automatic detection of onset and cessation of tree stem radius increase using dendrometer data
TRANSCRIPT
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Neurocomputing 73 (2010) 2039–2046
Contents lists available at ScienceDirect
Neurocomputing
0925-23
doi:10.1
� Corr
E-m
journal homepage: www.elsevier.com/locate/neucom
Automatic detection of onset and cessation of tree stem radius increase usingdendrometer data
Mikko Korpela a,�, Harri Makinen b, Pekka Nojd b, Jaakko Hollmen a, Mika Sulkava a
a Aalto University School of Science and Technology, Department of Information and Computer Science, P.O. Box 15400, FI-00076 Aalto, Finlandb Finnish Forest Research Institute - Southern Finland Regional Unit, P.O. Box 18, FI-01301 Vantaa, Finland
a r t i c l e i n f o
Available online 7 March 2010
Keywords:
CUSUM chart
Mann-Kendall test
Time series segmentation
Markov switching autoregressive model
Wood formation
12/$ - see front matter & 2010 Elsevier B.V. A
016/j.neucom.2009.11.035
esponding author.
ail address: [email protected] (M. Korpe
a b s t r a c t
Dendrometers are devices, which measure continuously the stem radius of a tree. We studied the use of
four data analysis methods for automatically and, thus, objectively determining the onset and cessation
dates of radial increase based on dendrometer data. The cumulative sum chart, time series
segmentation, Mann-Kendall test, and a Markov switching autoregressive model were compared. We
used data measured in two stands in southern Finland to test the performance of the methods. Once
configured properly, the cumulative sum chart produced results similar to those determined by an
expert. The overall performance of the other methods was lower.
& 2010 Elsevier B.V. All rights reserved.
1. Introduction
Wood formation is restricted to a certain period in the year,e.g., [1]. It is regulated by various endogenic and exogenic factorsincluding genotype, site, silviculture, and climatic variation. Inspite of the basic nature of the underlying process, our presentknowledge concerning the timing of the various phases and therate of increment during a growing season is still far fromcomplete. This lack of knowledge is largely due to difficulties inmeasuring wood formation at short intervals.
Dendrometers have traditionally been used for measuring theintra-annual wood formation of trees with high precision, e.g.,[2,3]. However, changes in stem dimensions are not solely a resultof wood formation; they are often caused by other processes,especially changes in stem hydration, for example, [4]. Because ofthe large and frequent changes in stem radius associated withfluctuations in stem water potential, it is difficult to usedendrometer measurements to determine the onset, cessation,and rate of wood formation, i.e., radial increment due toformation of new cells, e.g., [5–7].
In recent years, few studies have been published describing thetiming of radial increase or weather-growth relationship based ondendrometer data, for instance, [7–9]. Reversible and irreversiblechanges have been separated from each other based on daily orseasonal reference values identified from the dendrometer data.Further transformations are then used to calculate differentphases of stem radius change. However, the definitions andapproaches for identifying the onset and cessation of radial
ll rights reserved.
la).
increase have been different between the studies. Thus, criteriaand methods for determining onset and cessation dates are notyet generally accepted.
The aim of this study is to objectively and automatically detectthe onset and cessation of radial increase period based ondendrometer data. We compared the suitability of the cumulativesum (CUSUM) charts [10,11], time series segmentation models[12], Mann-Kendall test [13,14], and Markov switching autore-gressive models [15] for performing this determination auto-matically solely based on the data, cf. [16].
2. Dendrometer data
Dendrometer data were collected from two stands located300 m from each other in southern Finland (Ruotsinkyla, 603210N,253000E). In both stands, sample trees without visible damagewere selected from the dominant tree layer. In the first stand(altitude 45 m a.s.l.), the Norway spruce trees were growing in apure spruce stand on fertile mineral soil (H100 ¼ 30 m, dominantheight at age 100 years) classified as Oxalis-Myrtillus forest type[17]. Mean stem diameter of the sample trees at breast height was27 cm and relative crown length was 68%. The sample trees weremonitored during the growing seasons of 2001–2005. In thesecond stand (altitude 60 m a.s.l.), four Norway spruce and fourScots pine trees were monitored during the growing seasons of2002–2003, and another four spruce and four pine trees duringthe growing seasons of 2004–2005. They were growing in a mixedspruce-pine stand on a relatively fertile mineral soil classified asMyrtillus forest type (H100 ¼ 27 m) [17]. The total number ofobservations (year � tree combinations) was, thus, 57. Mean daily
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temperatures and precipitation sums were obtained from ameteorological station of the Finnish Meteorological Institutelocated about 5 km from the study stands.
