New Procedure for Change Detection Operating on Rényi Entropy with Application in Seismic Signals Processing Theodor D. Popescu & Dorel Aiordǎchioaie
Circuits, Systems, and Signal Processing ISSN 0278-081X Circuits Syst Signal Process DOI 10.1007/s00034-017-0492-y
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Author's personal copy Circuits Syst Signal Process DOI 10.1007/s00034-017-0492-y
New Procedure for Change Detection Operating on Rényi Entropy with Application in Seismic Signals Processing Theodor D. Popescu1 · Dorel Aiordˇachioaie2
Received: 7 September 2016 / Revised: 6 January 2017 / Accepted: 6 January 2017 © Springer Science+Business Media New York 2017
Abstract Reliable characterization of different signals is essential for better understanding of their generating and propagation phenomena. Many works in this area have been based on detecting special patterns or clusters in data, and event detection using parametric models. In this paper, we present an approach making use of the short-term time–frequency Rényi entropy and an algorithm to discriminate between the model parameter and noise variance changes, operating on Rényi entropy, as a new space of decision. This method enables a simpler analysis and interpretation of signals behavior. The experimental results obtained by Monte Carlo simulations for a multi-component synthetic signal, embedded in additive white Gaussian noise of different levels, proved the effectiveness of the procedure. Also, the procedure is used, with good results, in the analysis of a seismic signal with two components, during a strong to moderate ground motion. Finally, a comparison of the obtained results with those offered by other change detection approaches is presented. Keywords Change detection · Signal segmentation · Time–frequency analysis · Rényi entropy · Monte Carlo simulation · Seismic signal analysis
B
Theodor D. Popescu
[email protected] Dorel Aiordˇachioaie
[email protected]
1
National Institute for R&D in Informatics, 8-10 Averescu Avenue, 011455 Bucharest, Romania
2
“Dunˇarea de Jos” University of Galati, 47 Domneasca Street, 800008 Galati, Romania
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1 Introduction The problem of change detection and diagnosis has gained considerable attention during the last two decades in a research context and appears to be the central issue in various application areas [4,14,16,19,33], etc. Different approaches are reported in the literature using distance measures, artificial intelligence, fuzzy logic, statistical differences, etc. Some features, such as amplitude levels in the time domain, are easily extracted and classified, but are susceptible to noise. Others, such as energy concentration in the time–frequency domain [31], even though require more involved operations, can lead to more robust change detection [7]. Parametric signal processing algorithms can be used for change detection if an accurate model of the signal exists in a selected representation space. However, such modeling techniques have limitations as well. Modeling of nonstationary signal is more difficult and consistent parametric models often do not exist, except in very few special cases. Most of the signal encountered in practice do not satisfy the stationary conditions, which explains the growing interest in nonstationary signal processing. The time–frequency analysis (TFA) [9], compared with the classical one (formulated in the time-domain in general), usually provides a simpler interpretation and comprehension of nonstationary signals. The proposed solution in the literature, in this sense, consists in applying a transform on the time–frequency representation of the analyzed signal, and using a test on this new space of decision. One of the simplest feature-based signal processing procedures in time–frequency analysis is via energy concentration [31]. The idea is to analyze behavior of the energy distribution, i.e., the concentration of energy at a certain instant or a certain frequency band or more generally, in some particular time and frequency region. Recently, entropy-based measures have been applied to the time–frequency plane to quantify the information content of signals. Different distance measures, such as Kullback–Leibler distance, Rényi distance, and Jensen difference-based measures have been adapted to the time–frequency plane [2]. The paper presents a new approach for change detection in time–frequency information content with application in signal processing. It makes use of an algorithm to discriminate between the model parameter and noise variance changes, operating on the short-term Rényi entropy of the signal in the time–frequency plane, as a new space of decision. This approach enables more robust change detection. The approach is illustrated by Monte Carlo simulations for a multi-component synthetic signal, embedded in additive white Gaussian noise of different levels, and by a case study in the analysis of a seismic signal with two components, during a strong to moderate ground motion. The paper is organized as follows: Sect. 2 presents time–frequency information content measuring using the Rényi entropy. Section 3 has as object an original algorithm able to discriminate between the model parameter and noise variance changes, operating on Rényi entropy, as a new space of decision. Section 4 presents the results obtained by Monte Carlo simulations for a multi-component synthetic signal, embedded in additive white Gaussian noise of different levels, while Sect. 5 proves the effectiveness of the procedure when applied in the analysis of a seismic signal with two components, during a strong to moderate ground motion. Section 6 presents some
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comparisons of the experimental results obtained with the proposed method, and those resulted when other change detection and segmentation methods have been used, in simulation and in seismic signal analysis.
