Hindawi Publishing CorporationEURASIP Journal on Advances in Signal ProcessingVolume 2007, Article ID 24748, 7 pagesdoi:10.1155/2007/24748

Research ArticleAutomatic Threshold Determination for a Local Approach ofChange Detection in Long-Term Signal Recordings

Wassim El Falou,1, 2 Mohamad Khalil,2 Jacques Duchene,1 and David Hewson1

1 Institut des Sciences et Technologies de l’Information, Universite de Technologie de Troyes, France2 Faculte de Genie I, Universite Libanaise, Tripoli, Lebanon

Received 18 October 2006; Revised 26 January 2007; Accepted 27 April 2007

Recommended by Gloria Menegaz

CUSUM (cumulative sum) is a well-known method that can be used to detect changes in a signal when the parameters of this signalare known. This paper presents an adaptation of the CUSUM-based change detection algorithms to long-term signal recordingswhere the various hypotheses contained in the signal are unknown. The starting point of the work was the dynamic cumulativesum (DCS) algorithm, previously developed for application to long-term electromyography (EMG) recordings. DCS has been im-proved in two ways. The first was a new procedure to estimate the distribution parameters to ensure the respect of the detectabilityproperty. The second was the definition of two separate, automatically determined thresholds. One of them (lower threshold)acted to stop the estimation process, the other one (upper threshold) was applied to the detection function. The automatic deter-mination of the thresholds was based on the Kullback-Leibler distance which gives information about the distance between thedetected segments (events). Tests on simulated data demonstrated the efficiency of these improvements of the DCS algorithm.

Copyright © 2007 Wassim El Falou et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

1. INTRODUCTION

Change detection and segmentation are the first steps ofmany signal processing applications (see, e.g., speech pro-cessing [1–4], video tracking [5], ergonomics [6], biomed-ical applications [7–9], seismic applications [10]). Most de-tection and segmentation algorithms are based on the theoryof statistical detection and hypothesis testing [10–12].

In such an approach, a change occurs when the statisti-cal properties of the signal are modified. Roughly speaking,this can be expressed either by a different distribution func-tion before and after the change time, or by a modification ofthe parameter value of the same distribution. For the lattercase, when the parameter values are a priori known, an effi-cient algorithm to solve the detection problem is the CUSUM(cumulative sum) algorithm based on the log-likelihood ra-tio [10, 13]. CUSUM algorithm is optimal in the sense thatit optimizes the worst detection delay when the mean timebetween false alarms goes to infinity [10].

In many applications, modifications can affect energy,frequency, or both [14, 15]. Detection of a change in the fre-quency content can be performed using the CUSUM algo-rithm applied on the innovation of an AR (auto regressive)or ARMA (auto regressive moving average) modeling [4, 10],

the AR (or ARMA) coefficients carrying information aboutthe frequency content of the signal [14].

In usual applications, the parameters corresponding tothe segments to be detected are often unknown and otheralgorithms have to be applied for change detection. Such al-gorithms can be found in [9, 16], based on the computationof a dynamic cumulative sum (DCS) of the likelihood ratiobetween two locally estimated distributions. These distribu-tions are estimated at each time t using two sliding windowsbefore and after the current time t.

In this paper, we propose a modified method of DCS thatcan be adapted to long duration signals. This modificationis achieved on windows length and thresholding. The mainapplication of our study is to detect fatigue of in posturalmuscles during driving. For that purpose, electromyography(EMG) signals are acquired continuously during a long-termdriving task and the first step of the analysis is to detect seg-ments of the signal that contains EMG with a reasonable sig-nal to noise ratio.

The first part of this paper provides an overview of theCUSUM algorithm, focusing on the dynamic cumulativesum to describe its main properties and limits. Then a mod-ified detection algorithm is proposed to go beyond these

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Figure 1: EMG signal and its contents. Contractions have to be de-tected (segmented) and eliminated. x-axis: time in secondS. y-axis:arbitrary unit.

limits. An automatic determination of the thresholds is pre-sented in the third part of the paper.

2. PROBLEM STATEMENT

The fatigue that can be produced during driving can be de-tected by studying the EMG signal of the active muscles. Inour work, this signal is acquired on the muscles during 2.5hours of driving, the global aim being to detect the level ofthe fatigue during driving.

