#  Assessment of background noise properties in time and time-frequency
domains in the context of vibration-based local damage detection in real
environment

Katarzyna Skowronek  Tomasz Barszcz  Jerome Antoni  Radosław Zimroz  Agnieszka
Wyłomańska  Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center,
Wroclaw University of Science and Technology, Wyspianskiego 27, 50-370
Wroclaw, Poland  Faculty of Mechanical Engineering and Robotics, AGH
University of Science and Technology, Al. Mickiewicza 30, 30-059 Krakow,
Poland  Univ Lyon, INSA Lyon, LVA, EA677, 69621 Villeurbanne, France  Faculty
of Geoengineering, Mining and Geology, Wroclaw University of Science and
Technology, Na Grobli 15, 50-421 Wroclaw, Poland

###### Abstract

Any measurement in condition monitoring applications is associated with
disturbing noise. Till now, most of the diagnostic procedures have assumed the
Gaussian distribution for the noise. This paper shares a novel perspective to
the problem of local damage detection. The acquired vector of observations is
considered as an additive mixture of signal of interest (SOI) and noise with
strongly non-Gaussian, heavy-tailed properties, that masks the SOI. The
distribution properties of the background noise influence the selection of
tools used for the signal analysis, particularly for local damage detection.
Thus, it is extremely important to recognize and identify possible non-
Gaussian behavior of the noise. The problem considered here is more general
than the classical goodness-of-fit testing. The paper highlights the important
role of variance, as most of the methods for signal analysis are based on the
assumption of the finite-variance distribution of the underlying signal. The
finite variance assumption is crucial but implicit to most indicators used in
condition monitoring, (such as the root-mean-square value, the power spectral
density, the kurtosis, the spectral correlation, etc.), in view that infinite
variance implies moments higher than $2$ are also infinite. The problem is
demonstrated based on three popular types of non-Gaussian distributions
observed for real vibration signals. We demonstrate how the properties of
noise distribution in the time domain may change by its transformations to the
time-frequency domain (spectrogram). Additionally, we propose a procedure to
check the presence of the infinite-variance of the background noise. Our
investigations are illustrated using simulation studies and real vibration
signals from various machines.

###### keywords:

vibration signal , non-Gaussian distribution , heavy-tailed distribution ,
identification , infinite variance

##  1 Introduction

In many technical systems, the measurement of any physical variable is
performed to acquire some important information about object or process. Let
us call it signal of interest (SOI). Although measurement systems are very
advanced, the informative components may be noisy because of the presence of
other stronger sources, non-informative in a given context. Thus, a natural
concept of signal analysis is related to pre-processing (de-noising) to
improve the signal-to-noise ratio (SNR) and to better detect the SOI. Signal
de-noising can be done directly in the time domain (TD), in the frequency
domain (FD) or time-frequency domain (TFD). It is intuitive, that before
removing the noise from the signal, one needs to identify the noise component
properties as many techniques are developed under the specific assumptions of
the corresponding distribution [1, 2, 3]. One of the most common assumptions
for de-noising techniques is the Gaussian distribution of the background noise
[3]. However, in everyday practice, various techniques for de-noising are used
without checking if this assumption is fulfilled. On the other side, this may
bring significant consequences because applying methods dedicated for specific
cases (i.e. assuming Gaussian distribution) may return inaccurate results for
signals not satisfying the assumed properties. Some researchers indicated this
problem and discuss the limitations of classical methods for signals not
satisfying the assumed properties, see [4, 5, 6, 7, 8].

The other issue that needs to be highlighted is related to the properties of
the signal distribution after its transformation to other domains, like TFD.
Also, some of the de-noising techniques act on the signals in other domains
(time domain, spectral, wavelet coefficients, bi-frequency map) [9, 1, 5, 10,
6], so such issue is important also in this context. Even if the signal in TD
has required properties, after the transformation to other domains they may
change drastically. The perfect example is the transformation of Gaussian
distributed signal to TFD (spectrogram), see [11]. After this transformation,
we obtain the signal with generalized $\chi^{2}$ distribution, which has
different properties from the Gaussian one. Thus, the de-noising techniques
dedicated to the signals with Gaussian properties may not give the expected
results when applying them to time-frequency representation. In the
literature, one may find some interesting research where this issue is
considered and the distribution properties change (after signal
transformations to other domains) are discussed [11, 12, 13, 14, 15].

Let us move one step forward and consider the problem of periodic/cyclic
behavior identification of the signal. Here, we assume that the signal is a
mixture of SOI and the background noise. This is a typical model used for
local damage detection in rotating elements such as bearings or gears. The
target is to detect the SOI hidden in background noise. In the case of finite-
variance distributed noise (e.g. Gaussian), the analysis of random signals can
be done by using classical auto-dependence measures. The most common example
is the auto-covariance (ACVF) or auto-correlation function (ACF) and the
classical approaches for periodic/cyclic behavior identification utilize such
functions. There are techniques where signals are analyzed in TD as well as
TFD [16, 17, 18], see also [19, 20] for new approaches.

However, when applying such measures, one needs to take into account they are
properly defined only for finite-variance distributed signals. Applying their
sample versions to the signals from infinite-variance distribution is
inappropriate, and the obtained results may not give expected information.
This problem was discussed in our previous research [21, 22, 23, 24] but also
other authors analyze this issue and propose dedicated techniques for
impulsive signals [25, 26, 27, 28, 29] and highlight the small efficiency of
the classical methods, see e.g. [30, 31, 32, 33]. There were also proposed
transformations that can help to make the non-Gaussian signals closer to
Gaussian, see for instance [34] where the authors showed that a simple
logarithmic transform on the squared envelope had an excellent stabilizing
effect before computing its Fourier transform. Preliminary knowledge about the
noise properties can help to avoid inappropriate conclusions resulting from
the use of wrong tools for signal analysis and may help to select more
adequate techniques, like robust estimators of auto-dependence measures
dedicated for impulsive signals [35, 36, 37, 38, 39, 40] or auto-dependence
measures defined for signals with some specific non-Gaussian distributions
[41, 42, 43, 44, 45]. We note, similar as for the de-noising techniques, some
authors test the classical methods for local damage detection also for signals
with impulsive behavior [46, 9, 6]. The analysis presented in the mentioned
above bibliography positions clearly indicates the limitations of the
classical techniques for extreme cases.

In view of the above discussion, we note that the preliminary analysis of the
background noise properties is extremely important for selection of
appropriate tools for signal analysis which, in turn, is necessary to obtain
reliable results. As in this paper we discuss the problem in the context of
periodic/cyclic behavior identification for signal-based local damage
detection, our attention is paid to the identification if the distribution of
the signal has finite or infinite variance. More precisely, we assess the
probabilistic properties (in the mean of finite or infinite variance) of the
background noise that affects the properties of the signal (being a mixture of
the SOI and the noise) itself. We note, the considered problem is much more
general than the classical goodness-of-fit testing [47, 48, 49], i.e. testing
if the underlying signal has a given theoretical distribution. It is worth
noting that the distribution identification is the last step of the analysis
and may have less importance than the preliminary knowledge of the
distribution category (here in the context of finite and infinite variance).