Stainless-steel band-dendrometers were installed on each treeat a height of about 2 m. Before installing the band, the outer barkunder the band was lightly brushed to ensure smooth contactwith the trunk. The dendrometer consists of three basic elements:(1) a stainless-steel band encircling a tree, (2) a sensor (rotatingpotentiometer) reacting to movements of the stainless-steel bandand (3) an aluminum-body and fastening mechanism (Fig. 1). Thefastening mechanism consists of three parts: (1) a constant force
Fig. 1. The dendrometer mounted on a tree.
Fig. 2. Structure of the dendrometer: A ¼ stainless-steel band; B ¼ rotating pot
0 100 200 3000
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1
Julian day
radi
us (m
m)
2002
Fig. 3. Examples of the dendrometer data; a Norway spruce tree in Ruotsinkyla mea
resolution of the measurements is one hour.
spring, (2) a fastening arm, and (3) an adjustable foot (Fig. 2). Thespring stretches the steel band around the tree with a force ofabout 3 N, making the band capable of reacting to small variationsin girth but without damaging the tree.
Changes in tree girth were measured at a resolution of 0.1 mm,corresponding to diameter change of about 0.03 mm. The outputof the dendrometers was stored as hourly averages. Examples ofdendrometer data showing the change in the stem radius duringtwo years are shown in Fig. 3. From these measurements, thedaily values of stem circumference were calculated as the mean ofhourly values and the circumference changes were converted toradial changes assuming a circular stem cross-section. Thedendrometer has been described in more detail in [3]. Datamissing due to technical problems were replaced by linearlyinterpolated values. The first (last) available value was extendedto cover any missing leading (trailing) values of the annual series.
We visually evaluated the methods used in the previousstudies (cited in the introduction) for determining the onset andcessation dates of radial increase from dendrometer data. All themethods had difficulties in identifying the onset and cessationdates because of stem hydration changes, i.e., they were notsufficient for identification of the crucial dates for all trees andyears. It may happen, for example, that during spring stem radiusstarts to increase, but then stays constant for a rather long time,before the increasing trend reappears (Fig. 4). It is difficult to say,whether new xylem has actually been formed or whether thestem has just swollen.
We ended up, therefore, determining the onset and cessationdates visually from the dendrometer data to obtain labeling fortraining purposes. It should be emphasized that manual determi-nation of the crucial dates is laborious and is likely to result inobserver-related subjective differences and human errors. The
entiometer; C ¼ fastening arm; D ¼ adjustable foot; E ¼ cable; F ¼ spring.
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radi
us (m
m)
2003
sured in 2002 and 2003. The zero level is the first observation of the year. The
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us (m
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Fig. 4. Example of a difficult growth pattern. It is difficult to distinguish when the
actual radial increase starts: on around day 115 or around day 150. Also, only
based on the curve it is difficult to tell whether the increase ceases on around day
190, 235, or 340.
dk
Fig. 5. Parameters d and k of the CUSUM chart define the shape of the V-mask. The
shape affects how soon and how large changes are detected.
M. Korpela et al. / Neurocomputing 73 (2010) 2039–2046 2041
dates identified visually are called below as the ‘expert choice’.These expert choices were acquired retrospectively by looking atgraphs showing the diameter data recorded by the dendrometerduring a calendar year.
The tests can be done with all the data in order to assess theperformance of different methods. Alternatively, difficult cases(Fig. 4) can be left outside the data in order to study if jump-likechanges caused by changes in stem hydration are the mainobstacle for obtaining better results with the automatic methods.We labeled an onset or cessation date of radial increase as adifficult case if there was a clear jump-like increase of at leastabout 0.07 mm in stem radius before the onset date or after thecessation date or if the minimum achievable difference betweenthe date predicted by the CUSUM method and the ‘expert choice’was at least 10 days, cf. [16]. These kinds of cases are also difficultfor the expert. Altogether, there were 22 and 16 difficult cases foronset and cessation, respectively.
