Signal Segmentation in Time-Frequency Plane using Renyi Entropy -

Application in Seismic Signal Processing

Theodor D. Popescu 1 and Dorel Aiordachioaie2

Abstract— Reliable seismic waves characterization is essentialfor better understanding wave propagation phenomena, pro-viding new physical insight into soil properties. Many worksin this area have been based on detecting special patternsor clusters in seismic data, event detection using parametricmodels and time-frequency analysis. In this paper we presenta new approach making use of the short-term time-frequencyRenyi entropy and maximum a posteriori probability (MAP)estimator, operating on time-frequency Renyi entropy, as a newspace of decision; this method enables more robust featureextraction and a more accurate classification. This approachwas used in simulation and in the analysis of the earthquakerecords during the Kocaeli, Arcelik seism, Turkey, August 1999,a strong to moderate ground motion.

I. INTRODUCTION

The problem of change detection and diagnosis has gained

considerable attention during the last three decades in a

research context and appears to be the central issue in various

application areas. From statistical point of view, change de-

tection tries to identify changes in the probability distribution

of a stochastic process. In general, the problem involves both

detecting whether or not a change has occurred, or whether

several changes might have occurred, and identifying the

times of any such changes.

The analysis of the behavior of real data reveals the

most of the changes that occur are either changes in the

mean level, variance, or changes in spectral characteristics.

In this framework, the problem of segmentation between

”homogenous” parts of the data (or detection of changes in

the data) arises more or less explicitly.

A coherent methodology is now available to the designer,

together with the corresponding set of tools, see [1], [2]

among others. These enable him to solve a large variety

of applications in different fields: mechanical engineering,

industrial process monitoring, civil infrastructure, medical

diagnosis and treatment, speech segmentation, underwater

sensing, video surveillance and driver assistance systems.

The detection of events in seismic signals has been a

subject of great interest during the last thirty years. Most

of the methods in this area has been based on detecting

special patterns or clusters in seismic data, [3], [4], [5], [6],

[7]. Other approaches make use of AR and ARMA models,

used in conjunction with the Akaike information criterion

*This work was not supported by any organization1Theodor D. Popescu is with the Research Department, National Institute

for Research and Development in Informatics, 8-10 Averescu Avenue,011455 Bucharest, Romania pope at ici.ro

2Dorel Aiordachioaie is with the ”Dunarea de Jos” University of Galati,47 Domneasca, 800008 Galati, Romania Dorel.Aiordachioaieat ugal.ro

(AIC) method for change detection and isolation, as well

as to detect the primary (P-waves) and secondary (S-waves)

waves, [8], [9], [10], [11], [12].

Significant efforts have been made in order to represent the

temporal evolution of non-stationary spectral characteristics

in seismic signal analysis. A new method based on a time-

frequency analysis through the Wigner Distribution (WD) is

presented and applied in [13] and [14]. In [15], the seismic

signal analysis is performed by Smoothed Pseudo Wigner-

Ville Distribution (SPWVD) in order to obtain instanta-

neous frequency (IF) information. Two novel algorithms

to select and extract separately all the components, using

time-frequency distributions (TFD), of a multicomponent

frequency-modulated signal is presented in [16]. A new

signal analysis technique based on harmonic wavelet analysis

was used to look at the earthquake induced accelerations in

[17].

Starting from the experience in the field, the present

paper introduces a new approach for events detection in

seismic signals using maximum a posteriori probability

(MAP) estimator, operating on short-term time-frequency

Renyi entropy. The problem is transferred from the space of

original measurements to the space of time-frequency Renyi

entropy, where some measures of time-frequency distribu-

tion concentration and MAP algorithm for segmentation are

introduced and used.

II. TIME-FREQUENCY ANALYSIS

A. Preliminaries

The basic objective of time-frequency analysis is to de-

velop a function that may enable us to describe how the

energy density of a signal is distributed simultaneously at

time, t, and frequency, ω.