2 Time–Frequency Information Content Measuring Using the Rényi Entropy 2.1 Time–Frequency Analysis The basic objective of time–frequency analysis (TFA) is to develop a function that may enable us to describe how the energy density of a signal is distributed in the joint time, t, and frequency, ω, domain. A time–frequency transform maps one-dimensional time domain signal into a two-dimensional representation of energy versus time and frequency. The time–frequency representations (TFRs) can be classified according to the analysis approaches [30]. 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 TF x (t, ω) =
+∞
−∞
∗ x(τ )φt,ω (τ )dτ = x, φt ,ω ,
(1)
where φt,ω represents the basis functions (also called the TF atoms) and ∗ represents the complex conjugate. They are linear with respect to the signal. The basis functions are assumed to be square integrable, φt,ω ∈ L2 (R), i.e., they have finite energy [21]. The second category of time–frequency distributions, known as Cohen’s shift invariant class distributions, are based on the signal energy distribution in the time–frequency domain. They are characterized by a kernel function. The properties of the representation are reflected by simple constraints on the kernel that produces the TFR with prescribed, desirable properties, [6,9]. A mathematical description of these TFRs can be given by TFDx (t, ω) =
1 4π 2
+∞ +∞ +∞
−∞
−∞
−∞
1 x u+ τ × 2
(2)
1 x u − τ φ(θ, τ ) exp− jθt− jτ ω+ jθu dudτ dθ 2 ∗
where φ(θ, τ ) is a two-dimensional kernel function, determining the specific representation in this category, and hence, the properties of the representation. Basic distribution in this approach is the Wigner distribution (WD). WDx (t, ω) =
+∞
−∞
1 1 x t + τ x ∗ t − τ exp− jτ ω dτ. 2 2
(3)
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Its kernel is φ(θ, τ ) = 1. Also spectrogram (SP), Choi–Williams distribution (CWD) and binomial distribution (BD) are some of the methods used for obtaining the TFDs [9]. The Choi–Williams distribution (CWD) [17] and the binomial distribution (BD) [18] suppress cross-terms interference to a large extent, but some time–frequency resolution is lost. These distributions belong to the so-called reduced interference distribution (RID). They also belong to the Cohen’s class [6,9,20]. Every member of Cohen’s general class may be interpreted as two-dimensional filtered WD. In the sequel, we will use the reduced interference distribution (RID).
2.2 Rényi Entropy of a Time–Frequency Representation As we mentioned before, one of the simplest feature-based signal processing procedures in TFA is via energy concentration. The idea is to analyze the concentration of energy at certain time instant or certain frequency band or more generally, in some particular time and frequency region. Such analysis is capable of revealing more information about a particular phenomenon. An overview of time–frequency representation using energy concentration makes the object of [30]. Also, a review of some existing measures and their comparison are given in [31], where norms with distributions raised to powers equal and lower than one are used. All these measures provided good quantitative measure of the auto-terms concentration. Norm themselves failed to behave in the desired way when the cross-terms appeared. Various and efficient modifications were used in order to take into account the appearance of nondesirable oscillatory zero-mean distribution values. The distribution norm has been divided by a lowerorder norm, while some strict constraints were imposed on the kernel forms in [28]. However, even the normalized forms of the norm-based measures are not quite appropriate for the cases where are two or more components (or regions in time–frequency plane of a single component) of approximately equal energies whose concentration are very different. The norm-based measures, due to raising of distribution values to a high power (four or third), will favor distributions with “peaky” components. It means that if one component (region) is “extremely highly” concentrated, and all the others are “very poorly” concentrated, then they will not look for a compromise, for example, when all components are “very well” concentrated. We will briefly review in the following the Rényi entropy measures, able to quantitative analysis in describing the amount of information encoded in time–frequency distribution, as an approach for event detection in vibrating signals. Once the local frequency content has been obtained, using one of the time– frequency distributions presented above, a new entropy measure can be evaluated for extracting the information containing in a given position of t = n. The Rényi entropy measures class has been introduced in time–frequency analysis by Sang and Williams [28], with significant contributions of Flandrin et al. [12], establishing the properties of this measure. The generalized entropies of Rényi inspired new measures for estimating signal information and complexity in the time–frequency plane. Local properties of the Rényi entropies were studied by Sucic et al. [32]. These measures possess several additional interesting and useful properties, such as account-
