Wavelet Fisher’s Information Measure of 1 / f α Signals

Abstract

This article defines the concept of wavelet-based Fisher’s information measure (wavelet FIM) and develops a closed-form expression of this measure for 1 / f α signals. Wavelet Fisher’s information measure characterizes the complexities associated to 1 / f α signals and provides a powerful tool for their analysis. Theoretical and experimental studies demonstrate that this quantity is exponentially increasing for α > 1 (non-stationary signals) and almost constant for α < 1 (stationary signals). Potential applications of wavelet FIM are discussed in some detail and its power and robustness for the detection of structural breaks in the mean embedded in stationary fractional Gaussian noise signals studied.

Full text

Entropy 2011,13, 1648-1663; doi:10.3390/e13091648
OPEN ACCESS
entropy
ISSN 1099-4300
www.mdpi.com/journal/entropy
Article
Wavelet Fisher’s Information Measure of 1/fαSignals
Julio Ram´
ırez-Pacheco 1,2,?, Deni Torres-Rom´
an 1, Luis Rizo-Dominguez 2, Joel Trejo-Sanchez 2
and Francisco Manzano-Pinz´
on 2
1Department of Electrical Engineering, CINVESTAV-IPN Unidad Guadalajara, 45015 Av. Cient´
ıfica
1145, Col. El Baj´
ıo, Zapop´
an, Jalisco, M´
exico; E-Mail: dtorres@gdl.cinvestav.mx
2Department of Basic Sciences and Engineering, University of Caribe, 77528, SM-78, Mza-1, Lote-1,
Esquina Fracc. Tabachines, Canc´
un, Q.Roo, M´
exico; E-Mails: lrizo@ucaribe.edu.mx (L.R.-D.);
jtrejo@ucaribe.edu.mx (J.T.-S.); fmanzano@ucaribe.edu.mx (F.M.-P.)
?Author to whom correspondence should be addressed; E-Mail: cramirez@gdl.cinvestav.mx;
Tel.: +52-9988814400; Fax.: +52-9988814451.
Received: 6 July 2011; in revised form: 30 July 2011 / Accepted: 18 August 2011 /
Published: 6 September 2011
Abstract: This article defines the concept of wavelet-based Fisher’s information measure
(wavelet FIM) and develops a closed-form expression of this measure for 1/fαsignals.
Wavelet Fisher’s information measure characterizes the complexities associated to 1/f α
signals and provides a powerful tool for their analysis. Theoretical and experimental studies
demonstrate that this quantity is exponentially increasing for α > 1(non-stationary signals)
and almost constant for α < 1(stationary signals). Potential applications of wavelet FIM are
discussed in some detail and its power and robustness for the detection of structural breaks
in the mean embedded in stationary fractional Gaussian noise signals studied.
Keywords: 1/fαprocesses; structural breaks; Fisher information; fractal index estimation;
fractional Gaussian noise
1. Introduction
1/fαsignals, also known as scaling processes, model a large variety of phenomena occurring in
diverse fields of science and engineering [1–4], from voltage fluctuations in resistors, semiconductors and
vacuum tubes [5] to Internet traffic [6], laser propagation [7,8], turbulence, financial time series, among
others. In recent years, considerably effort has been undertaken to the robust estimation of the parameters