The time-frequency representations (TFRs) can be clas-

sified according to the analysis approaches [18]. In the

first category, the signal is represented by time-frequency

(TF) functions derived from translating, modulating and

scaling a basis function having a definite time and frequency

localization. For a signal, x(t), the TFR is given by

TFx(t, ω) =

∫ +∞

−∞

x(τ)φ∗t,ω(τ)dτ =< x, φt, ω >, (1)

where φt,ω represents the basis functions(also called the TF

atoms) and ∗ represents the complex conjugate. The basis

functions are assumed to be square integrable, φt,ω ∈L2(R),i.e. they have finite energy [19]. Short-time Fourier transform

2013 Conference on Control and Fault-Tolerant Systems (SysTol)October 9-11, 2013. Nice, France

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TABLE I

KERNELS USED FOR MAIN COHEN’S CLASS TIME-FREQUENCY

Name Kernel φ(θ, τ)

SP∫

h∗(u −

1

2τ) exp−jθu h(u + 1

2τ)du

WVD 1

CWD exp−θ2t2/σ2

RID 2d Low pass filter in θ, τ space

(STFFT) [20], wavelets [19], [21], and matching pursuit

algorithms [19], [22] are typical examples in this category.

The second category of time-frequency distributions,

known as Cohen’s shift invariant class distributions, charac-

terizes the TFR by a kernel function. Because we use some

of these distributions in our approach for events-detection in

seismic signals, some specific elements are introduced in the

next section.

B. Cohen’s Class Time-Frequency Distributions

As we mentioned before, this approach characterizes the

TFR by a kernel function. The properties of the represen-

tation are reflected by simple constraints on the kernel that

produces the TFR with prescribed, desirable properties [23].

A mathematical description of these TFRs can be given by

TFDx(t, ω) =1

4π2

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞

x(u +1

2τ)× (2)

×x∗(u −1

2τ)φ(θ, τ) exp−jθt−jτω+jθu dudτdθ

where φ(θ, τ) is a two-dimensional kernel function, deter-

mining the specific representation in this category, and hence,

the properties of the representation. The kernels used for

main Cohen’s class time-frequency distributions are given in

TABLE I.

The last two distributions, Choi-Williams distribution

(CWD) and Reduced Interference Distribution (RID), belong

to the so-called reduced interference distribution, which is by

itself an extension of the Wigner-Vile distribution (WVD),

[24]. The WVD has been of special interest since it satisfies

a large number of important properties. Every member of

Cohen’s general class may be interpreted as two-dimensional

filtered WVD. The spectrogram (SP) lacks the time resolu-

tion, even though the frequencies are fairly well defined.

Strong connected with WVD, pseudo-Wigner distribution

(PWD), has an important feature providing a straightforward

instantaneous-wise time-frequency representation of a given

signal, especially suitable for non-stationary signal analysis,

[25].

Even if all TDFs tend to the same goal, each representa-

tion has to be interpreted differently, according to its own

properties. For example, some of them present important

interference terms, other are only positive, other are perfectly

localized on particular signals, etc. The extraction of infor-

mation has to be done with care, from the knowledge of these

properties. In order to do this we need to have a ”clean” TFD.

That is, we need a distribution that can reveal the features

of the signal as clearly as possible without any ”ghost”

component. For that we need to apply a TFD that can get

rid of the cross-terms while preserving a high time-frequency

resolution. In the sequel, we will use a reduced interference

distribution (RID) for the reasons presented above.

III. MEASURING TIME-FRQUENCY

INFORMATION CONTENT

One of the simplest feature based signal processing proce-

dures in TFA is via energy concentration. The idea is to ana-

lyze behavior of the energy distribution, i.e. the concentration

of energy at certain time instant or certain frequency band

or more generally, in some particular time and frequency

region. An overview of time-frequency representation using

energy concentration makes the object of [18]. Also, a review

of some existing measures and their comparison are given

in [26]. A such measure for distribution concentration is

Renyi entropy, [27]. Once the local frequency content has

been obtained, using one of the time-frequency distributions

presented above, an entropy measure can be evaluated for

extracting the information containing in a given position of

t = n.