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ing and cross-component and transformation invariances, that make them natural for time–frequency analysis. For a generic time–frequency distribution, Px (n, k), the Rényi entropy measure has the following form: 1 α log2 Rα = Px (n, k) 1−α n
(4)
k
where n is the temporal discrete variable and k the frequency discrete variable, with α ≥ 2 being values recommended for time–frequency distribution measures [12]. 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 does not contribute to this measure. These were the reasons for the introduction of normalized Rényi entropy measures, that will be described next. The normalization can be done in various ways, leading to a variety of possible measure definitions. So in [31] and [12] some normalization schemes of the Rényi entropy, with the signal energy and with the distribution volume, are proposed. Eisberg and Resnik [11], assimilate the time–frequency distributions at a given instant t = n to a wave function and the general case in (4) for α = 3, gives 1 3 Px (n, k) R3 = − log2 2 n
(5)
k
The normalizing stage affects 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 (5) can be rewritten for a given n as follows: 1 3 R3 (n) = − log2 Px (n, k) 2
(6)
k
Empirically, the normalization proposed in [11] had shown 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 in (5), and it can be shown that they are within the interval 0 ≤ R3 (n) ≤ log2 N . Hence, the measure can be normalized by applying Rˆ 3 (n) = R3 (n)/ log2 N .
3 Change Detection in Rényi Entropy Change detection algorithms in signal characteristics can be used starting from the time–frequency distributions concentration. Such an approach, operating on shortterm Rényi entropy as a new space of decision, instead of on the original time signals, leads to more robust change detection in signals making the object of the analysis. In this framework, we use an algorithm [25], able to discriminate between the model
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parameter and noise variance changes, operating on Rényi entropy. This is of great practical importance in many applications, where the classical algorithms are unable to discriminate between the changes in the system and in the environment. The presented case study, having as object seismic signal analysis, will point out this facility: the algorithm will be able to determine the change points in seismic source and the change points in soil dynamics, of great interest for the geophysicists, to evaluate the effects of the seismic motion, especially to know when the vibration source or/and the soil dynamics are changed. The detection algorithm is based on likelihood. We give only the conceptual description of the algorithm (for details see [25]). The following signal model is used, for change instant, t0 : yt =
φtT θ0 + et , E[et ] = λ0 Rt φtT θ1 + et , E[et ] = λ1 Rt
if t ≤ t0 if t > t0
(7)
where yt is the observed signal, φt is a vector containing the previous values of the input and/or output, θ is the parameter vector, and et is the measurement noise, assumed to be Gaussian with a known time-varying noise variance Rt , for generality, and λ is either a scaling of the noise variance or the variance itself (Rt = 1). Neither θ0 , θ1 , λ0 nor λ1 are known. The assumption on the regression models in (7) is not too restrictive since many stationary processes encountered in practice can be closely approximated by such models. The identification and parameters estimation methods represent only tools to perform change detection and segmentation. Good and precise models offer high performance in these schemes, but also biased parametric models can be used for change detection and segmentation. This bias decreases, but does not annihilate the performance of the detection and segmentation procedures. The following hypotheses are used: H0 : θ0 = θ1 and λ0 = λ1 H1 : θ0 = θ1 and λ0 = λ1 H2 : θ0 = θ1 and λ0 = λ1
(8)
The sufficient statistics from the filters are given below: Data y1 , y2 , . . . , yt−L , yt−L+1 , . . . , yt Model M0 M1 Time interval T0 T1 θˆ1 , P1 RLS quantities θˆ0 , P0 Loss function V0 V1 Number of data n 0 = t − L n1 = L
(9)
where P j , j = 0, 1 denotes the covariance of the parameter estimate achieved from the RLS algorithm. The loss functions are defined by
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V j (θ ) =
yk − φkT θ
T
(λ j Rk )−1 yk − φkT θ , j = 0, 1.