Entropy 2011,13 1649
which describe their properties [2,9,10], the most important, by far, the scaling index α. In this context,
an estimator is considered robust if it is capable of providing minimum-biased estimates of the scaling
index αfor 1/f signals subject to trends, non-stationarities, missing values, etc., and independent of
signal type. As a matter of fact, the presence of trends, non-stationarities, etc., impact significantly
the estimation process leading to biased estimates of α[10,11]. On the other hand, recently, the use
of information theory quantifiers is playing an increasing role in the analysis of 1/fαsignals [7,8,12].
Wavelet entropies and q-entropies characterize the complexities associated to 1/fαsignals and have
recently been used for detecting structural breaks in the mean in stationary and non-stationary signal
analysis frameworks [13]. Motivated by this, this work extends the concept of time-domain Fisher’s
information measure (FIM) to the wavelet or time-scale domain and investigates its use in 1/fαsignal
analysis. Wavelet Fisher’s information measure, tantamount to computing FIM in a wavelet spectrum
representation of a signal, is shown to describe accurately the complexities associated to 1/fαsignals
and to provide a robust tool for their analysis. Complexities in this article account for measuring the
irregularities present in a random signal, i.e., a deviation from pure randomness. Therefore, wavelet
FIM will be able to report different quantities for regular and irregular random signals. A closed form
expression of wavelet FIM is derived for scaling signals and based on this, its values for anti-correlated
and correlated stationary/non-stationary 1/fαsignals found. Potential applications of wavelet FIM are
highlighted and an example application for the detection of structural breaks in the mean embedded in
fractional Gaussian noise (fGn) signals studied. The remainder of this article is structured as follows.
Section 2briefly recalls the definition of 1/fαsignals, their properties and wavelet analysis. It also
describes how probability densities can be constructed from the wavelet spectrum representation of
signals. Section 3introduces the concept of Fisher’s information measure and defines wavelet FIM
based on the wavelet spectrum based probability mass function (pmf) of signals. The properties and
applications of wavelet FIM for scaling signals are also pointed out in this section. Section 4overviews
the level-shift detection process and provides description of the methodology for detecting structural
breaks in the mean in stationary 1/fαsignals with wavelet FIM. Section 5presents results of the
detection capabilities of wavelet FIM in synthesized fGn signals with level-shifts and finally, Section 6
concludes the paper.
2. 1/fαSignals: Definition and Wavelet Analysis
2.1. 1/fαProcesses
Processes with 1/fαspectral behaviour, also called power-law or scaling processes, are random
signals for which their power spectral density (PSD) behaves as a power-law, i.e., as
S(f)∼cf|f|−α(1)
in a range of frequencies f∈(fa, fb)[14,15], where cfis a constant and fa,fbare the lower and upper
frequencies upon which the power-law scaling holds. Parameter αdetermines among other properties
stationarity and/or long-range correlations in the signal. When α < 1, the process is stationary and
when α > 1, the process is regarded as non-stationary. Many phenomena in nature exhibit power-law
behaviour in certain features and thus can be efficiently described in terms of 1/fαsignals properties.

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Several stochastic processes have been proposed in the literature to model the observed 1/f behaviour. In
particular, fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) have been extensively
used to model phenomena with 1/f power spectra and long-range correlations [16]. A Gaussian,
zero-mean and self-similar with stationary increments process (Hsssi), BH(t), is said to be a fBm. The
covariance structure of fBm (and of all Hsssi processes) is given by
EBH(t)BH(s) = σ2
2|t|2H+|s|2H− |t−s|2H(2)
where 0< H < 1is the Hurst parameter. Fractional Gaussian noise, GH,δ(t), is obtained from a fBm
process by sampling it at time instants tk, k = 0,1,2, . . . and computing increments of the form
GH,δ(t) = 1
δ{BH(tk+δ)−BH(tk)}(3)
for a fixed integer δ∈Z+. Fractional Brownian motion is non-stationary while fGn is stationary.
Both processes are characterized by the Hurst index Hand can alternatively be expressed in terms of
the scaling index α. In this framework, fBm attains a value 1< α < 3and fGn of −1< α < 1.
Generalizations of fBm and fGn have been proposed in the literature under the names of extended
fractional Brownian motions and extended fractional Gaussian noises, see for example [9] and [17]
for further details on these signals. Extended fBm, obtained from a cumulative sum of a fBm signal, are
non-stationary signals for which 3< α < 5and extended fGn, derived from a first-order difference of
fGn, are stationary signals for which −3< α < −1. Signals with power-law spectra also possess
self-similar characteristics and based on this property, several estimation methodologies have been
proposed. Although there exists several methodologies in the literature to estimate α, none is robust
to trends, non-stationarities, missing values and independent of signal class. Methodologies which
combine several techniques to attempt to estimate αin these conditions are now preferred. For a complete
understanding of scaling signals, self-similarity, long-range dependency and their relationships, analysis
and estimation issues, the reader is referred to [1,4,14,15,18,19].
2.2. Wavelet Analysis of 1/fαSignals
The analysis of signals by wavelet transforms is playing a significant role in many branches of
science and engineering [20,21]. In the context of 1/fαsignals, wavelet transforms are currently
being used with great success for their analysis, estimation and synthesis [16,22–24]. In [23,24], a
powerful non-parametric estimator of the scaling index αwas proposed. The so-called Abry–Veitch
estimator provides unbiased estimates of αfor scaling signals with embedded polynomial-type noise.
The Abry–Veitch estimator, however, is subject to biases when estimating αfor 1/fαsignals [10,11]
resulting in estimates of H= (α+ 1)/2>1. Novel techniques which combine different methodologies
to enhance the estimation process are now preferred [2,18,25]. This article proposes to use information
theory quantifiers and wavelet-based methods to enhance the analysis and estimation process of 1/f
signals. Let Xtbe a random signal satisfying ER|X(u)|2du<∞, Equation (1) and let ψ(t)be an
orthonormal analyzing wavelet satisfying the admissibility condition. The discrete wavelet transform
(DWT) of Xtis given by the inner product:
dX(j, k) = Z+∞
−∞
Xtψj,k(t)dt (4)