3. Methods
3.1. Cumulative sum chart
The cumulative sum (CUSUM) chart [10,11] is a statisticalprocess control tool, which detects small changes in the mean m ofa signal. It monitors the cumulative sum of previous observationsx, which are assumed to follow a normal distribution:
X �Nðm,s2Þ ð1Þ
Sm ¼Xm
i ¼ 1
ðxi�mÞ ð2Þ
The cumulative sum Sm at time step m is the sum of thedifferences between the previous observations and m, theestimated value of m. The CUSUM chart detects up or down driftsand abrupt changes in m. The chart has two parameters: d and k,which depend on standard deviation s, size of detectable changeDX, and probabilities of type I (a) and type II ðbÞ errors. Theparameters are conventionally determined based on the levels ofrisk that the decision maker is willing to tolerate [11]. Theparameters define the shape of a so-called V-mask (Fig. 5), whichis used to detect the changes in the mean. The shape of the V-mask affects how soon and how large changes are detected. TheV-mask is positioned such that the middle point of the verticalline of the mask coincides the current value of Sm. In case a
previous value of Sm is outside the mask, it is concluded that themean has changed (see Fig. 6).
We studied how to set the parameters of the chart to produceresults as similar as possible to the ‘expert choice’. One has tochoose suitable values for DX, s, a, and b. In addition the onset orcessation levels m for radial increase have to be determined.Cessation of radial increase can be detected by running the chartin reverse time.
The correct values of suitable detectable size of change andmagnitude of noise were not known in advance. Therefore,parameters d and k were varied in a wide range: d between 1and 50 and k between 0.01 and 0.1.
Performance of the CUSUM chart can be improved by focusingon only upward (onset) or downward (cessation) shifts in mean.Using only one-half of the V-mask for detecting the changesreduces the number of false alarms. Therefore, this procedure wasused in this study.
3.2. Time series segmentation
Time series segmentation can be used to divide time series intohomogeneous parts [12]. Dynamic programming [18] is acomputer programming and mathematical optimization method,which is guaranteed to produce the globally optimal segmenta-tion solution. We use dynamic programming to divide annualdendrometer time series into three linear segments. The predictedonset and cessation dates are the beginning and end of the secondsegment, respectively.
3.3. Mann-Kendall test
The Mann-Kendall test for trend is a nonparametric method fordetecting a trend in a time series [13,14]. Given the ranking of thesamples in the time series, the test is immune to their magnitude.The test statistic was computed in a sliding window. The onset ofgrowth was assigned to the middle position of the first windowwhere a positive trend was detected. In order for a positive trendto be detected, the p-value of a false alert had to undershoot acertain limit.
The model was applied on hourly data. For the length of thewindow, values approximately corresponding to odd number ofdays from 5 to 91 were tested. For the threshold, 20 valuesbetween 10�8 and 10�1 were tested. The threshold valuecandidates were logarithmically equally spaced. The detection ofcessation was performed in the same way, but with thedendrometer data negated and turned around in time.
3.4. Markov switching autoregressive model
The increment phase of a tree might be considered as a hiddenstate. The onset and cessation dates would then correspond to
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4
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16data
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−20
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40CUSUM
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40CUSUM
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40CUSUM
Fig. 6. Artificial data with constant mean (left column) and abrupt (middle column) and gradual change in mean (right column) and the respective CUSUM charts (bottom
row). When m stays the same, the previous values of cumulative sum Sm stay inside the V-mask. A change in m is detected, when a previous value of cumulative sum is
outside the V-mask.
M. Korpela et al. / Neurocomputing 73 (2010) 2039–20462042
changes in the state. The hidden Markov model (HMM) [19]assumes that the unobservable state sequence follows a Markovchain. Each state has a specific model for an observation. Byoptimizing the parameters of the HMM, the likelihood of theobserved data can be improved. Given the model class, theparameters, and the data, the most likely state at each time pointcan be computed. From this sequence, it is possible to deriveestimates for the onset and cessation dates.
Example 3 in [15] describes a HMM where the observationsfollow an autoregressive (AR) model with state-dependentparameters. The AR model is used to describe the dependencybetween consecutive values of the time series. Another descrip-tion of the class of models is found in [20]. The model is time-homogeneous, i.e., the transition probabilities P between hiddenstates are independent of time. Considering the annual cycle oftree growth, the assumption of constant transition probabilitiesover time does not seem suitable. Our approach builds on thismodel, but makes use of time-inhomogeneous transition prob-abilities. In other words, there is a separate matrix P(t) for eachtime step t, with the steps corresponding to calendar dates. Weuse pijðtÞ ¼ Pðstþ1 ¼ jjst ¼ iÞ to denote the probability of state j attime t + 1 conditional on state i at time t.