(10)
k∈T j
It makes sense to compute V1 (θˆ0 ) in order to test how the first model performs on the new data set. The maximum likelihood approach is stated in the slightly more general maximum a posteriori approach, where the prior probabilities qi for each hypothesis can be incorporated [16]. The exact a posteriori probabilities li = 2 log p(Hi | y1 , y2 , . . . , yt ), i = 0, 1, 2
(11)
Assuming that Hi , i = 0, 1, 2 is Bernoulli distributed with probability qi , i.e., Hi =
does not hold, hold,
with probability 1 − qi with probability qi
(12)
log p(Hi ) is given by log p(Hi ) = log(qi2 (1 − qi )n 0 +n 1 −2 ) = 2 log(qi ) + (n 0 + n 1 − 2) log(1 − qi ), i = 0, 1, 2.
(13)
For the signal model (7) where e ∈ N (0, λ), the prior distribution for λ can be taken as inverse Wishart distribution (or gamma distribution in scalar case). The inverse Wishart distribution has two parameters, m and σ , and is denoted by W −1 (m, σ ). Its probability density function is given by σ
σ m/2 e− 2λ p(λ) = m/2 2 Γ (m/2)λ(m+2)/2
(14)
The expected mean value of λ is E(λ) =
σ m−2
(15)
and the variance is given by Var(λ) =
2σ 2 (m − 2)2 (m − 4)
(16)
The mean value and the noise variance are design parameters, and from these the Wishart parameter m and σ can be computed. For the signal model (7) and the hypotheses given in (8), let the prior for λ as in (14) and the prior for the parameter vector be θ ∈ N (0, P0 ). With the loss function (10) and the least squares estimation, the a posteriori probabilities are approximately given by [16]:
Author's personal copy Circuits Syst Signal Process Table 1 Epochs of multi-component signal
Epoch Signal component 1
1.5 cos(2π t) + 4 cos(14π t)
2
cos(π t) + 4.5 cos(7π t)
3
0.5 cos(π t) + 1.5 cos(2π t) + 0.8 cos(6π t) + 3.5 cos(16π t)
4
0.5 cos(6π t) + 2.5 cos(16π t)
l0 ≈ (n 0 + n 1 − 2 + m) log
V0 (θˆ0 ) + V1 (θˆ1 ) + σ n0 + n1 − 4
+ log det P0−1 + P1−1 + 2 log(q0 ) V0 (θˆ0 ) + V1 (θˆ1 ) + σ l1 ≈ (n 0 + n 1 − 2 + m) log n0 + n1 − 4 − log detP0 − log detP1 + 2 log(q1 ) V0 (θˆ0 ) + σ l2 ≈ (n 0 − 2 + m) log n0 − 4 V1 (θˆ0 ) + σ + (n 1 − 2 + m) log n1 − 4 − 2 log detP0 + 2 log(q2 )
(17)
The last three equations are used in decision making concerning one of the three hypotheses presented above. So, all the probabilities in (17) are computed and the probability with the greatest value is chosen, resulting which of the hypotheses H0 , H1 , H2 check. On the data segment between two consecutive changes, model parameters or noise variance are estimated, according with the respective hypothesis: H1 or H2 . If no changes appear in model parameters and noise variance, the procedure continues until it is obtained a new maximum value for l1 or l2 in (17).
4 Monte Carlo Simulation Results In order to assess performance of the proposed approach, it was applied on a multicomponent synthetic signal, xt , embedded in additive white Gaussian noise of different levels. The signal contains four multi-component epochs with the duration of 6 seconds, when a sampling frequency of 100 Hz was used (see Table 1). It can be noted that the model parameters change and the noise variance is constant for each signal epoch, so in this case H1 : θ0 = θ1 and λ0 = λ1 hypothesis is true. A white Gaussian noise was added to the total signal to give yt = xt +et , with E[et2 ], having the values 0.1, 0.3 and 0.5, for the Case 1, Case 2 and Case 3, respectively. This example aims also to prove the robustness of the procedure based on the Rényi entropy segmentation, when the assumption concerning the regression hypothesis for the analyzed signal does not check.
