PACS: 89.75.Da; 89.75.Cf; 61.43.-j; 05.20.-y; 05.10.Gg

1.Introduction

In statistical mechanics, the concept of entropy has traditionally been employed to measure the information content or uncertainty of a random signal or system ^{1}^{,}^{2}. In the current literature, entropic functionals, such as the Shannon, Rényi and Tsallis entropies, have been used to quantify the complexities associated to random and nonlinear phenomena ^{3}^{,}^{4}. Moreover, more elaborate information functionals, such as the so-called information planes ^{5}, which consist of the product of entropy functionals and the Fisher information (and also of entropy and dissequilibrium), are currently being applied in numerous fields, for instance, in analyzing two-electron systems ^{6}, many particle systems ^{7}, randomness and localization of molecules ^{8} and x-ray astrophysical sources ^{9}. In order to obtain an estimate of such an information functional within a signal or system, a probability density function (pdf) for continuous signals or probability mass function (pmf) for sampled data is required. Traditionally, pmfs in time and frequency domain are used, however, in recent years, with the advent of multiscale analysis and time-frequency distributions, wavelet-domain pmfs are utilized giving rise to the so-called wavelet entropies ^{10}, generalized wavelet Fisher informations ^{11}, among others. The advantages of extending spectral and time entropies to the wavelet domain are numerous including the possibility to analyze nonstationary signals or time-varying behaviour. Wavelets, also permit to compute entropies for specific details or resolutions of the signal in order to capture particular behaviour. Applications of such wavelet entropy functionals are diverse, from electroencephalogram (EEG) and electrocardiogram (ECG) signal analysis ^{12}^{,}^{13} to laser propagation ^{14} and characterization of complexity within random signals ^{15}. Scale-invariant or 1/f signals, on the other hand, have been used to model a variety of phenomena in Physics and many other areas of science ^{16}. For instance, Gilmore and co-workers ^{17} found evidence of scaling behaviour in plasma turbulence. Moreover, ^{18} and this behaviour seems to be ubiquitous since it has been found in disciplines as diverse as Chemistry ^{19}, Physiology ^{20}, Psychology ^{21}, Biomedical Engineering ^{22} among others. Within scale-invariant signal analysis, the estimation of the scaling parameter, ^{23} , the shape and behaviour of sample paths, the stationarity and nonstationarity of realizations^{24} , among other properties^{25} . Many techniques for estimating α have been proposed, however, nowadays, no single technique is able to accurately estimate α under the variety of complexities found in real measured data^{23}^{,}^{24}^{,}^{25} . Recently, wavelet-based information tools have found application in the analysis of ^{10}^{,}^{15}^{,}^{23}^{,}^{26}. Wavelet information tools characterize adequately the theoretical complexities of these signals and as a consequence may help the signal analyst to propose tools or methodologies for their analysis/estimation. For instance, in ^{11}, a novel technique based on generalized wavelet Fisher information allowed to detect level-shifts in fractional Gaussian noise (fGn) signals of parameter ^{27}. The entropy planes for this two-parameter entropy are obtained for a variety of values of the parameter α and their relationship with the standard Shannon and Tsallis entropies is also found. The motivation behind the wavelet

2. Wavelet analysis of 1/𝑓^{α} signals

^{28}, heart-beat time series ^{29}, mood and self-steem ^{30}. *i.e*., as,

where ^{31}. Depending upon *e.g.*, when ^{24}. The well-known fractional Brownian motion (fBm), a Gaussian, nonstationary and self-similar signal with parameter *H* whose autocovariance is given by,

with

and thus is a ^{24}^{,}^{25}. Fractional Gaussian noise (fGn), which is obtained from a fBm process via a differencing operation, is stationary, self-similar, Gaussian and has a PSD of the form ^{16}^{,}^{24}^{,}^{25}:

for *H* ϵ (0, 1). In the limit of 𝑓

Wavelet analysis permits to represent a signal as a sum of small waves called wavelets. It has been employed for the analysis of complex time series ^{32} and ^{10}^{,}^{15}^{,}^{31}. Wavelet analysis can be computed in two different ways allowing to capture different behaviour within a signal. The discrete wavelet transform (DWT) is primarily used for computing wavelet variance, entropy, etc. The continuous wavelet transform (CWT), on the other hand, is more convenient for quantifying the cycles, synchronization as well as the correlation within one or more time series via the wavelet cross-correlation and wavelet coherence . In this article, the DWT is employed and the wavelet spectrum is computed. The wavelet spectrum obtains the variances of DWT coefficients at each wavelet scale j and allows not only to estimate the scaling parameter α but also to obtain a pmf which in turn can be used to compute entropic functionals. In the work of Abry ^{31}, the wavelet spectrum was studied and a formula for computing the wavelet spectrum of random signals was given by,

where ^{31}. Using the well-known PSD of

where *C* is a constant. The wavelet spectrum obtained in Eq.(6) has been used to esitmate α ^{31} and also for computing wavelet-based information tools^{10}^{,}^{11} . For further information on wavelets, either continuous or discrete and in the wavelet analysis of ^{31}^{,}^{32} and references therein.

3. Wavelet

In this article, a two parameter wavelet

where *j*. Equation (7) is indeed a pmf since ^{10}, quantifies the energy of a random signal per resolution level j. The RWE for the class of 1/ f α signals is therefore given by substituting(6) into (7) ^{10}^{,}^{11}^{,}^{15}*i.e*.,

where *N* and *j* represent the (logarithmic) length of the signal and the wavelet scale respectively. Many wavelet-based information tools have been obtained using Eq.(8) , the wavelet Fisher informations of ^{11} and ^{26}, the wavelet Tsallis q-entropy of ^{15}, among others. The article proposes a novel wavelet *q*. This means that Tsallis q-entropies can, for example, classify signals using a specific value of q but no other value of q can perform the same classification. There is, thus, a need for an information functional to provide alternative configurational parameters to analyze the same problem. The *q*´ and the question is if it is able to provide alternative configurations of their parameter values to analyze a given problem. In this article, this ^{34} is generalized to the wavelet domain and closed-form formulas of this entropy for ^{34},

where *q*-entropy results and when

where *q*-entropy and Tsallis

Substituting Eq.(8) into (10) and using the results ^{23} of the wavelet

where *i.e*.,

At this point and based on the results of Eqs.(11) and (12), an interesting question is how the wavelet *q* and a *q*´ and to investigate if different sets of values of *q* and *q*´ may permit to analyze the same problem, say detecting specific behaviour within a signal. The first question is answered if the wavelet *q* and *q*´ are obtained and the second by identifying planes which in principle may be different but provide a similar description of a given phenomena. In the following, the entropy planes obtained for particular values of the nonextensivity parameters *q* and *q*´ are studied. Wavelet entropy planes can also be used for identifying potential applications of this entropy for the analysis/estimation of *q* =7 and *q*´ = 4 and which is a typical behaviour of wavelet *q* and *q*´. The length of the signal *N* and the nonextensivity parameter *q*´ have the effect of increasing the rate at which entropies increase and in the same way of reducing the range of zero entropies as shown in Fig. 1. Increasing *N*, increases the rate at which entropies increase. Parameter *q*´ also increases this rate, however follows an interesting behaviour which depends upon the positivity or negativity of *q*´. To investigate further the effect of *q*´ on the shape of wavelet (*q*, *q*´) entropy planes, Fig. 2 displays a particular example when *q* = 13, *N* = 10 and negative *q*´. Note that as *q*´ decreases, the entropies increase more rapidly and the range of constant entropies decreases. Figure 3 displays another example when *N* =12, *q* = 7but using positive values of the nonextensivity parameter *q*´. Note that for this case, entropies increase more rapidly with higher values of q′ and the range of zero entropies decreases. From this, it is concluded that the rate at which entropies increase is boosted whenever *q*´ becomes more negative or *q*´ becomes more positive (mantaining *N* and *q* fixed). Figure 4 displays another wavelet entropy plane obtained when q=9 and q′→1. Note that in this case, wavelet entropies are monotonically decreasing and normalized to 1. This behaviour is similar to the one observed for the wavelet Tsallis *q*-entropy and the parameter *q*, in this case, permits to stretch the range over which constant entropies are observed. In contrast to Fig. 1, the wavelet (*q*, *q*´)-entropy plane of Fig. 4 is not sensitive to the length of the signal. Note that Fig. 1 and Fig. 4 provide alternative ways of characterizing complexity. The configuration of Fig. 1 treats purely random *q* and *q*´ values of Fig. 4 assigns unity entropies to these random signals.