Our model uses three states (before (1), during (2), and after(3) growth) to describe dendrometer data cut at the beginningand end of a calendar year. We assume that transitions 1-3,2-1, 3-1, and 3-2 (‘back in time’ or skipping growth season)are close to impossible, i.e., have a very small constant probability.That means state 3 is almost terminal. The estimation of theremaining transition probabilities builds on the assumption of theonset and cessation dates each following a discretized truncatednormal distribution, where the truncation occurs at the first and365:th day of the year. We use the labeled training data to fit the
distributions. The fitting is done with the mle function of theStatistics Toolbox in Matlab. The function iteratively computesthe maximum likelihood (ML) estimates of the distributionparameters. We start the iteration from the ML parameterestimates of the regular, non-truncated normal distribution. Inour experiments, the model operates on daily dendrometer data.
Assuming that Fo and Fc are the cumulative distributionfunctions of the onset and cessation dates, respectively, theprobabilities p12 and p23 can be computed as defined in Eq. (3). Forp12 (p23), set i¼1 (i¼2) and z¼o (z¼c). Note that z just indicatesthe identity of the probability distribution: either onset orcessation. The formulas apply for t ¼ 1,y,365. For leap yearsand t ¼ 366, we use the same probabilities as for t ¼ 365.The remaining probabilities are computed with the ruleP3
j ¼ 1 pijðtÞ ¼ 1 8i,t. Due to the effect of the small constant non-zero probabilities and small numerical work-arounds, Eq. (3) onlyholds approximately in our experiments. The numerical work-arounds to the transition probabilities are summarized as follows:use upper limit of 0.99, add a constant 0.01 to everything, andfinally normalize the probability sums to unity.
piðiþ1ÞðtÞ ¼ ðFzðtþ1:5Þ�Fzðtþ0:5ÞÞ=ð1�Fzðtþ0:5ÞÞ ð3Þ
When m denotes the order of the AR model, we assume thatthe state sequence starts at time point m+1 from the pre-growthstate with a probability of almost one, with only small non-zeroprobabilities given to the other states.
The AR coefficients of the model are learned from the trainingdata, using multiple linear regression on lagged values of thetraining growth patterns. The particular sections (pre-growth,during growth, post-growth) of the patterns corresponding toeach state are used to train the parameters of that state. The
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standard deviation common to all states is estimated as the rootmean square error of the regressions.
In order to get the final predictions for the onset and cessationdates, we take the sequence consisting of the most likely state ateach time point in the dendrometer data time series, and filter itwith a mode filter using a sliding window. The filter evens outsmall irregularities by taking the most common value inside asymmetric window around a given time point. The onset andcessation dates are the first and last occurrence of the growthstate, respectively. In the case that the growth state is notdetected, a growing season of one day at the end of the year isassumed as a fallback. The resulting large error ensures that thepoor performance of the predictor is duly noted.
The values tested for the order of the AR model were 1–5 days,and even numbers from 6 to 20 days. The length of the mode filterwindow ranged from 1 to 15 days, odd numbers.
Table 1Change point detection error (RMSE) in the validation set for CUSUM chart, time
series segmentation, Mann-Kendall test, Markov switching autoregressive model
(raw dendrometer data and daily differences), and mean predictor.
RMSE All data Without difficult cases
Onset Cessation Onset Cessation
CUSUM chart 13.8 12.9 4.4 13.2
Segmentation 20.5 18.6 8.3 17.6
Mann-Kendall test 22.2 62.9 29.7 57.4
Autoregressive model 35.6 26.5 24.4 27.3
Autoregressive model, diff. 11.0 21.5 8.9 16.0
Mean predictor 8.3 18.2 7.5 17.4
Results that are statistically significantly better than those of the mean predictor,
after correcting for multiple tests with the Holm–Bonferroni method [22], are
shown in boldface. Italic shows the results that failed the significance test.
