised Entropy (RE) by Wessel et al.

The renormalised entropy is a measure of the complexity of cardiac periodograms based on a fixed reference. It isdesigned in a way that its value is positive for healthy recordings and negative for pathological ones. [29]

First, the tachogram is resampled to equidistant time steps through interpolation. Then an autoregressive esti-mation of the spectrum is calculated. Then a reference is chosen as the most disordered frequency distribution ofa healthy tachogram. This reference spectral distribution is renormalised to a given energy. Consider the densitydistributions f0(x) and f1(x) with f0(x) being the reference. Its renormalised density distribution is:

f̄0 =f0(x)T

∫

f0(x)T dx(7)

15

with T being the numerically determined solution of the following integral:∫

lnf0(x)(f̄0−f1(x))dx = 0 (8)

The renormalised entropy is then defined by the following algorithm, with S(f(x)) being the Shannon entropy:

S(f(x)) = −

∫

f(x)lnf(x)dx (9)

Algorithm:

1. compute 41 = S(f1(x)) − S(f̄0(x)) with f0(x) as the reference. The value of T is denoted with T1,2. compute 42 = S(f0(x)) − S(f̄1(x)) with f1(x) as the reference. The value of T is denoted with T2,3. if T1 > T2, then distribution f0(x) is more disordered in the sense of renormalised entropy. Then RE = 41.

Otherwise f1(x) is more disordered and RE = 42. [29]

This renormalised entropy has been applied successfully in the classification of infarct patients vs. controls [29]and also for risk stratification [50].

A disadvantage of this approach is that the data goes through several processing steps which introduce errorsand alterations. First, the tachogram has to be resampled. Then, the periodogram is estimated. Then the Shannonentropy is computed from the probabilities. The exact probabilities are however difficult to determine. A very longinput signal or a probability model is required. Another disadvantage is that a reference signal has to be selected,which is difficult in practice.

1.5 Research Objective

Motivation The concept of HRV is very similar to that of the entropy. An in-health tachogram with consider-able variability is unordered and non-repetitive; its entropy would be high. A pathological tachogram with a lowvariability would have a lower entropy.

Existing entropy-based HRV-Analysis methods focused on highlighting specific aspects of the complexity ofthe signal, such as certain frequencies and pattern lengths. The overall entropy in its information theoretic formwas only used rarely, even though there are a number of effective methods of computing it. Additionally, duringthe computation the input signal went through several processing steps that required interpolation and estimation.This introduces a certain degree of error or deviation.

Another limitation of previous studies is that the comparison of multivariate classification strengths was of-ten not done systematically. However, pattern recognition theory provides a range of optimised classification ap-proaches.

Objectives The primary objective of this research is to evaluate the usefulness of the signal entropy in HRVanalysis of patients with myocardial infarction. The entropy is compared to the standard parameters [33].

The secondary objectives are (1) to compare the different entropy estimation methods in Table 3 for this taskand (2) to identify necessary signal preprocessing steps, such as signal quantisation, signal length, and artifactfilters.

Name TechniqueNGRAM [42] statistical finite context modelALRM [18] adaptive linear regression modelLZ77 [53] adaptive dictionary codingGZIP [13] LZ77 + Huffman codingBZIP2 [44] Burrows-Wheeler block sorting [7]

+ Huffman coding

Table 3: Entropy estimation methods to compare

The classification task will be performed by multivariate decision trees, which are automatically generated toachieve optimal discrimination. The classification rates will be determined using cross validation.

The criteria for assessing the usefulness of the entropy in HRV analysis are:

16

1. statistical significance of entropy mean differences between infarct and control groups,2. infarct classification rate of HRV entropy and gain by adding the entropy to standard

parameters,3. content and complexity of the entropy-based decision trees,4. consistency and reliability of the entropy estimation,5. robustness of the entropy estimation and decision tree against input signal fluctuations.

The rationale of this investigation is to extend the toolset of HRV analysis. The information gained from theECG signal its HRV signal aid the decision making process in the clinical setting.

Recent results by Baumert et al. [3] on the correlation of the LZ77 entropy with life threatening arrhythmiasshowed encouraging results. This research directly continues and extends their investigations.

In the following section related research is summarised. The data set and concrete methods applied are de-scribed in Section 3. Section 4 presents the results. Finally, the findings are discussed in Section 5.

17

2 Materials and Methods

This section describes research design and data processing techniques. Figure 12 gives an overview.

MCG+ECG

template matching

artefact rejection

dataset"Cut"

dataset"Interp"

dataset"Diff"

Length Normalisation

cut to same length

resample time signal

cut to same length

quantisation (except for ALRM)

LZ77

NGRAM

GZIP

ALRM

BZIP2

Entropy Estimation

ccncept analysis

compute alternative HRV parameters

standard HRV parameters

subject characteristics

population subgroups

range of configu-rations each

standardsclassifiers

standards+entropyclassifiers

Preprocessing

non-linear HRV parameters

entropy parameters

broadclassifiers

compare subgroups with Wilcoxon-test

generate & test classifiers

statistically significantentropy distributions

selectcandidates

age, smoker,medication ..

tachogram successive differences

entropy for all subjects

suitability of subjects(only sinus rhythm)

generate & test classifiers

generate & test classifiers

Classification

Gain?

Statistics

Figure 12: Overview of the data processing steps

2.1 Data Acquisition

The raw data set consists of 180 MCG and ECG recordings made at the Biomagnetic Center Jena at the Friedrich-Schiller-University in Jena. 107 recordings are from myocardial infarction patients at the University Hospital Jena.The remaining 73 are test persons. This data set been used for previous studies.

18

Each recording has a length of 10 minutes with a sampling frequency of 1000Hz. The MCG has 31 channels.Figure 4 shows the arrangement. The chest lead after Wilson (V1-V6) was used as well as a bipolar vector ECG(X,Y,Z). X and Y form a horizontal plane at the height of the lower end of the sternum. X points form back to frontand Y points from right to left. Z points from below the rib cage on the left to the jugulum. The preparation tooktwo to three minutes, in which the persons could settle.

The patient population consists of cases of myocardial infarction over a four year period from 1997 to 1999.Persons with pace makers were excluded, because the pace maker would have interfered with the MCG equipment.

The data set was initially collected for another study under supervision of Dr. U. Leder at the Friedrich-Schiller-University Jena [24]. The classification of subjects into control or infarcts and the classification into anterior or pos-terior infarcts has been adopted here. Additional information, e.g. the existence of risk factors and the medicationwere provided as well.

2.2 Data Preprocessing

2.2.1 Template Matching

In a first step, the heart rate is extracted from the ECG recording using a template matching approach.For each recording one of the MCG or ECG channels is selected. A single PQRST segment is selected as

template. The correlation between the template and the signal in each time step is computed. If the correlation fora time step exceeds a given threshold, a match is found. Additionally, a lower and upper threshold for the signalamplitude can be defined. This has been done with the software CURRY4.6 [31].

2.2.2 Artifact Filtering

In some instances, the accurately detected RR intervals contain artifacts, in particular extrasystoles. These artifactsare removed using the adaptive filtering technique introduced by Wessel et al. [30, p. 161]. Artifacts are detectedbased on their deviation from an adaptive mean and standard deviation. They are replaced by a random value thatmatches the adaptive mean and standard deviation.

2.2.3 Length Normalisation

Each recording has a length of 10 min. Depending on the heart rate of the subject, the number of heart beats andthus intervals within this time varies significantly.

Two alternatives to normalising the signal are taken. Firstly, all signals are cut to the length of the shortest one.The number of RR intervals is then equal, but large parts of the already short signals are wasted.

The second approach is to interpolate a continuous-time HRV signal and sample it. This approach makes useof the full 10 minutes. However, the signal is not the original measurement. This interpolated signal is also used tocompute standard HRV parameters [33, p. 357] in the frequency domain.

2.2.4 Quantisation

All entropy estimations methods except the adaptive linear regression models are based on symbolic input con-structed from an alphabet of symbols. The RR intervals are expressed in milliseconds. This quantisation is toofinely grained, because it is unlikely that a specific RR interval length appears repeatedly.

The RR intervals are resampled with a smaller rate, such as 128Hz. This is much like sorting the RR intervalsinto bins of a histogram with the size 1000 ms/128 = 7.8125 ms. This is the default bin size for standard HRVanalysis [33, p. 356].

The RR intervals fall within a range from 400 ms to 1400 ms. Rare outliers are replaced with the 400 ms or1400 ms respectively. A specific bin size then implies a specific number of symbols in the alphabet: the possibleinterval lengths. Therefore different bin sizes need to be tried.

2.3 Entropy Estimation

The following five entropy estimation techniques are applied and compared.Table 4 also shows their options andthe resulting configurations. A number of configurations is then selected for the further process. The selection

19

criteria are firstly that the estimated entropies are low and secondly that the estimation is reliable. Low estimatedentropies indicate that the real unknown entropy is approached better.

Method Technique Parameters Configurations #LZ77 [53] adaptive dictionary coding bufferLength(bL),

maxWordLength[1..30, 1..bL-1],[100,1..30]

495

NGRAM [42] finite context model contextLength 1..6 6

ALRM [18] adaptive linear regressionmodel

contextLength 1..32 32

GZIP [13] LZ77 (bL=33026b, mWL=258b) + Huffman coding

quality fast (−1), default,best (−9)

3

BZIP2 [44] Burrows-Wheeler blocksorting [7] + Huffman coding

blockSize 100kb (−1) 1

Table 4: Alternative entropy estimation methods

The NGRAM and ALRM method are statistical ones, while the remaining three are compression methods. Thecomparison between both groups is particularly valuable for establishing the consistency of the entropy estimation.

2.3.1 NGRAM

An Ngram is a probabilistic model, in which the probability of one symbol in a sequence depends on the previous(N-1) symbols. Ideally, the joint probability of a sequence of symbols is the expressed as:

P̂ (s1..sT ) =

T∏

t=1

P (st|s1..st−1) = P (s1) ∗ P (s2|s1) ∗ .. ∗ P (sT |s1..sT−1) (10)

Here each symbol depends on all its predecessors. Because this is not practical, the Ngram approach approximatesthis probability by only using the previous (N-1) symbols. [42]

The most basic solution to compute probabilities is to use the maximum likelihood estimator:

P (sN |s1..sN−1) =#(w1..wN )

#(w1..wN−1)(11)

However, this approach has some disadvantages. For example, if s1..sN does not appear in the training sample,then its probability is zero. More sophisticated approaches have been developed. The particular technique used inthis study is the rational interpolation approach by Schukat-Talamazzini et al. [43]. An concise overview can befound in [42]. The entropy per bit of the input signal is then estimated by:

E ≈overall entropy

# input values * # bit per value[bit] (12)

The larger the N is chosen, the more exact the probability estimation becomes. A 1gram approach means thatthe probability of each symbol only depends on its overall frequency, not on the previous symbols. This would bea very rough estimate. The (N-1) can be called the context length. The context length is the option of this entropyestimation method.

2.3.2 ALRM

The adaptive linear regression method (ALRM) is similar to the Ngram method in that each value is predictedbased on a fixed number of its predecessors. However, it is a numerical prediction, not a symbolic one. A linearmodel is assumed as regression function:

y′ = f (x1, x2, .., xn) = α + f1(x1) + f2(x2) + .. + fn(xn) = α + β1x1 + β2x2 + .. + βnxn (13)

The xi are the predictors, the predecessors, and y’ is the prediction. Each predictor contributes to the predictionthrough a linear function. This is a specialisation of the general additive model with arbitrary fi [18, p. 257]. Therecordings are used without quantisation.

20

The regression model is adaptively fitted to all preceding values using minimum least squares. The predecessorsare unweighted. The model is then used to predict the next value. The deviation of the prediction y’ from theoriginal value y determines the entropy for each value. If the deviation is below a certain error quantum δ, theprediction is correct and the entropy is 0. Otherwise the entropy is computed as:

Ei = log2(|y′ − y|/δ) + 1 [bit] (14)

One bit is required for the sign and the rest for encoding the quantised deviation. A large deviation for one valuemeans that it is less predictable and thus has a higher entropy. As error quantum δ 1000/128 ms = 7.8125 ms wasused. The entropy for each of the input values was added up. The resulting entropy is the number of bits requiredto transmit the input with accuracy δ and the adaptive linear regression model. [40]

To yield the entropy per bit of the input signal Equation 12 is applied. The context length n is the option forthis entropy estimation method.

2.3.3 LZ77

One of the most important all-purpose compression methods, called LZ77, was introduced by Ziv and Lempel [53].It is a sequential dictionary method. The compression mechanism is depicted in Figure 13. The key idea is toencode future sections of the input through a reference to an earlier occurrence of the same section in the input. Ofcourse, always the longest match is used. The output consists of code words. These are triplets of the address ofthe earlier occurrence, the length and the next symbol. The next symbol is required for the case that no match atall is found. [53]

buffer

look aheadcontext

input

output [3, 4, 8]

current

code word =[pointer, length, next]

1 2 76543

... ...

... ...1 8 9 3247 52 347 5576

Figure 13: Example of a coding step using LZ77

In this study the ratio of code words to input symbols is used as estimation of the entropy.

ELZ77 =# code words

# input symbols(15)

This is not quite the compression rate because the size of the code words is not taken into account. But ELZ77 isan interesting measure for research purposes. It has been defined as a complexity measure by Lempel and Ziv [25]and has been applied before to forecasting ventricular tachycardias [3].

The options for this entropy estimation method are the buffer length and the maximum word length, which isthe length of the look ahead part. The context length equals to buffer length - max. word length. LZ77 has beenimplemented in Matlab for this study on the basis of the original publication of Ziv and Lempel [53].

2.3.4 GZIP

Although the LZ77 technique produces a symbol sequence that is shorter than the original one, the code wordsare still not stored in an optimal way. Each code word would have a fixed size. In order to achieve maximumcompression, the output has to be optimally coded. This is done by GZIP. It encodes the output of LZ77, moreexactly a variant of it, in two Huffman trees, one for the pointers and one for the lengths of the earlier occurrences.The next symbol is only used when no match is found. This is done for blocks of the LZ77 output. [13]

21

GZIP thus provides an optimised implementation of LZ77. Each byte is considered a symbol. Therefore eachbeat-to-beat interval needs to be encoded as one byte = 8 bit in the file, even though only 7 = log2(128) bit arerequired due to the quantisation. The file is then compressed. The entropy per bit of the input signal is estimatedby:

E ≈compressed file size

# input symbols * # bit per symbol[bit] (16)

The compressed file size divided by the number of input symbols, the intervals, yields the entropy per interval.Because each interval was encoded with 7 bit during quantisation, we normalised by dividing by 7. The resultingvalue can then be interpreted as proportion of information in the quantised tachogram.

For the first sample inFigure 8, the entropy was 320*8 bit / 480*7 = 0.76 bit ≈ 3/4. However, the post-infarctpatient only had an entropy of 189*8 bit / 480*7 = 0.45 bit ≈ 2/4.

The option for this method is the quality. A higher compression speed can be achieved with a less than optimalcompression.

2.3.5 BZIP2

A different approach to compression is Burrows-Wheeler block sorting, which is implemented in the Bzip2 [44]format. Its key idea is to transform the input into a representation which is much easier to compress with simpleexisting compression algorithms. [7]

Figure 14 illustrates this transformation process. The algorithm works block-wise not sequentially. The al-gorithm constructs a matrix of all cyclic permutations. The rows, which are the permutations, are then orderedlexicographically. The last column then contains the same characters as the input, but in a different order. Thiscolumn and the number of the row where the original input is located form the new representation. An importantobservation is that this transformation is reversible. [7]

input

output

... ...1 8 9 3247 52 347 5576

L = 442557775382

I = 10

all cyclic permutations in lexicographic order

0

1

2

3

4

6

7

5

8

9

10

11

8247 52 3 47 55782 47 52 347 557

8247 5 23 47 5578247 52 34 7 557

824 7 52 347 5578247 52 34 75 57

824 75 2 347 5578247 52 347 55 7

8247 52 347 5 578247 5 2 347 557

8247 52 347 5578 247 52 347 557

move-to-front coding

Huffman or arithmetic coding

Figure 14: Example of a coding step using the Burrows-Wheeler compression

The idea behind this transformation is to group characters together so that the probability of finding a characterclose to another instance of it is maximised. In Figure 14 the characters in L are much more grouped than inthe input sequence. This works because the occurrence of one character is not independent from its precedingcharacters. For example, in the input the ’5’ is often preceded by a ’7’. The sorting brings together all permutationsthat start with a ’5’. Because the permutations are cyclic, the predecessor of the initial character is in the lastcolumn. Therefore also the ’7’ characters are grouped together in the last column. [7]

For this study, the entropy is again estimated by Equation 16. The healthy subject (Figure 8) has an entropy of323*8 bit / 480*7 = 0.77 bit ≈ 4/5 and the post-infarct patient produces 161*8 bit / 480*7 = 0.39 bit ≈ 2/5.

The option for this estimation method is the block size ranging from 100kb to 900kb. However, since thetachogram used here is far below 100kb, only 100kb are used. Bzip2 is executed on a command line. Each valueof the tachogram is coded as one byte of the input, because the algorithms treats each byte as a character. [44]

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2.4 Alternative HRV Parameters

A number of existing parameters are determined from the data set in order to firstly compare them with the param-eter entropy and secondly to conduct a multivariate analysis.

Most of the standard parameters [33] are included. The time-domain parameters are summarised in Table 5.SdaNN is the standard deviation of NN intervals for 5 min segments of a long-term ECG. Since only 10 minrecordings are available here, sdaNN1 is defined which uses 1 min segments.

Parameter Description UnitsmeanNN Mean of NN intervals over entire recording mssdNN Standard deviation of NN intervals over entire recording mscvNN Ratio sdNN / meanNN -sdaNN1 Standard deviation of NN intervals averaged over periods of 1 min msrmssd Root of the mean squared differences of successive NN intervals mspNN50 Proportion of the number of interval differences of successive intervals larger than 50 ms

to the total number of NN intervals%

pNNi10, pNNi20 Proportion of the number of interval differences of successive intervals smaller than 10(20) ms to the total number of NN intervals

%

Table 5: Time-domain parameters

The frequency-domain parameters are listed in Table 6. For their computation the Fast Fourier Transform wasused with a Blackman Harris window to the leakage effect.

Parameter Description UnitsVLF Power in very low frequency range (≤ 0.04Hz) ms2

LF Power in low frequency range (0.04Hz − 0.15Hz) ms2

HF Power in high frequency range (0.15Hz − 0.4Hz) ms2

P Power in the frequency range (0 − 0.4Hz) ms2

LF/HF Ratio LF/HF -LF/P Ratio LF/P -HF/P Ratio HF/P -

Table 6: Frequency-domain parameters

Other non-linear parameters that are considered are the Renyi entropy, which was introduced in earlier studiesby Voss et al. [49, p. 431]. Hoyer et al. [20] applied the concept of mutual information, which is related to that of theentropy. They defined the auto mutual information measure, which is comparable to the auto correlation functionbut is more suitable for HRV signals. beats. The alternative non-linear parameters are summarised in Table 7.