These signals contain a background (low-level) activitycorresponding to the postural maintaining (what is the partof interest for the study) as well as high level epochs corre-sponding to muscle contractions related to voluntary mo-tions. These events have to be eliminated from the signal inorder to keep the only muscle activity corresponding to the“resting” state (postural activity: Figure 1).

To eliminate the voluntary contractions from the signal,we developed a new method of detection (MDCS) that canbe adapted to long duration signals. After change detectionand signal segmentation, the next step (not presented in thispaper) would be to compute indices like the median fre-quency of the resting segments to quantify the fatigue. Inthis paper we only focus on the first problem of detection-segmentation.

3. DCS AS AN EXTENSION OF THECUSUM ALGORITHM

3.1. Overview of the CUSUM algorithm

Let (x1, x2, . . . , xn) be a sequence of observed randomvariables with conditional probability density fθ0 (xk/xk−1,. . . , x1) before the change time t0, θ0 being the parameter vec-tor of the segment S0 before t0, and with conditional prob-ability density fθ1 (xk/xk−1, . . . , x1) after this change time, θ1

being the parameter vector of segment S1 after t0.

Let Sk1 be the sum of the logarithms of the successive like-lihood ratios [10]:

Sk1 =k∑

i=1

si =k∑

i=1

logfθ1

(xi/xi−1, . . . , x1

)

fθ0

(xi/xi−1, . . . , x1

) . (1)

The decision function is defined as

gk = Sk1 − min1≤ j≤k

Sj1 (2)

and the corresponding stopping time is

ts = min{k : gk ≥ h

}, (3)

where h is a given threshold.Given that Eθ0 [si] < 0 and Eθ1 [s j] > 0 (detectability prop-

erty), an estimated value of change time t0 can be obtainedby the relation

t0 = max{k : gk = 0

}. (4)

The CUSUM algorithm can be written in a recursive wayas [10]

g0 = 0,

gk = max(0, gk−1 + sk

).

(5)

In the case of independent zero mean Gaussian sequencesand when the point is to detect a change of variance, the ex-pression of the likelihood ratio becomes [10]:

si = 12

lnσ2

0

σ21

+ x2i

(1

2σ20− 1

2σ21

), (6)

(Because: fθ0

(xi)= 1√

2πσ0e−x

2i /2σ

20 ; fθ1

(xi)= 1√

2πσ1e−x

2i /2σ

21

).

(7)

3.2. Application after autoregressive modeling

A signal is AR-modeled if it can be written as

xi = −p∑

n=1

an · xi−n + εi, (8)

where εi are the innovations or prediction errors of thesignal (white noise). The terms ai are the coefficients ofthe model and contain frequency information of the sig-nal. The variance σ2

0 of the innovations gives the energy ofthe signal. In general, detection cannot be applied on de-pendent signals. Therefore the change detection algorithmis applied on the sequences of prediction errors deducedfrom AR (autoregressive) modeling for S0 (before changetime, θ0 = (a0

1, . . . , a0p, σ2

0 )) and S1 (after change time, θ1 =(a1

1, . . . , a1p, σ2

1 )) [10, 14]:

si = 12

lnσ2

0

σ21

+

(ε0i

)2

2σ20−(ε1i

)2

2σ21

, (9)

where

εli = xi +p∑

k=1

akxi−k; l = {0, 1}. (10)

Wassim El Falou et al. 3

3.3. The DCS algorithm

Many algorithms can be found that detect spectral changeswhen the parameters are unknown (see, e.g., the Brandt al-gorithm [10], the divergence Hinkley algorithm [14], DCSalgorithm [9, 16]).

The latter (DCS) was developed for detection of changesin signals of long duration. It was based on local cumulativesums of likelihood ratios computed between two local win-dows estimated around the current time t. The parameters ofthe two segments, Stb (“b” for “before”) and Sta (“a” for “af-ter”), were estimated using two estimation windows Wa andWb of identical length N before and after the current time t:

(i) Wtb : xi; i = {t −N , . . . , t − 1} used to estimate the pa-

rameter θb of the probability function before the cur-rent time t,

(ii) Wta : xi; i = {t + 1, . . . , t + N} used to estimate the pa-

rameter θa of the probability function after the currenttime t.