For generic time-frequency distribution, Px(n, k), the

Renyi entropy measure has the following form:

Rα =1

1 − αlog2

(∑

n

∑

k

Pαx (n, k)

)(3)

where n is the temporal discrete variable and k the frequency

discrete variable, with α ≥ 2 being values recommended

for time-frequency distribution measures, [28]. For the case

α = 2 (distribution energy) oscillatory cross-terms would

increase the energy leading to false conclusion that the

concentration improves. The case α = 3 fails to detect the

existence of oscillatory zero-mean cross-terms (which do not

overlap with auto-terms), since for odd α do not contribute

to this measure. These were the reasons for the introduction

of normalized Renyi entropy measures, that will be described

next. The normalization can be done in various ways, leading

to a variety of possible measure definition. So in [26] some

normalization schemes of the Renyi entropy are proposed:

Normalization with the signal energy

REα =1

1 − αlog2

(∑n

∑k Pα

x (n, k)∑n

∑k Px(n, k)

)with α ≥ 2

(4)

Behavior of this measure is quite similar to the non-

normalized measure form, except in its magnitude. This

kind of normalization is important for comparison of various

distributions, or the same distribution when it is not energy

biased, by definition.

Normalization with the distribution volume

RV3 = −1

2log2

( ∑n

∑k P 3

x (n, k)∑n

∑k |Px(n, k)|

)(5)

If the distribution contains oscillatory values, then sum-

ming them in absolute value means that the large cross-

terms will decrease measure RVα. This indicates smaller

313

concentration, due to cross-terms appearance. The volume-

normalized form of measure has been used for adaptive

kernel design in [27].

Because the positivity of Px(n, k) ≥ 0 will not be

always preserved, along with the unity energy condition,∑n

∑k Px(n, k) = 1, some kind of normalization are

considered in [27] and [28].

Quantum normalization

Eisberg and Resnik, [29], assimilate the time-frequency

distributions at a given instant t = n with a wave function

and the general case in (3) for α = 3, gives

R3 = −1

2log2

(∑

n

∑

k

P 3x (n, k)

)(6)

The normalizing stage is affecting exclusively to index k,

when the operation is restricted to a single position n to

satisfy the condition∑

k Px(n, k) = 1 in such position.

The measure (6) can be rewritten for a given n as follows:

R3(n) = −1

2log2

(∑

k

P 3x (n, k)

)(7)

Empirically the normalization proposed in [29] has show

to be most suitable for an application in seismic signal

analysis, [13]. The values of R3(n) depend upon the size

N of the window N in (7) and it can be shown that they are

within the interval 0 ≤ R3(n) ≤ log2 N . Hence, the measure

can be normalized by applying R3(n) = R3(n)/ log2 N in

(7).

IV. MAXIMUM A POSTERIORI PROBABILITY

(MAP) ESTIMATOR

A. Problem Formulation

We introduce now the general segmentation problem for

linear regression model with piecewise constant parameters,

[2], which will be applied on short-term time-frequency

Renyi entropy. In segmentation the goal is to find a sequence

of time indices kn = k1, k2, . . . , kn, where both the number

n and the locations ki are unknown, such that a linear

regression model with piecewise constant parameters,

yt = φTt θ(i) + et, E(e2

t ) = λ(i)Rt (8)

when ki−1 < t ≤ ki is a good description of the observed

signal yt. Here θ(i) is the d-dimensional parameter vector

in segment i, φt is the regressor and ki denotes the change

times. The noise et is assumed to be Gaussian with variance

λ(i)Rt, where λ(i) is a possibly segment dependent scaling

of the noise and Rt is the nominal covariance matrix of the

noise. We can think of λ either a scaling of the noise variance

or variance itself (Rt = 1). Neither θ(i) or λ(i) are known.