Parameter Description Unitsrenyi0.25, renyi2, renyi4 Renyi entropy from histogram of order 0.25, 2 and 4 -AMIFdecay Auto mutual information decay over τ = 1 beat period -AMIFarea Area under auto mutual information curve -AMIFpdLF Auto mutual information decay to prominent peak of resampled, LF-filtered

intervals (5 min)-

AMIFpdHF Auto mutual information decay to prominent peak of resampled, HF-filteredintervals (5 min)

-

Table 7: Non-linear parameters

All of these parameters are computed with the HRV research software produced by Voss et al. [49] and Hoyeret al. [20] for their respective investigations.

2.5 Data Set Reduction

The data set is heterogeneous, because recordings were non-selectively collected over a longer period of time. Theheart rate variability is influenced by complex autonomous mechanisms as well as different pathological conditions.However, we are interested in the effect of myocardial infarction on the HRV only.

The data set is therefore reduced by rejecting recordings with other dominant influences. A recording wasrejected, if it contained:

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1. atrial fibrillation,

2. sinus arrhythmia (including strong respiratory ones),

3. any type of bundle branch block,

4. more than 10% extrasystoles, bigeminy or trigeminy,

5. a sudden change in characteristic, suggesting a response to some event.

The remaining 150 recordings showed a sinus rhythm. Of these 150 recordings 87 are from infarct patients and 63from test persons.

2.6 Population Subgrouping

The data set is heterogeneous in terms of age, gender and a range of other characteristics. Sub groups need tobe defined. In particular the numerical characteristics have to be categorised.Table 8 gives an overview of thesecategorisations.

The heart rate variability is known to decrease with the age of a person. The age therefore needs to be cate-gorised. However, the number of categories should be small in order to obtain sufficiently large subgroups. Below30 years there are no infarcts. Above 70 years there are no controls, except for one. The remaining age interval issplit into 30-50 years and 50-70 years. Both contain infarct patients as well as controls. 50-70 years is the mostcommon age range for infarction.

The second important characteristic in this study is the infarct stage at time of recording expressed in days afterthe infarct. In the first two weeks reactive processes in the cardiac muscle dominate. In the following two weeks,the functional condition of the heart stabilises. These intervals will be two categories for the infarct stage. Onlyfew recordings were made more than 30 days after the infarct. These will be put into a separate category.

Characteristic Categoriesage (< 30 years,) 30-50 years, 50-70 years, (≥ 70 years)gender female, maleinfarct stage none, 1.-14. day, 15.-30. day (, > 30. day)infarct location none, anterior, posteriormedication none, betablocker, betablocker + digitoxin, (digitoxin,

betablocker + other antiarrhythmicum)no. risk factors none, 1, 2, 3, 4

Table 8: Population characteristics and sub categories

The location of the infarct is categorised in anterior and posterior. The infarct treatment sometimes requiredheart medication, such as betablocker and/or digitoxin. Only in few cases just digitoxin was given.

For the infarct patients, the existence of the risk factors smoking, diabetes, hypertension and hypolipoproteine-mia was recorded. However, each combination of risk factors is only fulfilled by a small number of patients each.Therefore the number of risk factors for each patient is used as categorisation. Risk factors were not collected forthe control group.

To visualise and comprehend the multidimensional characteristics Formal Concept Analysis (FCA) [14] isused. A brief introduction to the FCA basics is given in Appendix section A. An open source tool that implementsthe FCA functionality is ToscanaJ [11].

2.7 Statistical Tests

The entropy and other HRV parameters for the each of the sub groups of Section 2.6 are statistically tested fora difference in mean using the Wilcoxon Test. The advantage of the Wilcoxon Test over the t-test is that it doesnot make the assumption that the distributions are normal distributions. Additionally, it is more suitable for smallsamples.

In particular, all combinations between age and infarct stage are investigated. The Wilcoxon test result betweenthe subgroups that are to be classified can be interpreted as measure of each parameter’s discrimination strength.

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2.8 Infarct Classification

The benchmark task in this study is to classify infarct patients vs. healthy individuals based on the HRV parameters.Decision trees are chosen as classification model, because they explicitly state which parameters are chosen andhow they are combined. They can be directly interpreted.

Decision Tree Generation The classification task is stated using a composite classifier approach.Figure 15shows the four tasks. In the first one a decision tree is generated based on all recordings. In the second one, aclassifier is generated for both age groups respectively. Both decision trees are combined to a composite decisiontree, in order to compare the classification strength to the first tree. The third and forth approach generate four treeseach, which are again combined to composite trees.

age

27 vs. 21 28 vs. 38

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55 vs. 59

no. of controlsno. of infarcts

(1)

(4)

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Figure 15: Four (composite) classification tasks

The advantage of this composite approach is that a priori information can be imposed on the tree, such as thedistinction of age groups or gender. Additionally, the classification rates within the sub groups are obtained fromthe sub trees. At the same time the classification rate of all four trees remains comparable.

The decision trees are generated using the CART algorithm [6] implemented in MATLAB. The options for thegenerator are stated in Table 9. The gender is marked as categorical parameter. In a second step the trees are prunedto the size with the minimal cost during cross-validation.

Option Value Meaningmethod classification a finite number of classes are predictedsplitmin 2 impure nodes must have 2 or more observations to be split (terminal

nodes can not be impure)prune off no pruning in training stepcost [0 1; 1 0] all misclassifications have cost 1 (balanced)splitcriterion gdi Gini’s diversity indexpriorprob - no prior probabilitiesnsamples 10 use 10-fold cross-validation for pruningtreesize min pick treesize with minimum cost in pruning

Table 9: Decision tree generation and pruning options

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Maximum Classification Rate The disadvantage of decision trees is that they can not model higher order sepa-rating hyperplanes in the multidimensional parameter space. Each decision only involves one parameter. To ensurethe classification rate is not affected, a Gaußclassifier and 1..4-Nearest-Neighbour classifiers are also applied to thesame tasks. The maximum of the statistical classification rate during leave-one-out validation is used for compari-son.

Validation The classification rates are obtained using leave-one-out (jackknife) validation. One recording istaken away, a decision tree is generated and pruned based on the remaining recordings and applied to the one leftout. This is done for each recording.

The classification rates of the decision trees are computed 10 times and averaged. The reason is that in thecross-validation during pruning the test recordings are picked randomly, but preserving class ratios. Because theclass sizes are small, this has an influence on the classification rate.

Parameter Set Table 10 shows the three parameter sets: standard parameters, standard with entropy parametersand the full set of parameters. Classifiers are generated for each of the parameter sets. The performance of the stan-dard parameters is the baseline. By comparing it with the performance of standard+entropy set, the gain throughthe entropy parameters is obtained. This gain is not only expressed in the classification rate, but also in the structureof the decision tree. With the third parameter set the performance of the entropy parameters can be compared tothe other non-linear parameters.

Set Parameters Std. Std.+Entropy Allpopulation age, gender X X X

time meanNN, sdNN, sdaNN1, rmssd, pNN50, pNNi10, pNNi20 X X Xfrequency VLF, LF, HF, P, LF/HF, LF/P, HF/P X X X

entropy [LZ77,Cut], [BZIP2,Diff], [LZ77,Cut]/m,[BZIP2,Diff]/m

X X

other non-linear

renyi025, AMIFdecay, AMIFarea, AMIFpdLF,AMIFpdHF X

Table 10: Parameter sets for classification

The four tasks are combined with the three parameter sets which results in 3*4=12 decision trees. Within onetask the gain in classification strength through the entropy parameters is shown. Between tasks the best decompo-sition of the data set can be determined.

In the classification tasks, the number of samples is often low. This is particularly true, if the data set is splitinto subgroups. The parameter set in Table 10 is therefore reduced by eliminating redundancies. From the 10highly correlated entropy parameters [LZ77,Cut] and [BZIP2,Diff] are chosen and from the entropy-meanNNratios [LZ77,Cut]/m and [BZIP2,Diff]/m. From the three renyi entropies renyi025 is chosen. This decision is alsobased on the results of the entropy estimation and statistical tests.

Data set For the classification, only the age range from 30 to 70 years is considered. Below 30 years there areno infarcts and above 70 years there is only one control recording. Additionally, only infarct recordings that weremade between the 1. and 30. day after the infarct are considered. Figure 15 shows the class sizes for the respectiveclassification tasks.

Trivial Classifiers In some classification tasks, the number of infarcts and controls is unbalanced. For example,for ages 50-70 there are 28 controls, but only 8 recordings 15.-30. day infarcts. A trivial classifier would simplyclassify all inputs as controls and already reach a classification rate of 78%. Such trivial classifiers are disregarded,because they are only due to unbalanced class sizes.

2.9 Limitations

The key limitation is that the data set was not initially collected for this research design. The data set was created tobe as broad as possible. No selection criteria were applied. Therefore the suitable subgroups need to be identified

26

first. In particular, the time between infarct and recording varies considerably. Only very few were made betweenthe 15. and 30. day, while most recordings were made between the 1. and 14. day. This results in a some subgroupscontaining only very little recordings.

Another limitation is the recording length of only 10 minutes. Each recording therefore only contains around500-950 beats. When normalising the signal length, every recording is shortened to the length of the shortest one.It is not possible to select a stationary signal piece. However, 500 beats are not necessarily to little to obtain aconsistent entropy estimation.

The recording length may also be a problem for the auto mutual information parameters from Hoyer et al. [20].Their algorithm usually works on blocks of 1024 beats of a 24 h ECG. Here we only used 512 beats. This is notcrucial to this study however, because these parameters are only used for comparison.

A third limitation is that the risk factors smoking, diabetes, hypertension and hypolipoproteinemia were onlyrecorded for the infarct patients, but not for the control group. It is therefore not possible to match these risk groupsduring classification. However, this would have led to more and even smaller subgroups.

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3 Results

3.1 Entropy Estimation Methods

Each entropy estimation method was applied with a range of configurations according to Table 4. The followingsections compare the results for each method individually across all three data set types. Section 3.1.6 comparesthem within data sets.

All results in this section are based on a quantisation step width of 1/128 = 7.8125 ms. All 180 recordings wereused because the entropy estimation for HRV signals in general is investigated. In order to visualise the results,20 recordings are chosen randomly in several places. The results have however been derived by examining allrecordings.

3.1.1 LZ77

From the LZ77 compression technique the ratio between the number of code words in the compressed sequenceand the number of symbols in the original sequence is taken as estimate of the entropy.

The two options are the bufferLength and the maxWordLength. The buffer is the adaptive window that LZ77operates in. BufferLength-maxWordLength is the length of the context that earlier occurrences of a signal piececan be drawn from.

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Figure 16: LZ77 entropy of one sample with varying bufferLength and maxWordLength

Figure 16 shows the LZ77 entropy of a sample recording with different bufferLengths and maxWordLengths.The entropy converges with increasing bufferLength and saturates at a bufferLength of around 400 beats. LargermaxWordLengths produce lower entropies. For this sample recording a maxWordLengths of more than 6 producenearly identical results. This indicates that the recurrences have limited length.

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Figure 17: LZ77 entropy of 20 random samples with varying bufferLength and fixed maxWordLength=12

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In order to visualise the influence of the bufferlength, Figure 17 shows the LZ77 entropy of 20 random record-ings for maxWordLength=12. The bufferLength dependency is very similar for all recordings. Figure 18 showsthe entropy of these 20 random recordings for a range of maxWordLengths and fixed maximum bufferLengths.The bufferLength is set to be equal to the length of the input signal. The entropy converges for increasing max-WordLengths. For most recordings the saturation is reached below a value of 8.

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Figure 18: LZ77 entropy of 20 random samples with varying maxWordLength and fixed maximum bufferLengths

The mean and standard deviation of entropies for the three data sets is given in Table 11. The Interp data setproduces particularly low entropies, which is caused by the regularity introduced during interpolation. The threedata sets are highly correlated.

Data Set Mean Std. Dev. Correlation&ConfidenceCut Interp Diff

Cut 0.3487 0.0799 1 0.0000 0.0000Interp 0.2691 0.0731 0.9664 1 0.0000Diff 0.3263 0.0875 0.9688 0.9690 1

Table 11: Statistics of LZ77 entropies with bufferLength=100 and maxWordLength=5 (Cut, Diff) or 7 (Interp)

For the further process, maxWordLength = 12 is used. The bufferLength for Cut and Diff will be 400 andfor Interp 1000. The shortest tachogram in this data set contained 480 beats, the equivalent of a pulse of 48/min.Assuming that a person has a pulse of at least 40 bpm, 400 beats can be recorded less than 10 min. A 10 minuterecording is then sufficient for this entropy analysis. 1000 values of the Interp data set are equal to 1000/2 = 500 s< 10 min.

3.1.2 NGRAM

The NGRAM technique estimates the overall entropy in bit. To obtain the entropy, the absolute one is divided bythe number of bits of the input. The option for NGRAMS is the context length N, which ranges from 1 to 6.

Figure 19 shows the entropy for 20 random recordings. The entropy estimation converges towards a specificvalue with increasing context size. For Cut and Diff this value is reached at N=3. For Interp convergence is slowerfor some samples.

Table 12 shows that the mean of the Interp entropy is lower than for Cut and Diff. This may be an effectof the interpolation. Diff entropies have a slightly lower mean than the Cut ones. This may be explained with theadditional recurrence produced by taking the difference signal. Entropies for all three data sets are highly correlatedagain.

For the further process, the 6GRAM is chosen for Cut, Diff and Interp. This contextLength is sufficient for thistype of signal.

3.1.3 ALRM

The adaptive linear regression model technique also estimates the overall entropy, but based on non-quantised data.It’s option is the contextLength ranging from 1 to 32.

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Figure 19: NGRAM entropies for 20 random recordings for contextLength 1..6

Data Set Mean Std. Dev. Correlation&ConfidenceCut Interp Diff

Cut 0.5715 0.1590 1 0.0000 0.0000Interp 0.4587 0.1552 0.9700 1 0.0000Diff 0.5286 0.1730 0.9704 0.9740 1

Table 12: Statistics of 6GRAM entropies

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Figure 20: ALRM entropies for 20 random recordings for contextLength 1..32

Figure 20 shows the entropy of 20 random sample recordings. Similarly to the NGRAM results, the entropyestimation converges for context sizes until about 10. But the entropy then slowly increases again with growingcontext size. Context sizes 10 and 32 are considered in Table 13. The difference in mean between context size 10and 32 is particularly strong for Diff and Cut. The mean for Interp is significantly lower. This can be explainedthrough the interpolation step. The regression works particularly well in the interpolated signal.

The correlation between the Cut and Diff is very high with a correlation coefficient of 0.994. Interp has a lowercorrelation to Cut and Diff, which is again due to it’s nature. The correlation coefficients between contextSize 10and 32 were more than 0.998 for each data set respectively. The gain in entropy from contextSize 10 to 32 doestherefore not strongly affect the characteristic of the result.

For the further process, the contextSize 10 is chosen. The Interp data set interferes with the ALRM techniqueand should not be used. Cut and Diff can be used.

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Data Set Mean Std. Dev. Correlation&ConfidenceCut Interp Diff

contextSize=10Cut 0.5673 0.2458 1 0.0000 0.0000Interp 0.2750 0.1783 0.9148 1 0.0000Diff 0.5208 0.2537 0.9937 0.9207 1

contextSize=32Cut 0.5826 0.2508 1 0.0000 0.0000Interp 0.2848 0.1805 0.9133 1 0.0000Diff 0.5526 0.2663 0.9937 0.9129 1

Table 13: Statistics of ALRM entropies with contextSize=10 and 32

3.1.4 GZIP

The GZIP compression technique produces a compressed file from an original file containing the signal. FirstLZ77 compression is a applied and then Huffman coding. The entropy is obtained by dividing the compressed sizeby the number of input bits. The option of GZIP determines the compression quality, by moderating the way thehash-table is built internally.

Figure 21 shows that there are only little differences between the quality options. The default and best (-9)option produce identical results.

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Figure 21: GZIP entropies for 20 random recordings for options fast (-1), default and best (-9)

The comparison in Table 14 shows that the mean of the Interp data set is considerably lower. This effect alsoshowed for the LZ77 approach. Here the Huffman coding eliminates additional redundancies and thus increases theeffect. Surprisingly, the correlation between Interp and Cut or Diff is higher than between Cut and Diff. Howeverall correlations are high again.

Data Set Mean Std. Dev. Correlation&ConfidenceCut Interp Diff

Cut 0.6145 0.1092 1 0.0000 0.0000Interp 0.5027 0.1184 0.9723 1 0.0000Diff 0.5762 0.1168 0.9462 0.9655 1

Table 14: Statistics of GZIP entropies with highest compression (-9)

For the further process, the best (-9) option should be chosen. This option does not compromise the systematicmethod for a benefit in runtime.

3.1.5 BZIP2

The BZIP2 compression technique uses Burrows-Wheeler block sorting and Huffman coding. Just like with GZIPeach signal has been saved as file and compressed. The ratio of compressed file size to original number of input

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bits is taken as entropy estimate. The option here is the block size. Because the signals always fit into the smallestblock size, there is only one configuration. This also means that the whole signal is processed at once in one block.

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Figure 22: BZIP2 entropies with blocksize=100kb for 20 random recordings

Figure 22 shows 20 random samples. The statistics are provided in Table 15. Again Interp has a lower entropymean than Cut and Diff. The entropies for the different data sets are all very highly correlated.

Data Set Mean Std. Dev. Correlation&ConfidenceCut Interp Diff

Cut 0.5981 0.1224 1 0.0000 0.0000Interp 0.4730 0.1090 0.9774 1 0.0000Diff 0.5746 0.1420 0.9811 0.9715 1

Table 15: Statistics of BZIP2 entropies with blockSize=100kb

For the further process, BZIP2 with a block size of 100kb should be considered for all three data sets.

3.1.6 Cut, Interp and Diff

The previous sections evaluated configurations for each entropy estimation technique individually. In order todetermine the consistency of the different entropy estimations, they need to be compared for each data set respec-tively.

Table 16 shows the correlation coefficients for the Cut data set. All techniques are highly correlated withcorrelation coefficients larger than 0.94. The correlation between LZ77, 6GRAM and GZIP is particularly high.One reason is that GZIP only differs from LZ77 through an additional Huffman coding step. The ALRM approachis least correlated with the other techniques.

LZ77400-12 6GRAM ALRM10 GZIP-9 BZIP2-100kbLZ77400-12 1 0.0000 0.0000 0.0000 0.00006GRAM 0.9944 1 0.0000 0.0000 0.0000ALRM10 0.9432 0.9552 1 0.0000 0.0000GZIP-9 0.9903 0.9893 0.9245 1 0.0000BZIP2-100kb 0.9650 0.9796 0.9673 0.9653 1

Table 16: Correlation coefficients (lower half) and confidence for Cut (upper half)

Table 17 shows the correlation coefficients for Interp. Again, the correlation between LZ77, GZIP and 6GRAMare very high. But ALRM is much less correlated with 0.87-0.94. This is due to the interference of interpolationand regression already mentioned.

As shown in Table 18, the correlation coefficients for Diff are the highest compared to Cut and Interp. Thelowest correlation coefficient is 0.9755 between LZ77 and ALRM.

The Diff data set differs from Cut in that successive differences between beat-to-beat intervals are used. Forinstance a drop from 950 ms to 900 ms is equal to a drop from 900ms to 850ms. This recurrence would not havebeen recognised in Cut. The Cut data set also presents variations in heart rate with a lower frequency.