Entropy 2011,13 1651
for some dyadically dilated (by a factor j∈Z) and integer translated (by a factor k∈Z) analyzing
wavelet ψj,k(t) = 2−j/2ψ(2−jt−k). DWT is related to the notion of multiresolution analysis (MRA) of
signals in which functions of various resolutions are constructed by projections of the original signal Xt
on the related spaces Vj. According to MRA, a signal Xtcan be represented by
Xt=
L
X
j=1
∞
X
k=−∞
dX(j, k)ψj,k (t)(5)
i.e., by a sum of different signals representing different levels of detail. In the context of 1/f αsignal
analysis, the wavelet spectrum, defined as the variance of wavelet coefficients, has shown to be very
useful. Wavelet spectrum permits to define estimators of index α, probability densities, and consequently,
information theory quantifiers. From its definition and Equation (1), it follows that the wavelet spectrum
of stationary 1/fαsignals is given by [16,23,24]:
Ed2
X(j, k) = Z∞
−∞
SX(2−jf)|Ψ(f)|2df (6)
where Ψ(.)is the Fourier integral of ψ(.),SX(.)the PSD of the process and f, the Fourier frequency.
For scaling signals, the wavelet spectrum takes the form [11]
Ed2
X(j, k)∼2jαC(ψ, α)(7)
where C(ψ, α) = cγR|f|−α|Ψ(f)|2df and cγis a constant.
2.3. Wavelet-Based Probability Densities
Wavelet spectrum, as noted above, allows among other properties to construct probability mass
functions of random signals and to detect the presence of non-stationarities embedded in stationary
data [11]. Wavelet-based pmfs provide the ability to define time-scale information theoretic quantifiers
which in turn allow to study the dynamics and complexities associated to random signals and systems.
The dynamics are studied by the use of sliding windows and the complexities by the values reported by
the Fisher information for regular and irregular signals. The pmf derived from the wavelet spectrum is
computed as:
pj=
1
NjPkEd2
X(j, k)
Plog2(N)
i=1 n1
NiPkEd2
X(i, k)o(8)
where Nj(respectively Ni) represents the number of wavelet coefficients at scale j(respectively i) and
Nis the length of the data. For 1/fαsignals, the wavelet-based pmf is determined by direct application
of Equation (7) into Equation (8) which results in
pj= 2(j−1)α1−2α
1−2αM (9)
where M=log2(N)and j= 1,2,...M represent the number of levels of detail. The density
pjrepresents the probability that the energy of the fractal signal is located at scale j. Several
information theoretic quantifiers have been defined using (9), examples include Shannon (called wavelet