At time t, DCS was defined as:

DCS(Ht

a,Htb

) =t∑

j=1

logfja(xj)

fjb

(xj) =

t∑

j=1

s j . (11)

When a change occurs at tM it has been demonstrated [9]that DCS reaches a maximum at this time tM .

The detection function was expressed as

g(t) = max1≤ j≤t

[DCS

(H

ja ,H

jb

)]−DCS(Ht

a,Htb

)(12)

and the stopping time was:

ts = inf{t : g(t) ≥ h

}, (13)

where h was a given threshold.When applying the DCS algorithm after AR (autoregres-

sive) modeling, a third window Wtp was necessary to com-

pute the prediction error after AR parameter estimation.Figure 2 illustrates the window definition (Wt

b for AR pa-rameter θtb estimation, Wt

a for AR parameter θta estimation,Wt

p for prediction error estimation), the evolution of DCSaround the change time t, and the corresponding evolutionof the decision function.

This change detection method has proved to be efficientwhen applied to uterine EMG [9] or postural muscle activ-ity [17]. However, some limitations of DCS can be under-lined that are related to its use in specific configurations:(i) as the estimation windows are used to estimate locallythe distribution parameters before and after the current timet, the choice of the window width has a great influence onthe detection process; (ii) the detectability property is nolonger preserved in the DCS algorithm. Therefore detectionfails when the two distributions are very close together (seeFigure 3). In fact, the detection function stabilizes after Npoints beyond the change time without reaching the thresh-old.

Based on the same basic concept, a modified algorithmwas developed to overcome these problems and to ensure thedetectability property.

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Figure 2: Upper tracing: position of the estimation and predictionwindows. Middle tracing: evolution of DCS around the change timetM . Lower tracing: detection function. For all tracings, x-axis: num-ber of points, y-axis: arbitrary units.

4. THE MODIFIED DYNAMIC CUMULATIVE SUM(MDCS) ALGORITHM

4.1. Variable window width

The algorithm is still based on two sliding windows Wtb and

Wta that are used to estimate θtb and θta at each time t. As for

DCS, Wta has a constant length N , but Wt

b now includes allsamples from 1 to t − 1. Hence, when both windows corre-spond to the same distribution (no change in the segment),the parameter estimation is always better for Wt

b than for Wta,

leading to Eθ0 [si] < 0.θtb and θta are estimated using these new windows:

Wtb : 1 · · · t − 1 −→ θtb,

Wta : t + 1 · · · t + N −→ θta.

(14)

The definitions of the log-likelihood ratios, the cumulativesum, and the detection function remain the same as for DCS.In addition, when the signal samples are dependent, it is stillpossible to perform AR modeling and to introduce an inter-mediate prediction window Wt

p.Figure 4 illustrates this new approach. MDCS is now de-

creasing before the change time and continuously increasingafter that, if the process is not stopped by a threshold cross-ing.

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Figure 3: An illustration of a change that was not detected bythe DCS algorithm. Upper tracing: signal segment. Middle tracing:DCS evolution. Lower tracing: detection function evolution. For alltracings, x-axis: number of points, y-axis: arbitrary units.

4.2. Double thresholding

One of DCS drawbacks was the fact that, between the changetime and the stopping time, Wt

b kept increasing, hence in-cluding samples taken after the change time to update θ0

estimates. To solve this problem, the idea was to apply twothresholds (hL and hH) to the detection function:

(i) the lower threshold hL stops θ0 estimate updating,(ii) the higher threshold detects the change hH .

This double thresholding allows a limitation in the bias of θ0

estimation without increasing the false alarm rate, as was thecase before when a threshold that was too low was applied tothe detection function.

5. AUTOMATIC CHOICE OF THE THRESHOLDS

One of the most crucial issues in change detection is thechoice of the detection threshold h. It mainly depends onthe signal characteristics and is generally adjusted by expe-

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Figure 4: (a) Simulated signal containing a change in variance(from 1 to 2) at point 10000. (b) Evolution of DCS before and af-ter the change time (change not detected). (c) Evolution of MDCSbefore and after the change time (change detected). For all tracings,x-axis: number of points, y-axis: arbitrary units.

rience or by using a training set of data. Methodologies canbe found in the literature to choose the threshold accordingto the probability of false alarm, and the mean time betweenfalse alarms [10, 18]. However, the formulation is asymptoticand difficult to apply in practical use.

