Chaos and Nonlinear Time-series Analysis of Finger Pulse Waves

for Depression Detection

Tuan D. Pham

, Truong Cong Thang

, Mayumi Oyama-Higa

, Hoc X. Nguyen

, Hameed Saji

and Masahide Sugiyama

Aizu Research Cluster for Medical Engineering and Informatics, CAIST, The University of Aizu,

Aizuwakamatsu, Fukushima 965-8580, Japan

School of Computer Science and Engineering, The University of Aizu, Aizuwakamatsu, Fukushima 965-8580, Japan

Chaos Technology Research Laboratory, 5-26-5 Seta, Otsu-shi, Shiga 520-2134, Japan

Aizu Research Cluster for Environmental Informatics, CAIST, The University of Aizu,

Aizuwakamatsu, Fukushima 965-8580, Japan

Keywords:

Chaos, Lyapunov Exponents, Nonlinear Dynamical Analysis, Sample Entropy, Photoplethysmograph,

Depression Detection, Biosignal Classiﬁcation.

Abstract:

Depressive disorders are mental illnesses that can severely affect one’s health and well-being. If depression is

not early detected and left untreated, it can consequently lead to suicide. This paper presents for the ﬁrst time

a novel combination of chaos theory and nonlinear dynamical analysis of signal complexity of photoplethys-

mography waveforms for detection of depression. Experimental results obtained from the analysis of mentally

disordered and control subjects suggest the potential application of the proposed approach.

1 INTRODUCTION

It has been known that depression has been a highly

prevalent, worldwide problem with multiple social

and health consequences (Waza et al, 1999). Sad feel-

ing, emotional indifference, and lack of interest re-

duce one’s ability to meet responsibilities and to enjoy

life. The causes of depression are complex resulting

in biological, psychological, and social dysfunctions

or a combination of these problems. Although de-

pression is the frequent mental disorder among older

people, its adverse impact on young people is much

greater than the elderly. The effects of depression

are not limited to a depressed person but they can be

far-reaching. Needless to say, the treatment for de-

pression is essential, because the consequences of un-

treated depression may be fatal.

The literature on improving detection of these is-

sues in primary care settings has predominantly fo-

cused on physician skills in assessing mental health

problems. While physician factors undoubtedly play

a role, it is believed that the detection of mental health

problems has not been widely investigated (Marcus et

al., 2011). Regarding the computerized detection of

depression, photoplethysmography (Allen, 2007) has

recently been realized as a useful biomedical technol-

ogy for studying mental disorders (Hu et al., 2011).

While speech (Low et al., 2011), image (Cohn et al.,

2009; Soennesyn, et al.), and other biosignals (Chen

et al., 2011; Kemp et al., 2012) have been used for

depression detection; photoplethysmography wave-

forms, which are generated from the measurements

of blood volume changes in the microvascular bed

of tissue at the skin surface and found as effective as

ECG (electrocardiography) in measuring the parame-

ters of heart rate variability (Russoniello et al., 2010),

provide simple and low-cost optical data for the same

study. For the ﬁrst time, this paper presents the com-

bination of extractionsof the Lyapunovexponentsand

sample entropy values of the photoplethysmography

waveforms measured at the ﬁnger tips for depression

detection.

2 LYAPUNOV EXPONENTS FOR

DISCRETE-TIME SYSTEMS

A Lyapunov exponent is a real number that measures

the average rate of divergenceor convergenceover the

entire attractor which is the phase-space point or set

of points representing various possible steady state of

298

D. Pham T., Cong Thang T., Oyama-Higa M., X. Nguyen H., Saji H. and Sugiyama M..

Chaos and Nonlinear Time-series Analysis of Finger Pulse Waves for Depression Detection.

DOI: 10.5220/0004222302980301

In Proceedings of the International Conference on Bio-inspired Systems and Signal Processing (BIOSIGNALS-2013), pages 298-301

ISBN: 978-989-8565-36-5

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

a dynamical system (Williams, 1997; Sprott, 2003).