In the literature, one can find interesting approaches when some specific
properties of given data are tested [50, 51]. In our previous research we also
analyzed the problem of heavy-tailed behavior recognition [52, 53, 54] but it
was considered for specific classes of distributions. In this paper, we
present the broader perspective and discuss the problem in the context of any
non-Gaussian distributions with possible infinite variance.

In this paper, we recall three most popular non-Gaussian distributions with
possible infinite-variance (depending on the parameters). We explain the
selection of these distributions in the context of local damage detection in
rotating machines. We discuss the problem of finite- and infinite-variance
distribution of the signals in TD and TFD. More precisely, we highlight that
finite-variance property of the corresponding distribution in TD may not be
transferred to TFD (here spectrogram), that may have the significant
importance for further analysis. We propose a visual test for checking if the
variance is finite. The variance is a key parameter for classical auto-
dependence measures applications. As we are working with real vibration
signals with complex spectral content, all analyses are performed in TFD.
Thus, the assessment of the probabilistic properties of a random noise
component is done for some wider frequency range, not just an arbitrary
selected sub-signal for a given frequency band taken from the spectrogram. In
order to achieve this, we propose an objective, automatic procedure based on
the mentioned visual test. To illustrate the problem and results of our
investigations, we present several exemplary real vibration measurements from
different machines and demonstrate their probabilistic properties in TFD
(spectrogram). We also provide deep simulation study to highlight the
importance of the research topic presented in this paper.

The rest of the paper is organized as follows. In Section 2 we formulate the
considered problem indicating two perspectives, practical problem of local
damage detection and probabilistic point of view. Next, in Section 3 we recall
three considered non-Gaussian distributions considered here as the general
classes with possible finite and infinite variances. Then, we discuss the
distribution of the signals transferred to TFD (spectrogram). Here, we
indicate four separate categories of distributions that are crucial for
selection of the appropriate tools for signals analysis in TD and TFD. In
Section 3 we also propose an automatic procedure for the infinite variance
behavior analysis. In Section 4 we analyze the simulated signals from three
considered distributions and demonstrate their probabilistic properties in TD
and TFD. Moreover, for the simulated signals, we demonstrate the procedure for
infinite variance testing. In Section 5 we analyze four real signals from
different machines and demonstrate their differences in the context of the
probabilistic properties by using the proposed methodology. The last section
concludes the paper.

##  2 Problem formulation

Let us assume that acquired signal consists of two main components:
informative signal (SOI) and non-informative component (called simple noise).
If SNR is high, the presence of noise may be neglected. However, in real
applications, especially in local fault detection problems, the SOI may be
completely hidden in the noise. As the SOI (in our case) has two specific
properties (impulsiveness and periodicity), there are plenty of techniques
that allow its detection, even under strong domination of the noise. However,
in most of the cases the crucial assumption needs to be fulfilled, namely the
noise should be Gaussian distributed. Unfortunately, in various applications,
the noise exhibits non-Gaussian impulsive behavior. What does it exactly mean?
Each noise that is not Gaussian distributed simply may be considered as non-
Gaussian. However, in local damage detection, a very important properties of
the SOI is its impulsive character and many techniques are based on this
property. If we consider non-Gaussian heavy-tailed distributed noise, the
situation becomes very complicated. Heavy-tailed distribution means that in
the realization of the process some very large (positive and negative) values
(i.e. outliers) may appear. An example of such a heavy-tailed noise is an
impulsive process. According to our research, sources of such impulsive
behavior may be related to specific processes performed by machine (cutting,
crushing, milling, drilling, compression, etc) [55, 23, 34, 56, 57, 58], may
be related to external completely random disturbances, disturbances during
data transmission or even numerical problems during data processing [59]. In
other words, the problem of impulsive noise may appear in practice in many
situations.

However, the mixture of the SOI and impulsive (non-Gaussian infinite-variance)
noise excludes impulsive criteria that involve moments equal or greater than
$2$ (such as kurtosis, smoothness index, etc.) [60, 61, 6, 21, 62]) as SOI
detectors. It is worth highlighting that infinite variance of a given
distribution implies the higher moments are also infinite. Thus, for instance,
the infinite-variance distributed signal has also infinite kurtosis (which is
based on the second and third moment). The other possibility to detect SOI is
to measure the auto-dependence of the signal. However, as it was mentioned, in
some cases the classical auto-dependence measures (ACF, ACVF) cannot be used.
According to Wiener-Khinchine theorem, the infinite second moment also does
not allow using of the power spectral density (PSD), which is one of the most
versatile tools in condition monitoring. Thus, the important issue is to
identify the probabilistic properties of the noise and confirm that the
classical methods can be applied. But how to check the nature of the noise
component? In real applications, the spectral content of the acquired signal
may be complicated. Some sources may be deterministic, so the signal should be
decomposed first to obtain only the random part. Thus, we suggest not testing
properties of the signal in TD, but in TFD and to use spectrogram for signal
decomposition. It is already known that if signal in TD has a Gaussian
distribution, in TFD (spectrogram) it has a generalized $\chi^{2}$
distribution [11]. The situation is much complicated when the nose is non-
Gaussian heavy-tailed distributed. In that case, after transformation to TFD,
the distribution is unknown. Moreover, even if the level of non-Gaussianity in
TD is negligible (signal is almost Gaussian), after transformation to TFD,
properties of the signal can be very different. It may happen, that even if we
could apply the classical methods (like ACVF or ACF) in TD, we should not use
them for signal in time-frequency representation.

In order to deal with this problem, one needs appropriate tools to investigate
properties of the noise and constrains regarding usage of classical methods.
We propose to use a visual test to check if the variance of the corresponding
distribution is finite and, therefore, if classical methods can be used. As we
are working in TFD, we analyze so called sub-signals - narrowband signals for
each frequency bin. Decision-making based on arbitrary selected sub-signal may
be significantly biased. Thus, we adopted the visual test for finite variance
and proposed an automatic procedure applied to wide frequency band. As test
based on empirical cumulative fourth moment results in the plot of statistics
that should converge to constant value for finite-variance distributed
signals. The analysis of simulated signals from non-Gaussian distributions
confirms the efficiency of the proposed algorithm for the considered issue.

##  3 Theoretical background

As it was mentioned, the background noise of the real vibration signals very
often exhibits non-Gaussian heavy-tailed behavior. There are plenty of
distributions belonging to this class, however in this paper we consider three
general exemplary cases.

###  3.1 Non-Gaussian distributions of random signals

The first considered distribution is the $\alpha$-stable one
$\mathcal{S}(\alpha,\sigma)$. This distribution has been successfully used in
condition monitoring applications by various authors, see e.g. [58, 63, 64,
65, 66, 67, 68]. In this paper, we consider the symmetric version of the
$\alpha-$ stable distribution defined through the characteristic function [69,
70]

$\displaystyle\Phi_{X}(x)=\mathbf{E}\exp\left\\{iXx\right\\}=\exp{\left(-\sigma^{\alpha}|x|^{\alpha}\right)},\quad\quad
x\in\mathbb{R}$ (1)

where $0<\alpha\leq 2$ is the stability index and $\sigma>0$ is the scale
parameter. For $\alpha=2$ the $\alpha-$stable distribution reduces to the
Gaussian one, and thus it can be considered as a generalization of this
classical distribution. The symmetric $\alpha-$stable distribution has no
closed-form probability density function (PDF) and cumulative distribution
function (CDF). The only exception is the Gaussian distribution (that is, for
$\alpha=2$) and the Cauchy distribution (that is, for $\alpha=1$). The
stability index is responsible for the heaviness of this distribution’s tail,
$1-F_{X}(x)\sim x^{-\alpha}$ ($F_{X}(\cdot)$ is a CDF of random variable $X$),
i.e. the smaller $\alpha$ the probability of large values is much higher. For
$\alpha<2$, the variance of $\alpha-$stable distribution is infinite.