Entropy 2011,13 1652
entropy) [8,26], Renyi and Tsallis q-entropies [12,13]. This article extends the Fisher’s information
concept to the wavelet domain using the density given in (9). Recall that Fisher information provides a
complementary characterization of a probability density. Unlike entropies that provide a global measure
of the spreading of a density, Fisher information provides a local characterization and is therefore more
sensitive to local re-arrangements of the pmf. This article shows that wavelet FIM characterizes (in the
equivalent way wavelet entropies do) the complexities associated to scaling signals. Wavelet FIM is
minimum for white noise while wavelet entropies are maximum. Accordingly, wavelet entropies are
exponentially decreasing for smooth fractal signals while wavelet FIMs are exponentially increasing.
3. Wavelet-Based Fisher’s Information Measure
3.1. Time-Domain Fisher’s Information Measure
Fisher’s information measure (FIM) has recently been applied in the analysis and processing of
complex signals [27–29]. In [27], FIM was applied to detect epileptic seizures in EEG signals recorded
in human and turtles, later the work of Martin [28], reported that FIM can be used to detect dynamical
changes in many non-linear models, such as the logistic map, Lorenz model, among others. The work of
Telesca [29] reported on the application of FIM for the analysis of geoelectrical signals. Recently, Fisher
information has been extensively applied in quantum mechanical systems for the study of single particle
systems [30] and also in the context of atomic and molecular systems [31]. Fisher’s information measure
has also been used in combination with Shannon entropy power to construct the so-called Fisher-Shannon
information plane/product (FSIP) [32]. The Fisher-Shannon information plane was recognized in that
work to be a plausible method for non-stationary signal analysis. In this work, the notion of Fishers’
information measure is extended to the wavelet domain and then a closed-form expression for this
quantifier is derived for the case of 1/f signals. Let Xtbe a signal with associated probability density
fX(x). Fisher’s information (in time-domain) of signal Xtis defined as
IX=Z∂
∂x fX(x)2dx
fX(x)(10)
Fisher’s information, IXis a non-negative quantity that yields large (possibly infinite) values for
smooth signals and small values for random disordered data. Accordingly, Fisher’s information is large
for narrow probability densities and small for wide (flat) ones [5]. Fisher information is also a measure
of the oscillatory degree of a waveform; highly-oscillatory functions have large Fisher information [30].
Fisher’s information has mostly been applied in the context of stationary signals using a discretized
version of Equation (10)
IX=
L
X
k=1 ((pk+1 −pk)2
pk)(11)
for some pmf {pk}L
k=0. Equation (11) can be computed in sliding windows resembling a real-time
computation. In this case, Fisher’s information is often called FIM.

Entropy 2011,13 1653
3.2. Wavelet-Based Fisher’s Information Measure
The wavelet spectrum-based probability density of Equation (9) allows to extend the Fisher’s
information quantifier defined in Equation (11) to the wavelet domain. The novel information theory
quantifier is named wavelet Fisher’s information (abbrev. wavelet FIM). Wavelet FIM, basically
equivalent to computing a FIM quantifier on a time-scale representation of the data, allows among other
applications to describe the complexities of 1/f signals. With the Fisher’s information computed in
this way, the information content of a signal is followed with optimal time-frequency resolution and
independently of stationarity assumptions. Wavelet FIM inherits all the properties associated to wavelets
and wavelet transforms. For instance, wavelet FIM is blind to polynomial trends of order Kembedded
in signals if the analyzing wavelet has Kvanishing moments. Wavelet FIM for 1/fαsignals is thus
given by
I1/f =(2α−1)21−2α(M−1)
1−2αM (12)
= 2α
2+2 sinh2(αln 2/2).PM
num (2 cosh (αln 2/2))
PM+1
den (2 cosh (αln 2/2))(13)
where PM
num(.)and PM+1
den (.)denote polynomials of argument 2 cosh(αln 2/2) that are given by
PM
num(.) = (2 cosh u)M−2(M−3)
2! (2 cosh u)M−2
+3(M−4)(M−5)
3! (2 cosh u)M−4−. . . (14)
PM+1
den (.) = (2 cosh u)M+1 −(M−2)
1! (2 cosh u)M−1
+(M−3)(M−4)
2! (2 cosh u)M−3−. . . (15)
where u=αln 2/2. The relation PM
num/P M+1
den (.)is zero for |α|>4and non-zero for |α|<4. Wavelet
FIM is thus a non-negative quantity nearly independent of length M=log2(N). Note that as Mlarge
and α1(the case of non-stationary signals), wavelet FIM behaves approximately as
I1/f ≈22α−1(16)
Therefore for large α, wavelet FIM increases exponentially fast with increasing α. In the limit of
α→ ∞, wavelet FIM I1/f → ∞ and as α < 0and Mlarge it is given by
I1/f ≈2α+3 {cosh(αln2) −1}(17)
which gives rise as α→ −∞ to I1/f →1. Wavelet FIM achieves a minimum for α= 0. The minimum
corresponds to white noise and is given by
I1/f = 0
Based on these results, the complexities associated to 1/fαsignals are theoretically derived. For
non-stationary signals (α > 1), wavelet Fisher’s information achieves large values and as αincreases