For the further process, the data sets Cut and Diff should be considered, because they represent different aspectsof the signal. The interpolated data set has a lower quality, because additional information were introduced duringinterpolation. It is therefore not used further.

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LZ771000-12 6GRAM ALRM10 GZIP-9 BZIP2-100kbLZ771000-12 1 0.0000 0.0000 0.0000 0.00006GRAM 0.9965 1 0.0000 0.0000 0.0000ALRM10 0.9041 0.9188 1 0.0000 0.0000GZIP-9 0.9945 0.9888 0.8731 1 0.0000BZIP2-100kb 0.9793 0.9854 0.9405 0.9669 1

Table 17: Correlation coefficients (lower half) and confidence for Interp (upper half)

LZ77400-12 6GRAM ALRM10 GZIP-9 BZIP2-100kbLZ77400-12 1 0.0000 0.0000 0.0000 0.00006GRAM 0.9955 1 0.0000 0.0000 0.0000ALRM10 0.9755 0.9858 1 0.0000 0.0000GZIP-9 0.9895 0.9933 0.9841 1 0.0000BZIP2-100kb 0.9817 0.9914 0.9897 0.9888 1

Table 18: Correlation coefficients (lower half) and confidence for Diff (upper half)

The selected entropy estimations for the analysis part of this study are LZ77400-12, 6GRAM ALRM10, GZIP-9 and BZIP2100kb for Cut and Diff respectively. The first selection criterion was a low entropy estimation. Thesecond one was estimation reliability. Both have been achieved through the selection of configurations in theprevious sections and this one.

A key result of the is analysis is that all entropy estimation techniques produce consistent results for HRVsignals. The entropies are highly correlated. A second result is that using the equal-length tachogram (Cut) or theequal-length successive differences (Diff) are prospective normalisation methods and are better than resamplingthe signal to equidistant time steps (Interp). The interpolation during resampling introduces additional regularity.However, the resampled tachogram has it’s value in computing the frequency domain HRV parameters.

3.1.7 Convergence with Signal Length

One of the possible limitations in this study is the length of the input sequence for the entropy estimation. To testhow the entropy estimation depends on the input length and whether the current length is sufficient, the entropy of20 random recordings has been computed for lengths of 10 values, 20 values, 30 values and so on.

Figure 23 shows this progressive LZ77 entropy. The entropy decreases with increasing input length. It con-verges towards some value for each recording. The figure also shows that the estimated entropy only saturatesfrom lengths of about 250 beats or 200 seconds onwards. The convergence is not smooth, because different varia-tions in the input signal alter the entropy estimation once they are included. The vertical line marks the length thatall inputs are normalised to. It is the length of the shortest input. For Interp all inputs have a length of 540 seconds.

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0.4

0.6

0.8

1

beat interval

entr

opy

[LZ77400−12,Diff] progressive

Figure 23: LZ77 entropy of 20 random samples for different signal lengths

The convergence characteristics for the 6GRAM technique are shown in Figure 24. The initial entropy estima-tion is closer to the final value, but varies largely. After 250 beats or 200 s the estimation is steady. However, someinputs produce changes in the entropy estimation towards the end of the input. This indicates that the input signaldoes not contain all types of variability in their natural distribution.

33

0 500 10000

0.5

1

1.5

beat interval

entr

opy

[6GRAM,Cut] progressive

0 200 400 6000

0.5

1

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time in sec.

entr

opy

[6GRAM,Interp] progressive

0 500 10000

0.5

1

1.5

beat interval

entr

opy

[6GRAM,Diff] progressive

Figure 24: 6GRAM entropy of 20 random samples for different signal lengths

The ALRM technique in Figure 25 converges similarly to LZ77 and 6GRAM. But for Interp the entropy de-creases even after 250 beats. It does not saturate within the length of the recording. This is because the ALRMmethod is sensitive to the additional regularity introduced during interpolation.

0 500 10000

0.5

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1.5

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entr

opy

[ALRM10,Cut] progressive

0 200 400 6000

0.5

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entr

opy

[ALRM,Interp] progressive

0 500 10000

0.5

1

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entr

opy

[ALRM10,Diff] progressive

Figure 25: ALRM10 entropy of 20 random samples for different signal lengths

The GZIP and BZIP2 based entropies in Figure 26 and Figure 27 do not saturate completely within the nor-malisation length for Cut and Diff. A longer recording would help to estimate the final value. However, the relationbetween the recordings does not change significantly after the normalisation length. The entropies can thereforestill be used to discriminate groups of recordings.

0 500 10000

0.5

1

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entr

opy

[GZIP−9,Cut] progressive

0 200 400 6000

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entr

opy

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0 500 10000

0.5

1

1.5

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entr

opy

[GZIP−9,Diff] progressive

Figure 26: GZIP-9 entropy of 20 random samples for different signal lengths

The result of this test shows that the recording length of 10 minutes is sufficient for the classification task.Saturation of the estimated entropy is achieved with lengths of at least 250 beats for the LZ77, 6GRAM and

34

0 500 10000

0.5

1

1.5

beat interval

entr

opy

[BZIP2−100kb,Cut] progressive

0 200 400 6000

0.5

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opy

[BZIP2−100kb,Interp] progressive

0 500 10000

0.5

1

1.5

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entr

opy

[BZIP2−100kb,Diff] progressive

Figure 27: BZIP2 entropy of 20 random samples for different signal lengths

ALRM method for Cut and Diff. GZIP and BZIP2 do not saturate completely within the maximum of 480 beats,but enough to be used for classification.

35

3.2 Population Overview

The concept graph in Figure 28 gives an overview of the recorded population. Each node is a concept, which isdefined by a number of attributes, such as ’infarct’ or ’50-70 years’. The bottom label of each node shows thenumber of recordings for that concept. For example, there are 63 control recordings and 87 infarct recordings. 43of the infarcts were in patients between 50 and 70 years of age. And from those 26 had two or more of the riskfactors. The technique of creating concept graphs is introduced in Appendix A.

Figure 28: Concept graph of the data set using age, infarct vs. control and risk factor counts (all matches)

The concept graph shows that below 30 years of age, only controls were recorded. On the other hand above70 years all but one recording are from infarct patients. About half of the infarcts happened at the age of 50-70years. This is the most common infarct age. From the location of the ’≥ 2 risk factors’ attribute we can see that norisk factors were recorded for control recordings. Within the infarct patients about half had ≥ 2 risk factors in therespective age groups. This indicates a relation between the existence of risk factor and infarct.

Figure 29 shows the entropy for each of the 150 recordings spread across the age. The entropy is the measureof heart rate variability under investigation in this study. A high entropy means a high complexity or variation inthe tachogram. The figure shows that the average entropy decreases with increasing age. The 15.-30. day infarctsin red have a particularly low entropy as well.

0 20 40 60 800.3

0.4

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entr

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0 20 40 60 800.3

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entr

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Age vs. [BZIP2−100kb,Cut] for Men

0 20 40 60 800.3

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0.8

0.9

1

age in years

entr

opy

Age vs. [BZIP2−100kb,Cut] for Women

control1.−14.d15.−30.d>30.d

Figure 29: Age versus BZIP2 entropy

While male infarcts appear from 40 years onwards, most infarcts of women happened at ages of more than60 years. The entropy for 15. -30. day infarcts deviates stronger from the controls in women than it does in men.However the number of infarcts in women is very low.

36

3.3 Statistical Tests

The previous section examined the entropy estimation of HRV signals in general. For the classification task thecharacteristics of this data set are important. This section provides an overview and explains the important factorsin choosing the class members for the classification task. Statistical tests on difference in mean entropy are made.

From this point on only the recordings that are suitable according to Section 2.5 are used. These are 150 outof 180. The parameters in this multivariate analysis are the 10 combinations of entropy estimation techniques anddata sets, the time and frequency domain parameters and the other non-linear parameters.

Additionally, the ratio of each entropy estimation to meanNN is used as parameter, e.g. denoted as ’[LZ77,Cut]/m’.The reason is that the entropy itself is independent from the mean. If the input signal is shifted in mean, the entropyremains the same. Using the entropy-mean ratio combines both information. The same rationale stands behind theparameter cvNN.

3.3.1 Infarct Stage

Overview In Section 2.6 age groups and infarct stages were introduced. Table 19 shows the recording counts forthe individual subgroups. There are more infarcts in men than there are in women. It should therefore be noticedthat the control group for 50-70 years contains more women than men. The size of each subgroup should alwaysbe taken into account when interpreting the diagrams in this section.

Infarct Stage < 30y 30-50y 50-70y > 70y Allm f m f m f m f m f

control 2 5 16 11 11 17 1 - 30 331.-14. day - - 13 1 19 11 9 5 41 1715.-30. day - - 5 2 5 3 2 2 12 7> 30. day - - 1 - 6 - 1 2 8 2all 2 5 35 14 41 31 13 9 91 59

Table 19: Age and gender distribution in population

Entropy Parameters The next step was to compare the entropy between the various subgroups. The Wilcoxontest was used to determine whether two entropy means have a significant difference. This was done for eachage group within each infarct stage (row in Figure 30) and each infarct stage within each age group (column).Additionally, men and women were compared within each subgroup and inter-subgroup comparisons have beenrepeated for just men and just women as well. Subgroups with only one recording were considered too small.

Figure 30 visualises the results. Each box is a subgroup with a given size. Each arrow is a significant differencein mean entropy. For each arrow the one entropy parameter with the lowest p-value is attached along with theprobability of the null hypothesis (p-value*100 in %). In order to save space, the age group < 30 years was leftout. However, there is one significant difference in mean between control of ages < 30 years and 30-50 years.

The entropy parameters show significant age dependencies in the rows of the graph. For 1.-14. day infarctsthis could be established particularly well. Multiple arrows indicate that the difference is significant for separatedgenders as well. The significance is very high between 30-50 years and 50-70 years or > 70 years respectively for1.-14. day infarcts. The entropy- meanNN ratio dominates the pure entropy parameters. The age dependency is notsignificant in the control groups, except for < 30 years and 30-50 years. This can be explained by the fact that withincreasing age primarily the ability of the heart to adapt to events such an infarct is reduced. Figure 31 shows boxplots for the age groups in each infarct stage. The entropy statistically decreases with increasing age.

Figure 32 shows the trend of the entropy across the stages of an infarct. Infarct patients with ages 30-50 yearsshow little variation in mean, also compared to the controls. The absence of arrows in the first column ofFigure 30shows this as well. For 50-70 years the controls and 1.-14. day infarcts have a similar mean, but the 15.-30. dayand > 30. day infarcts have a reduced entropy. This results in the significances in the second column ofFigure 30.

Significant difference between men and women within subgroups could not be established. The strongest dif-ference showed for 15.-30. day infarcts for ages 50-70 years with a 7% probability that the null hypothesis is true.The number of recordings used is however very small. When considering just men, many of the previous significantdifferences are found as well, in particular the age dependency.

A range of entropy estimation techniques has been applied. However, in Figure 30 only the one with the lowestWilcoxon test result is given. Table 20 compares the test result of each entropy estimation technique for three

37

>70 years50-70 years30-50 years

1.-14. day

15.-30. day

control # = 27

> 30. day

# = 14

# = 4

# = 3# = 6

# = 8

# = 30

# = 1

# = 7

# = 14

# = 280.1% [ALRM,Cut]

0,8% [LZ77,Cut]

0.9% [BZIP2,Cut]

0.02%[BZIP2,Cut]/m

0.17% [LZ77,Cut]/m

0.001%[LZ77,Cut]/m

1.4%[BZIP2,Diff]

M: 0.14%[BZIP2,Cut]/m

M: 1.6% [LZ77,Cut]

M: 0.8%[ALRM,Cut]

M: 1.9% [ALRM,Cut]F: 0.8% [BZIP2,Cut]

M: 0.9% [GZIP,Diff]

F: 2.2% [LZ77,Cutf]

1.9%[GZIP,Diff]/m

0.02% [GZIP,Cut]/m

F: 2.75% [BZIP2,Cut]/m

M: 3.9% [GZIP,Cut]/m

F: 0.4% [GZIP,Cut]/m

Figure 30: Significant differences in entropy mean using a Wilcoxon test between age, infarct stage and gendergroups (F=female, M=male)

<30y 30−50y 50−70y >70y

5

6

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Figure 31: Entropy statistics across ages

control 1.−14.d15.−30.d >30.d0

0.2

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M10

,Cut

]

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control 1.−14.d15.−30.d >30.d0

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]

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control 1.−14.d15.−30.d >30.d0

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M10

,Cut

]

>70y

Figure 32: Entropy statistics across infarct stages

key subgroup distinctions. All entropy estimation techniques produce similar results. For the first distinction allentropies are not significant. But for the remaining two, they all show a significant difference in mean. The choiceof entropy estimation technique is therefore not crucial. The results are rather a characteristic for the entropy itself.

The entropy-meanNN ratios on the other hand produce significant differences for the first two distinctiontasks. Dividing by meanNN increases the overall discrimination strength, even though the particular discrimination

38

Control vs. LZ77 400-12 6GRAM ALRM10 GZIP-9 BZIP2-100kbCut Diff Cut Diff Cut Diff Cut Diff Cut Diff

1.-14. day 43.9 51.8 51.8 53.9 64.9 42.9 29.2 52.3 43.9 51.315.-30. day 0.1 0.1 0.1 0.2 0.1 0.1 0.1 0.1 0.1 0.2> 30. day 0.8 1.1 1.6 1.0 2.3 1.3 1.2 4.0 0.9 3.2

<entropy>/m1.-14. day 0.1 1.8 0.4 3.2 5.9 4.2 0.0 0.2 0.1 0.715.-30. day 0.7 1.7 0.5 1.0 0.3 0.3 5.0 5.0 3.8 2.6> 30. day 7.4 10.9 11.9 9.9 3.6 5.5 21.4 19.8 24.9 21.4

Table 20: Entropy across stages as Wilcoxon test in % (50-70 years)

strength for control vs. 15.-30. day is not as high. The effect of dividing by meanNN for age differences is shownin Table 21. The entropy-meanNN ratios produce much more significant differences between age groups than theplain entropy estimations.

1.-14. day with Cut LZ77 400-12 6GRAM ALRM10 GZIP-9 BZIP2-100kb*1 /m *1 /m *1 /m *1 /m *1 /m

30-50y vs. 50-70y 6.58 0.36 6.05 0.36 7.16 1.40 3.02 0.04 3.88 0.0250-70y vs. > 70y 0.98 0.17 1.22 0.31 5.08 3.54 1.05 3.54 1.97 3.1230-50y vs. > 70y 0.21 0.00 0.19 0.00 0.63 0.03 0.14 0.00 0.38 0.00

Table 21: Entropy vs entropy-meanNN-ratios in Wilcoxon tests in %

Time and Frequency Parameters For comparison all Wilcoxon tests have been computed for each of the timeand frequency parameters that are part of the taskforce standard. Figure 33 visualises the results. This time eacharrow has attached all parameters that have a p-value of less than 0.05 ranked according to the p-value. Again,a significant difference between < 30 years and 30-50 years for controls exists. It appears in HF, HF/P, LF/HF,cvNN, pNN50, rmssd, pNNi20, pNNi10, sdNN and sdaNN1. It has been omitted to save space.

>70 years50-70 years30-50 years

1.-14. day

15.-30. day

Control # = 27

> 30. day

# = 14

# = 4

# = 3# = 6

# = 8

# = 30

# = 1

# = 7

# = 14

# = 28

cvNN,meanNN,

LF,sdaNN1

F: meanNN, VLF,HF,pNNi20,sdNN,rmssd

pNNi10,rmssd,pNNi20,HF,sdNN,pNN50,LF, cvNN

pNNi10, rmssd, pNN50,pNNi20

meanNN,pNNi20,rmssd,pNNi10,HF,sdNN

HF, HF/P, LF

sdaNN1, cvNN,sdNN, LF, LF/P

LF, pNNi10,pNNi20,HF

cvNN,HF,LF,sdNN,pNNi10,VLF,rmssd,

pNNi20,sdaNN1,LF/P

pNN50,LF,pNNi20,sdNN,cvNN,rmssd,

pNNi10,HFLF,pNN50,

sdNN,cvNN,pNNi20

F vs M: HF/P, LF/HF, LF

F vs M: sdNN

F: HF/P

cvNN,sdaNN1,sdNN

M:sdNN,cvNN

M: sdNN,rmssd,pNN50,pNNi10,pNNi20,HF,meanNN,cvNN,LF

F: cvNN,sdaNN1,sdNN, LF

M: pNNi10,rmssd,HF,pNN50,pNNi20

F: sdNN,rmssd,VLF,HF,pNNi10,pNNi20,cvNN,LF,meanNN,sdaNN1

M: rmssd,pNN50,pNNi10,pNNi20

M: meanNN

M: LF/HF

Figure 33: Significant (< 5%) differences of time and frequency parameter means for age, stage and gender groups.Parameters are ranked according to the Wilcoxon test result (F=female, M=male)

39

Similarly to the entropy results in Figure 30 the infarct stages for ages 50-70 years show significant differencesin mean. The same age group differences are found. An additional difference in mean between men and women inthe 30-50 years control group was found in the frequency parameters.

In both, the time and frequency parameters and the entropy ones, no significant difference between controlsand respective infarct stages could be found in the 30-50 year group. This indicates that the heart rate in these agesis not affected severely by a myocardial infarct.

Figures Figure 34 and Figure 35 show the standard deviation as parameter. The same age dependency and stagecharacteristics seen for the entropy show.

<30y 30−50y 50−70y >70y0

20

40

60

80

sdN

N

sdNN for control

30−50y 50−70y >70y0

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sdNN for 1.−14.day

30−50y 50−70y >70y0

20

40

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N

sdNN for 15.−30.day

Figure 34: sdNN statistics across ages

control 1.−14.d15.−30.d >30.d0

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sdNN for 30−50y

control 1.−14.d15.−30.d >30.d0

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control 1.−14.d15.−30.d >30.d0

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NsdNN for >70y

Figure 35: sdNN statistics across infarct stages

In Figures Figure 36 and Figure 37 the parameter pNNi is pictured. pNNi10 is the proportion of the number ofinterval differences of successive intervals smaller than 10 ms to the total number of NN intervals. With increasingage, pNNi10 increases. This indicates that the variability of the heart rate decreases, which conforms with thegeneral understanding of HRV. Across stages pNNi10 behaves inversely to the entropy. For 15.-30. day infarctspNNi10 is increased, which indicates a reduced variability.

<30y 30−50y 50−70y >70y0

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l10

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Figure 36: pNNi10 statistics across ages

Non-linear Parameters The results of the other non-linear parameters are depicted in Figure 38. The Renyientropies show similar significant differences to the entropy parameters developed here. A difference between

40

control 1.−14.d15.−30.d >30.d0

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Figure 37: pNNi10 statistics across infarct stages

control and > 30. day infarcts for 50-70 years of age could not be established. This not the case for the entropyparameters. On the other hand a significant difference between control and 1.-14. day infarcts is found in renyi025.The probabilities for each of the distinctions are similar to the ones of the entropy parameters.