As the second example of non-Gaussian heavy-tailed distribution, we consider
the symmetric Pareto one. We selected this distribution because, similar as
$\alpha-$stable one, it can have finite- and infinite-variance and exhibits
the power-law behavior. On the other side, in contrast to $\alpha-$stable
distribution, its PDF and CDF are given in explicit forms which makes
statistical inference simpler in this case. The PDF of the symmetric Pareto
distribution $\mathcal{P}(\gamma,\lambda)$ is given by [71]

$f_{X}(x)=\frac{\gamma\lambda^{\gamma}}{2(|x|+\lambda)^{\gamma+1}},\quad\quad
x\in\mathbb{R}.$ (2)

Similarly, as for the stability index in the $\alpha-$stable case, the
$\gamma>0$ parameter is responsible for the heaviness of this distribution’s
tail. The $\lambda>0$ parameter is responsible for the scale of a random
variable $X$. When $\gamma>2$, the double Pareto distribution has finite
variance. In that case $\gamma\leq 2$, the variance is infinite.

The last considered distribution is the t location-scale
$\mathcal{T}(\nu,\delta)$ defined through the PDF [72, 73]

$f_{X}(x)=\frac{\Gamma\left(\frac{\nu+1}{2}\right)}{\delta\sqrt{\nu\pi}\Gamma\left(\frac{\nu}{2}\right)}\left[\frac{\nu+\frac{x^{2}}{\delta^{2}}}{\nu}\right]^{-\left(\frac{\nu+1}{2}\right)},\quad\quad
x\in\mathbb{R},$ (3)

where $\Gamma(\cdot)$ is the gamma function, $\nu>0$ is a shape parameter and
$\delta>0$ is a scale parameter. The variance for t-location scale
distribution is only defined for $\nu>2$. Otherwise, it is infinite. When
$\nu$ tends to infinity, then t-location scale distribution tends to a
Gaussian distribution.

To demonstrate differences between the analyzed distributions, in Fig. 1 we
present their probability density functions (for positive arguments) and the
corresponding distributions’ tails (in log-log scales). The green lines
correspond to the $\alpha-$stable distribution, the red lines to the symmetric
Pareto distribution, while the blue lines to the t location-scale
distribution. In each case, the solid lines correspond to the finite-variance
cases (i.e., for $\alpha=2$, $\gamma=\nu=6$) while the dashed lines correspond
to the infinite-variance cases (i.e., for $\alpha=\gamma=\nu=1.5$). The
intermediate cases, i.e. $\alpha=1.9$, $\gamma=\nu=3$ are marked in dotted
lines. In the right panel one can see that the tails of non-Gaussian
distributions are "heavier" than in the Gaussian case (i.e. for
$\alpha-$stable with $\alpha=2$) where the large observations can occur with
smaller probability than for other cases. Moreover, for the finite-variance
cases (solid lines) the distributions’ tails are "lighter’ than for infinite-
variances (dashed lines and dotted green line).

Figure 1: The PDFs (for positive arguments) and the corresponding
distributions’ tails (in log-log scales) for $\alpha-$stable (green lines),
symmetric Pareto (red lines) and t location-scale (blue lines) distributions.
The solid lines correspond to the finite-variance cases (i.e. for $\alpha=2$,
$\gamma=\nu=6$). The dashed lines correspond to the infinite-variance cases
(i.e. for $\alpha=\gamma=\nu=1.5$). The dotted lines correspond to the
intermediate cases (i.e. $\alpha=1.9$, $\gamma=\nu=3$).

###  3.2 Distribution of non-Gaussian signals in time-frequency representation

Since most methods for local damage detection operate in time-frequency
representation, in this part we discuss the distribution of the signals
transformed into time-frequency map. It is recalled that the signal is a
mixture of the SOI – assumed nonstationary when analyzed in the time-frequency
domain – and of background noise (that is reasonably assumed stationary), and
the latter only is the subject of analysis of the present paper. . As for the
time-frequency representation, we propose to use the spectrogram (the square
of the absolute value of short time Fourier transform, STFT) in view of that
many methods in condition monitoring are rooted on this representation, e.g.
the spectrogram and its reassigned versions such as the synchrosqueezing
transform, the Welch’s estimator of the power spectral density and of the
spectral correlation, etc.

We recall for the signal $x_{1},x_{2},\cdots,x_{n}$ the STFT is defined as
follows

$STFT(t,f)=ST_{R}(t,f)+iST_{I}(t,f)=\sum_{m=1}^{n}x_{m}w(t-m)e^{-i2\pi
f\frac{m}{n}},$ (4)

where $ST_{R}(t,f)$ and $ST_{I}(t,f)$ are real and imaginary parts of STFT
respectively, $w(\cdot)$ is window, $t\in T$ is time point and
$f\in\mathcal{F}$ is frequency. Real and imaginary parts of STFT can be
expressed as

$ST_{R}(t,f)=\sum_{m=1}^{n}x_{m}w(t-m)\text{cos}(-2\pi
f\frac{m}{n}),~{}~{}ST_{I}(t,f)=\sum_{m=1}^{n}x_{m}w(t-m)\text{sin}(-2\pi
f\frac{m}{n}),$ (5)

where $ST_{R}(t,f)$ and $ST_{I}(t,f)$ are real and imaginary parts of STFT,
respectively. The spectrogram $S(\cdot,\cdot)$ is a square of the absolute
value of STFT

$S(t,f)=|STFT(t,f)|^{2}=ST_{R}(t,f)^{2}+ST_{I}(t,f)^{2}.$ (6)

As it was highlighted in [11] that if the signal $x_{1},x_{2},\cdots,x_{n}$
represents samples of independent observations from the zero-mean Gaussian
distribution, then for any $f\in\mathcal{F}$ the vectors $ST_{R}(\cdot,f)$ and
$ST_{I}(\cdot,f)$ comes also from a Gaussian distribution with mean equal to
zero. The exact formulas for the variances of the Gaussian distributions
corresponding to the samples given in (5) and their theoretical covariance are
presented in [11].

