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frequency domain, Spike energy [8, 9], high frequency demodulation [10], acoustic emission [11], adaptive filtering, artificial neural networks, time-frequency [12, 13], etc). The application of ANNs has been gaining importance in the area of automated fault detection and diagnosis of rotating machinery [14-16]. The neural networks have the advantages of adaptive learning, nonlinear generalization, fault tolerance, resistance to noisy data, and parallel computation abilities.

A method to optimize the Levenberg-Marquardt algorithm is presented in this paper. For this purpose, there are several ways such as Genetic algorithm [17], Ant Colony [18], Particle Swarm Optimization [19], etc. Another effective algorithm is the Bees Algorithm. This algorithm is used to optimize various problems [20-22].

In the proposed method, time domain features extracted from the vibration signal and pattern recognition using ANN is used for bearing fault diagnosis. Vibration signals measured from a single location are used in the proposed method. Some of the previous works dealt with signals from multiple locations for fault detection [23-24]. The number of input parameters used in the proposed algorithm is less than that of previous works [14-16] and hence, training speed is high, the performance of the optimum Levenberg-Marquardt algorithm is compared with standard Levenberg-Marquardt algorithm. Difference training algorithms were used for comparison and it was observed that trainlm has minimum epoch toward other training in this context.

2. Feature Extraction

2.1 Descriptive Statistics Data samples can have thousands (even millions) of values. Descriptive statistics are a way to

summarize this data into a few numbers that contain most of the relevant information. The Following time domain statistical parameters are used to detect incipient bearing damage.

Where Xi (i=1, …, N) is the amplitude at sampling point i and N is the number of sampling points. µ is the mean of X, σ is the standard deviation of X, and E(t) represents the expected value of the quantity t.

( )2

1

1 , N

rms ii

Root Mean Square y XN =

= ∑ (1)

( ) , iqrInterquartile Range y iqr X= (2)

( )3

3, skw

E XSkewness y

µσ−

= (3)

( )4

4, kur

E XKurtosis y

µσ−

= (4)

( )1

1, N

mea ii

Mean y XN =

= ∑ (5)

1

1

, N N

geo ii

Geometric Mean y X=

⎡ ⎤= ⎢ ⎥⎣ ⎦∏ (6)

1

, 1har N

ii

NHarmonic Mean y

X=

=∑

(7)

( ) , ,triMean Excluding Outliers y trimmean X percent= (8)

( ) , maxmax iLargest Element y X= (9)


3

( ) , minmin iSmallest Element y X= (10)

( ) , mod iMost FrequentValue y mode X= (11)

12 2

1

1 , N

std ii

Standard Deviation y X XN =

⎛ ⎞⎛ ⎞= −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠∑ (12)

( ), varVariance y variance X= (13)

( ), medMedian y median X= (14)

( ) , ranSample Range or Peak y range X= (15)

1

, N

sum ii

Sum y X=

=∑ (16)

( ) , trpTrapezpidal Numerical Integration y trapz X= (17)

( ) , madMean Absolute Deviation y mad X= (18)

( ), ,prcPercentiles y prctile X p= (19)

, maxcsf

rms

yCrest Factor yy

= (20)

, rmsshp

mea

yShape Factor yy

= (21)

, maximp

mea

yImpact Factor yy

= (22)

3. Feature Selection Feature selection has a significant impact on the success of pattern recognition. In this section

is computed the Euclidean distance between pairs of classes. Given an m-by-n data matrix X, which is treated as m (1-by-n) row vectors x1, x2, …, xm, the Euclidean distances between the vector xr and xs is defined as ( )( )'2

rs r s r sd x x x x= − − (23)

Therefore, sum of Euclidean distances and the maximum summations is selected for input features of ANN.

4. Fault Diagnosis Using Neural Networks An ANN is composed of nodes arranged in input, hidden and output layers, with all the nodes

in each layer having weighted inter-connections with all the nodes in the succeeding layer. Nodes in the hidden and output layers consist of artificial processing units called neurons. After training, neural network can recognize various conditions or states of a complex system. The number of nodes in the input layer is equal to the number of elements [25].

All of the train algorithms in this paper are faster training. These faster algorithms fall into two main categories. The first category uses heuristic techniques, which were developed from an analysis of the performance of the standard steepest descent algorithm. One heuristic modification

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5

different loads, (e.g. 0, 1, 2, and 3 hp). The motor speed during the experimental tests was 1720-1797 r/min. The bearing dataset was obtained from the experimental system under the four different operating conditions. The number of training samples is 20 samples and the number of testing samples is 10 samples.