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Multi−component signal and real change instants
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RIDH, Lg=120, Lh=300, Nf=2400, lin. scale, contour, Threshold=5% 10
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The Monte Carlo simulation consisted in simulation of the signal for the epochs given in Table 1, and the values of E[et2 ], in three cases, for 300 realizations of the random sequence, et , in computation of Rényi entropy, and in applying of the proposed segmentation algorithm presented in Sect. 3. First, we present for Case 1, the signal and real change instants, in Fig. 1. Figure 2 shows the RID of the synthetic signal, computed with a kernel based on the Hanning window [20]: RIDH x (t, ω) =
+∞ −∞
h(τ )Rx (t, τ ) exp− jωτ dτ,
(18)
with Rx (t, τ ) =
+ |τ2|
− |τ2|
2π u τ ∗
g(u) τ x t +u− du 1 + cos x t +u+ |τ | τ 2 2
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Renyi entropy of multi−component signal and real change instants 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
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Time [samples] Fig. 3 Rényi entropy with the real changes in multi-component signal—Case 1
for the number of frequency bins, Nf = 2400, identical with the time instants, time smoothing window g(u), with Lg = 120, frequency smoothing window h(τ ), with Lh = 300, and a threshold of 5%. The software support was assured by MATLAB Time–Frequency Toolbox, [3]. The next step of the analysis consisted in the evaluation of the short-term Rényi entropy as measure of time–frequency distribution concentration, computed for RID, using (9). It used a sliding window of N = 32 values and a constant bias to be added to signal of 0.3. The experimental results are given in Fig. 3 for the realization of the signal (Fig. 1). Similar signals, given in Figs. 1, 2 and 3, have been obtained for the Case 2 and Case 3 of the experiment. The Rényi entropy obtained in all cases, for 300 realizations of the noise, was made subject of proposed segmentation procedure. The segmentation procedure has been applied to an autoregressive model AR (2), model structure resulted after some experiments with AR models of different orders. yt = −φ1 (i) ∗ yt−1 − φ2 (i) ∗ yt−2 + et
(19)
The Monte Carlo simulation results, for all 300 realizations of the noise, are given in Figs. 4, 5 and 6 for the cases making the object of the experiment. The segmentation results, in all cases, are very closed to the real change instants of the signal (see Fig. 1), despite of the level of noise; some false change instants and delay in detection are present, but they are not significant for the experimental conditions.
5 Case Study The object of the present case study is the analysis of NS and WE seismic components of the ground motion during the Kocaeli seism, Arcelik station (ARC), Turkey, August 17, 1999 [15,34]. The data were sampled with a sampling frequency of 200 Hz, for
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MC change time in Renyi entropy − Case 1 300
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Fig. 6 Case 3: MC change detection time for Rényi entropy—Case 3
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NS earthquake component, Kocaeli, Arcelik, Turkey, 8/17/1999
NS Acc. (g units)
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Frequency [Hz] Fig. 7 NS seismic component and Fourier amplitude spectrum
around 30 seconds, and were previously corrected to remove the measurement noise effects.
5.1 Preliminary Seismic Data Analysis The present analysis will be useful in interpreting the results obtained in change detection in seismic signals, and in their validation. We present in Fig. 7 the NS seismic component and its Fourier amplitude spectrum. In the frequency domain representation of the NS component, at least six frequency components, located at 0.6, 2, 2.6, 2.9, 3.4 and 3.8 Hz, are clearly seen. These frequencies were labeled with NS1, NS2, NS3, NS4, NS5 and NS6, respectively. Of these frequency components, the dominant is located at around 2.6 Hz. Beyond 4 Hz, the spectral amplitudes are small. For the same component, the time domain representation indicates that the maximum amplitude is located at around 12 s. The frequency appears to change from high to low and then slightly high frequencies are observed in the time interval 8 to 14 s. A low-frequency component may be observed all along the record. However, in certain small time window interval, a high-frequency component is also evident. The same type of analysis has been performed for the WE seismic component. We present in Fig. 8 the seismic component making the object of investigation and its Fourier amplitude spectrum.
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WE earthquake component, Kocaeli, Arcelik, Turkey, 8/17/1999
WE Acc. (g units)
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Frequency [Hz] Fig. 8 WE seismic component and Fourier amplitude spectrum
A behavior similar to the NS seismic component can be noticed. Briefly, we present the result analysis. In this case, also, at least six frequency components, located at 0.6, 1.4, 1.9, 2.7, 3 and 3.5 Hz, are clearly seen. These frequencies were labeled with WE1, WE2, WE3, WE4, WE5 and WE6, respectively. Of these frequency components, the dominant is located at around 1.9 Hz. Beyond 5 Hz the spectral amplitudes are small. The time domain representation indicates that the maximum amplitude is located at around 9 s. As in the previous case, the frequency appears to change from high to low and then slightly high frequencies are observed in the time interval 8 to 14 s. Also, a low-frequency component may be observed all along the record. However, in certain small time window interval, a high-frequency component is also evident. We present and discuss the application of change detection and diagnosis algorithm on short-term Rényi entropy measure, for both seismic components.