For a discrete system, we consider how a 1-D map

k+1

= f(x

) evolves when it is started at two initial

states x

and (x

+ ε

), where ε

is a very small value

to indicate the two initial states are very close to each

other. The Lyapunov exponent is deﬁned when the

two trajectories are separated by a distance ε

after n

iterations of the map as

|ε

| ≈ |ε

nλ

(1)

where λ is the Lyapunov exponent.

Taking the natural logarithm of both sides of

Equation (1), the divergence of the two trajectories

can be approximated as (Dingwell, 2006)

λ ≈



+ ε

) − f

)



(2)

If the interest is the study of the effects of very

small perturbations, the limit of Equation (2) is taken

as ε

→ 0, then the remaining term inside the loga-

rithm is expanded using the chain rule:

+ ε

) − f

) = (f

)

′

)

n−1

∏

i=0

′

) (3)

The back substitution of Equation (3) into Equa-

tion (2) gives

λ ≈

n−1

∑

i=0

ln| f

′

)| (4)

Finally, the limit of Equation (4) is taken as

n → ∞, giving

λ = lim

n→∞

n−1

∑

i=0

ln| f

′

(5)

For M-dimensional mappings, Equation (5) is ex-

tended to yield a spectrum of Lyapunov exponents ar-

ranged in a decreasing order:

≥ λ

≥ · ·· ≥ λ

where λ

is known as the maximum or largest Lya-

punov exponent (LLE).

3 SAMPLE ENTROPY

Let x = {x

, . . . , x

}, and Q

be the set of

all subsequences of length m in x: Q

, . . . , x

(N−m+1)m

}, where x

= {x

, . . . , x

i+m−1

It is said that x

and x

are similar if and only if

i+k

− x

j+k

| < r, ∀k, 0 ≤ k < m, i 6= j (6)

Let L

= {x

, . . . , x

(N−m−1)m

}, the probability of

patterns of length m that are similar to the pattern of

the same length that begins at i is

(r) =

(r)

N − m− 1

(7)

where J

(r) is the number of subsequences in L

that

are similar to x

The total average probability B

(r) for all i, i =

1, . . . , N − m, is

(r) =

N − m

N−m

∑

i=1

(r) (8)

Finally, the value of SampEn, given m and r, can

be calculated by the following equation:

SampEn(m,r) = log



(r)

m+1

(r)



(9)

4 EXPERIMENT

We are interested in applying chaos and nonlinear

time-series analysis of ﬁnger pulse waves for depres-

sion detection by extracting the biosignal features us-

ing LLE and SampEn. The experimental data used in

this study is the same data recently studied by (Hu et

al., 2011). The dataset consists of 195 patients diag-

nosed with depression and 113 students considered as

the control subjects. Finger pulse waves were mea-

sured on the depressed and control subjects at various

number of times.

The ﬁrst step in the estimate of the LLE of the

ﬁnger pulse waves is the reconstruction of an appro-

priate state space for the nonlinear system. Takens’

embedding theory, which states that an appropriate

state space from a single original time series can be

reconstructed with a time delay (Takens, 1981), is ap-

plied here for the state-space reconstruction. The pur-

pose of selecting a time delay is to ﬁnd a value of the

delay which is large enough to ensure that the result-

ing individual coordinates are relatively independent;

however, it should not be so large to be completely

independent statistically (Abarbanel, 1996). In this

study, the embedding dimensions of the ﬁnger pulse

waves were chosen to be 4 for the reconstruction of

the state space. Because of the difﬁculty in the com-

putation of the Lyapunov spectrum and the interest in

ChaosandNonlinearTime-seriesAnalysisofFingerPulseWavesforDepressionDetection

299

estimating λ

(LLE) which is the most signiﬁcant in-

dicator of chaos, many algorithms have been devoted

to the calculation of the LLE. One of the most popular

methods for the LLE estimate is the one proposed by

Rosenstein et al (1993) and applied in this study.