Table 2. Description of the Experimental Dataset

Load (HP) Motor speed (rpm) Operating condition Label of class 0 1797 Normal baseline C1 1 1772 Normal baseline C1 2 1750 Normal baseline C1 3 1730 Normal baseline C1 0 1797 Inner race fault C2 1 1772 Inner race fault C2 2 1750 Inner race fault C2 3 1730 Inner race fault C2 0 1797 Ball Fault C3 1 1772 Ball Fault C3 2 1750 Ball Fault C3 3 1730 Ball Fault C3 0 1797 Outer race fault C4 1 1772 Outer race fault C4 2 1750 Outer race fault C4 3 1730 Outer race fault C4

In order to develop a robust fault diagnosis model that is able to identify the existence of different faults under varying load conditions, and to evaluate the proposed methods, this fault diagnosis problem is set as a four-class classification problem.

6.2 Feature Selection Result Sum of Euclidean distances and the maximum summations is selected for input features of

ANN. The pair of features selected as Table 3. Total pairs are 235 cases. A comparison on total distance between five the best of pairs is given in Table 4. Where (d1-2) is Euclidean distance between C1 and C2.

Table 3. Comparison on Euclidean distances between difference time-domain features

Feature Euclidean distance between two classes (d1-2) (d1-3) (d1-4) (d2-3) (d2-4) (d3-4)

Root Mean Square (rms) 0.36527 0.12165 0.84370 0.24362 0.47843 0.72205 Interquartile Range (iqr) 0.62092 0.31616 0.86678 0.30475 0.24587 0.55062

Skewness (skw) 0.48905 0.30709 0.34190 0.18196 0.14715 0.03481 Kurtosis (kur) 0.44903 0.00351 0.85616 0.44552 0.40713 0.85265

Mean Value (mea) 0.16971 0.19552 0.11276 0.02581 0.05695 0.08277 Geometric Mean (geo) 0.13479 0.02957 0.27366 0.10521 0.13888 0.24409 Harmonic Mean (har) 0.00010 0.00213 0.00266 0.00223 0.00276 0.00054

Mean Excluding Outliers (tri) 0.24693 0.22670 0.13923 0.02022 0.10770 0.08747 Maximum Value (max) 0.36053 0.07967 0.82030 0.28086 0.45977 0.74063 Smallest Value (min) 0.32131 0.08183 0.83252 0.23948 0.51121 0.75069

Most Frequent Value (mod) 0.13781 0.09184 0.09088 0.04596 0.04692 0.00096 Standard Deviation (std) 0.36654 0.12308 0.84441 0.24347 0.47787 0.72134

Variance (var) 0.17659 0.03425 0.74724 0.14234 0.57064 0.71299 Median Value (med) 0.29295 0.20717 0.17082 0.08577 0.12212 0.03635

Numerical Range (ran) 0.35127 0.08301 0.84973 0.26826 0.49845 0.76672 Sum of time series data (sum) 0.55116 0.56333 0.52371 0.01217 0.02745 0.03963 Trapezoidal Integration (trp) 0.55073 0.56281 0.52314 0.01209 0.02758 0.03967

Mean Absolute Deviation (mad) 0.43147 0.16840 0.85011 0.26307 0.41864 0.68171 Percentiles (prc) 0.33700 0.08401 0.84306 0.25299 0.50606 0.75905 Crest Factor (csf) 0.44491 0.00950 0.48584 0.45442 0.04092 0.49534

Shape Factor (shp) 0.23338 0.12034 0.55137 0.11304 0.31800 0.43103 Impact Factor (imp) 0.22834 0.08329 0.55426 0.14506 0.32591 0.47097


6

Table 4. Comparison on total distance between five the best of pairs

Input features Total distance Interquartile Range & Kurtosis 1.72294 Interquartile Range & Mean Absolute Deviation 1.71690 Interquartile Range & Numerical Range 1.71651 Interquartile Range & Standard Deviation 1.71120 Interquartile Range & Root Mean Square 1.71049

6.3 Neural Network Structure Interquartile range and root mean square are selected for input features of ANN. A

comparison on epoch mean, epoch STD, run-time mean, and run-time STD between difference network structures and activation functions are given in Table 5.

Table 5. Comparison on epoch, run-time between difference network structures and activation functions

Train Algorithm

Input Features

Network Structure

Activation Functions

Epoch Mean

Epoch STD

Run-Time Mean (sec)

Run-Time STD

trainlm iqr & rms 2-5-4 tansig-purelin 3.4 1.07 0.50972 0.25

trainlm iqr & rms 2-5-5-4 tansig-tansig-purelin 9.4 4.09 0.60380 0.27

trainlm iqr & rms 2-5-5-4 tansig-tansig-tansig 26.9 27.77 0.97810 0.87

trainlm iqr & rms 2-5-4 tansig-tansig 384.5 391.40 11.84917 11.62

6.4 Train Algorithm Selection Here a two-layer feed-forward network is created with two-element input, five hidden tansig

neurons, and four purelin output neurons. A comparison on epoch mean, epoch STD, run-time mean, and run-time STD between difference train algorithms is given in Table 6.