3.5. Model fitting and assessing model performance
For each method, the accuracy of the predicted dates wasverified using cross-validation [21]. In r-fold cross-validation, thedata set is divided into r disjoint sets of equal size. The parametersof the prediction method yielding the minimum error are selectedr times and each time one of the sets is held out as a validationdata set. Cross-validation was not used for determining theparameter values. The parameters were determined in eachtraining set without using the validation data. For each methodand training set, parameter values in an appropriate range weretested. The parameter value ranges are described in the precedingsections. Parameter values, which yielded the minimum meansquare error (MSE) e in days (Eq. (4)) for a training data set, werechosen. The performance of the model was estimated as thesquare root of the mean square errors (RMSE) obtained using thevalidation data sets.
e¼1
n
Xn
i ¼ 1
ðaðiÞe �aðiÞc Þ2
ð4Þ
Above, n is the number of onset or cessation dates in the trainingor validation set, ae
(i) is the ‘expert choice’ and ac(i) is the date
predicted by the change detection method for the i th dendrom-eter band. All cross-validation tests for all methods were of theleave-one-out type.
For choosing the optimal Markov switching autoregressivemodel, the average of onset and cessation errors, each computedwith Eq. (4), was used. That is the equivalent of including bothonset and cessation errors inside the sum in the equation.
With the Mann-Kendall and AR methods, some choices ofparameters may result in a failure to detect the onset or cessation.There, the best parameters were chosen by computing the error(Eq. (4)) over the successful cases (i.e., detection achieved), only.In addition, parameters that caused at least 60% of training casesto fail were discarded. The model selection was performed in theremaining group of parameter combinations, where the errorestimates were reasonably confident, i.e., covering over 40% oftraining cases. In this group, parameter choices were dropped andthe best remaining selected until a detection was reached withthe test case. The AR method proved to be quite unreliable in ourapplication, and the initially discarded uncertain (Z60% trainingfail) cases had to be reconsidered in the possible case of failure todetect the onset or cessation with any of the choices in the othergroup (o40% fail) of parameter value combinations.
A mean predictor was used as a baseline method. The meanpredictor computes the average value of the labels in the trainingset and uses it as the predicted date of onset or cessation for the
test case. Also the mean predictor was used in a leave-one-outsetting.
4. Results
4.1. CUSUM chart
It was visually found that the average stem radius duringspring or fall is a better value for m than the average of wholewinter. For example, swelling and shrinkage of the stem oftenmakes the radius drift in one direction or the other during thelong winter. Therefore, average over a shorter time interval waspreferred in determining the value of m, since that correspondsbetter to the actual starting or ending level of stem radius. Thespring was defined as the two week period before the first daywith average temperature 43 3C. Similarly, fall was defined as thetwo week period after the first day with average temperatureo3 3C after summer. The average stem radii of spring and fallwere used as values of m for detecting the onset and cessation ofgrowth, respectively.
For the CUSUM chart the lowest RMSEs obtained with separateparameter values k and d for each tree and year were 5.8 days foronset and 5.9 days for cessation, respectively, cf. [16]. The onsetand cessation dates of other trees cannot, however, be detectedwith the same accuracy using these parameter values, because theoptimal parameters differed between the trees and years. Thesame parameter values were, therefore, used for all the trees andyears. In leave-one-out cross-validation the training RMSEs were12.1 and 11.9 days for onset and cessation, respectively. In thevalidation data set, the RMSEs were 13.8 and 12.9 days for onsetand cessation, respectively (Table 1). The former value is higherthan that of a mean predictor (RMSE 8.3) and thus not acceptable.The latter value, 12.9, was found to be significantly better than18.2, the result of the mean predictor.
The significance of the 12.9 vs. 18.2 result was assessed with apermutation test using one million random partitions (i.e.,random labelings) of the individual errors produced by the twomethods. A full permutation test would have been infeasible dueto the large total number of partitions. Only 1.3% (p-value 0.013)of the random partitions showed a performance difference at leastas great as what was observed when the errors were attributed tothe correct method. This meets the standard 5% false positiveconfidence level.
The relatively high errors for onset of growth were mainlycaused by some difficult cases. The difficulty was caused byincreasing stem radius related to stem hydration before the actualradial increase begins (Fig. 4). These kinds of curves were also
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Mean
AR diff
AR
Mann−Kendall
Segmentation
CUSUM
0 100 200
Fig. 7. Box plots of prediction errors, the whole data set included. Left: error of onset date, right: error of cessation date. Outliers (+) are points that have a distance larger
than 1.5 times the width of the interquartile range to both the first and the third quartile. The whiskers cover the data that are outside the interquartile range but not
outliers.