Entropy 2011,13 1654
I1/f increases exponentially (as 22α−1) fast. When α < 1(the case of stationary signals), wavelet FIM
increases either when scaling signals become more persistent or anti-persistent. Wavelet FIM achieves a
minimum value when α= 0,i.e., for completely random signals (white noise). For extended fractional
Gaussian noises, wavelet FIM presents minimum variation and for increments of order M≥1of
these processes, wavelet FIM is constant. Figure 1depicts the theoretical behaviour of wavelet Fisher’s
information for different scaling signals. The length axis in this figure should be understood in powers
of two. Observe from figure that wavelet FIM describes efficiently the complexities associated to 1/f
signals. Wavelet FIM, therefore, provides an alternative for characterizing the information content of
scaling signals.
Figure 1. Theoretical plane of wavelet Fisher’s information for 1/fαsignals. When α > 1
increases exponentially and when α < −1it converges to 1.
6
8
10
12
14
16 −4
−2
0
2
0
1
2
3
4
5
6
7
8
9
Scaling−Index
Time Series Length
Wavelet Fisher’s Information
3.3. Applications of Wavelet Fisher’s Information Measure
Because wavelet FIM describes properly the characteristics and complexities of fractal 1/fαsignals,
many applications can be identified using this complexity-based framework. As a matter of fact, based on
the fact that wavelet FIM achieves large values for non-stationary signals and small values for stationary
ones, a potential application area of wavelet FIM is in the classification of fractal signals as fractional
noises and motions. Classification of 1/f αsignals as motions or noises remains as an important,
attractive and unresolved problem in scaling signal analysis [18,25,33] since the nature of the signal
governs the selection of estimators, the shape of quantifiers such as qth order moments, the nature of
correlation functions, etc. [34]. Another important potential application of wavelet FIM, related to the

Entropy 2011,13 1655
classification of signals, is in the blind estimation of scaling parameters [35]. Blind estimation refers
to estimating α, independently of signal type (stationary or non-stationary). Wavelet FIM can also be
utilized for the detection of structural breaks in the mean embedded in 1/fαsignals. Structural breaks
in the mean affect significantly the estimation of scaling parameters leading to biased estimates of αand
consequently in misinterpretation of the phenomena. As a matter of fact in [11], it was demonstrated
that the well-known Abry–Veitch estimator overestimates the scaling index αin the presence of a single
level-shift leading to values of H= (α+ 1)/2>1, which in principle is not permissible in the theory.
In the following, the paper concentrates on the detection of structural breaks in the mean embedded in
synthesized stationary fGn signals by the use of wavelet Fisher’s information measure. The work studies
anti-correlated and correlated versions of fractional Gaussian noises and the power of wavelet FIM in
detecting simple structural breaks in the mean in these signals.
4. Detection of Structural Changes in the Mean
4.1. The Problem of Level-Shift Detection
Detection and location of structural breaks in the mean (level-shifts) has been an important research
problem in many areas of science [36,37]. In the Internet traffic analysis framework, detection, location
and mitigation of level-shifts significantly improves on the estimation process. As a matter of fact the
presence of a single level-shift embedded in a stationary fGn results in an estimated H > 1[11]. This is
turn, results in misinterpretation of the phenomena under study and inadequate construction of qth-order
moments. Let B(t), t ∈Rbe a 1/f signal with level-shifts at time instants {t1, t1+L,...tj, tj+L}.B(t)
can be represented as
B(t) = X(t) +
J
X
j=1
µj1[tj,tj+L](t)(18)
where X(t)is a signal satisfying Equation (1) and µj1[a,b](t)represents the indicator function of
amplitude µjin the interval [a, b]. The problem of level-shift detection reduces to identifying the
points {tj, tj+L}j∈Jwhere a change in behaviour occurs. Often, the change is visually perceptible,
but frequently this is not the case and alternative quantitative methods are preferred. In what follows,
description of the procedure for detecting level-shifts in 1/f signals by wavelet FIM is described.
4.2. Level-Shift Detection in 1/f Signals with Wavelet FIM
To detect the presence of level-shifts in fractal 1/f signals, wavelet Fisher’s information is computed
in sliding windows. A window of length w, located in the interval m∆≤tk< m∆ +wapplied to signal
{X(tk), k = 1,2,...N}is
X(m;w, ∆) = X(tk)Π t−m∆
w−1
2(19)
where m= 0,1,2,...mmax,∆is the sliding factor and Π(.)is the well-known rectangular function.
Note that Equation (19) represents a subset of X(tk)and thus by varying mfrom 0to mmax and
computing wavelet Fisher’s information on every window, the temporal evolution of wavelet FIM is