We will assume Rt to be known and the scaling as a possibly

unknown parameter.

One way to guarantee that the best possible solution is

found is to consider all possible segmentation kn, estimate

one linear regression model in each segment, and then choose

the particular kn that minimizes an optimality criteria:

kn = arg minn≥1,0<k1<...<kn=N

V (kn) (9)

For the measurements in the ith segment, that is yki−1+1,

. . . yki= yki

ki−1+1, the least square estimate and its covari-

ance matrix are denoted:

θ(i) = P (i)

ki∑

t=ki−1+1

φtR−1t yt, (10)

P (i) =

ki∑

t=ki−1+1

φtR−1t φT

t

−1

. (11)

The following quantities, V - the sum of squared residuals,

D - − log det of the covariance matrix P and N - the number

of data in each segment, are given by

V (i) =

ki∑

t=ki−1+1

(yt − φTt θ(i))T R−1

t (yt − φTt θ(i))

(12)

D(i) = − log det P (i) (13)

N(i) = ki − ki−1 (14)

and represent sufficient statistics in each segment. The data

and quantities used in segmentation procedure are shown in

TABLE II.

TABLE II

DATA AND QUANTITIES USED IN SEGMENTATION PROCEDURE

Data y1, y2, . . . , yk1. . . ykn−1+1, . . . , ykn

Segment Segment 1 . . . Segment n

LS estimation θ(1), P (1) . . . θ(n), P (n)Statistics V (1), D(1), N(1) . . . V (n), D(n), N(n)

Note that the segmentation kn has n − 1 degrees of

freedom. Two types of optimality criteria have been mainly

proposed in this field: statistical criteria (Maximum Likeli-

hood (ML) or Maximum A posteriori Probability estimate

(MAP)) and information based criteria. The main problem

in segmentation is the dimensionality. The number of seg-

mentations kn is 2N (can be a change or no change at each

time instant).

B. Maximum a posteriori probability (MAP) estimator

We give in the following the conceptual description of the

Maximum a posteriori probability (MAP) estimator, [2], for

the data and quantities given in TABLE II:

1) Examine every possible segmentation, parameterized

in the number of jumps n and jump times kn, sepa-

rately.

2) For each segmentation, compute the best models in

each segment parameterized in the least square esti-

mates θ(i) and their covariance matrices P (i).3) Compute in each segment the sum of squared predic-

tion errors V (i) and D(i) = − log detP (i).

314

4) The MAP estimate of the model structure for the

three different assumptions on noise scaling ((i) known

λ(i) = λ0, (ii) unknown but constant λ(i) = λ and (iii)

unknown and changing λ(i) is given by the following

equations, [2]:

kn = arg minkn,n

n∑

i=1

(D(i) + V (i)) + 2n log1 − q

q(15)

kn = arg minkn,n

n∑

i=1

D(i) + (Np − nd − 2) ×

× logn∑

i=1

V (i)

Np − nd − 4+ 2n log

1 − q

q(16)

kn = arg minkn,n

n∑

i=1

(D(i) + (N(i)p − d − 2) ×

× logV (i)

N(i)p − d − 4) + 2n log

1 − q

q(17)

respectively.

The last two a posteriori probabilities (16) and (17) are

only approximate; the exact expressions can be found in [2].

The evaluations involved statistics (12), (13) and (14). In all

cases, constants in the a posteriori probabilities are omitted.

The difference in the three approaches is thus basically only

how to treat the sum of square prediction errors. A prior

probability q causes a penalty term increasing linearly in nfor q < 0.5. The derivations of (15) to (17) are valid only

if all terms are well-defined. The condition is that P (i) has

full rank for all i, and that the denominator under V (i) is

positive. That is, Np − nd − 4 > 0 and N(i)p − d − 4 > 0,

in (16) and (17), respectively. The segments must therefore

be forced to be long enough.