According to [11], under the assumption the signal $x_{1},x_{2},\cdots,x_{n}$
of independent observations comes from the centered Gaussian distribution, the
$S(\cdot,f)$ for any $f\in\mathcal{F}$ is a sample from the so-called
generalized $\chi^{2}$ distribution defined thought the PDF, [74]

$\displaystyle
p_{\theta,\beta}(x)=\frac{1}{(2\beta)^{\frac{\theta}{2}}\Gamma\left(\frac{\theta}{2}\right)}x^{\frac{\theta-2}{2}}\text{exp}\left(-\frac{x}{2\beta}\right),~{}x>0,$
(7)

where parameter $\theta\in\mathbb{N}$ is called a number of degrees of
freedom, $\beta>0$ is the scale parameter and $\Gamma(\cdot)$ is a gamma
function. The generalized $\chi^{2}$ distribution has also strong connection
with gamma distribution, see e.g., [75, 74]. One can show that under the
assumption of Gaussian distributed signal, the spectrogram given in Eq. (6),
is a quadratic form of Gaussian variables. Based on the theory of Gaussian
quadratic forms [74], in [11] the authors calculated the parameters of
generalized $\chi^{2}$ distribution of $S(\cdot,f)$ for any $f\in\mathcal{F}$.
The parameters $\theta$ and $\beta$ are expressed in means of variances of
$ST_{R}(\cdot,f)$ and $ST_{I}(\cdot,f)$ and their covariance. If we denote the
corresponding theoretical variances as $\sigma_{R}^{2}$ and $\sigma_{I}^{2}$,
respectively and covariance as $\sigma_{ST}$, then the theoretical values of
the parameters of $\chi^{2}$ distribution corresponding to $S(\cdot,f)$ (for
any $f\in\mathcal{F}$) are given by

$\displaystyle\theta=\frac{(\sigma_{R}+\sigma_{I})^{2}}{\sigma_{R}^{2}+\sigma_{I}^{2}+2\sigma_{ST}^{2}},~{}\beta=\frac{\sigma_{R}+\sigma_{I}}{2}.$
(8)

The situation is much more complicated if the signal
$x_{1},x_{2},\cdots,x_{n}$ is not Gaussian distributed. In the case of
symmetric double Pareto or t-location scale distribution, there is no
analytical formula describing the distribution of the noise in the real and
imaginary part of the STFT or the spectrogram. We only highlight that
depending on the values of parameters responsible for the heavy-tailed
behavior (i.e. $\gamma$ and $\nu$ for symmetric Pareto and t location-scale
distributions, respectively), we can obtain finite- or infinite-variance
distributed samples in time-frequency representations of the signal
(spectrogram).

For the $\alpha$-stable distributed signals, $ST_{R}(\cdot,f)$ and
$ST_{I}(\cdot,f)$ are also $\alpha$-stable distributed with the same stability
index $\alpha$ as the signal $x_{1},x_{2},\cdots,x_{n}$. This follows from the
probabilistic properties of the $\alpha-$stable random variables [76] and the
generalized central limit theorem [77]. Using the results presented in [78,
79], where the distribution of the squared Fourier transform (periodogram) for
$\alpha$-stable linear processes is discussed, we may conclude the
distribution of the series $S(\cdot,f)$ for any $f\in\mathcal{F}$ belongs to
the domain of attraction of $\alpha/2-$stable distribution if the random
signal $x_{1},x_{2},\cdots,x_{n}$ is $\alpha-$stable distributed (with
stability index $\alpha$).

In this paper, we highlight the important role of variance of corresponding
theoretical distribution for selection of appropriate tools for signal
analysis. The information about variance existence (i.e. if the variance is
finite of infinite for corresponding distribution) is crucial for further
steps. Thus, in Fig. 2 we demonstrate possible types of distributions (in the
categories of finite and infinite-variance case) of signals in TD and TFD. We
highlight that the finiteness of the variance of the signal distribution in TD
does not guarantee its finiteness after its transformation to TFD
(spectrogram). We remind, in the class denoted as (4) there is included the
$\alpha-$stable distribution with $\alpha<2$. In the schema, we do not
highlight this distribution as a separate category. In Table 1 we present the
considered distributions and ranges of their parameters (responsible for non-
Gaussian behavior, i.e. $\alpha,\gamma$ and $\nu$ for $\alpha-$stable,
symmetric Pareto and t location-scale distributions, respectively)
corresponding to cases (1)-(4) of the schema presented in Fig. 2. We note, the
parameters $\sigma,\lambda$ and $\delta$ for $\alpha-$stable, symmetric Pareto
and t location-scale distributions, respectively, do not have influence on the
non-Gaussian behavior. Thus, we do not include them in the table. Table 1 may
be useful for real signal analysis when identification of the distribution
type for the background noise may be important for the selection of
appropriate tools for local damage detection.

Figure 2: Possible types of distributions (in the categories of finite and infinite-variance distribution) of signals in TD and TFD. Distribution | Parameters | Variance | Distribution | Variance | Category
---|---|---|---|---|---
TD | Parameters | TD | TFD | TFD | (see Fig. 2)
$\mathcal{S}(\alpha,\sigma)$ | $\alpha=2$ | finite | Generalized $\chi^{2}$ | finite | (1)
$\mathcal{P}(\gamma,\lambda)$, $\mathcal{T}(\nu,\delta)$ | $\gamma>4,\nu>4$ | finite | unknown | finite | (2)
$\mathcal{P}(\gamma,\lambda)$, $\mathcal{T}(\nu,\delta)$ | $2<\gamma\leq 4,2<\nu\leq 4$ | finite | unknown | infinite | (3)
$\mathcal{P}(\gamma,\lambda)$, $\mathcal{T}(\nu,\delta)$ | $0<\gamma\leq 2,0<\nu\leq 2$ | infinite | unknown | infinite | (4)
$\mathcal{S}(\alpha,\sigma)$ | $0<\alpha<2$ | infinite | domain of attraction | infinite | (4)
|  |  | of $\alpha/2-$stable |  |
Table 1: The considered distributions and ranges of their parameters
corresponding to the categories (1)-(4) presented in Fig. 2 of the signals’
distribution in TD and TFD.

###  3.3 Assessment of probabilistic properties of random signals in time and
time-frequency domains

As it was noted, the problem considered here is much more general than the
classical goodness-of-fit testing if real signal can be modeled by a given
theoretical distribution. In this paper, we test if the signal belongs to the
finite- or infinite-variance class of distributions without specification of
the distribution, as in many cases this identification may be very difficult
or even impossible. To the assessment of the infinite-variance behavior we
propose to use the simple statistic, called empirical cumulative fourth moment
(ECFM) that was analyzed in our previous research in the similar context. The
selection of this statistic for the discussed problem is related to the fact
that ECFM has a simple form, and it exhibits completely different behavior for
finite- and infinite-variance distributed signals that is a crucial point in
the testing procedure.

In our previous research [52, 53] we have discussed the problem of
discrimination between Gaussian and near-Gaussian distributions for which the
variance may be infinite. The perfect example was the $\alpha-$stable
distribution with the stability index close to $2$. However, this methodology
may also be applied to other distributions considered in this paper, and it
can be extended for the assessment of the probabilistic properties of the
signal also in TFD. In [52] to distinguish the Gaussian and infinite-variance
distribution of given data, the authors proposed to use the ECFM statistic

$\displaystyle
C(k)=\frac{1}{k}\sum_{i=1}^{k}(x_{i}-\overline{x})^{4},\quad\quad
k=1,2,\cdots,n,$ (9)

where $x_{1},x_{2},\cdots,x_{n}$ is the considered signal of independent
identically distributed (i.i.d.) observations and $\overline{x}$ the
corresponding sample mean.

In [52, 53] it was highlighted that the statistic given in Eq. (9) converges
to a constant for the Gaussian distribution (or any other distribution with
finite fourth moment). In practice, given a finite sample, one observes that
ECFM exhibits irregular chaotic behaviour only for distributions with infinite
fourth moment. In this paper, the ECFM statistic is applied to confirm or
reject the finite-variance distribution of the signal in TFD.