M. Korpela et al. / Neurocomputing 73 (2010) 2039–20462044
difficult for the CUSUM chart, i.e., it detected the onset date tooearly.
When the difficult cases were left out, the RMSE of the CUSUMchart in the prediction of onset clearly decreased. However, thevalidation error in the prediction of cessation showed noimprovement. For CUSUM the RMSEs in the training data setwere 3.8 days and 9.6 days for the onset and cessation,respectively. The RMSEs in the validation data set were 4.4 daysand 13.2 days for the onset and cessation, respectively. These twoare lower errors than those produced by the mean predictor. Thesignificance of the differences was again tested with one millionpermutations of the errors of the two comparable methods. In theonset case, the advantage of the CUSUM chart was found to bestatistically significant: the p-value was 0.006. However, in thecessation case the p-value was 0.148, which does not meet the 5%(0.05) significance level. It should be noted that results withoutthe difficult cases can be used for studying the effect of jump-likechanges to the performance of the methods. They should not beused for assessing the relative performance of the methods. Thiscan only be done using the results with all data.
4.2. Time series segmentation and Mann-Kendall test
The time series segmentation had higher RMSE than theCUSUM chart (Table 1). The error in the detection of cessation wasat a similar level as with the mean predictor, both with all thecases and without the difficult cases. When the whole data setwas considered, time series segmentation had a larger RMSE thanthe mean predictor in the detection of onset. However, when thedifficult cases were omitted, the RMSE of the two methods wasalmost equal.
The Mann-Kendall test produced high RMSE values (Table 1).Leaving the difficult cases out did not change the resultmuch—the RMSE of onset prediction actually increased. AllRMSEs were clearly higher than those of the mean predictor.
The parameters chosen in cross-validation tests were 1.0�10�8 for the decision threshold and approximately 19 or 21days for the window size of the Mann-Kendall test. Notably theparameters were the same for both onset and cessation.
Furthermore, the chosen threshold value represents the minimumend of the parameter range. While extending the parameter rangemight have improved the overall error numbers a little, some ofthe extreme errors (Fig. 7), particularly in the prediction ofcessation, would have been even more extreme. The chosenparameters did not change much when the tests were repeatedwithout the difficult cases. However, the second smallest decisionthreshold option, about 2.3 �10�8, also appeared among thechosen parameters in the onset prediction task.
4.3. Markov switching autoregressive model
The Markov switching autoregressive model operating on rawdendrometer data (‘autoregressive model’ in Table 1) haddifficulties in detecting the onset of stem radius increase. Thedetection of the onset in the whole data set was notably moreaccurate with daily differences (‘autoregressive model, diff.’ inTable 1) than the raw dendrometer data.
Most of the large performance discrepancy in the prediction ofonset can be explained by the total failure of the raw data modelto detect the onset and cessation in one of the cases (the largesterrors in Fig. 7). The onset and cessation RMSEs of the modeloperating on raw data were 19.2 and 15.0, respectively, when thesingle total failure was excluded from the results. The results ofthe model without the difficult cases were worse than the‘exclude-one’ errors, the onset and cessation RMSEs being 24.4and 27.3, respectively. Here the model clearly suffered from thedifferent subset of training data: the RMSEs were only 4.4 (onset)and 12.5 (cessation), when the errors for the ‘non-difficult’ caseswere picked from the results of the test with the whole data set.However, this does not mean that the method is better than theCUSUM chart. The removal of difficult cases for the AR modelinvolved both difficult onset and difficult cessation. This was donebecause a single model, i.e., a single set of parameters, was usedfor detecting both onset and cessation. When the difficult onsetswere removed independently from the results acquired with thewhole data set, the onset RMSE of the AR model (raw data) was38.7 vs. 11.8 for CUSUM. The corresponding cessation errors,
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i.e., only difficult cessations removed from the complete dataresults, were 14.1 (AR model, raw data) and 11.5 (CUSUM).
The RMSEs of the AR model operating on daily differenceswere quite close to those of the mean predictor in every errorcategory of Table 1. The RMSE of the model in the first category(onset, all data) was right between the errors of the meanpredictor and the CUSUM chart, but in other categories, theCUSUM chart performed better. In the final column of Table 1, theAR model operating on daily differences seems to be a little betterthan the mean predictor (RMSE 16.0 vs. 17.4), but the difference isnot statistically significant: a p-value of 0.387 was obtained in apermutation test consisting of a million random re-labelings ofthe errors.