The auto mutual information parameters only yielded one significant difference in mean between controls and15.-30. day infarcts for men of ages 50-70 years. One explanation could be the small length of the recordings. Thecomputation of these parameters was originally based on more than 1024 beats.

>70 years50-70 years30-50 years

1.-14. day

15.-30. day

Control # = 27

> 30. day

# = 14

# = 4

# = 3# = 6

# = 8

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# = 1

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F: renyi025,renyi2,renyi4

renyi025

renyi025,renyi4,renyi2

renyi4,renyi025,

renyi2

renyi025,renyi2 renyi2,renyi4,renyi025

renyi025,renyi2,renyi4

renyi025,renyi2,renyi4

renyi025,renyi2,renyi4

M vs F: renyi025,

renyi2

M: renyi025 M: renyi2,renyi4,renyi025

M: renyi025,renyi2,renyi4

F: renyi025 F: renyi025,renyi2,renyi4

M: AMIFpdLF

Figure 38: Significant (< 5%) mean differences of the other non-linear parameters means for age, stage and gendergroups. Parameters are ranked according to the Wilcoxon test result. (F=female, M=male)

Wilcoxon Tests for LZ77 Configurations The last step in this statistical analysis was to determine the Wilcoxontest result as a function of the configuration of the LZ77 entropy estimation technique. This breaks the entropyconcept, because not the best entropy estimation is used. But the configuration with the best Wilcoxon test resultis obtained, which is a good candidate for a discriminator. This was done for the four classification tasks controlvs. 1.-14. day and control vs. 15.-30. day for 30-50 years and 50-70 years respectively.

Figure 39 shows a selection of the interesting results. The plot in the middle shows the typical performance.Very small bufferLengths and maxWordLengths tend to either lead to very low or very high probabilities. Thesemarginal probabilities should not be regarded as candidate. The saturated entropy also shows a low probability.Here the best entropy estimation is also the best discriminator.

41

Figure 39: Wilcoxon test probability for LZ77 configurations for selected problems

The left plot shows the probability for control vs. 1.-14. day for 30-50 years. The probability is lowest forbufferLengths around 50. However, the probabilities are always above 40%, so the distinction is not significant.The third plot of controls vs. 15.-30. day for 50-70 years shows that the probability is particularly low for max-WordLength=4. There is a valley throughout the bufferLengths. In this one case the best discriminator is not thebest entropy approximation. The difference is only about 0.05%, but this is half of the probability of the bestentropy estimation. It is important to note that in this case the Diff data set was used, which breaks long term recur-rences into shorter term ones. For the further process, the configuration [LZ77400-4,Diff] is added to the parameterset.

Parameter Comparison The key differences between controls and 1.-14. day or 15.-30. day or > 30. dayrespectively are found to be significant by several parameters. Table 22 compares the Wilcoxon test results ofa number of these parameters. The LZ77 entropy performs similarly to rmssd, pNN50 and pNNi10. Only thedifference between control and 1.-14. day infarcts is not found to be significant. MeanNN shows the inverse:control vs. 1.14. day is significant, whereas control vs. > 30. day is not significant.

control vs 1.-14. day 15.-30. day > 30. day[LZ77,Cut] 49.84 0.12 0.76[LZ77,Cut]/m 0.12 0.73 7.43meanNN 1.25 6.50 87.44sdNN 11.42 0.28 19.79cvNN 0.73 0.81 35.44rmssd 68.01 0.06 1.56pNN50 47.65 0.31 1.75pNNi10 77.34 0.04 1.21renyi025 4.56 0.11 11.91

Table 22: Key parameters across stages as Wilcoxon test in % (50-70 years)

The entropy-meanNN-ratio combines the strengths of the entropy and meanNN. The first two distinctions arefound to be highly significant. The difference between control and > 30. day is not significant at the 5% level,but has a very low probability of 7.43%. Figure 40 shows a boxplot of the subgroups. The number of elementsin each subgroup should be considered when interpreting this plot. A similar effect can be observed in cvNN.The significance increases if sdNN is divided by meanNN. The entropy-meanNN-ratio also outperforms the renyientropies in terms of overall significance.

As a result, the entropy estimations are shown to have a similar discrimination strength to many of the time do-main parameters. However, the entropy divided by the meanNN produces the strongest significances. MeanNN isa moderating variable for the entropy and other parameters. Secondary results are that the entropy estimations pro-duce consistent significances and that in most cases the best entropy estimation also produces the best significancesfor the key subgroup distinctions.

42

control 1.−14.d15.−30.d >30.d2

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7740

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]/m

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]/m

>70y

Figure 40: Statistics of entropy-meanNN-ratios across infarct stages

3.3.2 Infarct Location

Overview An alternative categorisation of the infarcts is the location of the infarct. In this data set only anteriorand posterior are differentiated. Table 23 shows the sizes of the resulting subgroups.

Infarct Location < 30y 30-50y 50-70y > 70y Allm f m f m f m f m f

control 2 5 16 11 11 17 1 - 30 33anterior - - 12 2 14 8 7 6 33 16posterior - - 7 1 16 6 5 3 28 10all 2 5 35 14 41 31 13 9 91 59

Table 23: Age and infarct location distribution in population

Entropy Parameters Wilcoxon tests have been applied in the same manner that was used in the previous section.The results for the entropy parameters are displayed in Figure 41. There significant entropy differences betweenage groups of anterior and posterior infarcts. The entropy difference between controls and anterior infarcts andcontrols and posterior ones is highly significant for 50-70 years.

>70 years50-70 years30-50 years

anterior

posterior

control # = 27

# = 13

# = 8# = 22

# = 22

# = 8

# = 14

# = 1# = 28

4,3%[BZIP2,Cut]/m

0.3% [LZ77,Cut]/m0.03% [GZIP,Cut]/m

0.01%[ALRM,Cut]/m

0.01%[6GRAM,Cut]/m

0.2%[BZIP2,Cut]/m

4.1%[ALRM,Cut]

0.2%[BZIP2,Cut]/m

M: 0.02%[ALRM,Cut]/m

M: 3.0%[GZIP,Cut]

M: 0.1%[GZIP,Cut]/m

F: 2.4%[LZ77,Cut]

F: 0.6% [LZ77,Cut]/m

M: 4.1% [GZIP,Cut]/m

F: 0.9% [GZIP,Cut]/m

Figure 41: Significant differences of entropy mean according to a Wilcoxon test between age subgroups, infarctlocation and gender (F=female, M=male)

The important observation is however that anterior and posterior infarcts do not have a significant entropydifference. Statistically, the entropy does not vary significantly between anterior and posterior infarcts.Figure 42

43

on the left shows a corresponding boxplot for ages 50-70 years. The boxplot in the middle and on the right showthe entropy across ages.

control anterior posterior

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posterior

Figure 42: Entropy statistics for infarct locations

Table 24 compares the Wilcoxon test results of the different entropy estimations. The plain parameters have avery similar characteristic. However, the significances are not very high for control-anterior and not even below the5% threshold for control-posterior and anterior-posterior. The entropy-meanNN-ratios also show consistent results,but with much stronger significances for control-anterior and control-posterior.

LZ77 400-12 6GRAM ALRM10 GZIP-9 BZIP2-100kbCut Diff Cut Diff Cut Diff Cut Diff Cut Diff

control vs. anterior i. 3.7 3.0 3.4 2.5 2.2 2.2 2.9 2.5 4.0 4.5control vs. posterior i. 11.1 20.7 17.4 20.0 30.5 23.7 8.9 18.1 13.7 21.5anterior vs. posterior i. 37.9 43.1 29.6 39.2 25.5 24.5 42.5 32.4 38.5 34.8

<entropy>/mcontrol vs. anterior i. 0.3 1.0 0.3 1.3 0.6 0.4 1.0 0.8 0.9 1.3control vs. posterior i. 0.1 2.1 0.5 2.5 3.9 4.1 0.03 0.4 0.2 1.0anterior vs. posterior i. 89.7 87.8 87.8 99.0 46.0 44.6 76.9 95.3 71.6 84.2

Table 24: Entropy across locations as Wilcoxon test in % (50-70 years)

Time and Frequency Parameters The performance of the time and frequency parameters is shown in Figure 43.The age difference is confirmed by a range of parameters for anterior infarcts. However, the age differences forposterior infarcts are not reflected as well as with the entropy parameters. Again the control-anterior and control-posterior distinction is found, but no significant difference between anterior and posterior infarcts.

The parameter cvNN is a strong predictor for control vs. anterior and control vs. posterior. It is depicted inFigure 44 on the left. PNNi10 is a good predictor for control vs. anterior only.

Non-linear Parameters A final look at the remaining non-linear parameters inFigure 45 shows an interestingresult: the auto mutual information (AMIF) parameters are sensitive to posterior infarcts. AMIFarea and AMIFde-cay produce significant differences for control vs. posterior infarcts and anterior vs. posterior ones for 30-50 yearsof age. This is interesting, because significant differences for early infarcts could only rarely be established usingother parameters. Additionally, no other parameters yield significant differences between anterior and posteriorinfarcts.

Figure 46 shows boxplots for the AMIF parameters. These three AMIF parameters seem to be sensitive to pos-terior infarcts. AMIFpdLF however has a very high variance and does therefore not yield a significant difference.But to show the tendency, the AMIF parameter results between 5% and 10% were included in Figure 45. Moresamples could help establish a significant difference. A second approach to reduce the variance of AMIFpdLF isto use a longer tachogram.

Parameter Comparison Table 25 compares the key parameters for the distinctions of infarct location. Theentropy-meanNN-ratio outperforms all other parameters for the first two distinctions. The distinction betweenanterior and posterior infarcts is only found to be significant using the AMIFpdLF parameter for 50-70 years ofage. For 30-50 years AMIFarea and AMIFdecay are good discriminators.

44

>70 years50-70 years30-50 years

anterior

posterior

control # = 27

# = 13

# = 8# = 22

# = 22

# = 8

# = 14

# = 1# = 28

F: meanNN,rmssd,pNNi10,pNNi20,LF/P

meanNN

HF/P

cvNN,pNNi10,LF,rmssd,sdNN,HF,

pNN50,pNNi20,VLF

cvNN,LF,meanNN

sdaNN1

HF,HF/P,LF

sdNN,pNNi10,rmssd,cvNN,HF,LF,pNNi20,pNN50,VLF,sdaNN1

cvNN,SdNN,LF,pNNi20,HF,rmssd,pNNi10,

sdaNN1,VLF,pNN50meanNN,

pNNi10,pNNi20

M vs F: HF/P,LF/HF,LF

M vs F: LF,cvNN,sdNN,sdaNN1

F vs M:HF/P

F: HF/P

M: rmssd,pNNi10,pNNi20,sdNN,pNN50,

HF,cvNN,LF,VLF

M: sdNN,cvNN,LF,sdaNN1,pNNi10

M: meanNNF: cvNN,sdaNN1,sdNN,LF F: cvNN

Figure 43: Significant (< 5%) differences of time and frequency parameter means between infarct location, ageand gender groups. Parameters are ranked according to the Wilcoxon test result (F=female, M=male)

control anterior posterior

0.02

0.04

0.06

0.08

0.1

cvN

N

cvNN for 50−70y

control anterior posterior

0.2

0.4

0.6

0.8

1

pNN

l10

pNNi10 for 50−70y

control anterior posterior10

20

30

40

50

60

70

80

sdN

N

sdNN for 50−70y

Figure 44: Time and frequency parameter statistics for infarct locations

The result of this section is that there is no significant difference between anterior and posterior infarcts in en-tropy and time and frequency parameters. Only the AMIF parameters yield a significant differences. The distinctionbetween controls and anterior or posterior infarcts however found to be significant.

45

>70 years50-70 years30-50 years

anterior

posterior

control # = 27

# = 13

# = 8# = 22

# = 22

# = 8

# = 14

# = 1# = 28

AMIFarea,AMIFdecay

AMIFarea,AMIFdecay

renyi025,renyi4,renyi2

renyi025

AMIFpdLFrenyi025,renyi2,renyi4,AMIFarea

renyi025,renyi2,renyi4,AMIFarea,AMIFpdLF

M vs F: renyi2,renyi025

F vs M: AMIFdecay

M: renyi025,renyi2,renyi4,AMIFarea,

AMIFdecay

M: renyi4,renyi2,reny025,AMIFpdLF

F: AMIFdecay

M: AMIFarea F: renyi025,renyi2,renyi4

F: renyi025

7.1% AMIFpdLF

8,7%AMIFdecay

M: 5.8%AMIFpdLF

M: 5.8% AMIFpdLF

AMIFpdLF

Figure 45: Significant (< 5%) mean differences of the other non-linear parameters means for infarct location, ageand gender. Parameters are ranked according to the Wilcoxon test result (F=female, M=male)

control anterior posterior0.4

0.5

0.6

0.7

0.8

0.9

1

AM

IFde

cay

AMIFdecay for 30−50y

control anterior posterior0

2

4

6

8

10

AM

IFar

ea

AMIFarea for 30−50y

control anterior posterior

0.7

0.75

0.8

0.85

0.9

AM

IFpd

LF

AMIFpdLF for 50−70y

Figure 46: AMIFdecay, AMIFarea and AMIFpdLF across infarct locations

control-anterior control-posterior anterior-posterior[LZ77,Cut] 3.73 11.11 37.86[LZ77,Cut]/m 0.29 0.12 89.73meanNN 91.44 4.73 12.42sdNN 3.08 5.42 50.35cvNN 1.20 0.86 89.73pNNi10 1.34 41.73 23.59renyi025 0.91 2.40 50.35AMIFpdLF 48.78 7.06 2.65

Table 25: Key parameters across locations as Wilcoxon test in % (50-70 years)

46

3.3.3 Risk Factors

The categorisation of the patients could also be made using the risk factors smoking, hypertension, diabetes andhypolipoproteinemia. Because risk factors are only moderating variables they are summarised more briefly in thissection.

Overview Figure 47 on the left shows a concept graph of the risk factors. The top node represents the patients withno risk factors, the layer below the ones with one risk factor and so on. The numbers are counts of exact matches.For example, there are 7 patients which are just smokers, 11 patients that smoke and have hypolipoproteinemia,but no patient which is a smoker and has hypertension, but no other risk factors. The right side shows the samegraph with nested age groups.

Figure 47: Concept graph of risk factors for infarct patients with exact counts (left) and with nested age groups(right).

The concept graphs show that the risk factors are spread relatively evenly across the subgroups. Each subgroupsonly contains a relatively small number of patients. This is particularly the case if age groups are taken into account.Therefore patients with the same number of risk factors are grouped together. Table 26 shows the counts for thisgrouping. The existence of risk factors in the control group is not available.

Risk Factors < 30y 30-50y 50-70y > 70y Allm f m f m f m f m f

control 2 5 16 11 11 17 1 - 30 330 factors - - 2 2 6 3 1 2 9 71 factor - - 5 1 6 2 3 3 14 62 factors - - 8 - 12 3 8 4 28 73 factors - - 4 - 5 6 - - 9 64 factors - - - - 1 - - - 1 -All 2 5 35 14 41 31 13 9 91 59

Table 26: Age and risk factor count distribution in population

Entropy Parameters The Wilcoxon test results between these subgroups are depicted in Figure 48. For ages 50-70 years significant differences between controls and all risk factor groups are found in the entropy-meanNN-ratio

47

parameters. Between the risk factor groups several but not all differences are found to be significant. Figure 49shows how the entropy and entropy-meanNN-ratios decrease with increasing number of risk factors. Significantage group differences are found for one and two risk factors.

>70 years50-70 years30-50 years

0 factors

1 factor

control # = 27

2 factors

# = 3

# = 6

# = 12# = 15

# = 8

# = 9

# = 8

# = 6

# = 4

# = 28

3 factors # = 11# = 4

2.7%[LZ77,Cut]/m

3.0% [GZIP,Cut]/m

0.1%[BZIP2,Cut]/m

1.2% [GZIP,Cut]/m

4,9% [BZIP2,Cut]/m

1.8%[BZIP2,Cut]/m

0.2%[LZ77,Diff]/m

0.04%[BZIP2,Cut]/m

0.1% [6GRAM,Cut]/m

1,8%[BZIP2,Cut]/m

M vs F: 2.4%[GZIP,Cut]/m

F vs M: 0.8%[BZIP2,Cut]/m

M: 3.6%[LZ77,Cut]/m

M: 0.1%[BZIP2,Cut]/m

M: 0.1% [LZ77,Cut]

M: 1,5% [6GRAM,Cut]/m

M: 1.8%[BZIP2,DIff]/m

M: 1.0% [BZIP2,Cut]/m

M: 2.4%[GZIP,Cut]/m

F: 2.6%[LZ77,Cut]

F: 3.4% [LZ77,Cut]/m

F: 1.2%[LZ77,Cut]

F: 1.9% [GZIP,Cut]/m

Figure 48: Significant (< 5%) differences of entropy mean for risk factor counts, age and gender. Parameters areranked according to the Wilcoxon test result (F=female, M=male)

control 0 1 2 3

0.4

0.5

0.6

0.7

0.8

0.9

[GZ

IP−

9,C

ut]

50−70y

control 0 1 2 3

5

6

7

8

9

10x 10

−4

[GZ

IP−

9,C

ut]/m

50−70y

Figure 49: Entropy statistics for risk factor counts

Time, Frequency and Nonlinear Parameters The time, frequency and AMIF parameters capture similar groupdifferences. CvNN and renyi025 produce the highest significances between risk factor counts for ages 50-70 years.Figure 51 shows how both parameters statistically decrease with increasing risk factor counts.

For 30-50 years of age the parameters LF/P and AMIFpdLF produce significant differences between risk factorcounts. However there is not a continuous trend across risk factor count groups as Figure 52 shows. The differencesin LF/P and AMIFpdLF could therefore also be the result of an incidental grouping of individuals.

48

>70 years50-70 years30-50 years

0 factors

1 factor

control # = 27

2 factors

# = 3

# = 6

# = 12# = 15

# = 8

# = 9

# = 8

# = 6

# = 4

# = 28

3 factors # = 11# = 4

LF/P,HF/P

AMIFpdHF

HF

AMIFpdHF

LF/P

LF/P,AMIFpdHF

cvNN,LF,renyi025,sdNN,renyi2,

renyi4

renyi025,pNN50,sdNN,cvNN,renyi2,renyi4,pNNi10,rmssd

AMIFarea

cvNN,LF

AMIFpdHF

HF,HF/P, LF

pNNi20,sdaNN1,mssd,pNNl10,HF,sdNN,renyi025,

cvNN,renyi2

cvNN,LF,renyi025,sdaNN1,renyi2,renyi4,LF/P,VLF,HF

meanNN,AMIFpdHF

cvNN,renyi025,HF,LF,sdNN,VLF, renyi2,

renyi4,pNNi10,pNNi20,rmssd,

sdaNN1,pNN50,AMIFarea

cvNN,renyi025

M vs F: HF/P,LF/HF,LF

F: HF/P

M: sdaNN1

M: cvNN,sdNN,renyi025,LF,renyi2,renyi4,sdaNN1,VLF,HF,

rmssd,pNNi20,pNN50 M:meanNN,AMIFpdHF

M: HF,renyi025,cvNN,sdNN,pNNi10,LF,renyi2,VLF,renyi4,pNNi20,rmssd,

pNN50,meanNN,sdaNN1,AMIFpdHF,

M: LF/P

M: AMIFpdHF

M: AMIFpdHF

M: LF,cvNN,renyi025

M: HF/P

M: cvNN, sdNN,sdaNN1,renyi2,reny025, VLF,

renyi4

F: pNNi20

F: sdNN,pNNi10,VLF,LF,renyi025,renyi2,renyi4,cvNN,sdaNN1,rmssd,

pNNi20,HF

F: LF/HF,cvNN

F: cvNN, sdNN,renyi025,VLF,renyi2,LF, renyi4

Figure 50: Significant (< 5%) differences of time, frequency and other non-linear parameter means for risk factorcounts, age and gender. Parameters are ranked according to the Wilcoxon test result (F=female, M=male)

control 0 1 2 3

0.02

0.04

0.06

0.08

0.1

cvN

N

cvNN for 50−70y

control 0 1 2 3

3

3.5

4

4.5

5

5.5

reny

i025

renyi025 for 50−70y

Figure 51: cvNN and renyi025 statistics for risk factor groups and 50-70 years

Parameter Comparison Table 27 compares the key parameters for the distinction of risk factor counts. TheWilcoxon test result of the entropy, sdNN, cvNN and renyi025 clearly decreases with increasing risk factor count.This indicates that the difference in these parameters between control and a risk factor group increases with thenumber of risk factors. The entropy-meanNN-ratios show significant or near significant results for all distinctions.The entropy-meanNN-ratio has the best overall performance.