In the following part of this section, we show how to parameterize the chaotic
behavior of the ECFM statistic for infinite-variance distributed signals. As
it was mentioned, the methods for local damage detection are mostly based on
the analysis of the signals in TFD. Thus, in this paper, the procedure
presented below is applied to the time-frequency representation (spectrogram)
of a given signal. Precisely, we measure the chaotic behavior of ECFM
statistic for vectors $S(\cdot,f)$ for all $f\in\mathcal{F}$ and analyse the
distribution of this measure along the frequencies. However, this algorithm
can be also applied for identification of infinite-variance behavior for
signals in any other domains.

The procedure consists of the following steps:

  1. 1.

The signal $x_{1},x_{2},\cdots,x_{n}$ first is transformed to TFD. In this
paper, we use the spectrogram defined in Eq. (6).

  2. 2.

For each $f\in\mathcal{F}$ we normalize the vector $S(\cdot,f)$ by subtracting
its sample mean and by dividing by the sample conditional standard deviation
on the quantile levels $0.1$ and $0.9$ (see [80] for more details). The
normalisation was performed in order to standardise the output and reduce the
influence of the mean and scale parameters. Note that we followed empirical-
based standardisation due to the heavy-tail nature of the data linked e.g. to
infinite variance.

  3. 3.

For each vector $S(\cdot,f)$ we calculate the ECFM statistic according to Eq.
(9).

  4. 4.

For each $f\in\mathcal{F}$ we identify the segments of the ECFM statistic
between the jumps. In order to identify the segments, first we calculate the
increments of the ECFM statistic and then identify their peaks, considering
them as the points separating the segments.

  5. 5.

For each $f\in\mathcal{F}$ we select the segments of the ECFM statistic that
are long enough. In our analysis, we selected the segments of minimum $10\%$
of the length of $S(\cdot,f)$. For further analysis, we take the last long
segment. The corresponding vector we denote as $D(\cdot,f)$.

  6. 6.

For each $f\in\mathcal{F}$ we fit the straight line to the vector $D(\cdot,f)$
using the least squares method. The estimated value of the slope parameter for
frequency $f$ we denote as $a_{f}$.

  7. 7.

We analyze the distribution of the estimated slopes along the frequencies. If
the signal in TFD has finite-variance distribution, we expect the ECFM
statistic calculated for vectors $S(\cdot,f)$ stabilizes. Thus, in this case,
the $a_{f}$ parameters are close to zero. More precisely, the distribution of
$a_{f}$ is concentrated around zero. We expect here that the median of the
slopes is close to zero and the interquartile range, IQR, is small. On the
other side, if the distribution of the signal in TFD has infinite variance,
then the ECFM statistic for each sub-signal $S(\cdot,f)$ exhibits chaotic
behavior and estimated $a_{f}$ parameters are non-zero. We expect here the
median of the estimated slopes (in absolute values) is significantly higher
than zero and their IQR is high.

The pseudo code of the described above procedure in presented in A, see
Algorithm 1.

After the verification if the corresponding theoretical distribution is
finite- or infinite- variance, the next step is the identification of the
theoretical distribution corresponding to the signal. This point may be
crucial (especially for testing the Gaussian or generalized $\chi^{2}$
distributions) as some of the local damage detection methods are dedicated
only for special distributions of given signal. To fit the proper distribution
(or to select which one is the best choice from the tested ones), we propose
to use the simple visual test based on the comparison of the empirical tail
and the theoretical one corresponding to the tested distribution (with the
estimated parameters from considered signal). The empirical tail is defined
similarly as the theoretical one, however the theoretical CDF is replaced by
the empirical one, i.e. $1-\hat{F}_{n}(x)$. We recall, the empirical CDF for
i.i.d. signal $x_{1},x_{2},\cdots,x_{n}$ is defined as follows [81]

$\displaystyle\hat{F}_{n}(x)=\frac{1}{n}\sum_{j=1}^{n}1\\{x_{j}\leq x\\},$
(10)

where $1\\{A\\}$ is the indicator of a set $A$.

By comparing theoretical and empirical tails, one can conclude which
distribution from tested ones is more proper for the considered signal. To
confirm that the tested distribution is the best choice, we propose to use the
Kolmogorov-Smirnov (KS) statistic defined as

$\displaystyle KS=\sup_{x}\left|{F}_{X}(x)-\hat{F}_{n}(x)\right|,$ (11)

where $F_{X}(\cdot)$ is the CDF of the tested theoretical distribution with
the parameters estimated from the signal and $\hat{F}_{n}(\cdot)$ is the
empirical CDF given in Eq. (10). Small value of KS statistic indicates that
the empirical distribution is close to the tested one. The KS statistic can
also be used for the given distribution testing. Based on Monte Carlo
simulations, we can calculate the corresponding $p_{value}$.

##  4 Simulated signals analysis

In this section we analyze the simulated signals, and we present how to assess
the finite and infinite-variance distribution for the i.i.d. observations in
time and time-frequency domains. The methodology is demonstrated for three
distributions described in Section 3.

In Fig. 4 we demonstrate the exemplary simulated signals from
$\mathcal{S}(\alpha,\sigma)$, $\mathcal{P}(\gamma,\lambda)$ and
$\mathcal{T}(\nu,\delta)$. For each of the considered distribution we consider
three cases of parameters responsible for heavy-tailed behavior, namely for
$\alpha-$stable case we take $\alpha=2$ (Gaussian distribution), $\alpha=1.9$
and $\alpha=1.5$; for symmetric Pareto distribution we take $\gamma=6$,
$\gamma=3$ and $\gamma=1.5$; for t location-scale we take $\nu=6$, $\nu=3$ and
$\nu=1.5$. The other parameters are assumed to be one. Let us emphasize that
$\alpha=2$ and $\gamma=\nu=6$ in $\alpha-$stable, symmetric Pareto and t
location-scale distributions, respectively, correspond to the finite-variance
cases while for $\alpha=\gamma=\nu=1.5$ we have infinite-variance
distributions. In the middle row of Fig. 4 we present the intermediate cases.
More precisely, for $\gamma=\nu=3$ the symmetric Pareto and t location-scale
distributions have finite variances while $\alpha=1.9$ for $\alpha-$stable
distribution corresponds to the infinite-variance case. These cases will be
discussed in depth.

In Fig. 4 one can see the significant differences between finite- and
infinite-variance cases. For the signals presented in the bottom row and
middle-left panel, the large observations are clearly visible.

In Fig. 4 we demonstrate the ECFM statistic calculated for the samples
demonstrated in Fig. 4. One can see the clear differences between top and
bottom rows. For the finite-variance cases (top row), the ECFM statistics tend
to some constants. In the bottom row, the ECFM statistics exhibit chaotic
behavior which clearly confirms the infinite-variance distributions. The
middle row—left column indicates that the considered sample is infinite-
variance distributed, while the middle and right columns demonstrate the
finite-variance behavior of the sample. However, it is not as clear as in the
top row for the corresponding distributions.