Entropy 2011,13 1656
followed. Suppose the wavelet FIM at time m(for sliding factor ∆) is denoted as IX(m), then a plot of
the points
{(w+m∆, IX(m))}mmax
m=0 := IX(20)
represents such time-evolution. In [11], it was demonstrated that the presence of a sudden jump in a
stationary fractal signal will cause the estimated ˆ
H > 1. The level-shift, thus, causes the signal under
observation become non-stationary. In the wavelet Fisher’s information framework, this sudden jump
will cause its value to increase suddenly. Therefore a sudden jump increase in the plot of Equation (20)
can be considered as an indicator of the occurrence of a single level-shift in the signal. These theoretical
findings are experimentally tested by the use of synthesized scaling signal with level-shifts. The
synthesized signals corresponds to fGn signals generated using the circular embedding algorithm [38,39]
(also known as the Davies and Harte algorithm).
5. Results and Discussion
The previous section stated the methodology for detecting structural breaks in the mean embedded
in fGn signals. A sudden increase (or peak) in the computed wavelet FIM is interpreted as a possible
level-shift in the signal. Figure 2displays the detection capabilities of wavelet FIM in an anticorrelated
fractional Gaussian noise signal with Hurst-index H= 0.1and length 215. Top left plot shows the signal
with a single level shift added to its structure at tb= 32768. For illustrative purposes, the level-shift is
also plotted in white. The amplitude of the level-shifts studied in this paper are one-half the standard
deviation of the signal √VarXt/2. The selection of this amplitude guarantees that the level-shift is not
perceptible by eye and as a matter of fact is difficult to visualize with a standard level-shift detection
tools [36,37]. Bottom left plot of Figure 2displays the wavelet FIM of the single level-shift fGn signal
computed in sliding windows of length w= 4096 and shifts of ∆ = 700. Note that the plot of
wavelet FIM of this signal present a sudden peak at tp∼35000. This sudden peak (of high amplitude)
corresponds to the level-shift added to the signal and thus can be interpreted as a result of a level-shift
present in the studied signal. The above suggests that wavelet FIM is capable of detecting this weak
level-shift with plausible results. Top right plot of Figure 2displays a fractional Gaussian noise signal
(H= 0.1) with more elaborate level-shifts embedded on its structure. The level-shifts were placed at
t1= 11384,t2= 21384,t3= 44152 and t4= 54152. The amplitude, again, is √VarXt/2. Wavelet FIM
was computed in this case in windows of length w= 4096 and sliding factor of ∆ = 625. Note that the
level-shifts are effectively detected with wavelet FIM method. Increasing the amplitude of the level-shifts
in the form µ > √VarXt/2increases the detection capabilities of wavelet Fisher’s information. Unlike
traditional level-shift detection methodologies, wavelet FIM allows to detect level-shifts in anticorrelated
and correlated fGn signals. The method proposed by Rea [36,37], known as atheoretical regression trees,
detects level-shifts in white noise signals. To our knowledge, this article presents for the first time wavelet
FIM and the application of wavelet FIM as a level-shift detection methodology.

Entropy 2011,13 1657
Figure 2. Wavelet Fisher’s information of a fractional Gaussian noise with H= 0.1and
embedded jumps. Top left plot displays the fGn signal with a single level-shift and top right
plot with a more elaborate combination of jumps. Bottom plots represent their corresponding
wavelet Fisher’s information.
time−index
Values
0 10000 30000 50000
−4 −2 0 2 4
time−index
Values
0 10000 30000 50000
−4 −2 0 2 4
time−index
Wavelet FIM
10000 30000 50000
0 100 200 300 400 500 600
time−index
Wavelet FIM
10000 30000 50000
0 1000 2000 3000 4000
Figure 3depicts the wavelet Fisher’s information measure for anticorrelated fGn signals with a
single level-shift at tb= 32768. Recall that anticorrelated fGn signals share the property that if a
signal displays a positive value it is likely to be followed by a negative value. Figure 3a displays the
wavelet Fisher’s information for a fGn signal with Hurst-index H= 0.2. Figure 3b the wavelet FIM
of a fGn with H= 0.3, Figure 3c for H= 0.4and finally Figure 3d for a signal with Hurst-index
H= 0.5. The last case (H= 0.5) corresponds to a completely disordered signal (white noise). The
analyses were performed with w= 4096 and ∆ = 700. Note from these results that wavelet FIM
allows to effectively detect the single level-shift embedded in the signal’s structure. Wavelet FIM, thus,
provides for anticorrelated signals, the ability to effectively detect single weak level-shifts. Figure 4
presents the detection capabilities of wavelet FIM in correlated stationary fGn signals with a single
jump located at tb= 32768. Recall that correlated fGn signals share the property that positive values
are likely to be followed by positive values and negative values by negative values, a property known
as persistence. Fractional Gaussian noise signals with H > 1/2are long-range dependent and thus
standard statistical methodologies no longer hold [1,9,17]. Within Figure 4, plot (a) represents the
wavelet FIM of a fGn with H= 0.6, plot (b) for H= 0.7, plot (c) for H= 0.8and finally plot
(d) for a fGn signal with Hurst-index H= 0.9. From these plots it can noted that wavelet FIM provides
appropriate detection of the jumps embedded in these correlated fGn signals. With these results, it is
noted that wavelet FIM detect appropriately a single jump embedded in the structure of anticorrelated and
correlated fractional Gaussian noise signals. For the case of correlated fractional Gaussian noises, small
peaks are also generated for H > 0.7, however, there exists a single high-amplitude peak which suggest
the presence of a level-shift. The small peaks are the consequence of computing wavelet FIM in sliding
windows and, in the results presented above, do not influence or affect the detection process. Detection
of a single level-shift may be of interest in many areas of science [36,37]. Since in many fields of science,