The result of this section is that the entropy and other parameters statistically decrease with increasing numberof risk factors. The difference between consecutive groups could not always be found significant. This is partly dueto the small group sizes. However, the difference between control and risk factor groups increases with increasingnumber of risk factors in that group. The entropy-meanNN-ratio shows the highest significances.

49

control 0 1 2 3

0.1

0.2

0.3

0.4

0.5

LF/P

LF/P for 30−50y

control 0 1 2 3

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

AM

IFpd

HF

AMIFpdHF for 30−50y

Figure 52: LF/P and AMIFpdHF statistics for risk factor groups & 30-50 years

c↔0 c↔1 c↔2 c↔3 0↔2[LZ77,Cut] 62.00 15.90 7.23 2.36 38.70[LZ77,Cut]/m 3.84 6.50 0.08 1.70 8.38meanNN 29.64 89.41 7.24 88.83 51.19sdNN 98.59 17.68 0.77 1.85 6.45cvNN 55.92 14.29 0.06 2.01 3.18renyi025 71.01 17.68 0.38 0.39 7.36

Table 27: Key parameters across risk factor counts as Wilcoxon test in % (50-70 years, c= control)

50

3.3.4 Medication

Overview The heart medication of infarct patients may have an influence on the heart rate variability as well andis therefore analysed in this section. The drugs considered are betablockers, digitoxin and other antiarrhythmica.There are two main groups. The first are 20 infarct patients without any of those drugs. The second are 61 patientstreated with just betablockers. Only very few individuals had digitoxin or other antiarrhythmica or combinations.Figure 53 shows the corresponding concept graph. In this section only the two main subgroups are considered.Table 28 provides the counts as reference.

Figure 53: Concept graph of types of medication within infarcts. The counts show exact matches.

Medication < 30y 30-50y 50-70y > 70y Allm f m f m f m f m f

control 2 5 16 11 11 17 1 - 30 33no med. - - 3 3 6 2 2 4 11 9betablocker - - 15 - 23 10 8 5 46 15all 2 5 34 14 40 29 11 9 87 57

Table 28: Age and risk factor count distribution in population

Entropy Parameters Figure 54 shows that in the entropy parameters a significant difference between controlsand betablocker-patients can be established for ages 50-70 years. However the difference between controls andpatients with no heart medication was not found significant. The difference between betablocker-patients andpatients without medication is also only found significant for men between 50-70 years.Figure 55 shows that thesubgroup with no medication lies between the controls and the betablocker-subgroup. This explains why onlybetween opposite ends a significant difference could be established.

The degree to which the difference in entropy between betablocker patients and patients with no medicationcan be attributed to the medication is however not clear. The reason why a patient did not have heart medication isnot known.

Time, Frequency and Nonlinear Parameters Figure 56 shows the Wilcoxon test results for the alternative pa-rameters. The differences that are found significant are very similar to the ones of the entropy parameters. CvNN,renyi025 and sdNN produce the highest significances. They are pictured inFigure 57. All three parameters are lowfor the betablocker group and high for the control group with the no-medication-patients in between.

Parameter Comparison Table 29 again compares the discrimination strength of the key parameters. All parame-ters show a higher significance for control vs. betablocker. The entropy-meanNN-ratios, sdNN, cvNN and renyi025are below the 5% threshold. However, the significance is highest with the normalised entropy. It also shows muchlower probabilities for the other two distinctions. It thus outperforms the other parameters.

The result is that the entropy-meanNN-ratio is the best discriminator for medication groups. The particulareffect of a drug on the HRV can not be exactly determined the patients with no medication are not a control groupin this sense. The decision of not giving any heart medication was not made for the purpose of this study.

51

>70 years50-70 years30-50 years

no med.

beta-blocker

control # = 27

# =6

# = 13# = 33

# = 8

# = 15

# = 6

# = 1# = 28

M: 2,9%[GZIP,Cut]/m

2.2%[GZIP,Diff]/m

0.02% [GZIP,Cut]/m

2.9%[LZ77,Cut]/m

4.1%[GZIP,Diff]/m

0.02%[BZIP2,Cut]/m

1.2%[LZ77,Cut]/m

0.0002%[LZ77,Cut]/m

M vs F: 3.0%[GZIP,Cut]/m

M: 2,4%[LZZ77,Cut]

M: 0.1%[LZ77,Diff]/m

F: 1.3%[BZIP2,Cut]/m

M: 0.03%[LZ77,Cut]/m

M: 3,6% [GZIP,Cut]/m

F: 0.6% [GZIP,Cut]/m

Figure 54: Significant (< 5%) differences of entropy mean for medication types, age and gender. Parameters areranked according to the Wilcoxon test result (F=female, M=male)

control none betablocker

0.4

0.5

0.6

0.7

0.8

0.9

[GZ

IP−

9,C

ut]

50−70y

control none betablocker4

5

6

7

8

9

10x 10

−4

[GZ

IP−

9,C

ut]/m

50−70y

Figure 55: Entropy statistics for medication groups in ages 50-70 years

c↔none c↔betablocker none↔betablocker[LZ77,Cut] 32.22 8.24 76.71[LZ77,Cut]/m 14.29 0.04 10.34meanNN 74.64 10.98 63.34sdNN 48.15 3.05 52.12cvNN 39.19 0.35 24.29renyi025 23.07 1.02 75.46

Table 29: Key parameter across medications as Wilcoxon test in % (50-70 years)

52

>70 years50-70 years30-50 years

no med.

betablocker

control # = 27

# =6

# = 13# = 33

# = 8

# = 15

# = 6

# = 1# = 28

LF/P

cvNN,LF,renyi025,sdNN,sdaNN1

AMIFpdHF

HF, HF/P,LF

AMIFpdHF

renyi025,cvNN,sdNN,LF,renyi2,renyi4,sdaNN1,HF,rmssd,LF/P,

pNN50,VLF,pNNi20,pNNi10

renyi025,HF,cvNN,LF,sdNN,rmssd,VLF,renyi2,pNNi10,pNNi20,renyi4,LF/P,pNN50

M vs F: HF/P,LF/HF,LF

F: HF/P

M: rmssd,pNNi10,pNNi20,cvNN,

pNN50,renyi025,renyi4

M: renyi025,cvNN,sdNN,LF,HF,rmssd,

renyi2,renyi4

M: HF,renyi025,cvNN,pNNI10,sdNN,rmssd,LF,pNNi20,renyi2,renyi4,pNN50

F: LF/HF,AMIFdecay

F: cvNN,renyi025,sdNN,LF,sdaNN1,renyi2

Figure 56: Significant (< 5%) differences of time, frequency and other non-linear parameters for medication, ageand gender. Parameters are ranked according to the Wilcoxon test result (F=female, M=male)

control none betablocker10

20

30

40

50

60

70

80

sdN

N

50−70y

control none betablocker

0.02

0.04

0.06

0.08

0.1

cvN

N

50−70y

control none betablocker

3

3.5

4

4.5

5

5.5

reny

i025

50−70y

Figure 57: Statistics of key time and non-linear parameters for medications and 30-50 years

53

3.4 Infarct Classification

This section aims to generate classifiers for infarct vs. controls. Decision trees are used as classifier model, becausethey show concrete classification rules that can be interpreted in the clinical context. It is possible to see whichparameters are chosen and how they are combined. The classification rate is used as a criterion to assess the gainthat is achieved by adding the entropy parameters. The setting of the classification experiments are described indetail in Section 2.8.

3.4.1 Statistical Classifiers

A disadvantage of decision trees is that they can not model highly dimensional separating hyperplanes very well.This may result in lower classification rates than theoretically possible. Therefore their classification rate is com-pared to the maximal classification rate achieved by a statistical classifier, such as a Gaußor Nearest-Neighbour-classifier. If the classification rate of a tree matches the statistical one, the performance loss by using decision treesis minimal.

An initial examination of the statistical classification results shows that the classification rate in each of therespective classification tasks is virtually the same for the three parameter sets fromTable 10. Figure 58 shows thisfor three selected tasks.

1 2 3 4 5 6 7 80

20

40

60

80

1003070:c,mi

clas

sific

atio

n ra

te in

%

statistical classifier1 2 3 4 5 6 7 8

0

20

40

60

80

1005070,m:c,mi

clas

sific

atio

n ra

te in

%

statistical classifier1 2 3 4 5 6 7 8

0

20

40

60

80

1005070,f:c,mi

clas

sific

atio

n ra

te in

%

statistical classifier

Std

Std+Entropy

Std+Entropy+Other

Figure 58: Classification rates of statistical classifiers for selected tasks. 1 = euclidean, 2= balls, 3 = diagonal, 4 =gaussian with regularisation parameter 10, 5 = 1NN-rule, 6 = 2NN-rule, 7 = 3NN-Rule, 8 = 4NN-rule.

This is interesting because it indicates that each set contains the same amount of information. The entropyparameters and other non-linear parameters therefore do not add information. They may however express highlycomplex information in just one variable and thus aid classification. Subsequently, only the maximum classificationrate across all techniques and all three parameter sets will be used for comparison.

3.4.2 Without Grouping

At first no subgrouping is imposed on the data set. A decision tree is generated that classifies between infarcts andcontrols for ages 30-70 years. To validate the tree and it’s classification rate, the leave-one-out technique is used.Thus, a whole series of trees is generated in each of the 12 experiments. Of course these trees are very similar,because the input only varies in one missing and one extra recording. Therefore only one or two representativedecision trees are pictured for each experiment. The maximum, minimum and average number of leaves is used asa measure of size of the trees. The classification rate in the gray boxes is the result of the leave-one-out validation.

Standard Parameters The standard parameters produce a number of similar trees. They all make use of theparameters meanNN, HF, LF and age, but in different arrangements. The thresholds for each of these parametersis always the same across the trees. Figure 59 shows a representative decision tree. The decision trees based on thestandard parameters yielded a classification rate of 65% in leave-one-out validation.

Figure 60 visualises the decision tree. Each line maps to one decision node in the tree. Each scatter plot canonly show two dimensions. Therefore the last two decision nodes are shown in a new plot on the right.

54

65%

meanNN < 928.4

infarct

control

control

control

HF < 138.5

LF < 23.1

infarct

age < 33.4# leavesmin = 3 max = 26aver. = 5.4

Figure 59: Decision trees without grouping based on standard parameters

0 200 400 600600

700

800

900

1000

1100

1200

1300

LF

mea

nNN

all recordings

0 50 100 150 20025

30

35

40

45

50

55

60

65

70

HF

age

remaining recordings

control

infarct

control

infarct

controlcontrol

infarct

control

Figure 60: Visualisation of decision tree in Figure 59

The average number of leaves is 5.4. There are however a few larger trees with up to 26 leaves. Table 30 showshow often each of the parameters was used in a tree node. The trees are fairly heterogeneous. This indicates someinfluence of the sample on the tree. The tree in Figure 59 is just one of several possible trees. The key result israther the combination of meanNN, HF, LF and age.

Parameter Occurrences Parameter OccurrencesmeanNN 144 LF/HF 10HF 111 VLF 8LF 87 LF/P 7age 46 pNNi10 5sdaNN1 23 pNNi20 5sdNN 22 HF/P 1rmssd 20 pNN50 1cvNN 16 gender 0

Table 30: Ranked standard parameters in trees without grouping

Standard and Entropy Parameters If the entropy parameters are added, the result is a different one. All trees inthe leave-one-out validation are identical. This decision tree is depicted in Figure 61 and visualised in Figure 62. Asimple threshold of the parameter [BZIP2,Diff]/m is used to classify. The classification rate is 70%. The parameter[BZIP2,Diff]/m by itself has a higher discrimination strength than combinations of the standard parameters.

Because all trees are identical during validation, the structure of the tree and its classification rate can beassumed to be largely independent of the particular sample.

55

70%

infarct control

[BZIP2,Diff]/m < 0.00075

# leavesmin = 2 max = 2aver. = 2

Figure 61: Decision trees without grouping based on standard and entropy parameters

30 40 50 60 704

5

6

7

8

9

10

11

12x 10

−4

age

[BZ

IP2,

Diff

]/m

control

infarct

control

infarct

Figure 62: Visualisation of decision tree in Figure 61

Standard, Entropy and other Non-linear Parameters In the last step a decision tree is generated based onall parameters. The result is identical to previous one. Each tree only consists of the [BZIP2,Diff]/m threshold inFigure 61. This shows that none of the other non-linear parameters, even with combinations of any other parameter,produces a higher classification rate.

Performance Comparison Figure 63 shows summarises the classification performance for the three parametersets. The horizontal line marks the maximum classification rate obtained with any of the statistical classifiers.The decision trees for each parameter set match the statistical classification rate. This indicates that there is nosignificant loss in performance by choosing decision trees as classifier model.

standard parameters standard and

entropy parameters

standard, entropy and other non-linear parameters

best statistical classifierclassificationrate

Figure 63: Performance of tree and best statistical classifier without grouping

3.4.3 Within Age Groups

The second step is to impose a division between the two age groups. The statistical investigations have shown thata number of parameters are dependent on the age. This results in a composite decision tree.

Standard Parameters Representative decision trees based on the standard parameters are shown in Figure 64.For 30-50 years, the first decision node is frequently ’age < 33.9’. This node simply uses the fact that there

are no infarcts below 33.9 years of age. The second node is frequently the gender. Subsequent nodes use meanNN,LF/HF, LF/P and pNNi20 as a criterion. Table 31 shows that in some trees also meanNN, LF/HF, LF/P and pNNi20were used. The classification rate that can be achieved within 30-50 years is only 55%.

56

age

30-50 years 50-70 years

55% 42%

47%

control

age < 33.9

# leavesmin = 3 max = 8aver. = 4.4

infarct control

gender = male

control

age < 33.9

gender = male

infarct

pNNi20 < 0.91meanNN < 871.6

control

controlinfarct

infarctLF/P < 0.3# leavesmin = 2max = 12aver. = 3.6

infarct

cvNN < 0.0395

control

infarct

cvNN < 0.0293

control infarct

meanNN < 928.5

Figure 64: Decision trees for age groups based on standard parameters

Parameter Occurrences Parameter Occurrencesage 48 LF 3gender 43 VLF 1meanNN 28 pNNi10 1LF/HF 13 sdaNN1 1LF/P 11 sdNN 1pNNi20 10 HF/P 0

Table 31: Ranked standard parameters in trees for 30-50 years

For 50-70 years, the parameter cvNN, which is sdNN/meanNN, is used as key criterion. In some trees a combi-nation of cvNN and meanNN is used. Other parameters found in the trees are shown in Table 32. The classificationrate of 42% is however very low. In fact this is an instance in which a trivial classifier, such as ’every input is aninfarct’, would have performed better.

Parameter Occurrences Parameter OccurrencescvNN 51 rmssd 9meanNN 36 pNNi10 3LF 17 HF 2sdNN 15 age 1pNN50 13 HF/P 1VLF 11 LF/HF 1sdaNN1 11 gender 0

Table 32: Ranked standard parameters in trees for 50-70 years

This results in an overall classification rate of only 47%. A random number generator applied to equally sizedclasses would achieve a classification rate of 50%. Thus, the standard parameters itself applied within age groupsdo not produce a good classifier.

Standard and Entropy Parameters When the entropy parameters are added to the parameter set the classifica-tion rate significantly increases to an overall 74%. The structure of the partial decision trees simplifies, asFigure 65shows. The parameter [BZIP2, Diff]/m is used as the key criterion, independently of the age. During validation,the first decision is in nearly all trees a threshold of [BZIP2,Diff]/m.

Figure 66 shows how the tree performs. The [BZIP2,Diff]/m threshold is only a little bit lower for 50-70 years.The parameter sdaNN1 helps to isolate several more control cases for 50-70 years. The essential criterion remainsto be the [BZIP2,Diff]/m threshold.

For 50-70 years some trees contain the parameters sdaNN1, [LZ77400-12,Cut], LF and pNN50. Table 33 showsthis. The trees are therefore a bit larger than for 30-50 years.

57

age

30-50 years 50-70 years

71% 77%

74%

infarct

[BZIP2,Diff]/m < 0.00075

# leavesmin = 2 max = 9aver. = 2.5

control

# leavesmin = 2max = 10aver. = 3

control

sdaNN1 < 23.1

infarct control

infarct

[BZIP2,Diff]/m < 0.000729

control

[BZIP2,Diff]/m < 0.000729

Figure 65: Decision trees for age groups based on standard and entropy parameters

30 40 50 60 70

4

6

8

10

x 10−4

age

[BZ

IP2,

Diff

]/m

all recordings

0 20 40 60 80

4

6

8

10

x 10−4

sdaNN1

[BZ

IP2,

Diff

]/mremaining recordings

control

infarct

control

control

control

infarct

infarct

Figure 66: Visualisation of decision tree in Figure 65

Standard, Entropy and other Non-linear Parameters Adding the other non-linear parameters leads to thesame result as without them. Figure 67 shows that the classification rate is only marginally lower. The size of thedecision trees is lower than in the previous experiment. From the other non-linear parameters only AMIFpdLF andAMIFdecay are used in combinations with [BZIP2,Diff]/m, but only in two instances each.

age

30-50 years 50-70 years

70% 75%

73%

infarct

[BZIP2,Diff]/m < 0.000750

# leavesmin = 2 max = 6aver. = 2.3

control

# leavesmin = 2max = 7aver. = 2.5

control

sdaNN1 < 23.1

infarct control

infarct

[BZIP2,Diff]/m < 0.000729

control

[BZIP2,Diff]/m < 0.000729

Figure 67: Decision trees for age groups based on all parameters

58

Parameter Occurrences Parameter Occurrenceshline [BZIP2,Diff]/m 66 pNN50 12sdaNN1 22 rmssd 1[LZ77400-12,Cut] 14 cvNN 1LF 13 meanNN 1

Table 33: Ranked standard and entropy parameters in trees for 50-70 years

Performance Comparison The classification rate of decision trees based on standard parameters is lower thanthe ones achieved by statistical classifiers, whichFigure 68 shows. This indicates that within the standard parame-ters the classification rule is fairly complex, involving many parameters. The decision tree approach does not yielda good performance.

age

30-50 years50-70 years

Figure 68: Performance of tree and best statistical classifier with age grouping

When the entropy parameters are added, the classification rate exceeds the one of the statistical classifiers. Herethe classification rule can be modelled well with a decision tree. These decision trees mostly consist of a simpleentropy threshold. Thus, the entropy parameter expresses the classification rule much more adequately than thewhole set of standard parameters.