Computing the exact likelihood is computationally in-

tractable because of the exponential complexity. Some al-

gorithms implementing recursive local search techniques

and numerical searches based on dynamic programming or

MCMC (Markov Chain Monte Carlo) techniques are given

in [2].

V. EXPERIMENTAL RESULTS

To prove the validity of our approach we will carry it out

to a synthetic signal and to the earthquake records during

the Kocaeli, Arcelik seism, Turkey, August 1999.

A. Synthetic Signal

In order to assess performance of the proposed method,

it was applied on a synthetic multi-component signal, xt.

The signal contains seven multi-component epochs with the

duration of 6 seconds, excepting the last epoch of 5 seconds,

when a sampling frequency of 100 Hz was used, given in

TABLE III.

A white Gaussian noise, et, with zero mean and E(e2t ) =

0.1, was added to the total signal to give yt = xt + et. The

resulted signal and the real change instants are given in Fig.

1.

TABLE III

THE EPOCHS OF MULTI-COMPONENT SIGNAL

Epoch Signal component

1 0.5 cos(πt) + 1.5 cos(4πt) + 4 cos(8πt)2 1.5 cos(2πt) + 4 cos(14πt)3 cos(πt) + 4.5 cos(7πt)4 0.5 cos(πt) + 1.5 cos(2πt) + 0.8 cos(6πt) + 3.5 cos(16πt)5 0.5 cos(6πt) + 2.5 cos(16πt)6 0.5 cos(3πt) + 1.7 cos(8πt)7 0.8 cos(3πt) + cos(5πt) + 3 cos(8πt)

0 5 10 15 20 25 30 35 40 45−6

−4

−2

0

2

4

6

8 Original signal and real change instants

Time [seconds]

Fig. 1. Original signal and the real change instants

The RID of the synthetic signal, computed with a kernel

based on the Hanning window, [24], has been used in eval-

uation of the short-term Renyi entropy as measure of time-

frequency distribution concentration. It was used a sliding

window of N = 32 values and a constant bias to be added

to signal of 0.3, see [13].

The segmentation procedure has been applied for an AR(2)

model, model structure resulted after some experiments with

AR models of different orders.

yt = −φ1(i) ∗ yt−1 − φ2(i) ∗ yt−2 + et (18)

The segmentation results, under the form of the Renyi

entropy with the resulted change instants are presented in

Fig. 2. The MAP estimator with unknown and constant noise

scaling and MCMC numerical search procedure have been

used.

0 5 10 15 20 25 30 35 40 450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8MAP segmentation of Renyi entropy

Time [seconds]

Ren

yi en

tropy

Fig. 2. MAP segmentation of Renyi entropy

The following change instants resulted in segmentation:

[6.3 12.2 18.22 23.98 30.07 30.26 35.93 36.03]

The segmentation results are very closed with the real

change instants of the signal (see Fig. 1); two false change

instants are present, but they are not far away from the real

change detection instants: 30 s and 36 s.

315

B. Seismic Signal Segmentation

The problem of seismic signal segmentation constitutes a

very interesting and challenging task. The main difficulty in

solving this problem is attributed to the fact that both the sta-

tistical properties of seismic noise, as well the characteristics

of the recorded events are in general unknown.

Several methods for seismic signal segmentation have been

suggested earlier. Some of these methods typically employ

either an autoregressive modeling of the data and a general-

ized likelihood ratio test to detect the significant statistical

changes in the waveform, [30], [31], a best-basis searching

algorithm, [32], based on binary segmentation constructed

by Coifman and Wickerhauser [33], or by exploiting the

particular nature of the signals, and by using some interesting

properties that obeys difference based test statistic as well as

its ingredients, [34].

The earthquake accelerations used in this case study were

recorded during the Kocaeli, Arcelik seism, Turkey, August

17, 1999, a strong to moderate ground motion. The data

were sampled with a sampling period of 0.005 seconds, for

around 30 seconds, and were previously corrected to remove

the measurement noise effects. Only horizontal seismic com-

ponents: NS and WE made the object of the analysis.