Figure 3: Examples of simulated signals from $\mathcal{S}(\alpha,\sigma)$,
$\mathcal{P}(\gamma,\lambda)$ and $\mathcal{T}(\nu,\delta)$ distributions. The
green color (left columns) corresponds to the $\alpha-$stable distribution
with $\alpha=2$ (top panel), $\alpha=1.9$ (middle panel) and with $\alpha=1.5$
(bottom panel). The red color (middle column) corresponds to the symmetric
Pareto distribution with $\gamma=6$ (top panel), $\nu=3$ (middle panel) and
$\gamma=1.5$ (bottom panel). The blue color (right columns) corresponds to the
t location-scale distribution with $\nu=6$ (top panel), $\nu=3$ (middle panel)
and $\nu=1.5$ (bottom panel). The other parameters are assumed to be one.

Figure 4: The ECFM statistic calculated for the signals demonstrated in Fig.
4. The finite-variance cases are presented in the top row, while the infinite-
variance cases are demonstrated in the bottom row. In the middle row, we
demonstrate the intermediate cases.

The spectrograms of the signals presented in Fig. 4 are presented in Fig. 5.
To demonstrate the ECFM-based methodology, for further analysis, we select a
specific frequency $f=1.0963kHz$ and examine the corresponding sub-signals
from the spectrogram. Because the signals in TD are represented by independent
observations and do not contain any additional components, in this case the
selection of $f$ is not a critical issue. However, when we analyze the real
signals, the selection of the appropriate frequency bin corresponding to the
noise component is a crucial point. In Fig. 7 we demonstrate the sub-signals
from the spectrograms presented in Fig. 5 corresponding to the selected
frequency $f$. In the middle and bottom rows, one can notice the occurrence of
large values in the corresponding sub-signals. The impulsive behavior is not
visible in the top row.

Figure 5: The spectrograms (in log scales) for simulated signals from
$\mathcal{S}(\alpha,\sigma)$, $\mathcal{P}(\gamma,\lambda)$ and
$\mathcal{T}(\nu,\delta)$ distributions presented in Fig. 4. The left columns
correspond to the $\alpha-$stable distribution with $\alpha=2$ (top panel),
$\alpha=1.9$ (middle panel) and with $\alpha=1.5$ (bottom panel). The middle
column corresponds to the symmetric Pareto distribution with $\gamma=6$ (top
panel), $\nu=3$ (middle panel) and $\gamma=1.5$ (bottom panel). The right
columns correspond to the t location-scale distribution with $\nu=6$ (top
panel), $\nu=3$ (middle panel) and $\nu=1.5$ (bottom panel). The other
parameters are assumed to be one.

Figure 6: The sub-signals (corresponding to $f=1.0963kHz$) from spectrograms
presented in Fig. 5 for exemplary simulated signals from
$\mathcal{S}(\alpha,\sigma)$, $\mathcal{P}(\gamma,\lambda)$ and
$\mathcal{T}(\nu,\delta)$ distributions. The green color (left columns)
corresponds to the $\alpha-$stable distribution with $\alpha=2$ (top panel),
$\alpha=1.9$ (middle panel) and with $\alpha=1.5$ (bottom panel). The red
color (middle column) corresponds to the symmetric Pareto distribution with
$\gamma=6$ (top panel), $\nu=3$ (middle panel) and $\gamma=1.5$ (bottom
panel). The blue color (right columns) corresponds to the t location-scale
distribution with $\nu=6$ (top panel), $\nu=3$ (middle panel) and $\nu=1.5$
(bottom panel). The other parameters are assumed to be one.

Figure 7: The ECFM statistic calculated for the sub-signals demonstrated in
Fig. 7. The finite-variance cases (in TFD) are presented in the top row, while
the infinite-variance cases (in TFD) are demonstrated in the middle and bottom
row.

The ECFM statistics for the sub-signals presented in Fig. 7 are demonstrated
in Fig. 7. One can see in the top row that the ECFM stabilizes, which clearly
indicates the finite-variance cases. In the middle row for all considered
cases, the ECFM exhibits chaotic behavior, which indicates the infinite-
variance distributions. It should be noted that the middle row corresponds to
the $\alpha-$ stable distribution with $\alpha=1.9$ (left panel), the
symmetric Pareto distribution with $\gamma=3$ (middle panel), and the
location-scale distribution t with $\gamma=3$ (right panel). For the symmetric
Pareto and the t location-scale distributions, the signals in TD were
classified as finite-variance distributed; see the middle row (middle and
right panels) in Fig. 4. This is the case, when the characteristics of the
signal in TD are not transferred to TFD. In the bottom panels of Fig. 7 we
present the ECFM statistic for the signals classified in TD as infinite-
variance distributed. The similar property we observe for the signals in TFD.

In the next part, we present the application of the described in the previous
section procedure for testing of finite- and infinite variance for simulated
signals from three considered distributions. According to the algorithm, first
the signals are transformed to TFD (spectrograms). Then, the ECFM is
calculated for each sub-signal (see Fig. 7 for exemplary sub-signals). In the
next step for each sub-signal (after normalisation) the increments of ECFM
statistic are calculated (see examples in Fig. 9) to identify the segments of
the statistic between jums (see point 4. of the procedure). Finally, for each
sub-signal the last long segment of ECFM is selected, and the straight line is
fitted (see point 6 of the procedure). The selected segments of ECFM statistic
and the fitted lines (marked by black lines) with estimated slopes $a_{f}$ are
presented in Fig. 9 for sub-signals demonstrated in Fig. 7. For sub-signals
corresponding to the finite-variance distributions in TD (see top panels in
Fig. 9) the $\hat{a}_{f}$ values are close to zero. For the infinite-variance
and intermediate cases (middle and bottom panels of Fig. 9) the estimated
values (in absolute values) are higher than zero, however for the signals from
$\mathcal{S}(1.5,1)$, $\mathcal{P}(1.5,1)$ and $\mathcal{T}(1.5,1)$
distributions the obtained values are significantly higher than for the
intermediate cases (i.e. for $\mathcal{S}(1.9,1)$, $\mathcal{P}(3,1)$ and
$\mathcal{T}(3,1)$ distributions).

Figure 8: The increments of ECFM statistic calculated for the sub-signals
demonstrated in Fig. 7. The finite-variance cases (in TFD) are presented in
the top row, while the infinite-variance cases (in TFD) are demonstrated in
the middle and bottom row.

Figure 9: Selected segments of ECFM statistic calculated for the normalized
sub-signals demonstrated in Fig. 7 and the fitted straight lines (black line)
with the estimated slopes $a_{f}$. The finite-variance cases (in TFD) are
presented in the top row, while the infinite-variance cases (in TFD) are
demonstrated in the middle and bottom row.

In Fig. 11 we present the distribution of the slopes $a_{f}$ for simulated
signals. For each considered distribution and each set of parameters, we
simulate signals of length $10000$. For each simulated signal, we apply the
presented above procedure and calculate the median of the obtained $a_{f}$
values. In Fig. 11 we present the distribution of the medians calculated for
$1000$ simulated signals from the considered cases. In order to highlight the
differences between considered distributions, the results are presented in the
same scale on x-axis. In addition, in Fig. 11 we present the same results with
the same scales on x-axis in the rows. One can clearly conclude about the
significant differences between the values of $a_{f}$ estimated for
distributions with finite- and infinite- variances.