According to the cross-validation test of the Markov switchingautoregressive model with raw daily dendrometer data, 16 daysas the order of the AR model usually produced the lowest RMSEvalues in the training set, but other AR orders between 12 and 20sometimes resulted in the lowest error. For the length of the modefilter, 15 days typically resulted in the lowest RMSE. The otherappropriate choices for mode filter length were 11 and 13. In thefailure case, all parameter choices were tried in vain. Changing thelength of the filter window naturally cannot find the missinggrowth state, but none of the AR model orders improved themodel, either. In the tests without the difficult cases, the mostcommon parameter values were 18 and 3 days for the AR orderand filter length, respectively. Especially the amount of filtering(15 vs. 3 days) necessary for the best results depended heavily onthe data set used.
The Markov switching autoregressive model based on dailydifferences both performed better and was simpler than the oneoperating on raw data. In contrast to the experiments with themodel using raw data, where the erratic sequence of most probablestates required heavy filtering, the tests with daily differential datarevealed that no filtering was usually needed. In other words,according to the cross-validation, filter length 1 always resulted inthe lowest training RMSE when dealing with the whole data set. Afilter length of 11 days was once selected in the cross-validationtests run without the difficult cases. The AR order was always 1.
With the Markov switching model, we also ran tests where theparameters, excluding the transition probabilities, were iteratedwith the expectation–maximization (EM) algorithm [15]. In thiscase, the parameters described in Section 3.4 were considered asthe starting point. The procedure increased the likelihood of theobserved data. The prediction performance, however, degraded,often significantly.
5. Discussion
The development of stem radius or circumference can bemonitored with a high time resolution and summed up tolong-term results using dendrometers. Durability, automatedmeasurement, and low price make dendrometers suitable to beused in a growth monitoring network covering large regions.
Much of the variation in stem radius is, however, independent ofwood formation. In late winter and early spring, rising temperaturemay increase evapotranspiration, whereby this water loss cannot bereplaced by water uptake due to soil frost and it consequently causesa reduced stem radius, e.g., [7]. Later in spring, water uptake willresult in an increase of stem radius not related to the formation ofnew tracheids, e.g., [8,23–25]. Onset of wood formation may, thus,be masked by the rehydration of the stem, cf. [26].
Likewise, increases in stem radius in late summer may becaused by stem swelling rather than wood formation, making itdifficult to determine the cessation of wood formation fromdendrometer measurements. Especially in the slow-growing
boreal forests, the daily swelling and shrinkage of the stem isrelatively large compared to the radial stem increase caused bycell division and enlargement. In cold regions, such as Finland,wintertime ice formation in stems does further complicate theinterpretation of dendrometer data [27].
The CUSUM chart proved to be a good starting point for detectingautomatically the dates when the radial increase of trees begins andceases. However, suitable parameter values are needed to getaccurate results. Once configured properly, the CUSUM chartproduced results similar to the ‘expert choice’. In many cases, theresults of the CUSUM chart were accurate for the onset of radialgrowth. However, swelling and shrinkage of the stem beforethe onset of growth turned out to decrease the performance of themethod. To detect the cessation date is difficult, because during fallthe amount of stem radius increase is small compared to thereversible changes in stem radius. However, the CUSUM chart wasable to predict the cessation dates relatively well in all cases. Theswelling and shrinkage of the stem in the difficult cessation casesdid not affect the performance of the method much.
The results of the other prediction methods were lessimpressive. Mann-Kendall test was very inaccurate. In the testswith the Markov switching autoregressive models, it wasdiscovered that difference data worked better than rawdendrometer data. However, the model operating on differencedata only outperformed the mean predictor in the cessation taskwith the reduced data set. Even then the margin was small, andthe numbers are not completely comparable due to thedifferences in the data sets, explained in Section 4.3. The resultsmay be improved with a different assumed distribution of theonset and cessation dates, because the (truncated) normaldistribution used in our method cannot accurately model theskewed distributions of the expert choices. The simple 3-segmentmodel performed at the level of the mean predictor, whenthe difficult cases were omitted. The advantage of the model is thelack of the training or tuning stage, which may be adverselyaffected by atypical or unrepresentative samples.