In case of a CUSUM algorithm, a very useful factor tochoose the threshold h is the Kullback-Leibler distance be-tween two probability densities fθ0 and fθ1 of a random vari-able x, defined as

K(θ0, θ1

) =∫Ln

fθ0 (x)fθ1 (x)

fθ0 (x)dx. (15)

The Kullback-Leibler distance can be considered as a dis-tance between these two probability densities. In addition,it is known [10] that the delay for detection is inversely pro-portional to the Kullback-Leibler distance. If h is the thresh-old used in the detection algorithm, the relationship betweenh and the Kullback-Leibler distance can be expressed as

Eθ1 (s) = K(θ1, θ0

) = h

τ, (16)

where τ is the mean delay for detection. Hence the Kullback-Leibler distance can be used to choose the threshold h.

Wassim El Falou et al. 5

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kH

01

2

kL

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 5: Variation of the segmentation error with the low and highthreshold values. Thresholds are indicated as the factors kL (y-axis)and kH (x-axis) to apply to the mean square value of the Kullback-Leibler distance MSKL.

From (16) we can write h ≈ M · K(θ1, θ0) where M isthe number of points after time changes. So we can use anestimation of the Kullback-Leibler distance to calculate thethreshold h.

Now considering two AR models θ0 = (a01, . . . , a0

p, σ20 )

and θ1 = (a11, . . . , a1

p, σ21 ), the Kullback-Leibler distance be-

tween θ1 and θ0 can be expressed as [6]

K(θ1, θ0

) = −12− 1

2ln

σ21

σ20

+12σ2

1

σ20

[1 +

∞∑

k=1

(c0/1k

)2]

,

(17)

where the coefficients c0/1k are the coefficients of the following

Taylor expansion:

A0(z)A1(z)

= 1 +∞∑

k=1

c0/1k z−k. (18)

The following steps are proposed to choose the threshold hautomatically.

(1) The signal is first divided into successive segments ofequal length N .

(2) The AR model θ = (a1, . . . , ap, σ2) is estimated foreach segment.

(3) Then the Kullback-Leibler distance is computed be-tween each pair of successive segments, leading to asequence of values that are thus sorted in ascendantorder.

(4) The sequence is limited to the lowest 90% of valuesin order to suppress the influence of any possible verylarge value.

(5) The mean square value MSKL of the remaining distri-bution of the Kullback-Leibler distances is then com-puted, providing the low and high thresholds hL =N · kL · MSKL and hH = N · kH · MSKL, N being thewindow width.

The determination of kL and kH was performed by simula-tion with the same reference set as that used to build theROC curves in the previous paragraph. Segmentation wasachieved with successive values of kL and kH and the num-ber of nondetection and false alarms counted.

Figure 5 shows the variation of the segmentation error(sum of nondetections and false alarms) with respect to boththresholds using the simulation data. The surface presents aminimum at kL=1 and kH=3.

6. RESULTS AND DISCUSSION

This method was first tested on simulated signals generatedby concatenating segments of random noise filtered at differ-ent frequency bands, then to electromyographic recordings.

As an illustration, the segmentation was applied to elec-tromyographic signals recorded during a long term (2h30)experiment assessing the comfort of car seats through a mea-sure of local muscular fatigue. Each experiment was dividedinto 7 phases lasting from 10 minutes to 30 minutes. Figure 6shows one of those phases after MDCS segmentation (25 seg-ments).

This new technique of windowing—double thresholdsdecreases the probability of false alarm especially in the elec-tromyography signals which are long duration signals. Thisis coming from the fact that the detection function g(t) risesto the second threshold only when a real change occurs. Fur-thermore, The Kullback-Leibler distance is used to determinethese thresholds automatically because the characteristics ofthe electromyography signals change from person to anotherand depend on many other parameters. Finally, it is impor-tant to notice that this method can be applied to whateverkind of signals presenting changes in frequency or amplitude.