5.2 Time–Frequency Analysis The basic property required by our analysis is a good time localization of the signal spectral content. Thus, considering a RID from the Cohen class we choose the one satisfying the time and frequency support, among other desirable properties. Using the Hanning window as primitive function we get the distribution, according to [3]
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RIDH, Lg=204, Lh=512, Nf=4096, lin. scale, contour, Threshold=5% 6
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Time [seconds] Fig. 9 Reduced interference distribution for NS seismic component
RIDH x (t, ω) =
+∞ −∞
h(τ )Rx (t, τ ) exp− jωτ dτ,
(20)
with a generalized autocorrelation function 2π u g(u) 1 + cos |τ | τ − |τ2|
τ τ ∗
x t +u− du ×x t +u + 2 2
Rx (t, τ ) =
+ |τ2|
(21)
In addition, to the kernel based on the Hanning window 1 + cos(2π u/τ ) as a primitive function, we added time and frequency smoothing windows. Time smoothing window, g(u), has the length Lg = 204, while the frequency smoothing window, h(τ ), is of the width Lh = 512. A threshold of 5% is used. The number of frequency bins is Nf = 4096. It is identical with the time instants support. The RID of the NS seismic component computed with these parameters is shown in Fig. 9. In Fig. 9 at linear scale, for NS seismic component, at least five smooth welldefined frequency components can be observed. The strongest amplitude is located at 2.9 Hz and the component of large duration is located at 2.6 Hz. The large duration is in relative comparison with the frequency component located at 2.9 Hz. In this case, both the synchronization and cross-terms are smaller than observed in the case of other TFDs. This improvement facilitates the identification/interpretation of the frequency components of the seismic signals. It may be feasible to interpret some seismic signals
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Short−term Renyi entropy using RID for NS seismic component 0.25
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Time [seconds] Fig. 10 Renyi entropy for NS seismic component
in terms of frequency dispersive characteristics. It is also possible to observe that the times when maximum amplitudes occur in the time domain of the earthquake record also correspond to the times where several time–frequency characteristics of the signal “converge.” The short-term Rényi entropy as measure of time–frequency distribution concentration, computed for RID, for NS seismic component is presented in Fig. 10. We used a sliding window of N = 32 values and a constant bias to be added to signal of 0.3, see [13]. Results of the TFD analysis of the WE seismic component are presented in Fig. 11 for RID, with a Hanning window and the same parameters like for NS seismic component. Figure 11 shows at least six frequency components of approximate the same time duration. We remember that the frequencies of interest are located at 0.6, 1.4, 1.9, 2.5, 3 and 3.5 Hz. Similar conclusions, as in the previously analyzed case, concerning TFD properties, could be established. To evaluate the TFD for WE seismic component, we present in Fig. 12 the short-term Rényi entropy as measure of time–frequency distribution concentration, computed for RID. We used a sliding window of N = 32 values and a constant bias to be added to signal of 0.3, as in the previous case. Using the TFA, it was observed that in the Kocaeli earthquake ground motions, Arcelik station (ARC) acceleration with same frequency occurred at different time instants. The experimental results exhibit some interesting features from the interpretation point of view of the TFD used. Without being conclusive, some time–frequency distribution trends are possible to identify. The RID time–frequency distribution is the one that appears to offer the best trade-off between the signal time–frequency distribution and the suppression of numerical artifacts (such as cross-term interference and synchronization effects) of the algorithm. Both energy concentration spots and time–frequency trends seem to be somewhat clearer and also consistent with the partial information seen in other TFDs.
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RIDH, Lg=204, Lh=512, Nf=4096, lin. scale, contour, Threshold=5% 6
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5.3 Change Detection in Rényi Entropy After a preliminary analysis of the estimated Rényi entropies, for different lengths of the sliding window and different autoregressive model orders, we find the length of 32 data (0.16 s) as the best for this application, and a model AR (1) of the following form: (22) yt = −φ1 (i) ∗ yt−1 + et
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Model AR(1) parameter estimates of Renyi entropy Model parameters
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Using the procedure described in Sect. 3, we present in Fig. 13 the evolution of the parameter estimates and of the noise variance during the seismic motion on NS direction, for hypotheses H1 and H2 from (8). The evolution of the parameter estimates and of noise variance with the change detection instants in hypotheses H1 and H2 , for the WE direction is presented in Fig. 14. The variance traces of the parametric model show some significant jumps for both investigated directions, suggesting that main distinct rupture events occurred. These could not have been observed by TFA methods and Rényi entropy. So, it is quicker to recognize many characteristics of the signal with the proposed approach, operating on the new space of decision, provided by Rényi entropy. The procedure, enabling the energy distribution in the signal to be clearly observed, assures more robust feature extraction and a more accurate classification, in seismic engineering.