3.4.4 Within Age and Gender Groups

The third step is to further divide the data set according to the gender of the subjects. This leads to four subsetswith individual decision trees to be generated. These trees are again combined to a composite decision tree. Thiscomposite decision tree can then be directly compared to the previous ones. When interpreting the results it shouldbe kept in mind that the sizes of the subsets are naturally smaller than in the previous experiments.

Standard Parameters The standard parameters in Figure 69 lead to an overall performance of 68%. This com-parable to the 65% of the tree without grouping. The number of leaves is larger here. Compared to the tree with agegrouping, the performance has increased significantly. This is due to the new gender division. It allows for distincttrees to be built for males and females. This indicates a difference in infarct characteristics between the genders,which the tree generator does not model well by itself.

Figure 70 shows that for 50-70 year old men controls and infarcts are not well separable. The pNN50 thresholdis just very carefully chosen. For women instead, controls and infarcts are well separable using the cvNN parameter.

For 30-50 year old males the age threshold is always the first decision node in the trees. This is simply due tothe fact that there are no infarcts below 33.4 years. This also means that the other parameters do not provide strongenough discriminators for this subgroup.

For 30-50 year old females pNNi20 is the key discriminator. The parameters cvNN, sdNN and HF were onlyused as first decision node once each.

For 50-70 year old males the tree pictured in Figure 69 occurs most frequently. It always has this structureduring validation and is thus stable. However, it’s classification rate is fairly low. This is a problem because this isthe most common infarct patient group.

59

age

30-50 years50-70 years

gender

malefemale

gender

male female

68%

67%69%

87%

49%

64%

71%

control

age < 33.4

# leavesmin = 2 max = 6aver. = 2.5

infarct

control

age < 33.4

LF/P < 0.29

meanNN <871.6

control

infarct

infarct

# leavesmin = 2 max = 2aver. = 2

# leavesmin = 2 max = 5aver. = 3.2

# leavesmin = 2 max = 4aver. = 2.1

control infarct

pNNi20 < 0.91

infarct

pNN50 < 0.0013

meanNN <928.5

infarctcontrol

control infarct

cvNN < 0.0284

Figure 69: Decision trees for age and gender groups based on standard parameters

0 0.2 0.4600

700

800

900

1000

1100

1200

1300

pNN50

mea

nNN

50−70 years, male

50 55 60 65 700

0.02

0.04

0.06

0.08

0.1

0.12

age

cvN

N

50−70 years, female

control

infarct

control

infarct

infarctcontrol

infarct

Figure 70: Visualisation of right half of the decision tree in Figure 69

For 50-70 year old females the classification rate is very high with 87%. The key parameter is cvNN. All butone tree during validation were identical to the one in Figure 69.

Standard and Entropy Parameters When adding the entropy parameters, the overall classification rate shownin Figure 71 increases to 74%. This is a bit more than from the tree without grouping, but the same as from the treewith only age grouping. Thus imposing a division of males and females does not increase the overall performance.However, there are differences in the structure of the tree.

The scatter plots in Figures Figure 72 and Figure 73 show the performance of the tree. For 30-50 years the subtrees are effective. Although, there were only very few infarcts in women. For 50-70 year old men the separation

60

age

30-50 years50-70 years

gender

malefemale

gender

male female

74%

74%74%

87%

63%

86%

70%

control

age < 33.4

# leavesmin = 2 max = 6aver. = 2.5

infarct

control

age < 33.4

LF/HF < 6.02

[BZIP2,Diff]/m < 0.00072

control

control

infarct

# leavesmin = 2 max = 2aver. = 2

# leavesmin = 2 max = 5aver. = 2.3

# leavesmin = 2 max = 4aver. = 2.1

infarct control

[LZ77,Cut] < 0.288

infarct

[BZIP2,Diff]/m < 0.000639

infarct control

cvNN < 0.0284

control

Figure 71: Decision trees for age and gender groups based on standard and entropy parameters

of control and infarct is improved through [BZIP2,Diff]/m. However, both classes are much less separable than forwomen.

30 35 40 45 50

4

6

8

10

x 10−4

age

[BZ

IP2,

Diff

]/m

30−50 years, male

30 35 40 45 500

2

4

6

8

age

LF/H

F

remaining recordings

30 35 40 45 500.1

0.2

0.3

0.4

0.5

age

[LZ

77,C

ut]

30−50 years, female

control

infarct

control

control

controlcontrol

infarctinfarct

Figure 72: Visualisation of left half of the decision tree in Figure 71

For 30-50 year old males the age threshold still dominates, which shows that other parameters are not as strong.For females a simple threshold of the parameter [LZ77,Cut] performs very well with a classification rate of 86%.Here the entropy itself, not the entropy-meanNN ratio, is used.

For 50-70 year old males, the low performing standard parameter tree is replaced by a [BZIP2,Diff]/m threshold

61

50 55 60 65 70

4

6

8

10

x 10−4

age

[BZ

IP2,

Diff

]/m

50−70 years, male

50 55 60 65 700

0.02

0.04

0.06

0.08

0.1

age

cvN

N

50−70 years, female

control

infarctcontrol

control

infarct infarct

Figure 73: Visualisation of right half of the decision tree in Figure 71

with a significantly higher classification rate. This is important, because 50-70 year old males are the most commoninfarct patient group. For females the high performing cvNN threshold remains. It should also be noted that thediscrimination of infarct and controls works particularly well with females.

Standard, Entropy and other Non-linear Parameters Adding the other non-linear parameters results in aslightly reduced overall classification rate of 72%. As shown inFigure 74, the trees during validation tend to be-come larger. AMIFpdLF and AMIFarea are used in some instances. The key difference is that a renyi025 thresholdis used as tree for 50-70 year old males. The classification rate is however the same.

age

30-50 years50-70 years

gender

malefemale

gender

male female

72%

71%72%

81%

63%

79%

70%

control

age < 33.4

# leavesmin = 2 max = 4aver. = 2.2

infarct

# leavesmin = 2 max = 3aver. = 2.1

# leavesmin = 2 max = 5aver. = 2.3

# leavesmin = 2 max = 4aver. = 2.3

infarct control

[LZ77400-12,Cut] < 0.288

infarct

renyi025 < 4.74

infarct control

cvNN < 0.0284

control

Figure 74: Decision trees for age and gender groups based on all parameters

Performance Comparison Figure 75 shows that the performance of the decision trees matches the one of thestatistical classifiers. The decision trees model the classification rule adequately.

62

age

30-50 years 50-70 years

gender

male female

gender

male female

Figure 75: Performance of tree and best statistical classifier with age and gender gr.

63

3.4.5 Within Age Groups and Stages

The fourth and last approach is to impose a subgroups according to age and infarct stage. Again, this results in foursubtrees that are combined to a composite one.

Standard Parameters The standard parameters produce more complex and diverse decision trees than for ageand gender groups. Figure 76 shows two of the common trees during validation for each group. At the same timethe overall classification rate is with 61% lower.

age

30-50 years 50-70 years

stage stage

61%

64%57%

91%

50%

68%

49%HF/P < 0.19

# leavesmin = 2 max = 5aver. = 3.8

control

# leavesmin = 3 max = 7aver. = 4.1

# leavesmin = 2 max = 14aver. = 5.3

# leavesmin = 2 max = 4aver. = 2.4

infarct

cvNN < 0.029

infarct control

sdNN < 16.84

control

1.-14.day 1.-14.day 15.-30.day15.-30.day

control

sdNN < 44.9

infarct

HF/P < 0.19

control

meanNN< 1036.5

infarct

infarkt

LF/P < 0.288

control

LF/HF < 1.72control

meanNN < 871.6

control

control

age<40.9

infarct

cvNN < 0.042control

sdNN < 36.4

control

infarct

age<40.9

control

meanNN < 928.5

infarct

control

cvNN < 0.0386

control

meanNN < 853.6

infarct

infarct

sdNN < 16.84

control

HF/P < 0.033

infarct control

sdNN < 59.16

Figure 76: Decision trees for age groups and stages based on standard parameters

For 30-50 years and during 15.-30. day after infarct the first decision node uses the age threshold 40.9. This isdue to the fact that the 15.-30. day infarct cases tend to have a higher age.

For 50-70 years and during 1.-14. day after infarct the parameter cvNN is the key parameter, just like in theprevious experiments. The classification rate is however only 50%. For 15.-30. day the parameter sdNN shows tobe a good discriminator with a classification rate of 91%. However, the number of infarct recordings in this groupis only 8 compared to 28 controls.

64

Standard and Entropy Parameters When the entropy parameters are added the overall classification rate in-creases by 3%. As shown in Figure 77, the structure of the trees becomes simpler and the entropy parameters[BZIP2,Diff]/m, [LZ77,Cut]/m and [LZ77,Cut] are used as key decision nodes for all groups except for 50-70years during 15.-30. day. There the parameter sdNN outperforms the entropy.

age

30-50 years 50-70 years

stage stage

64%

68%57%

89%

58%

65%

52%[BZIP2,Diff]/m < 0.000750

# leavesmin = 2 max = 8aver. = 3.6

control

# leavesmin = 3 max = 5aver. = 3.4

# leavesmin = 2 max = 8aver. = 2.7

# leavesmin = 2 max = 4aver. = 2.3

infarct

[BZIP2,Diff]/m < 0.000729

infarct control

sdNN < 16.84

control

1.-14.day 1.-14.day 15.-30.day15.-30.day

control

[LZ77,Cut]/m < 0.000338

infarct

cvNN < 0.042control

sdNN < 36.4

control

infarct

age<40.5

control

control

[LZ77,Cut]/m < 0.0386

infarct

infarct

[LZ77,Cut] < 0.294

control

[BZIP2,Diff]/m < 0.000750

controlinfarct

[BZIP2,Diff]/m < 0.000731

control

control

[LZ77,Cut]< 0.3396

infarct

Figure 77: Decision trees for age groups and stages based on standard and entropy parameters

Standard, Entropy and other Non-linear Parameters Adding the other non-linear parameters only changesthe structure of the trees marginally, as shown in Figure 78. In few instances AMIFpdLF and AMIFarea were usedin decision nodes. The classification rate is 2% smaller.

For the grouping with age and infarct stages in general, the trees during validation are more heterogeneous thanthe ones from the age and gender grouping. This and the fact that the classification rate is lower indicates that thegrouping according to age and infarct stages is not as intrinsic to the infarct classification problem.

Performance Comparison A final comparison of the classification rate shows that the decision trees performworse than the statistical classifiers, no matter which parameter set. This indicates that the grouping using stagesis not as suitable as the one using genders.

65

age

30-50 years 50-70 years

stage stage

62%

68%55%

88%

58%

59%

52%[BZIP2,Diff]/m < 0.000750

# leavesmin = 2 max = 5aver. = 3.3

control

# leavesmin = 3 max = 7aver. = 3.6

# leavesmin = 2 max = 10aver. = 2.7

# leavesmin = 2 max = 4aver. = 2.3

infarct

[BZIP2,Diff]/m < 0.000729

infarct control

sdNN < 16.84

control

1.-14.day 1.-14.day 15.-30.day15.-30.day

control

[LZ77,Cut]/m < 0.000338

infarct

meanNN < 995.3control

AMIFarea < 3.167

infarct

control

age<40.5

infarct

infarct

[LZ77,Cut] < 0.294

control

[BZIP2,Diff]/m < 0.000750

controlinfarct

[BZIP2,Diff]/m < 0.000731

control

control

[LZ77,Cut]< 0.3396

infarct

[BZIP2,Diff]/m < 0.000731

control

[LZ77,Cut] < 0.353

infarct

infarct

LF < 21.84

control

Figure 78: Decision trees for age groups and stages based on all parameters

age

30-50 years 50-70 years

stage

1.-14.day 15.-30.day

stage

1.-14.day 15.-30.day

Figure 79: Performance of tree and best statistical classifier with age and stage grouping

66

3.5 Investigation of Infarct Stage

In this study the infarct stages 1.-14. day, 15.-30. day and > 30. day have been used to group the recordings.Even though this division seemed natural at first, the classification results inSection 3.4.5 indicates the opposite.The decision tree classification rate using stage distinctions is lower than the one of the statistical classifiers. Thisindicates that the infarct stage division complicates the classification task. The decision trees are then not able tomodel the distinction.

This section aims to investigate what the infarct stage actually means for the condition of the heart. The numberof days between infarct and recording is known as displayed in Figure 80. The [BZIP2-100kb,Cut] entropy hasbeen used in this diagram to visualise the dependency between entropy and recording delay. Most recordings weremade around 10 days after the infarct. Some however were recorded during a later stage. In particular, for ages of50-70 years the average entropy, a measure of the HRV, for 15.-30. day infarcts seems lower than for 1.-14. dayinfarcts.

0 10 20 30 40 50 60 700.3

0.4

0.5

0.6

0.7

0.8

0.9

1

days between infarct and recording

entr

opy

Recording Delay vs. [BZIP2−100kb,Cut] Entropy

infarct 30−50yinfarct 50−70yinfarct >=70

Figure 80: Recording delay vs. [BZIP2-100kb,Cut] entropy

The arising question is what influenced the delay between infarct and recording. On the one hand organisationalissues, such as measurement room bookings or transfers from another hospital may cause additional delays. Thiswould support the hypothesis that a high recording delay indicates a later healing stage.

On the other hand the reason for a delay may be the more severe condition of the patient that does not permit therecording until later. If this is the case then the 15.-30. day recordings may contain the more severe infarcts, ratherthan infarcts in an advanced healing stage. The MCG/ECG recordings were made at the end of the in-hospitaltreatment. But in general, the treatment length is not proportional to the severity of the infarct. It is rather governedby administrative and accounting procedures.

The individual reasons for the delayed recordings are not available. However, as indicators for the severity ofthe infarct the parameters in Table 34 can be used. All three parameters were collected for the infarct patients onlyduring in-hospital treatment. It is not known whether the infarct was a first infarct or a re-infarct.

Parameter DescriptionLVEF left ventricle ejection fraction; the ratio between stroke volume and end-diastolic volume;

normal: ≥ 65%LVDD left ventricle end-diastolic diameterCK creatine kinase concentration;normal: < 80 U/l for men, < 70 U/l for women; MB type

of CK is a key diagnostic enzyme for cardiac infarction

Table 34: Available indicators for infarct severity [15]

The ejection fraction has been measured either invasively through cardiac catheterisation or non-invasivelythrough sonography or using both techniques. The first step is therefore to check the comparability of both tech-niques. Figure 81 on the left plots both techniques against each other and shows a linear regression of their rela-tionship. The variation of the measurements is very high. One reason could be that the measurements were notperformed at the same time and day. Another problem is that the EF measures have a low accuracy in general. The

67

results of both EF measurements should not be compared with each other. Each of the techniques should be seenseparately.

0 50 1000

20

40

60

80

100

LVEF(catheter) in %

LVE

F(e

cho)

in %

30−50y

50−70y

>70y

30 40 50 60 7020

40

60

80

100

LVDD in mm

LVE

F (

cath

eter

) in

%

1.−14.d

15.−30.d

>30.d

30 40 50 60 7020

40

60

80

100

LVDD in mm

LVE

F (

echo

) in

%

1.−14.d

15.−30.d

>30.d

Figure 81: LVDD versus LVEF with colour-coded infarct stages

The ejection fraction indicates the ability of the heart to maintain sufficient blood circulation. An EF of > 60%is normal, < 50% is pathological and < 30% is severe. But the EF is also moderated by the size of the heart. AnLVDD up to 56mm is considered normal. An LVDD of more than 70 is pathologically dilated and > 80mm issevere. A large LVDD indicates a dilatation of the heart, which results in a reduced efficiency. The combinationof low ejection fraction and high LVDD indicates a more severe infarct situation. Figure 81 shows the LVEF andLVDD.

A Wilcoxon test between 1.-14. day and 15.-30. day infarcts for 50-70 years found no significant difference forthe ejection fraction. There was a significant difference in mean for LVDD with a probability of 2.9%. The meanof 15.-30. day infarcts is a bit higher than for 1.-14. day infarcts. However, the value of the mean for 15.-30. dayinfarcts is 56.6mm and the LVDD values do not exceed 65mm. The dilation is thus not pathological. Additionally,there are only 5 cases of 15.-30. day infarct with 50-70 years and a recorded LVDD. Figure 82 pictures the LVDDdistributions.

0 10 20 30 4030

35

40

45

50

55

60

65

70

recording delay in days

LVD

D in

mm

30−50y50−70y>70y

30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

entr

opy

[LZ

7740

0−12

,Cut

]

LVDD in mm

age 50−70 years

1.−14.d15.−30.d>30.d

Figure 82: LVDD versus infarct stages

The creatine kinase concentration is shown in Figure 83. Again, parameter has not been recorded for all infarctspatients. In the 15.-30. day group there are only two values. Thus, the Wilcoxon test between 1.-14. day and 15.-30.day did not result in a significant difference.

In conclusion, this analysis has not shown a significant difference in infarct severity between 1.-14. day and15.-30. day infarcts. Therefore, the infarct stages as they were introduced earlier may be interpreted as healingstages over time. The distinction of healing stages should be made during the analysis of this data set. Only for theconstruction of a decision tree it is not useful.

68

0 5 10 15 20 25 30 35

10

20

30

40

50

60

70

80

90

recording delay in days

CK

in U

/l

30−50y50−70y>70y

Figure 83: CK versus infarct stages

69

4 Discussion

The objective of this research is to evaluate the usefulness of the entropy in HRV analysis of patients with myocar-dial infarction. A large number of configurations of preprocessing techniques, entropy estimation techniques andclassification rules has been derived that aim to achieve this.

4.1 Interpretation of Classification Rules

4.1.1 Nature of Classification Problem

The problem of classifying between post-infarct vs. healthy subjects is a challenging one, because the HRV charac-terises the functional condition of the heart. A heart which recovered very well from an infarction, for example dueto a successful thrombolysis or sufficient collateral blood supply, may be in a good functional condition after theinfarct. Thus the control and infarct group can be expected to overlap in terms of the entropy and the other HRVparameters. The level of classification accuracy of 74% ≈ 3/4 achieved in this study matches this expectation.Wessel et al. [29] also achieved classification rates in this range.

The problem of classifying between high-risk vs. low-risk post-infarct subjects is more naturally solved usingthe entropy. The high-risk subjects have a reduced autonomic activity, which this is reflected by the entropy. Low-risk patients have a good functional condition, similar to healthy subjects, which is reflected by a high entropy.During risk stratification high-risk and low-risk subjects should overlap much less in terms of the entropy.