Entropy 2011,13 1658
the time series representing a particular phenomena may be subject to multiple level-shifts, it is also of
interest to study the detection capabilities of wavelet FIM in these frameworks. For this purpose, the
simulation of anticorrelated and correlated fGn signals with multiple level-shifts were performed. The
level-shifts are located at time-instants t1= 11384,t2= 22384,t3= 44152 and t4= 54152.
Figure 3. Wavelet FIM of anticorrelated fractional Gaussian noises with a single level-shift.
(a)fGn with H= 0.2; (b)fGn with H= 0.3; (c)fGn with H= 0.4; and (d)fGn with
H= 0.5.
time−index
Wavelet FIM
10000 30000 50000
0 100 200 300 400
(a)
time−index
Wavelet FIM
10000 30000 50000
0 100 300 500
(b)
time−index
Wavelet FIM
10000 30000 50000
0 50 150 250
(c)
time−index
Wavelet FIM
10000 30000 50000
0 500 1000 1500
(d)
Figure 4. Wavelet FIM of correlated stationary fractional Gaussian noises with a single
level-shift at tb= 32768. (a)fGn with H= 0.6; (b)fGn with H= 0.7; (c)fGn with
H= 0.8and the wavelet FIM of a fGn with H= 0.9is displayed in (d).
time−index
Wavelet FIM
10000 30000 50000
0 200 600 1000
(a)
time−index
Wavelet FIM
10000 30000 50000
0 20 60 100 140
(b)
time−index
Wavelet FIM
10000 30000 50000
0 200 400 600
(c)
time−index
Wavelet FIM
10000 30000 50000
0 10000 30000
(d)

Entropy 2011,13 1659
The selection of such level-shifts was motivated by experimental studies performed in [36,37].
Figure 5depicts the wavelet Fisher’s information of anticorrelated fGn signals with multiple mean
level-shifts. Figure 5a displays the information measure for a fGn signal with Hurst-index H= 0.2.
Note that in this case, the level-shifts are effectively detected as well as located in time, i.e., wavelet
FIM allows not only to detect the presence of level-shifts in anticorrelated signals but also to provide
a methodology for its location. Figure 5b,c presents the wavelet FIM of fGn signals (embedded with
multiple mean breaks) with H= 0.3and H= 0.4respectively. It is straightforward to note that
the wavelet FIM of these signals presents similar behaviour than the one observed in plot (a). The
location of the level-shifts are also effectively indicated by the use of wavelet FIM. Therefore, based on
these results the level-shifts embedded in anticorrelated fractional Gaussian noise signals (H < 0.5) are
effectively detected and located by the use of wavelet FIM. Figure 5depicts the dynamics of wavelet
FIM for a totally disordered signal, white Gaussian noise. White Gaussian noise’s (fGn with H= 0.5)
wavelet FIM presents four peaks which can be associated to the presence of the level-shift embedded to
their structure.
Figure 5. Wavelet FIM of anticorrelated stationary fractional Gaussian noises with multiple
mean level-shifts. (a)fGn with H= 0.2; (b)fGn with H= 0.3; (c)fGn with H= 0.4and
the wavelet FIM of a fGn with H= 0.5is displayed in (d).
time−index
Wavelet FIM
10000 30000 50000
0 200 600 1000
(a)
time−index
Wavelet FIM
10000 30000 50000
0 200 600 1000
(b)
time−index
Wavelet FIM
10000 30000 50000
0 200 600 1000
(c)
time−index
Wavelet FIM
10000 30000 50000
0 100 300 500
(d)
Note that the amplitudes are weaker than in the previous plots, however, they are sufficient to
detect the presence and location of a level-shift. Figure 6displays the wavelet Fisher’s information for
correlated fractal noises with multiple mean breaks. Figure 6a displays the wavelet Fisher’s information
for a fractional noise with H= 0.6. Note that it is straightforward to detect and locate the presence of the
level-shifts embedded in this signal. Note also that neighboring peaks are shown in the analysis, however
they play a minor role in affecting the detection process. Similar dynamics in the Fisher’s information
are obtained when analyzing a fractal noise with H= 0.7(Figure 6b). For a fractional Gaussian noise
with stronger persistence (H= 0.8), wavelet FIM detects the first two level-shifts added to the fGn
signal and then displays a peak which is not related to the presence of a level-shift in the signal. The