Table 6. Comparison on epoch, run-time between difference training algorithms

Training Algorithm

Epoch Min

Epoch Max

Epoch Mean

Epoch STD

Run-Time Min (sec)

Run-Time Max (sec)

Run-Time Mean (sec)

Run-Time STD

trainlm 2 5 3.4 1.07 0.38285 1.22765 0.50972 0.25

traincgf 35 60 42 7.83 0.74939 1.11952 0.86962 0.11

trainscg 30 58 46 9.53 0.74692 1.08240 0.89008 0.10

traincgp 32 54 44.2 7.39 0.66434 1.05609 0.89613 0.10

traincgb 24 134 47.2 31.45 0.63427 2.01249 0.93361 0.39

trainbfg 23 61 37.6 11.43 0.75528 1.40633 1.03299 0.22

trainoss 54 123 83.1 21.98 0.93726 1.67859 1.25111 0.24

trainrp 137 438 218.22 103.96 1.34747 3.45847 1.89990 0.75

traingdx 241 568 356.1 123.48 1.99374 4.49646 2.82216 0.92

6.5 Levenberg-Marquardt Optimization The Levenberg-Marquardt algorithm was designed to approach second-order training speed

without having to compute the Hessian matrix. The Levenberg-Marquardt algorithm uses this ap-proximation to the Hessian matrix in the following Newton-like update:

1

1T T

k kx x µ−

+ ⎡ ⎤= − +⎣ ⎦J J I J e (24)

Where J is the Jacobian matrix that contains first derivatives of the network errors with respect to the weights and biases, and e is a vector of network errors.

When the scalar µ is zero, this is just Newton’s method, using the approximate Hessian matrix. When µ is large, this becomes gradient descent with a small step size. Newton’s method is faster and more accurate near an error minimum, so the aim is to shift towards Newton’s method as quickly as possible. Thus, µ is decreased after each successful step (reduction in performance


7

function) and is increased only when a tentative step would increase the performance function. In this way, the performance function will always be reduced at each iteration of the algorithm.

The mu, mu_dec, and mu_inc are optimization variables. The parameter mu is the initial value for µ. This value is multiplied by mu_dec whenever the performance function is reduced by a step. It is multiplied by mu_inc whenever a step would increase the performance function.

The mu range is defined between 0 and 2, the mu_dec range is defined between 0 and 1, and the mu_inc range is defined between 1 and 20. The optimization fitness is defined as

: [ 1 0 ]Fitness minimization epoch mean of run (25)

Table 7. Bees Algorithm Parameters

Bees Algorithm Parameters Symbol Population 40

Number of selected sites 10 Number of top-rated sites out of m selected sites 5

Initial patch size 0.001 Number of bees recruited for best e sites 4

Number of bees recruited for the other selected sites 2

A comparison between optimum parameters and original parameters is given in Table 8.

Table 8. Comparison between optimum parameters and original parameters

Parameter Original Value Optimum value mu 0.001 1.473157563

mu_dec 0.1 0.019734974

mu_inc 10 1.417490935

A comparison on original and optimum Levenberg-Marquardt between difference input features is given in Table 9.

Table 9. Comparison on original and optimum Levenberg-Marquardt between difference input features

Input Features

Standard Levenberg-Marquardt Optimum Levenberg-Marquardt Epoch Mean

Epoch STD

Run-Time Mean (sec)

Run-Time STD (sec)

Epoch Mean

Epoch STD

Run-Time Mean (sec)

Run-Time STD (sec)

iqr & kur 109.40 87.38 1.94468 1.62788 17.50 9.24 0.66366 0.21133

iqr & mad 119.60 164.28 1.73469 1.90249 14.50 3.21 0.60395 0.09887

iqr & ran 76.60 73.68 1.25379 0.88302 16.40 6.60 0.61305 0.16675

iqr & std 120.80 82.48 1.85674 1.06475 17.90 3.51 0.67445 0.14919

iqr & rms 121.20 109.88 1.68635 1.22527 13.80 3.71 0.59305 0.11775

7. Conclusion In this work a method for optimizing Levenberg-Marquardt training was presented. The

method combines benefits of both Levenberg-Marquardt training and the Bees Algorithm to improve classifier accuracy and performance. A comparison between the presented optimum train and the original train showed that the presented method is exceptionally more efficient in classifier accuracy and performance.

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