Advances in Mathematical Physics

Volume 2015 (2015), Article ID 210592, 9 pages

http://dx.doi.org/10.1155/2015/210592

## Generalized Wavelet Fisher’s Information of Signals

^{1}Department of Basic Sciences and Engineering, University of Caribe, 77528 Cancún, QROO, Mexico^{2}Department of Sciences and Engineering, University of Quintana Roo, 77019 Chetumal, QROO, Mexico^{3}Department of Electronics, Systems and Informatics, ITESO Jesuit University of Guadalajara, 45604 San Pedro Tlaquepaque, JAL, Mexico^{4}Engineering and Technology Department, Sonora Institute of Technology, 85000 Ciudad Obregón, SON, Mexico

Received 22 October 2014; Accepted 18 March 2015

Academic Editor: Remi Léandre

Copyright © 2015 Julio Ramírez-Pacheco et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper defines the generalized wavelet Fisher information of parameter . This information measure is obtained by generalizing the time-domain definition of Fisher’s information of Furuichi to the wavelet domain and allows to quantify smoothness and correlation, among other signals characteristics. Closed-form expressions of generalized wavelet Fisher information for signals are determined and a detailed discussion of their properties, characteristics and their relationship with wavelet -Fisher information are given. Information planes of signals Fisher information are obtained and, based on these, potential applications are highlighted. Finally, generalized wavelet Fisher information is applied to the problem of detecting and locating weak structural breaks in stationary signals, particularly for fractional Gaussian noise series. It is shown that by using a joint Fisher/*F*-Statistic procedure, significant improvements in time and accuracy are achieved in comparison with the sole application of the *F*-statistic.

#### 1. Introduction

signals are used as models of several phenomena in a diversity of fields of science. From physics [1] to computer networking [2] and even in psychology [3], processes govern the observed characteristics of the phenomena within these fields. The parameter , which appears in several equations defining these processes, plays an important role since the behaviour and characteristics of the signals is dependent upon the values of . Consequently, the estimation of is of importance since the estimated not only describes the phenomena, but also allows to propose actions in accordance with the relation of and the process it describes. For instance, in computer networking, a value of gives rise to increased delays and packet losses in the network. In response, network analysts may increase network resources or modify network node algorithmics with the purpose of reducing the observed delays and losses and enhance network performance. This type of problem is also observed in other fields and, as a consequence, the efficient and robust estimation of plays a role of major importance [4, 5]. In the literature, several methodologies have been proposed for estimating but it is recognized that no technique can provide robust and efficient estimates for the variety of conditions observed in measured data [5]. Modern techniques, which may consist of combinations of single techniques, attempt to estimate under these conditions; however, they have shown limited performance. Current techniques utilizing signal processing techniques [5] and information theory approaches offer new possibilities of obtaining robust and efficient estimates. Information theory techniques have indeed been used to study the nature, structure, and complexity of many physical phenomena [6–9]. Fisher information, for instance, was used in [10] to study electroencephalogram (EEG) signals and later in [11] for characterizing nonlinear systems. Moreover, Fisher information in wavelet domain was used in [12] to detect weak structural breaks in signals. Tsallis entropies, on the other hand, have been used for studying a variety of signal characteristics [13] and its wavelet counterparts are currently being used from structural damage identification in [14], signal classification in [15], and for detecting structural breaks in the mean in pure-power law (PPL) signals [16] among others [17, 18]. In general, with the use of wavelet based information tools, significant improvements can be achieved in the overall analysis and estimation of signals. This paper extends the generalized Fisher information proposed by Furuichi in [19, 20] to the wavelet domain and then obtains a closed-form expression of this quantifier for signals. The relationship between generalized wavelet Fisher’s information and wavelet -Fisher information is discussed in detail since both quantifiers measure similar characteristics and are dependent upon . Furthermore, the construction of information planes for signals Fisher information allows to identify analysis/estimation applications. Finally, the application of this technique to the problem of detecting and locating structural breaks within signals is given along with statistical analyses of their results. Since the family of signals is of importance in physics, biology, physiology, medicine, and so forth, the results presented in this paper are relevant for enhancing the analyses and estimation procedures within these fields. The rest of the paper is structured as follows. In Section 2, a brief review of the properties, definitions, types, and open issues regarding signals is provided. Also, some important results of the wavelet analysis of these signals are also briefly outlined. Section 3, defines the generalized wavelet Fisher’s information concept, derives a closed-form expression of this quantifier for the family of signals, and studies the behaviour of the information planes of signals’s Fisher information. A comparison with the wavelet -Fisher information is discussed in detail in this section. Section 4 highlights the potential applications of generalized wavelet Fisher’s information for the class of signals and proposes a technique to detect and locate structural breaks in the mean within these processes. Section 5 presents the results of the level-shift detection capabilities of generalized wavelet Fisher’s information on synthesized fractional Gaussian noise (fGn) signals and finally; Section 6 draws the conclusions of the paper.

#### 2. Signals with Behaviour

signals are ubiquitous in science and engineering and model phenomena as diverse as DNA sequences [21, 22], heart-beat time series [23, 24], mood [25], self-steem [26], and so forth. An important feature of signals is their power-law behaviour of their power spectral density (PSD); that is, as,where is a constant, is the scaling parameter, and represent the power-law scaling interval [27]. Depending upon , several well-known processes are obtained; for example, when and , long-memory signals are obtained. Moreover, signals are self-similar in the sense that their distributional properties are invariant under dilations in time and space. signals may be regarded as stationary if and nonstationary if [4, 28]. The well-known fractional Brownian motion (fBm), a Gaussian nonstationary self-similar signal with parameter whose autocovariance given bywith , has a spectral density given byand thus it can be regarded as a signal with parameter . Fractional Gaussian noise (fGn), the first difference of fBm is a stationary Gaussian self-similar signal with PSD given by [27]and . As , and therefore is a signal. Many other signals exist and the interested reader is referred to [4, 27, 28] for further information.