Entropy 2011,13 1660
third level-shift is effectively detected and the fourth is no longer detected. Note, however, that the
level-shifts studied in this article are weak (√VarXt/2) and thus can be deduced that the correlation
structure of a fGn signal has an impact on detecting level-shifts. Incrementing the amplitude of the
level-shifts increments the detection capabilities of wavelet FIM. For a fGn signal which is in the border
of non-stationary behaviour (H= 0.9), wavelet FIM (Figure 6d) fails to detect level-shifts. From the
above, it is concluded that wavelet FIM allows to detect single weak level-shifts embedded in fGn signals
and for the case of multiple mean breaks, wavelet FIM effectively detects weak level shifts provided that
H≤0.8. For fGn signals with stronger correlation (H > 0.8), wavelet FIM fails in detecting weak
level-shifts but effectively detects stronger level-shifts (µ=√VarXt>1).
Figure 6. Wavelet FIM of correlated stationary fractional Gaussian noises with multiple
mean level-shifts. (a)fGn with H= 0.6; (b)fGn with H= 0.7; (c)fGn with H= 0.8and
the wavelet FIM of a fGn with H= 0.9is displayed in (d).
time−index
Wavelet FIM
10000 30000 50000
0 50 150 250
(a)
time−index
Wavelet FIM
10000 30000 50000
0 50 150 250
(b)
time−index
Wavelet FIM
10000 30000 50000
0 50 100 150 200
(c)
time−index
Wavelet FIM
10000 30000 50000
0 100 300 500
(d)
6. Conclusions
In this article, wavelet Fisher’s information measure (wavelet FIM) was defined and its properties
for 1/fαprocesses derived. It was shown that wavelet Fisher’s information measure describes the
complexities associated to scaling signals and provides a powerful tool for their analysis. Based on
a derivation of a closed form expression of Wavelet Fisher’s information, it was demonstrated that it
is exponentially increasing for non-stationary signals and almost constant for differences of fractional
Gaussian noises. Based on this behaviour, several potential application were highlighted and particular
emphasis was put on detecting mean level-shifts embedded in anti-correlated as well as correlated
fractional noises. Experimental studies demonstrate that wavelet FIM allows to effectively detect single
weak level-shifts embedded in stationary fractal noises independently of the degree of persistence or
anti-persistence. For the multiple mean level-shift case, wavelet FIM performs well for anti-correlated
signals but for correlated ones (H > 0.8) it is affected by the strong correlation of the signal and by
the fact that the signal is approaching non-stationary behaviour. However, by increasing the amplitude

Entropy 2011,13 1661
of the level shift (µ > √VarXt/2), the detection power of wavelet FIM is increased. Wavelet FIM
therefore provides a novel, alternative and robust tool for the analysis and estimation of anti-correlated
and correlated fractal noises.
Acknowledgements
The present article was jointly funded by the National Council of Science and Technology
(CONACYT) under grant 47609, FOMIX-COQCYT grant No 126031 and University of Caribe
internal funds for the support of research groups (CAs). J. Ram´
ırez-Pacheco thanks the support from
CINVESTAV-IPN Unidad Guadalajara.
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