Intervening Variables The aim of the classifier generation is to model the impact of the variable ’heart condition’on the variable ’entropy’ and the other parameters. However, there are a number of intervening variables that alsoinfluence the HRV. These are the age, the existence of risk factors, the heart medication, the infarct location andwhether there has been a previous infarct.

While the age could be taken into account in this study, the risk factors and medication were only be assessedusing statistical tests. The number of risk factors has a small degree of influence on the HRV. More risk factorsstatistically lead to a little lower entropy within the infarct group. The influence of betablockers within the infarctgroup is undetermined because the decision of giving betablockers is dependent on the infarct condition. However,previous studies [32] did not find a significant influence of betablockers on HRV except that the rise of LF in themorning hours is suppressed. Based on the statistical tests, the infarct location is an independent variable. There isno significant entropy mean difference between anterior and posterior infarcts, whichFigure 41 shows.

4.1.2 Standards-based Rule

In the classifier generation step, decision trees were generated to classify between infarct patients and controls.The kind of parameters used as tree nodes and their combination encapsulates the rule for classification.Figure 84shows the best standard parameter based tree. The best tree is the one with the highest classification rate and at thesame time a small size.

65%

meanNN < 928.4

HF < 138.5

LF < 23.1

age < 33.4

infarct

infarct

control

control

control

Figure 84: Best standard-parameter-based classification tree

70

MeanNN - LF (LF/P) Combination The parameters meanNN and LF, or LF/P respectively, appear in all threedecision trees. In the tree without grouping in Figure 59 the combination of both parameters is the centre of theclassification rule.Figure 85 on the left shows a scatterplot of this pair of decisions.

0 200 400 600600

700

800

900

1000

1100

1200

1300

LF

mea

nNN

all recordings

0 0.5 1600

700

800

900

1000

1100

1200

1300

LF/Pm

eanN

N

30−50 years

0 0.5 1600

700

800

900

1000

1100

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infarct infarct controlcontrol

Figure 85: Parameter combination meanNN - LF (LF/P) for different subgroups

When the two age groups are imposed in Figure 64, the combination of meanNN and LF/P can be foundfor ages 30-50 years. A close look reveals that this subtree is only applicable to male subjects. This finding ofthe generator is confirmed in the third tree inFigure 69: The meanNN-LF/P combination is only found for malesubjects of 30-50 years. Figure 85 visualises this. Therefore, the meanNN-LF/P combination is most appropriatefor this subgroup, but dominates if no grouping is imposed.

Little power in the LF band indicates an infarct. This shows that the autonomous activity which produces theLF power is reduced. This complies to the clinical observation that HRV is reduced after infarction [22, 5]. Inpost-infarct patients the autonomic activity is reduced and the sympathico-vagal balance is shifted towards thesympathicus. If the LF is normal, then low heart rate indicates an infarct.

PNNi20 The parameter pNNi20 is defined as the proportion of the number of interval differences of successiveintervals smaller than 10 (20) ms to the total number of NN intervals. A pNNi20 threshold was found to be the bestclassification rule for female subjects of 30-50 years, which Figure 86 illustrates. This was automatically foundin Figure 64 and confirmed inFigure 69. However, it should be kept in mind that there were only 3 female infarctsamples between 30 and 50 years.

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Figure 86: Parameter pNNi20 for females of 30-50 years

A pNNi20 value below 0.91 indicates a healthy subject and above indicates an infarct. A high pNNi20 valuemeans that a high proportion of successive differences was below 20 ms. This produces a smooth tachogram,rather than a chaotic one. This loss in variability matches to the common understanding that pathological statescause reduced heart rate variability.

CvNN The parameter cvNN is the ratio sdNN/meanNN. For 50-70 year old females a cvNN threshold emergedas best. In the tree with two age groups in Figure 64 it is part of the classification rule for 50-70 years.Figure 69

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however shows that this cvNN threshold is most suitable for the female subjects in the 50-70 years group. Figure 87visualises the cvNN threshold on the right.

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Figure 87: Parameter cvNN for females of 50-70 years

The parameters sdNN and meanNN are proportional to some degree. Subjects with a high meanNN, that meanswith a slow heart beat, tend to have a higher sdNN. Figure 87 on the left shows this. The figure also shows that thehealthy subjects have a relatively low meanNN and high sdNN. CvNN is the ratio sdNN/meanNN and each of itsvalues is a line in the meanNN-sdNN diagram, in particular on the right diagram.

The left diagram in Figure 87 is actually similar to the left diagram in Figure 85. The only difference is that LFwas replaced with sdNN. Both parameters are related, because according to Parseval’s Theorem sdNN 2 = P , thepower in the frequency spectrum, and LF is a portion of P.

Age The age as parameter has been used in all three types of trees. The question was always whether age<33.9(33.4) years. This expresses the fact that there are no infarct cases below this age. Even though this is an obviousstep, it improves the classification rate, in particular for 30-50 year old males in Figure 69. The presence of thisdecision node can be interpreted in the way that there is a lower bound to the age at which infarcts occur.

Summary The combination of meanNN with a parameter that expresses the variability, such as sdNN or LF,covers a large portion of the classification task. For individual subgroups individual parameters such as pNNi20can outperform this general approach. There is not one best parameter or classification rule, which is also indicatedby the variations of the decision trees during validation. In particular, only multivariate combinations of parametersshow a high classification rate. The highest classification rate is 68%.

4.1.3 Entropy-based Rule

In the second step the same classification tasks have been applied to the set of standard and entropy parameters.In fact the entropy parameters superseded the standard parameters in most instances. Figure 84 shows the beststandard parameter based tree. The most prominent parameter is the following:

[BZIP2,Diff]/m When the decision tree generator was applied without imposed grouping, it produced a simple[BZIP2,Diff]/m threshold, as in Figure 61. Values below this threshold are considered infarct and above controls.This conforms to the general understanding of the heart rate variability: A reduced variability characterises apathological condition, because it expresses a reduced autonomic activity. [22, 5, 34] This simple threshold alreadyexceeded the classification strength of the best standard-parameter-based tree. This tree was even very stable duringvalidation. Figure 89 visualises this threshold.

The right diagram in Figure 89 decomposes [BZIP2,Diff]/m into [BZIP2,Diff], the entropy, and meanNN. Thetwo classes control and infarct seem well separable compared to the previous results. But not the entropy by itselfor the meanNN by itself are able to separate the classes as well. It is the combination of both.

In fact this combination is similar to the one of cvNN = sdNN/meanNN [34]. The meanNN is combinedwith a parameter that expresses variation. However, the [Bzip2,Diff] entropy performs much better than sdNN. Theentropy captures the variability, not only the variance. For example, in the entropy the sequence of values in the

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age

30-50 y. 50-70 y.

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< 23.1

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74%

Figure 88: Best entropy-based classification tree

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Figure 89: Parameter [BZIP2,Diff]/m and [BZIP2,Diff] vs. meanNN

tachogram is accounted for. This is not the case for the standard deviation. Figure 91 and the associated text showthis in more detail.

When the age grouping was introduced a [BZIP2,Diff]/m threshold was found for both age groups, see Figure 65.For 50-70 years the threshold was slightly lower. This is natural since the entropy decreases with increasingage. [34, 27] For the age and gender grouping in Figure 71 more of the standard parameters discussed in theprevious section are involved, which complicate the tree, but do not improve the classification strength.

Other Entropy Estimations It is important to mention here, that the other entropy estimations are highly corre-lated to [BZIP2,Diff]. They classification strength is comparable. [BZIP2,Diff], however, is slightly better for thisclassification task and was thus chosen from the candidates.

Summary The [BZIP2,Diff]/m parameter expresses the heart rate variability as a whole. It combines the meanand variability of the input signal in one variable. In the infarct classification task it outperforms combinations ofstandard parameters with a rate of 74%. It provides the researcher with a simple, yet robust classification rule.

4.1.4 Partial Classifier for Risk Stratification

Another perspective on the classification problem is to try to classify with maximum reliability. The possible out-comes could be ’infarct’, ’control’ and ’undecided’. This is interesting for practical applications of the classificationrule. In this study the partial classifier inFigure 90 is an example. The classifier only labels the inputs with verylow entropy as ’infarct’ and the rest with ’undecided’. The reliability of this classifier is therefore high.

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Figure 90: Partial classifier for infarcts with very low entropy

Such a partial classifier could be used to stratify the post-infarction risk [22]. The entropy measures the au-tonomic activity. Post-infarct patients with a high entropy, thus high variability, similar to healthy subjects canbe regarded as low-risk patients. However, patients with an entropy lower than any other healthy subject can beregarded as high risk patients. Voss et al. [48] also used non-linear HRV parameters for risk stratification. Baumertet al. [3] applied the entropy concept for detecting arrhythmias in their onset.

4.2 Entropy as HRV Parameter

The primary objective of this research is to evaluate the usefulness of the signal entropy in HRV analysis of patientswith myocardial infarction. The criteria for assessing the usefulness were defined with the research objectives inSection 1.5 and can now be applied.

Statistical Significance of Entropy Mean Difference In extensive statistical tests between control and infarctgroups the entropy-meanNN-ratios have consistently produced stronger significances than the standard parameters.This can be seen from Table 22, Table 25, Table 27 and Table 29. The kinds of differences found in the entropyparameters are very similar to the ones found in the whole set of standard parameters. There is no key aspect thatis not covered by the entropy.

Gain in Classification Rate through Entropy The classification rate of decision trees based on the entropy iswith 74% higher than the one of decision trees based the whole set of standard parameters. Table 35 summarisesthe classification rates. In fact, when the entropy parameters were added to the standard parameters, the gain inclassification rate was achieved by only using one of the entropy parameters in most cases.

Decomposition Standard Std.+Entropy Allnone 65 70 70age grouping 47 74 73age and gender grouping 68 74 72age and stage grouping 61 64 62

Table 35: Classification rates of composite decision trees

Content and Complexity of Entropy-Based Decision Trees The decision trees based on standard parametersconsist of combinations of two to four parameters, such as in Figure 59. The choice of parameters changes withthe grouping. The entropy based decision trees are much simpler, such as in Figure 61. In most cases they consistof a simple [BZIP2,Diff]/m threshold. The entropy threshold was found to be a bit lower for higher ages, such as

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in Figure 65. This is natural, because the entropy decreases with increasing age. The entropy-based decision treesare therefore much simpler and more stable.

Consistency and Reliability of Entropy Estimation The number of different entropy estimation approacheshave all produced highly correlated results. For the Diff data set the correlations were above 97.5%, see Table 18.Additionally, the different entropy estimations performed very similarly in the statistical tests. The entropy param-eter is consistent across estimation methods. This also means that the entropy estimation is reliable.

Robustness Against Input Signal Fluctuations Fluctuations in the tachogram, such as a slope, or a drops inbeat interval length affect the value of several standard parameters. Figure 91 shows two examples: The secondsample signal has a drop in the middle. Although the variability of the signal is lower than the first one, the standarddeviation (sdNN) is identical. The third sample signal is a sine with some noise. The signal is highly repetitive,there is no localised adaptation. The standard deviation does not recognise this fact, but the entropy does. It istherefore more robust against input signal fluctuations.

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Figure 91: LZ77 entropy of simulated signals with identical sdNN

Evaluation as HRV Parameter The entropy has shown to be a very valuable parameter for HRV research. Itoutperforms multivariate standards classifiers in infarct classification. This confirms the hypothesis and extendsthe previous findings by Baumert et al. [3] on the detection of arrhythmias. A key characteristic of the entropy isthat it captures the variability in one parameter. The standard parameters each capture some aspect of the heart ratevariability. Several of these parameters need to be combined to obtain a classifier.

A second methodological advantage of the entropy is that the sequence of the beat intervals is recognised. Forexample, in Figure 91 signal 4 is the sorted version of signal 1. Between both there is no difference in sdNN but aclear difference in entropy.

In contrast to previous entropy-based approaches [37, 9, 29], [BZIP2,Diff] measures the entropy in its infor-mation theoretic sense. Not a subset or specific aspect is highlighted, but the overall variability. The issue withApEn [37] and SampEn for example is how to justify choosing a specific pattern length. The BZIP2 entropy esti-mation approach does not require complex preprocessing steps, such as interpolation and transformations, whichunavoidably introduce a certain degree of error or deviation. The BZIP2 entropy estimation is straight forward.

Another advantage of the BZIP2 approach is that the signal does not need to be as long as for ApEn and MSE.The BZIP2 estimation converges within 500-600 values. ApEn requires at the very least 1000 values. [37] In theMSE study [9] 20000 values were used. 20000 beats require about 5 hours of recording from a 24 h ECG. Thus,the environmental demands on the heart can not be controlled. 600 beats can be recorded within 10-12 min in acontrolled supine position. This is very practical for screening patients.

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4.3 Final Infarct Classification Process

The infarct classification process that results from the findings of this study is the one pictured inFigure 92. Thediagram was derived from Figure 12 by removing the remaining configurations and adding the classification rule.

10 min MCG/ECG

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...

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agesdaNN1

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...... ...

Figure 92: Possible HRV based infarct classification process

Data Acquisition The tachogram length should be 10-15 min. The reason is that the entropy estimation methodsaim to find recurrent sections or regularities. These regularities exist over long distances in the signal. Additionally,most entropy estimation methods work adaptively and therefore show a convergence towards the final entropyestimate. The absolute minimum is 250 beats, which require at least 5 min. The use of both an MCG and ECGsimplified the template matching task. The MCG provides another 31 channels to choose from during templatematching.

Artifact Filtering Artifact filtering is necessary for the entropy analysis, because artifacts such as extrasystolesincrease the complexity of the signal and thus increase the entropy. A high entropy is however associated with ahealthy subject.

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Length Normalisation A length normalisation is necessary for entropy estimation, because the estimated en-tropy converges with increasing length of the signal. Therefore each signal should have the same number of beats,480 in this case.

The alternative of resampling the signal to an equidistant time signal is less preferable, because additionalregularity is introduced that reduces the entropy and its discrimination strength

Tachogram vs. Successive Differences Both the tachogram (Cut) and its successive differences (Diff) signalare useful, but distinct inputs for HRV analysis. The key difference is that the successive differences emphasisethe higher frequencies, whereas the tachogram is the original signal including lower frequencies. In this study theentropy based on the successive difference signal yielded the best discrimination performance.

Quantisation The quantisation of the signal is a prerequisite to symbolic entropy estimation. A quantisation stepwidth of 1/128 s was found appropriate. The signal characteristics are preserved, while the number of values, orsymbols, is reduced to 128.

Entropy Estimation The entropy estimation using Burrows-Wheeler block sorting produces the highest com-pression rate and thus the closest entropy. This can be explained by the fact that with this method the whole signal iscompressed as one block. Therefore a maximum number of regularity can be utilised. Additionally, the entropy pa-rameter [BZIP2,Diff]/m is the highest performing configuration of an entropy estimation and a data preprocessingmethod.

The entropy estimation of BZIP2 is consistent with the other tested methods. All estimated entropies are veryhighly correlated with 92% for Cut in Table 16 and 97.5% for Diff in Table 18. This corresponds to the fact that thedifferent entropy estimation techniques performed very similarly in the various statistical tests, such as in Table 20.

Adaptive dictionary coding, such as LZ77, is well-suited for research purposes. The parameter maxWordLengthindicates the length of recurring signal pieces. With increasing maxWordLength the entropy decreases in a saturat-ing fashion, which Figure 18 showed. In this study, the length of recurrences is less than 8-10 values. The distanceat which signal pieces appear repeatedly can be estimated by the length of the context required by a compressionmethod. For LZ77 the context length saturates at about 400 beats. Recurring signal pieces are therefore found atdifferent locations of the signal. However, the strongest drop in entropy appears for a context of 100 beats. Thisshows that the signal is highly repetitive in the short to medium term.

Classification Rule The classification rule inFigure 92 had with 74% the highest classification rate for infarctvs. control and was at the same time one of the simplest trees.

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5 Conclusion

This study followed the objective of evaluating the usefulness of the entropy as HRV parameter. In contrast toprevious entropy-based approaches, the original information theoretic definition of entropy was used. The entropywas estimated using statistical and optimised compression-based methods, which were not modified from theiroriginal form. The consistency of the entropy across these different methods was assessed.

The entropy was compared to standard parameters using the task of infarct classification. To obtain optimalmultivariate classification rules, an automated decision tree generator was used. The resulting decision trees wereassessed in terms of their validated classification rate and simplicity. The gain in these two criteria by adding theentropy to the existing standard parameters was determined.

The key findings are:

1. One entropy parameter threshold outperforms the best multivariate standards-based infarct classifier by 5-10%. The highest achievable classification rate is 74%.

2. [Bzip2,Diff]/m, which is the compression rate of bzip2 applied to the successive differences normalised withmeanNN, had the highest discrimination strength.

3. There is a subgroup of post-infarct patients with significantly lower entropy than other post-infarct patientsas well as healthy individuals. These could be hypothesised as being high-risk patients.

4. Statistical and compression-based entropy estimations are highly consistent with a correlation of > 94% forthe tachogram (Cut) and > 97% for successive differences of the tachogram (Diff).

5. Bzip2, which implements Burrows-Wheeler compression, produced the highest compression rate and thusthe closest entropy estimation.

6. The required tachogram length for all entropy estimations is only 500-600 beats, which can be recorded in10-12 min. The quantisation step width of 1/128 s = 7.8125 ms is suitable for entropy estimation.

7. The original tachogram (Cut) and its successive difference (Diff) are distinct and both useful input data. Theinterpolated data (Interp) are less suitable, because distorting regularity is introduced.

8. Recurring patterns have a length of less than 8-10 values. But the instances of one pattern can be found overlong ranges.

These results prove that the entropy is a very useful and also natural measure of heart rate variability. Infact, choosing the entropy is an intuitive step. The concept of variability is very similar to that of the complexitymodelled in the entropy.

The next step in this research is to apply the BZIP2 entropy to the post-infarct risk stratification problem.Higher classification rates can be expected, because the distinction between low-risk and high-risk patients is muchclearer than between healthy and well-recovered infarction cases. Additionally, non-HRV-based risk stratificationparameters should be involved to obtain a full strength classification rule.

Acknowledgments

I (Stephan Lau) wish to thank my supervisors Prof. Dr. Haueisen, Prof. Dr. Schukat-Talamazzini and Prof. Dr.Voss for their support and expert advice. The implementations of the NGRAM and ALRM technique from Prof.Dr. Schukat-Talamazzini were very helpful. Dr. Goernig from the Clinic for Internal Medicine was an excellentcontact person.

The data set was provided by Dr. Leder, who is now with the Department of Accounting and Controlling at thehospital. The author is also grateful to Dr. Hoyer from the Institute for Pathophysiology and Dr. Baumert from theUniversity of Applied Sciences Jena for their suggestions and for providing access to their HRV research software.

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[52] B. Wormuth and P. Becker. Introduction to formal concept analysis. tutorial at the Int. Conference onFormal Concept Analysis (ICFCA), available online: http://www.wormuth.info/ifca04/materials.html, 2004.accessed 2. Nov. 2005.

[53] J. Ziv and A. Lempel. A universal algorithm for sequential data compression. IEEE Transactions on Infor-mation Theory, 23(3):337–343, 1977.

[54] U. Zwiener. Allgemeine und klinische Pathophysiogie, Teil 1. Gustav Fischer, Jena, Germany, 1993.