##### 2.1. Wavelet Representations of Signals

Wavelets and wavelet transforms have been extensively used for the analysis and estimation of signals [8, 9, 29–31]. In this paper, the wavelet spectrum [31] of signals is of interest since it allows constructing distributions from which information measures can be computed. For signals, the wavelet spectrum is computed aswhere is the Fourier integral of the mother wavelet , is the PSD of the process , is the expectation operator, and is the discrete wavelet transform of the process at time and wavelet scale [29–31]. Using the well-known PSD of signals, (5) now becomeswhere is a constant. For a more detailed discussion of the wavelet analysis and synthesis of scaling signals, please refer to [27–31].

#### 3. Generalized Wavelet Fisher’s Information Measure

The first uses of Fisher information were in the context of statistical estimation [32, 33]. Recently, however, Fisher information has been used to characterize nonstationarity in complex processes such as EEGs and other nonlinear signals [10, 11]. Fisher information quantifies the spreading of a probability distribution in the sense that is high for flat densities and low for narrow ones. Fisher information is also sensitive to local discontinuities in the density as reported in [34]. Furthermore, Fisher’s information also measures the oscillatory degree of a function and their smoothness character. In this context, Fisher’s information is large for smooth signals and small for highly oscillatory ones. In the same way Tsallis -entropies generalize Shannon entropy, generalizations of Fisher information have been proposed in the literature. Plastino and coworkers defined the -Fisher information in [6]; later, this measure was extended to the wavelet domain in [7]. Lutwak et al. defined the Fisher information in [35, 36] and later Furuichi defined the generalized Fisher information in [19, 20]. The wavelet domain generalization of Plastino’s (wavelet -Fisher information) was used to detect and locate weak structural breaks in signals using a joint wavelet and Bai & Perron technique [7]. Significant improvements were found specifically for long time series with LRD. This paper presents a wavelet-domain extension to the time-domain generalization of Fisher’s information of Furuichi [19]. To study in more detail the properties of generalized Fisher’s information for signals, a closed-form expression is first obtained. In addition, an in-depth discussion of their relationship with the wavelet -Fisher information is given. The generalized Fisher information of a probability density is computed as [19, 20]where is the -expectation operator defined as and is the -score function which is defined as the derivative of the -logarithm of a density; that is, aswhere is the -logarithm function. For discrete distributions, , the generalized Fisher information is computed using a discrete-time version of (7), which is given byfor some and . When is substituted by the relative wavelet energy (RWE), (9) results in the generalized wavelet Fisher information of parameter which measures the spreading of a distribution, the oscillatory nature of signals, among others. Unlike standard wavelet Fisher information measure, generalized wavelet Fisher information provides increased flexibility (with ) and the possibility to adapt the analyses to the characteristics of the data under study.

##### 3.1. Generalized Wavelet Fisher’s Information of Signals

Generalized wavelet Fisher’s information is obtained by substituting the RWE in (9). Recall that the of signals is obtained by means of the wavelet spectrum or wavelet variance of these signals. For stationary signals, the wavelet spectrum is , where is a constant depending on the mother wavelet and . The RWE of signals is thus given by [7–9, 15, 16],where and represent the (logarithmic) length of the signal and the wavelet scale, respectively. The generalized wavelet Fisher information of parameter is, thus,

Equations (11) are employed for the construction of information planes which permit to study in more detail the characteristics and behaviour of Fisher information for signals. Figure 1 displays signals Fisher information for , , and negative . It is interesting to note that under these conditions, the Fisher information planes display two different states. First state corresponds to Fisher information values of zero and the second to nonzero exponentially increasing Fisher information values . The exponential increase of Fisher information starts in and it is entirely controlled by . Observe that by setting , the two states lie in the intervals and and a maping of zero Fisher information values to stationary signals and of nonzero values to nonstationary signals is made. In the same way, by setting , a maping of zero Fisher values to nonstationary fBms is made and a maping of nonzero exponentially increasing Fisher values to extended fBms is performed. The case allows to distinguish among the stationary and nonstationary families of signals while the case permits distinguishing among nonstationary ones. Parameter controls the boundary ; decreasing further shifts the value of to the right. Figure 2 displays signals Fisher information for . As in Figure 1, the planes of Figure 2 were obtained for and . For the particular case , the Fisher information are exponentially decreasing in the interval and converging to zero for . As increases, the information planes converge to zero more quickly. Notice that, for , a two-state behaviour is observed as well. In contrast to the case, the displays an exponentially decreasing behaviour for the first state and zero values for the second one. Similar applications as those for negative can also be attached to the case. Generalized wavelet Fisher information therefore allows characterizing the complexities of signals. Fisher information for () are zero (exponentially decreasing) for random signals and exponentially increasing (converging to zero) for smooth correlated signals. The length of the intervals on which the Fisher information is zero (exponentially decreasing) and exponentially increasing (converging to zero) is controlled by the value of parameter . Because of the similarities in the information planes with those of the wavelet -Fisher information, an interesting question is how generalized wavelet Fisher information is related with wavelet -Fisher information. In the following, we elaborate further on this question and discuss the similarities and/or differences between both information measures in more detail.