4. Applications

Wavelet (*q*, *q*´)-entropy may have several applications not only for the analysis/estimation of 1/ f α signals but also for the characterization of the complexities of any random signal. For *q*, *q*´)-entropy may help increase the accuracy of estimating algorithms by identifying and eliminating level-shifts that bias the estimations. Moreover, based on the results of Sec. 3, wavelet (*q*, *q*´)-entropy can also be used for classifying *q*, *q*´)-entropy in two different ways using different configuration of values for the nonextensivity parameters q and q′. First, by adjusting the range of zero frequencies in Fig. 1 to *q*, *q*´)-entropy. Top plot represents a concatenated time series in which the first time points up to the middle of the duration are from a stationary signal and the rest come from a nonstationary signal. Bottom plot of Fig. 5 display the wavelet (*q*, *q*´)-entropy from the concatenated signal using sliding windows of length W=1024, *q* = 8 and *q*´= 6. Note that the entropies for the first part of the signal (stationary) are zero while entropies corresponding to the second half part vary. This simple example demonstrates that a robust and powerful technique for discriminating *q*, *q*´)-entropy can be obtained. Another interesting application of wavelet (*q*, *q*´)-entropy is in the field of level-shift detection and location. It has been shown in the work of Stoev that a single level-shift has the effect of overestimating *H* yielding H > 1, thus, wavelet (*q*, *q*´)-entropy applied to a segment with a single level-shift will result in an entropy value suddently decaying to zero (or sudently increasing above zero) resembling an impulse shaped form. Therefore, the wavelet (*q*, *q*´)-entropy of signal with level-shifts will result in a signal composed of impulses. The location and strength of the impulse is related to the location and amplitude of the level-shift. With the use of wavelet (*q*, *q*´)-entropy, a level-shift detection/location problem becomes in a peak detection and location problem. Figure 6 displays an example in which a signal with 3 level-shifts is detected by the use of wavelet (*q*, *q*´)-entropy. Therefore, with the use of wavelet (*q*, *q*´)-entropy, an efficient and fast methodology for detecting and locating weak level-shifts can be designed. Many other applications can be perfomed with the use of wavelet (*q*, *q*´)-entropy, the purpose of this article, however is not to investigate applications but to present the wavelet (*q*, *q*´)-entropy, their theoretical properties on

5. Conclusions

In this article, a novel wavelet (*q*, *q*´)-entropy for the analysis of *q*, *q*´) entropy of Borges and permits to quantify the complexities and information content of random signals and systems using the DWT. Closed-form expressions for this entropy are found for the class of *q*, *q*´)-entropy planes are obtained. Wavelet (*q*, *q*´)-entropy planes, as demonstrated, are not only useful for explaining the complexities of *q*, *q*´)-entropy planes was presented for a variety of values of parameters *q* and *q*´. Finally, two possible application areas of the wavelet (*q*, *q*´)-entropy, specifically for classifying *q*, *q*´)-entropy may provide promising and robust techniques for signal classification and level-shift detection/location.