81

Stephan LauInstitute of Computer ScienceFriedrich-Schiller-University JenaD-07740 JenaGermany

stephan.lau@cs.uni-jena.destephan_lau@yahoo.com.au

Jens HaueisenBiomagnetic Center JenaClinic of NeurologyFriedrich-Schiller-University JenaD-07740 JenaGermany

haueisen@biomag.uni-jena.dejens.haueisen@tu-ilmenau.de

Ernst Guenter Schukat-TalamazziniInstitute of Computer ScienceFriedrich-Schiller-University JenaD-07740 JenaGermany

schukat@informatik.uni-jena.de

Andreas VossDepartment of Medical EngineeringUniversity of Applied Sciences JenaD-07703 JenaGermany

voss@fh-jena.de

Matthias GoernigClinic of Internal Medicine IFriedrich-Schiller-University JenaD-07740 JenaGermany

matthias.goernig@med.uni-jena.de

Uwe LederClinic of Internal Medicine IFriedrich-Schiller-University JenaD-07740 JenaGermany

uwe.leder@med.uni-jena.de

Hans Reiner FigullaClinic of Internal Medicine IFriedrich-Schiller-University JenaD-07740 JenaGermany

hans.figulla@med.uni-jena.de

82

List of Figures

1 Structure of the heart and course of blood flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 The sinus node and the Purkinje system of the heart . . . . . . . . . . . . . . . . . . . . . . . . . 53 Relation between excitation and ECG with sinus rhythm . . . . . . . . . . . . . . . . . . . . . . 64 31-channel MCG of 1000 ms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Cardiac nerves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Preferred localisation of vascular obliteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 ECG characteristics of myocardial infarction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Sample tachogram of healthy and post-infarct individual . . . . . . . . . . . . . . . . . . . . . . 109 Chronotropic responses to graded efferent vagal stimulation . . . . . . . . . . . . . . . . . . . . . 1010 Power spectral density of a healthy and post-infarct recording . . . . . . . . . . . . . . . . . . . . 1211 Kaplan-Meier survival curves for all cause cardiac mortality . . . . . . . . . . . . . . . . . . . . 1312 Overview of the data processing steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1813 Example of a coding step using LZ77 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2114 Example of a coding step using the Burrows-Wheeler compression . . . . . . . . . . . . . . . . . 2215 Four (composite) classification tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2516 LZ77 entropy of one sample with varying bL & mWL . . . . . . . . . . . . . . . . . . . . . . . . 2817 LZ77 entropy of 20 random samples with varying bL & mWL=12 . . . . . . . . . . . . . . . . . 2818 LZ77 entropy of 20 random samples with varying mWL & max. bL . . . . . . . . . . . . . . . . 2919 NGRAM entropies for 20 random recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3020 ALRM entropies for 20 random recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3021 GZIP entropies for 20 random recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3122 BZIP2 entropies for 20 random recordings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3223 LZ77 entropy of 20 random samples for different signal lengths . . . . . . . . . . . . . . . . . . 3324 6GRAM entropy of 20 random samples for different signal lengths . . . . . . . . . . . . . . . . . 3425 ALRM10 entropy of 20 random samples for different signal lengths . . . . . . . . . . . . . . . . 3426 GZIP-9 entropy of 20 random samples for different signal lengths . . . . . . . . . . . . . . . . . 3427 BZIP2 entropy of 20 random samples for different signal lengths . . . . . . . . . . . . . . . . . . 3528 Concept graph of the data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3629 Age versus BZIP2 entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3630 Sign. entropy differences for infarct stage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3831 Entropy statistics across ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3832 Entropy statistics across infarct stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3833 Sign. time & frequency par. differences for infarct stage . . . . . . . . . . . . . . . . . . . . . . . 3934 sdNN statistics across ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4035 sdNN statistics across infarct stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4036 pNNi10 statistics across ages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4037 pNNi10 statistics across infarct stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4138 Sign. non-linear par. differences for infarct stage . . . . . . . . . . . . . . . . . . . . . . . . . . . 4139 Wilcoxon test for LZ77 configurations for selected problems . . . . . . . . . . . . . . . . . . . . 4240 Statistics of entropy-meanNN-ratios across infarct stages . . . . . . . . . . . . . . . . . . . . . . 4341 Sign. entropy differences for infarct location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4342 Entropy statistics for infarct locations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4443 Sign. time & frequency par. differences for infarct location . . . . . . . . . . . . . . . . . . . . . 4544 Time and frequency parameter statistics for infarct locations . . . . . . . . . . . . . . . . . . . . 4545 Sign. non-linear par. differences for infarct location . . . . . . . . . . . . . . . . . . . . . . . . . 4646 AMIFdecay, AMIFarea and AMIFpdLF across infarct locations . . . . . . . . . . . . . . . . . . 4647 Concept graph of risk factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4748 Sign. entropy differences for risk factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4849 Entropy statistics for risk factor counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4850 Sign. time, frequency & non-lin. par. differences for risk factors . . . . . . . . . . . . . . . . . . 4951 cvNN and renyi025 statistics for risk factor groups and 50-70 years . . . . . . . . . . . . . . . . . 4952 LF/P and AMIFpdHF statistics for risk factor groups & 30-50 years . . . . . . . . . . . . . . . . 50

83

53 Concept graph of types of medication within infarcts . . . . . . . . . . . . . . . . . . . . . . . . 5154 Sign. entropy differences for medications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5255 Entropy statistics for medication groups in ages 50-70 years . . . . . . . . . . . . . . . . . . . . 5256 Sign. time, frequency & non-lin. par. differences for medications . . . . . . . . . . . . . . . . . . 5357 Statistics of key time & non-lin. par. for medications & 30-50 years . . . . . . . . . . . . . . . . 5358 Classification rates of statistical classifiers for selected tasks . . . . . . . . . . . . . . . . . . . . 5459 Decision trees without grouping based on standard parameters . . . . . . . . . . . . . . . . . . . 5560 Visualisation of decision tree in Figure 59 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5561 Decision trees without grouping based on std. and entropy par. . . . . . . . . . . . . . . . . . . . 5662 Visualisation of decision tree in Figure 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5663 Performance of tree & best statistical classifier without grouping . . . . . . . . . . . . . . . . . . 5664 Decision trees for age groups based on standard parameters . . . . . . . . . . . . . . . . . . . . . 5765 Decision trees for age groups based on std. and entropy par. . . . . . . . . . . . . . . . . . . . . . 5866 Visualisation of decision tree in Figure 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5867 Decision trees for age groups based on all parameters . . . . . . . . . . . . . . . . . . . . . . . . 5868 Performance of tree & best statistical classifier with age grouping. . . . . . . . . . . . . . . . . . 5969 Decision trees for age & gender groups based on std. par. . . . . . . . . . . . . . . . . . . . . . . 6070 Visualisation of right half of the decision tree in Figure 69 . . . . . . . . . . . . . . . . . . . . . 6071 Decision trees for age & gender groups based on std. & entropy par. . . . . . . . . . . . . . . . . 6172 Visualisation of left half of the decision tree in Figure 71 . . . . . . . . . . . . . . . . . . . . . . 6173 Visualisation of right half of the decision tree in Figure 71 . . . . . . . . . . . . . . . . . . . . . 6274 Decision trees for age & gender groups based on all parameters . . . . . . . . . . . . . . . . . . . 6275 Performance of tree & best stat. classifier with age & gender gr. . . . . . . . . . . . . . . . . . . 6376 Decision trees for age groups & stages based on std. par. . . . . . . . . . . . . . . . . . . . . . . 6477 Decision trees for age groups & stages based on std. & entropy par. . . . . . . . . . . . . . . . . . 6578 Decision trees for age groups and stages based on all parameters . . . . . . . . . . . . . . . . . . 6679 Performance of tree & best statistical classifier with age & stage gr.. . . . . . . . . . . . . . . . . 6680 Recording delay vs. [BZIP2-100kb,Cut] entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 6781 LVDD versus LVEF with colour-coded infarct stages . . . . . . . . . . . . . . . . . . . . . . . . 6882 LVDD versus infarct stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6883 CK versus infarct stages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6984 Best standard-parameter-based classification tree . . . . . . . . . . . . . . . . . . . . . . . . . . 7085 Parameter combination meanNN - LF (LF/P) for different subgroups . . . . . . . . . . . . . . . . 7186 Parameter pNNi20 for females of 30-50 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7187 Parameter cvNN for females of 50-70 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7288 Best entropy-based classification tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7389 Parameter [BZIP2,Diff]/m and [BZIP2,Diff] vs. meanNN . . . . . . . . . . . . . . . . . . . . . . 7390 Partial classifier for infarcts with very low entropy . . . . . . . . . . . . . . . . . . . . . . . . . . 7491 LZ77 entropy of simulated signals with identical sdNN . . . . . . . . . . . . . . . . . . . . . . . 7592 Possible HRV based infarct classification process . . . . . . . . . . . . . . . . . . . . . . . . . . 7693 Context of a water habitat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8694 Concept lattice diagram for example context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8795 Simplified concept lattice diagram for example context . . . . . . . . . . . . . . . . . . . . . . . 8896 Examples of standard scales in this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

84

List of Tables

1 Locations of myocardial infarction and frequency . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Reflexes influencing heart rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Entropy estimation methods to compare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 Alternative entropy estimation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Time-domain parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Frequency-domain parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Non-linear parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Population characteristics and sub categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Decision tree generation and pruning options . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510 Parameter sets for classification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2611 Statistics of LZ77 entropies with bL=100, mWL=5 (7 for Interp) . . . . . . . . . . . . . . . . . . 2912 Statistics of 6GRAM entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3013 Statistics of ALRM entropies with contextSize=10 and 32 . . . . . . . . . . . . . . . . . . . . . . 3114 Statistics of GZIP entropies with highest compression (-9) . . . . . . . . . . . . . . . . . . . . . 3115 Statistics of BZIP2 entropies with blockSize=100kb . . . . . . . . . . . . . . . . . . . . . . . . . 3216 Correlation coefficients and confidence for Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . 3217 Correlation coefficients and confidence for Interp . . . . . . . . . . . . . . . . . . . . . . . . . . 3318 Correlation coefficients and confidence for Diff . . . . . . . . . . . . . . . . . . . . . . . . . . . 3319 Age and gender distribution in population . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3720 Entropy across stages as Wilcoxon test in % (50-70 years) . . . . . . . . . . . . . . . . . . . . . 3921 Entropy vs entropy-meanNN-ratios in Wilcoxon tests in % . . . . . . . . . . . . . . . . . . . . . 3922 Key parameters across stages as Wilcoxon test in % (50-70 years) . . . . . . . . . . . . . . . . . 4223 Age and infarct location distribution in population . . . . . . . . . . . . . . . . . . . . . . . . . . 4324 Entropy across locations as Wilcoxon test in % (50-70 years) . . . . . . . . . . . . . . . . . . . . 4425 Key parameters across locations as Wilcoxon test in % (50-70 years) . . . . . . . . . . . . . . . . 4626 Age and risk factor count distribution in population . . . . . . . . . . . . . . . . . . . . . . . . . 4727 Key parameters across risk factor counts as Wilcoxon test . . . . . . . . . . . . . . . . . . . . . . 5028 Age and risk factor count distribution in population . . . . . . . . . . . . . . . . . . . . . . . . . 5129 Key parameter across medications as Wilcoxon test . . . . . . . . . . . . . . . . . . . . . . . . . 5230 Ranked standard parameters in trees without grouping . . . . . . . . . . . . . . . . . . . . . . . . 5531 Ranked standard parameters in trees for 30-50 years . . . . . . . . . . . . . . . . . . . . . . . . . 5732 Ranked standard parameters in trees for 50-70 years . . . . . . . . . . . . . . . . . . . . . . . . . 5733 Ranked standard and entropy parameters in trees for 50-70 years . . . . . . . . . . . . . . . . . . 5934 Available indicators for infarct severity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6735 Classification rates of composite decision trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

85

A Formal Concept Analysis Basics

Formal Concept Analysis (FCA) is a data analysis and visualisation technique. Its key idea is to formalise thegeneral notion of a concept. This allows the researcher to identify and relate concepts in the data. A practical toolto apply formal concept analysis is ToscanaJ [11]. This appendix gives a concise introduction to FCA theory andit’s practical use.

A.1 Context and Concept

Definition 1 A formal context is a triple C:=(O, A, I) of a set O of objects, a set A of attributes and a binaryincidence relation I ⊆ O×A between O and A. An object o ∈ O has the attribute a ∈ A if (o, a) ∈ I , also written’oIa’ [14].

A cross table, like in Figure 93, can capture finite contexts intuitively by assigning objects to rows and attributesto columns. The objects (O) in this example are the life forms of a water habitat. The attributes (A) are theirbehaviour, anatomy and physiology. The crosses represent the incidence relation (I).

Figure 93: Context of a water habitat (reproduced and extended Figure 1.1 in [14])

Definition 2 For a set of objects O1 ⊆ O the set of common attributes is denoted by O′

1 := {a ∈ A| oIa for allo ∈ O1}. For a set of attributes A1 ⊆ A the common objects are denoted by A′

1 := {o ∈ O| oIa for alla ∈ A1} [14].

The character ’ can be interpreted as a derivation operator, which can be applied to sets of objects or at-tributes. In the example, O1 = {bream, frog, dog} would have the common attributes O′

1 = {needs water(a),moves around(g), has limbs(h)}. The notion of common attributes enables us to define a formal concept.

Definition 3 A formal concept within a context (O, A, I) is a pair (O1, A1) ∈ ℘(O) × ℘(A) with O′

1 = A1 andA′

1 = O1. O1 is called the extent and A1 is the intent of the concept. B(O, A, I) is the set of all concepts [14]. ℘denotes the power set.

In the cross table a concept forms a maximal rectangle of crosses taking row and column permutations into ac-count. In the example, O1 = {bream, frog, dog} and A1 = {needs water(a), moves around(g),has limbs(h)} form the concept (O1, A1). O1 is the extent and A1 the intent. A formal concept relates to anatural concept in that it groups objects with a specific set of attributes. A concept represents an abstraction. Inthe very limited context of the example in Figure 93, (O1, A1) represents the abstraction of life forms that movearound using limbs; a ’complex animal’. The derivation operator ’ has a number of properties which are useful forworking with concepts.

86

Theorem 1 For a context (O, A, I), O1, O2 ⊆ O and A1, A2 ⊆ A the following statements are true [14]:

1. O1 ⊆ O2 ⇒ O′

2 ⊆ O′

1, A1 ⊆ A2 ⇒ A′

2 ⊆ A′

1

2. O1 ⊆ O′′

1 , A1 ⊆ A′′

1

3. O′

1 = O′′′

1 , A′

1 = A′′′

1

4. O1 ⊆ A′

1 ⇔ A1 ⊆ O′

1 ⇔ O1 × A1 ⊆ I

A.2 Concept Lattice

The concepts of a context do not exist in isolation, but can be intuitively related to each other on the basis ofcommon objects or attributes. We define an order on the concepts.

Definition 4 On a set of concepts the hierarchical order relation ≤ is defined as: (O1, A1) ≤ (O2, A2) ⇔ O1 ⊆O2 (or equivalent A1 ⊇ A2). If (O1, A1) ≤ (O2, A2), then (O1, A1) is called subconcept of (O2, A2) and(O2, A2) is superconcept of (O1, A1). The ordered set of all concepts B(O, A, I) is called concept lattice [14].

In the example in Figure 93, concept (O2, A2) with O2 = {leech, bream, frog, dog} and A2 = {needswater(a), moves around(g)} is a superconcept of (O1, A1), because O1 ⊆ O2 (A2 ⊇ A1). (O2, A2) couldrepresent the natural concept ’animal’ in this context. ’complex animal’ would then be one of its subconcept. Theconcepts are arranged in an abstraction hierarchy. General concepts are located at the top and specific ones at thebottom.

The order relation ≤ imposes a hierarchy on the concepts. Figure 94 shows such a hierarchy for the exampleas an Hasse diagram. A Hasse diagram is a visualisation technique for finite ordered sets in general. To fullyunderstand how the diagram is constructed the following definition is needed.

Figure 94: Concept lattice diagram for example context (reproduced and extended Figure 1.2 in [14])

Definition 5 Let (S,≤) be a finite ordered set. s ∈ S is a lower neighbour of t ∈ S, if s < t and there is noelement u ∈ S with s < u < t, then t is an upper neighbour of s, denoted as a � b [14].

In the Hasse diagram concepts are represented as nodes. If concept (O2, A2) is upper neighbour of (O1, A1)then it is located above (O1, A1) and both are connected through a line. The subconcepts of a concept are found bytracing the links downwards, the superconcepts are found upwards. The labelling in the diagram can be simplifiedby naming each object and attribute only once at the node where it appears first.Figure 95 shows this simplificationfor the example.

Advanced FCA theory builds on the definitions in this section. The basic theorem is the one given below.FCA theory is concerned with inferring meaningful patterns in the concept lattice. FCA is a knowledge discoverytechnique.

87

Figure 95: Simplified concept lattice diagram for example context (reproduced and extended Figure 1.4 in [14])

Theorem 2 (Basic Theorem on Concept Lattices) The concept lattice is a complete lattice. The infimum and supre-mum are [14]:

∧

t∈T

(Ot, At) =

(

⋂

t∈T

Ot,

(

⋃

t∈T

At

)′′)

∨

t∈T

(Ot, At) =

((

⋃

t∈T

Ot

)′′

,⋂

t∈T

At

)

A.3 Conceptual Scaling

In practical applications the high number of attributes leads to a high number of nodes in the concept graph.However, the researcher would want to examine and visualise subsets of attributes and objects that are meaningful.This is called conceptual scaling. A conceptual scale is a concept graph based on a subset of the context.

There are basic scale types which occur within larger scales. Nominal scales are used to display symbolicattributes just like the gender in Figure 96. Ordinal scales display numerical values that have a linear order, suchas the risk factor count. An interordinal scale is the combination of two ordinal scales of the same attribute, forexample the age. Its nodes represent ranges of the attribute. Boolean scales show the combinations of booleancharacteristics. For example, Figure 47 on page 47 on the left is a boolean scale or the risk factors. [52]

Figure 96: Examples of standard scales in this study: left: nominal scale of gender (exact matches), middle: ordinalscale of risk factor count, right: interordinal scale of ages

88

Figure 28 on page 36 is an example of a custom conceptual scale. It includes attributes for infarct, age rangesand risk factors. Here the risk factor attribute only distinguishes between ≥ 2 risk factors and < 2. This a thirdway of using the risk factors in a concept graph. Depending on the intention of of the concept graph, the attributescan be chosen on different abstraction levels.

Another visualisation option are the node labels. Depending on the intention of the researcher, either the objectsthat fulfil at least the given attributes can be shown, or only the objects that fulfil only the given attributes. This isdenoted with the terms ’all matches’ and ’exact matches’. For large numbers of objects it can be suitable to showthe object counts or percentages rather than an object list.

In conclusion, formal concept analysis can be used for data exploration, visualisation and knowledge discovery.The concept graph is a good means of communication. A more detailed informal introduction is [52]. The formalfoundation is covered by Ganter and Wille [14]. A review of recent tools can be found in [47].

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