For the calculation of SampEn, m = 7 and r =

0.2σ, where σ is the standard deviation of the signals,

were speciﬁed.

Sensitivity and speciﬁcity are statistical measures

of the performance of a binary classiﬁcation test. Sen-

sitivity is the measure the proportion of actual posi-

tives which are correctly identiﬁed as such. In this

study, it is the percentage of the depressed patients

who are correctly identiﬁed as having depression.

Speciﬁcity is the measure of the proportion of neg-

atives which are correctly identiﬁed. Here it is the

percentage of the control subjects who are correctly

identiﬁed as not having depression.

The concept of autonomic nerve balance (Wilson,

2005) stems from the fact that the human nervous sys-

tem has two major parts: the voluntary and the auto-

nomic systems. The voluntary system is concerned

with movement and sensation and consists of motor

and sensory nerves.

The autonomic system regulates biological func-

tions such as blood pressure and heart rate over which

human beings have less conscious control. The auto-

nomic system has two states: sympathetic (stress) and

parasympathetic (healing). Computation of the auto-

nomic nerve balance (ANB), which is based on heart

rate per unit time measured from the end pulse wave ,

was patented and described in (Higa, 2011).

Figure 1 shows the LLE and autonomic nerve

balance (ANB) features extracted from the ﬁnger

pulse waves of the patients and control subjects, and

the graphical results obtained by the fuzzy c-means

(FCM) clustering algorithm. Figure 2 shows the LLE

and SampEn features extracted from the ﬁnger pulse

waves of the patients and control subjects, and the

graphical results obtained by the FCM algorithm. It

can be observed from the ﬁgures that the distributions

of the LLE and SampEn representing the depression

and control groups are presented in much more com-

pact and well deﬁned clusters than those of the LLE

and ANB. The k-nearest neighbor (k-NN) algorithm,

where k = 7, was also used to classify the depressed

and control subjects in the experiment.

The speciﬁcity and sensitivity results provided by

the FCM and k-NN using LLE and ANB are shown in

Table 1, and the speciﬁcity and sensitivity results pro-

vided by the FCM and k-NN using LLE and SampEn

are shown in Table 2. Based on the results shown

in Table 1 and Table 2, the FCM performs better than

the k-NN in classifying the depressed patients (sensi-

Figure 1: LLE and ANB features of ﬁnger pulse waves clas-

siﬁed by FCM.

Figure 2: LLE and SampEn features of ﬁnger pulse waves

classiﬁed by FCM.

Table 1: Sensitivity (%) and speciﬁcity (%) of depression

detection by LLE and ANB.

Classiﬁer Sensitivity Speciﬁcity

FCM 97.78 79.31

k-NN 92.24 94.44

Table 2: Sensitivity (%) and speciﬁcity (%) of depression

detection by LLE and SampEn.

Classiﬁer Sensitivity Speciﬁcity

FCM 1.00 87.07

k-NN 98.89 98.28

tivity), but the k-NN gives better results in detecting

the control subjects (speciﬁcity) than the FCM. For

practical purpose, the FCM would be preferred to the

k-NN because of the importance of the correct detec-

tion of depression. In general, these results show the

superior performance for depression detection of the

combination of the LLE and SampEn values using ei-

ther the FCM or k-NN classiﬁer.

BIOSIGNALS2013-InternationalConferenceonBio-inspiredSystemsandSignalProcessing

300

5 CONCLUSIONS

The extractions of the ﬁnger pulse waves using the

largest Lyapunov exponents and the sample entropy

values have shown to be a better combination of the

biosignal features than the coupling of the largest Lya-

punov exponents and autonomic nerve balance val-

ues. The improvement suggests the usefulness of

chaos and nonlinear dynamical analysis of the pho-

toplethysmography waveforms for depression detec-

tion, which can be useful for mental health care.

ACKNOWLEDGEMENTS

This work was supported by the FY 2012 University

of Aizu Competitive Research Funds for Revitaliza-

tion Category.

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