Figure 10: The distribution of the slopes $a_{f}$ for simulated signals from
three considered distributions. Here we present the medians of the $a_{f}$
values obtained based on $1000$ simulated signals. To highlight the
significant differences between the considered cases, the results are
presented on the same scales on x-axis.

Figure 11: The distribution of the slopes $a_{f}$ for simulated signals from
three considered distributions. Here we present the medians of the $a_{f}$
values obtained based on $1000$ simulated signals. The same scales on x-axis
are applied for the cases presented in the same rows.

In order to underline the differences between the considered cases, in Fig. 12
we present the comparison of the medians of the estimated slopes $a_{f}$
presented in Fig. 11 (left panel) and their IQRs, see right panel of Fig. 12.
Because the estimated slopes may be negative, here we present their absolute
values to demonstrate the plots in log scales. In each case, the first bars
correspond to the signals from finite-variance distribution in T-D, namely
$\mathcal{S}(2,1)$, $\mathcal{P}(6,1)$ and $\mathcal{T}(6,1)$, presented in
green, red and blue colors, respectively. The second bars correspond to the
intermediate cases, namely $\mathcal{S}(1.9,1)$, $\mathcal{P}(3,1)$ and
$\mathcal{T}(3,1)$ distributions. The last bars correspond to
$\mathcal{S}(1.5,1)$, $\mathcal{P}(1.5,1)$ and $\mathcal{T}(1.5,1)$
distributions.

It is clearly seen that the medians of the estimated $a_{f}$ values are
significantly smaller for the finite-variance distributed signals (first and
second bars). For $\mathcal{S}(2,1)$, $\mathcal{P}(6,1)$ and
$\mathcal{T}(6,1)$ distributions, they are negative (in log scales). For the
most extreme cases (last bars), the medians of the fitted slopes are
significantly higher than for other cases. The differences between the fitted
slopes are also visible in the IQR statistic, considered as the dispersion
measure. For the infinite-variance cases, the values of IQR are significantly
higher than for finite-variance distributed signals. The medians and IQR of
$a_{f}$ values clearly indicate the chaotic behavior of ECFM statistic for
infinite-variance cases.

Figure 12: The comparison of the medians of the estimated slopes $a_{f}$
presented in Fig. 11 (left panel) and their interquartile ranges (IQR), see
right panel. Here we present the absolute values and demonstrate the plots in
log scales. In each case, the first bars correspond to the signals from
finite-variance distributions, the second bars correspond to the intermediate
cases, and the last bars correspond to infinite-variance distributions in TFD.
The green color corresponds to $\alpha-$stable distribution, the red one to
the symmetric Pareto distribution, the blue one - to t location-scale
distribution.

##  5 Real signals analysis

In this section, we present the analysis for four real signals with potential
different distributions of the background noise, see Fig. 13. The signals
correspond to the healthy machines, thus we do not expect here any signal of
interest. Moreover, the high-pass filtering was applied for all considered
signals to remove the possible deterministic components.

To illustrate the problem, we propose to investigate four exemplary signals
denoted here sig. 1, sig. 2, sig. 3, and sig. 4. These signals come from
various machines, namely, vibration signal from rolling element bearings from
pulley used in belt conveyor system (sig. 1), acoustic signal (sound) measured
close to idler installed in belt conveyor (sig. 2), vibration signals from
hammer crusher used in copper ore processing plant for hard rock material
fragmentation (sig. 3 and sig. 4).

The first considered signal (denoted further as sig. 1) describes vibration
from healthy bearings, there is an amplitude modulation with cycle c.a. 1$Hz$
that is related to some minor shaft problems. No impulsive behaviour is
present in the time series.

The second signal (sig. 2) presents noise signal from healthy bearings in
idler. Minor impulses are visible in the time domain. However, in that
example, signature of the healthy element is contaminated by several high
amplitude impulses from moving clamp connecting two parts of the belt (clearly
seen on spectrogram presented in Fig. 14b).

Sig. 3 and sig. 4 describe vibration of crushers. The difference between
signals is related to operational conditions (load, i.e., material stream
coming to the crusher). As granulation of copper ore fed to machine may be
very different (from sand like material to pieces of rocks with a dozen of
kilograms) signals may contain nearly Gaussian noise or due to shocks - a
strongly impulsive components.

It was already mentioned, that first look at the real signal may be a bit
confusing. It may pretend to be Gaussian distributed, however wideband
impulsive behavior may be hidden in the measured data. Thus, to investigate
true properties of real signals we use time-frequency representation
(spectrogram), see Fig. 14. To calculate spectrograms we have applied the
’spectrogram’ Matlab procedure with the following parameters: for sig. 1 and
sig. 2 window $kaiser(500,5)$, overlap size $474$ and $512$ points to
calculate fast Fourier transform, for sig. 3 and sig 4 window
$kaiser(2000,5)$, overlap size $1896$ and $2048$ points to calculate fast
Fourier transform.

As it is clearly seen, time-frequency representations of the analyzed signals
are different, as machines are different. However, there are some important
features. Some frequency bands with high energy (expressed as red color) are
usually present at low frequencies. For high frequencies, the energy is
smaller (sig. 1), however some resonances may appear (sig. 3, sig. 4). Non-
Gaussian heavy-tailed behavior is related to vertical lines that means
wideband (impulsive) disturbance existing at some time instance. This is
clearly seen for sig. 2.

Figure 13: The analysed signals in TD.

Figure 14: Spectrograms of the signals demonstrated in Fig. 13.

In order to identify the possible infinite-variance distribution of the real
signals, we apply the procedure described in the previous section. More
precisely, for each signal, we analyze the ECFM statistics for sub-signals
taken from the spectrogram. Then, we parametrize the possible chaotic behavior
of ECFM statistics by analyzing the estimated slopes $a_{f}$ from the linear
regression applied for the last long segment of ECFM. In Fig. 15 we
demonstrate the distribution of the obtained $a_{f}$ values along frequencies.
To the analysis, we take frequencies from $4.5kHz$ to $9kHz$ for sig. 1,
$12.5kHz$ to $22.5kHz$ for sig. 2, $6kHz$ to $12.5kHz$ for sig. 3 and sig. 4.
It is worthy to note, that due to significant differences between sub-signals
associated with different frequency bins, we are not allowed to pick a single
vector to test the presence of non-Gaussianity. Thus, we are doing this for a
wide range of frequencies, and we have found similarities between these sub-
signals. In this way, we minimize the probability of misleading decisions.

Figure 15: The distribution of the $a_{f}$ values for four analyzed signals
along the frequencies.

Analyzing the results presented in Fig. 15, we can conclude that for sig. 2
and sig. 3 we expect the chaotic behavior of ECFM statistic applied for the
signals in TFD. Thus, we can assume the infinite-variance distributions of the
background noise, similar as for the signals presented in the bottom panel in
Fig. 4. Obviously, significantly smaller values of $a_{f}$ are obtained in
sig. 1, when we expect the finite-variance distribution of the signal in TFD.
This signal may correspond to the cases presented in the top panel of Fig. 4.
For sig. 4 the values of the estimated slopes are higher than for sig. 1 but
significantly smaller than for sig. 2 and sig. 3. Thus, one may conclude that
this signal corresponds to the intermediate case (corresponding to cases
presented in the middle panel of Fig. 4) or to the finite-variance
distribution but with a heavier tail than in sig. 1. To confirm our
preliminary assumption about the classes of distributions corresponding to the
analyzed signals in TFD, in Fig. 16 we demonstrate medians (left panel) and
IQRs (right panel) for the estimated values of $a_{f}$ calculated from the
estimated slopes for all frequencies. Similar as for simulated signals, here
we demonstrate the logarithm of the descriptive statistics applied for the
absolute values of $a_{f}$.