Both methods (DCS and MDCS) were tested on simu-lated data made of 1000 segments of white noise with a vari-ance change from 1 to 2 and 1000 segments without a change.To compare the results, we chose the ROC curves (receiveroperating characteristics) that plot the probability of detec-tion with respect to the probability of false alarms. In gen-eral, higher is the curve, better are the results. Figure 7 clearlyshows how the modified algorithm improves the overall de-tection quality.

In these curves presented on Figure 7, we can see that ifwe need a detection probability equal to 0.9, the false alarmprobability given by the DCS algorithm is about 0.1 but it isless than 0.02 for the MDCS method. MDCS decreases theprobability of false alarm for a given detection probability.

7. CONCLUSION

The local approach of change detection allows a local esti-mation of the distribution parameters before and after thecurrent time t. A change is detected in the same way as

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0

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Det

ecti

onp

roba

bilit

y

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Figure 7: Comparison of DCS and MDCS methods by ROC curvescomputed from simulated data. x-axis: false alarm probability, y-axis: detection probability.

for the classical CUSUM approach after parameter estima-tion. A first algorithm (DCS) had been successfully testedon long term recordings related to biomedical signals. How-ever, DCS presented some limitations in its ability to detectslow changes, the main of them being that it did not respectthe detectability property. In addition, the threshold of thedetection function had to be chosen by expertise or by us-ing reference data sets. The modified algorithm overcomesthese problems by a restriction of the estimation window forthe segment S0 (before change point) using a low thresh-old that is distinct from the detection function threshold it-self. In addition, these thresholds are learned automatically

by using the Kullback-Leibler distance. As a consequence,MDCS becomes an offline algorithm if applied extensively toeach recording to be segmented, since the Kullback-Leiblerdistance distribution must be computed first for each newrecording. However, it seems wise to imagine that the samethresholds could be applied to a class of similar signals suchas electromyograms recorded on various muscles and vari-ous subjects during the same experimental protocol. Never-theless this point has yet to be demonstrated.

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Wassim El Falou was born in Lebanonin 1975. He received the Diploma in en-gineering (electrical-electronics) from theLebanese University, and a M.S in mathe-matical modelling from Saint Joseph andLebanese Universities. He received his Ph.D.degree in 2002 from the University of Tech-nology of Troyes, in surface EMG signalprocessing. He is currently teaching at sev-eral universities in Lebanon, including theLebanese University. His main research interests are embedded sys-tems design, signal processing, classification methods, and voicerecognition.

Mohamad Khalil was born in Akkar Atika,in Lebanon, in 1973. He obtained the En-gineering degree in electrical and electricityfrom the Faculty of Engineering, LebaneseUniversity, Tripoli, Lebanon, in 1995. He re-ceived the D.E.A degree in biomedical engi-neering from the University of Technologyof Compiegne (UTC) in France, in 1996. Hereceived his Ph.D. degree from the Univer-sity of Technology of Troyes in France, in1999. He received his HDR (Habilitation a diriger des recherches)

degree from UTC in 2006. He is currently a Researcher at severaluniversities in Lebanon including the Lebanese University. His cur-rent interests are the signal and image processing problems: detec-tion, classification, analysis, representation, and modeling of non-stationary signals, with application to biomedical signals and im-ages.

Jacques Duchene received the Engineerdegree in electronics from the EcoleSuperieure d’Electricite, France in 1973,and the doctorat d’etat in sciences in 1983.He joined the University of Technology ofTroyes in 1994, where he is currently incharge of the Charles Delaunay Institute ofResearch. His main research interests aresignal processing, pattern recognition andclassification. He now focuses on signalsegmentation as well as signal decomposition. The main appli-cation fields in biomedical engineering are ergonomics (comfortin cars), biomedical monitoring (quality of balance for elderly),and EMG characterization and modelling (frequency parameters,conduction velocity distribution).

David Hewson received the BPhEd andMPhEd degrees from the University ofOtago in New Zealand in 1990 and 1993and a Ph.D. degree from the University ofAuckland in 2000. He worked as a ResearchPhysiologist for the Royal New Zealand AirForce between 1994 and 2000, before un-dertaking two years of postdoctoral studyat the University of Technology of Troyes inFrance. He is now an Associate Professor atthe University of Technology of Troyes. His research interests areergonomic and clinical applications of surface electromyography.