This seismic motion has constitute the subject of inves-

tigation in many papers which offer elements on how the

signals was acquired and preprocessed, as well as on the

main spectral components, and other specific characteristics,

see e.g. [35], among others.

The horizontal seismic components: NS and WE are

presented in Fig. 3 and Fig. 4, respectively.

0 5 10 15 20 25−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

NS A

cc. (

g un

its)

Time [seconds]

NS earthquake component, Kocaeli, Arcelik, Turkey, 8/17/1999

Fig. 3. NS seismic component

0 5 10 15 20 25−0.1

−0.05

0

0.05

0.1

0.15

WE

Acc.

(g u

nits

)

Time [seconds]

WE earthquake component, Kocaeli, Arcelik, Turkey, 8/17/1999

Fig. 4. WE seismic component

Fig. 5 and Fig. 6 shows the RID of NS and WE seismic

components, computed like in the analysis of synthetic

signal.

5 10 15 20 250

1

2

3

4

5

6RIDH, Lg=204, Lh=512, Nf=4096, lin. scale, contour, Threshold=5%

Time [seconds]

Freq

uenc

y [H

z]

Fig. 5. Reduced Interference Distribution for NS seismic component

5 10 15 20 250

1

2

3

4

5

6RIDH, Lg=204, Lh=512, Nf=4096, lin. scale, contour, Threshold=5%

Time [seconds]

Freq

uenc

y [H

z]

Fig. 6. Reduced Interference Distribution for WE seismic component

Based on RID resulted for both seismic components, the

short-term Renyi entropies were computed, for a sliding

window of N = 32 values and a constant bias to be added

to signal of 0.3. The resulted signals were used as input

information in MAP segmentation algorithm. The segmenta-

tion procedure has been applied for an AR(1) model, model

structure resulted after some experiments with AR models

of different orders.

yt = −φ1(i) ∗ yt−1 + et (19)

where et is a random sequence of zero mean and variance

E(e2t ) = λ(i)σ2

t , with σ2t = 1, for segment i.

Using the proposed procedure, we present in Fig. 7 and

Fig. 8 the MAP segmentation results of Renyi entropy for

seismic motion on NS and WE directions, respectively.

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25MAP segmentation of short−term Renyi entropy: NS direction

Time [samples]

Fig. 7. Segmentation of short-term Renyi entropy: NS direction

As in the case of the synthetic signal, the MAP estima-

tor with unknown and constant noise scaling and MCMC

numerical search procedure have been used. The following

values for change instants have been obtained for NS seismic

component,

316

0 5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16MAP segmentation of short−term Renyi entropy: WE direction

Time [samples]

Fig. 8. Segmentation of short-term Renyi entropy: WE direction

[4.69 8.22 12.20 14.20]

and WE seismic component,

[8.21 9.47 11.81]

The results can be compared with time-frequency analysis,

for the reduced interference distribution (RID) (see Fig. 5 and

Fig. 6), as non-parametric technique offering information on

energy and frequency content evolution in time of the seismic

motion, to evaluate, on this basis, if the change detection

instants resulted with MAP are correct.

The analysis results in both cases confirm the efficiency of

the segmentation approach used, the change instants resulted

by MAP appearing clear in energy and frequency contents

of time-frequency distribution.

VI. CONCLUSIONS

The presented approach could be considered as one of

interest in seismic engineering. The approach makes use of

the short-term time-frequency Renyi entropy and maximum a

posteriori probability (MAP) estimator, operating on Renyi

entropy, as a new space of decision, assuring more robust

feature extraction and a more accurate classification. This

approach has been used in simulation and in the analysis

of the earthquake records during the Kocaeli, Arcelik seism,

Turkey, August 1999, a strong to moderate ground motion,

proving its potential segmentation capability.

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