A relatively high number of cases (39% and 28% for the onsetand cessation, respectively) were labeled as difficult. The changesin stem radius caused by wood formation and other factors cannotalways be distinguished using the studied methods. It provednecessary to compare the dendrometer measurement with directmeasurements on tracheid formation on the stems. Thus, theresults should always be checked before using them in furtheranalysis. The amount of manual work needed for identifying thecrucial dates can anyway be reduced using the studied methodsfor automatic detection of changes.
The CUSUM chart and Mann-Kendall test can also be used foron-line detection of onset of radial increase, although the use ofthe latter cannot be recommended. In contrast, the cessation ofradial increase cannot be detected on-line using these methods. Inretrospective analysis of onset and cessation of radial increase—asin segmentation and Markov switching autoregressive mode-l—using observations on both sides of the change point has thepotential to improve the performance of the detection method.However, this benefit did not help the methods outperform theCUSUM chart.
6. Conclusions
The CUSUM chart, time series segmentation models,Mann-Kendall test, and Markov switching autoregressive modelswere used for automatic detection of onset and cessation dates ofradial increase of stems based on automated dendrometer data. Attheir best, the results of the CUSUM chart agreed rather well withdates determined by an expert. The overall prediction accuracy of
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M. Korpela et al. / Neurocomputing 73 (2010) 2039–20462046
the other methods was lower. Most probably, the CUSUM chartand the estimated parameters can also be used to assignpreliminary onset and cessation labels to trees in other stands.
In the future we will analyze the relationship betweenenvironmental factors and the onset and cessation dates of radialincrease. Also, the connection between tracheid formation anddendrometer data is a topic of further investigation.
Acknowledgements
This work has been funded by the Academy of Finland, in theresearch projects ‘Analysis of dependencies in environmental timeseries data’ (AD/ED), Grant no. 126853, and ‘Environmentalinformatics for analyzing the role of forests in the global carboncycle’, Grant no. 128079. We are grateful to the reviewers whoseinsightful comments helped us improve the paper.
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Mikko Korpela received the degree of M.Sc. (Tech.) atthe Department of Computer Science and Engineeringat the Helsinki University of Technology in 2006. He ispursuing the Ph.D. degree from Helsinki University ofTechnology with a dissertation on computationalmethods of data analysis with applications in ecology.Since 2006, he has been working as a researcher at theDepartment of Information and Computer Science(formerly Laboratory of Computer and InformationScience), Helsinki University of Technology. His re-search interests include data mining and high-dimen-sional data sets.
Harri Makinen received the degrees of M.Sc. (For.) in1991 at the Faculty of Forestry, University of Joensuu,Finland, and D.Sc. (For.) in 1996 at the Faculty ofForestry University of Freiburg, Germany. He hasworked at the Finnish Forest Research Institute since1996. Currently, he is a senior researcher in growthand yield. His research interests are the effects ofenvironmental factors and forest management on treegrowth and wood properties and formation, andmodeling of wood properties. He has also participatedand coordinated several EU and Nordic researchprojects.
Pekka Nojd received his D.Sc. (For.) in 1996 at theUniversity of Helsinki, Finland. Currently, he is a seniorresearcher in growth and yield. His research interestsare the effects of environmental and climatic factors ontree growth, including both favorable and detrimentaleffects of air quality.
Jaakko Hollmen received the degrees of M.Sc. (Tech.)in 1996, Lic.Sc. (Tech.) in 1999 and D.Sc. (Tech.) in2000, all at the Department of Computer Science andEngineering at the Helsinki University of Technology.He has worked at the Department of Information andComputer Science (formerly Laboratory of Computerand Information Science) at the Helsinki University ofTechnology (TKK) since the year 2000. Before joiningTKK, he worked at the Siemens Corporate Technologyin Munich, Germany. Currently, he is a Chief ResearchScientist at TKK. His research interests include theoryand practice of data analysis and data mining,
especially their applications in bioinformatics and environmental informatics. Jaakko Hollmen is a seniormember of IEEE.Mika Sulkava received an M.Sc. degree in EngineeringPhysics from Helsinki University of Technology, Fin-land, in 2003 and a D.Sc. in Computer Science andEngineering from Helsinki University of Technology in2008. He is a member of Parsimonious Modellinggroup, which aims at achieving maximally simple orcompact models as a result of the data analysisprocess. His main research interests are machinelearning, data mining, and exploratory data analysisin the field of environmental informatics.