Figure 16: Results for real signals. Medians (left panel) and IQRs (right
panel) for the estimated values of $a_{f}$ calculated from the estimated
slopes for all frequencies. Here we demonstrate the log of the descriptive
statistics applied for the absolute values of $a_{f}$.

Our preliminary assumptions about the identification of the distribution class
is confirmed by Fig. 16. Comparing the results presented for simulated
signals, see Fig. 12, we can classify sig. 1 as the finite-variance
distributed. Sig. 2 and sig. 3 are classified as infinite-variance distributed
signals. However, for sig. 3 the distribution tail is lighter than for sig. 2.
Sig. 4 corresponds finite-variance case. The last step of our analysis is the
comparison of the empirical tails of the selected sub-signals taken from the
spectrograms for real signals with the tails of the fitted generalized
$\chi^{2}$ distribution (see Eq. (7)). The parameters of the generalized
$\chi^{2}$ distribution taken to the comparison are estimated from the
analyzed sub-signals. The results for selected sub-signals from the
corresponding spectrograms are presented in Fig. 17. The green line
corresponds to the generalized $\chi^{2}$ distribution, while the black stars
present the empirical CDFs. The results are presented in log-log scales. For
the analysis, we selected the sub-signals arbitrary. However, for other sub-
signals we obtain similar results. To confirm that all sub-signals taken from
the spectrograms correspond to the same classes in Fig. 18 (see A) we
demonstrate the empirical tails for all sub-signals.

Figure 17: The comparison of the empirical tails of the selected sub-signals
taken from the spectrograms for real signals with the tails of the fitted
generalized $\chi^{2}$ distribution with estimated parameters. The green line
corresponds to the generalized $\chi^{2}$ distribution, while the black stars
present the empirical CDFs.

Analyzing Fig. 17, we can conclude that the selected sub-signal corresponding
to sig. 1 may be considered as generalized $\chi^{2}$ distributed, while the
sub-signals corresponding to sig. 2, sig. 3 and sig. 4 have heavier tails. We
note, sig. 4 however is finite-variance distributed in TFD. The assumption of
the generalized $\chi^{2}$ distribution for sig. 1 is confirmed by KS test.
The median of $p_{value}$ of KS test for generalized $\chi^{2}$ distribution
applied to all sub-signals from the spectrograms for sig. 1 is equal to
$0.27$. The $p_{value}$s for all sub-signals are higher than $10\%$. We
remind, the $p_{value}$ higher than the confidence level here $5\%$ indicates
the hypothesis of tested distribution can not be rejected. For other signals,
the $p_{value}$s of KS test are significantly smaller than $5\%$. The boxplots
of $p_{value}$s of the KS test for all considered signals are presented in
Fig. 19 in A. The practical aspect of the obtained results is simple, namely
for sig. 1 and sig. 4 the approaches based on the classical auto-dependence
measures can be applied for local damage detection, while for sig. 2 and sig.
3 they may be not useful. For this case, more appropriate tools, dedicated to
infinite-variance distributed signals, need to be applied for the analysis of
the signals in TFD.

##  6 Conclusions

In many cases, researchers working on local damage detection pay attention on
properties of the signal of interest, but not of the background noise. In this
paper, we have built another perspective. We note that almost all measured
signals are associated with some noise. Thus, before applying methods for
damage detection that are based on two of the most intuitive features of the
SOI, namely impulsiveness or cyclic/periodic nature, one has to check if the
use of a given algorithm is allowed by the properties of the background noise.
In this paper, we discuss the probabilistic properties of the background noise
and indicate that they are extremely important in the context of applying
classical methods for damage detection. The noise may have a Gaussian or non-
Gaussian distribution. Even if we identify the non-Gaussian distribution of
the background noise, this information may not be sufficient for selection of
proper tools for signal analysis. In this paper, we categorize the types of
noise depending on the existence of the variance of its distribution. The
selection of the variance as the criterion is related to the fact that most of
the classical techniques used for local damage detection actually require it
to be finite. The problem and the proposed methodology are demonstrated based
on three popular non-Gaussian distributions that are used as models of
impulsive noise. We discuss the problem of non-Gaussian heavy-tailed
distribution of the background noise in the time and the time-frequency
domains, as many techniques are applied in these domains. We have demonstrated
that the non-Gaussian character of the noise in the time domain is transferred
to the time-frequency domain, however the level of non-Gaussianity increases
through applying squared STFT, i.e. the spectrogram. In consequence, the
finite-variance distributed signal may become infinite-variance distributed
after the transformation to the other domain. This observation sheds new light
on the application of classical methods to signal analysis in the time and
time-frequency domains.

As a main solution, we have proposed an adaptation of visual test based on the
ECFM for variance presence testing. The methodology is based on the time-
frequency representation of a given signal and takes under consideration the
specific behavior of used statistics for infinite-variance distributed data.
The proposed methodology is intuitive, simple to interpret and efficient, i.e.
it provides clear information if classical methodology for SOI detection can
be applied to the underlying signal.

According to our previous research, even some classical techniques may be
useful for damage detection in the presence of non-Gaussian noise. However,
the classical methods fail if the non-Gaussianity level is significant, [6].
In this paper, we continue this research and demonstrate how to identify the
extreme cases when the classical approaches may not give reliable results. We
believe, that the proposed approach could be very helpful to all researchers
working with noisy signals when the noise is non-Gaussian and heavy-tailed
distributed.

## Acknowledgements

The work of KS, RZ and AW is supported by the National Center of Science under
Sheng2 project No. UMO-2021/40/Q/ST8/00024 "NonGauMech - New methods of
processing non-stationary signals (identification, segmentation, extraction,
modeling) with non-Gaussian characteristics for the purpose of monitoring
complex mechanical structures".

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##  Appendix A Additional figures

Figure 18: Empirical tails of the sub-signals taken from the spectrograms of
the real signals.

Figure 19: Boxplots of $p_{values}$s of KS test for $\chi^{2}$ distribution
applied for the sub-signals from the spectrograms of analyzed real signals.
Algorithm 1 Algorithm for identification of infinite variance of the signal

Take a signal in TD

Calculate $S(\cdot,\cdot)$

Normalize spectrogram for each $f\in\mathcal{F}$:
$S_{norm}(\cdot,f)\leftarrow(S(\cdot,f)-median(S(\cdot,f)))/\text{CondStd}(S(\cdot,f))$

for $f\in\mathcal{F}$ do

Calculate ECFM for each $f\in\mathcal{F}$ for $S_{norm}(\cdot,f)$

Calculate increments of ECFM for $S_{norm}(\cdot,f)$ using diff()

Identify segments of ECFM based on values of increments.

Choose last segment longer than $10\%$ of initial data.

Calculate $\hat{a}_{f}$.

end for

Analyze IQR and Median of calculated $\hat{a}_{f}$ for all $f\in\mathcal{F}$.