6 Comparisons with Other Methods Several methods for signal change detection and segmentation have been suggested earlier, see e.g. [4,16,23], among others. They typically employ multiple detection algorithms [29], hidden Markov models [5], explicit management of multiple model,
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Model AR(1) parameter estimates of Renyi entropy Model parameters
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AFMM (adaptive forgetting by multiple models) [1], or formulate the segmentation problem as a least squares problem with sum-of-norm regularization over the state parameter jumps, a generalization of l1 -regularization [23]. Different approaches for signal change detection and segmentation have been investigated by Monte Carlo simulations in [27], for different synthetic signals, with some filtering techniques using a whiteness test, Cumulative Sum (CUSUM) and Geometric Moving Average (GMA) criteria. Also, techniques based on sliding windows and distance measures, using Generalized Likelihood Ratio (GLR), Divergence Test (DIV), and maximum a posteriori probability (MAP) estimator with unknown and constant noise scaling, have been investigated. These change detection and segmentation algorithms have been directly applied on the original signals, making the object of the analysis. In [26] are presented some experiments in Monte Carlo simulation for a synthetic signal, containing seven multi-component epochs, similar with the signal presented in Sect. 4 , when the segmentation is performed on the short-term time– frequency Rényi entropy and the maximum a posteriori probability (MAP) estimator is used. It can be noted that some false change instants are present. All these approaches are unable to discriminate between changes in system dynamics (model parameters) and environment (noise variance). The approach presented in [25] offers this facility,
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and some Monte Carlo simulations for change detection in a second order FIR model are presented; as in [27] the procedure is directly applied on the original signals. 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 generalized likelihood ratio test to detect the significant statistical changes in the waveform [8,22], a best-basis searching algorithm [35], based on binary segmentation constructed by Coifman and Wickerhauser [10], or by exploiting the particular nature of the signals, and by using some interesting properties of the statistic test [24]. The same seismic signal, making the object of the analysis in Sect. 5, has been used in [27], for a segmentation method, based on maximum a posteriori probability (MAP) estimator for optimal segmentation, applied to the original signal; the results are compared with time–frequency analysis, offering information on the evolution in time of energy and frequency content of the seismic motion. Also, in [25] the same signal is analyzed by the change detection procedure described in Sect. 3, but applied directly on the original seismic signals. In [26], the segmentation method, based on maximum a posteriori probability (MAP) estimator for optimal segmentation, is applied on the short-term time–frequency Rényi entropy. The results can be compared with those from Sect. 5. It can be noted that the proposed approach, based on short-term time– frequency Rényi entropy and an algorithm able to discriminate between the model parameter and noise variance changes, assures a more robust feature extraction, using energy concentration and a classification of the change instants in system dynamics and noise variance.
7 Conclusions The paper presents a new and original approach making use of the short-term time– frequency Rényi entropy and of an algorithm for change detection and segmentation, able to discriminate between the model parameter and noise variance changes, operating on Rényi entropy. This new space of decision assures some new facilities for change detection in time–frequency information content, assuring a simpler analysis and interpretation of the signals behavior. Also, the approach offers a robust change detection scheme, able to work well and to separate the changes in the experimental conditions from the real changes in the system, especially for systems with arbitrary and nonstationary known or unknown inputs. The approach effectiveness is proved by the experimental results obtained by Monte Carlo simulations for a multi-component synthetic signal, embedded in additive white Gaussian noise of different levels, and in the analysis of seismic components of a strong to moderate ground motion, than other algorithms used in similar conditions, as it is presented in Sect. 6 of the paper. Acknowledgements The authors thank the Executive Agency for Higher Education, Research, Development and Innovation Funding (UEFISCDI) for its support under Contracts PN-II-PT-PCCA-2013-4-0044, Grant 224-2014.
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