Non-stationarity in time series data is one of the most important challenges in signal processing nowadays. One of the most often cases occurs when signal is a mixture of different processes that reveal different statistical properties. Common way to deal with is the data segmentation. In the following paper we propose an automatic segmentation procedure based on gamma distribution approach. In the algorithm we estimate the parameters of gamma distribution for subsequent batches of distance values between consecutive impulses (waiting times). Then we use Expectation-Maximization algorithm to classify estimated parameters. Obtained classes refer to particular signal segments. Procedure has been applied to real vibration signal from roadheader working in underground mining industry

Agnieszka Wyłomańska

Jacek Wodecki

Radosław Zimroz

Rafal Połoczański

English

Vibroengineering PROCEDIA

  © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 59 
A new segmentation method of roadheader signal based 
on the statistical analysis of waiting times 
Jacek Wodecki1, Agnieszka Wyłomańska2, Rafal Połoczański3, Radosław Zimroz4 
1, 4KGHM Cuprum Ltd. R&D, Wroclaw, Poland 
2, 3Faculty of Pure and Applied Mathematics, Hugo Steinhaus Center,  
Wroclaw University of Science and Technology, Wroclaw, Poland 
1Corresponding author 
E-mail: 1jwodecki@cuprum.wroc.pl, 2agnieszka.wylomanska@pwr.edu.pl, 3rafal.poloczanski@pwr.edu.pl, 
4rzimroz@cuprum.wroc.pl 
(Received 8 August 2016; accepted 10 August 2016) 
Abstract. Non-stationarity in time series data is one of the most important challenges in signal 
processing nowadays. One of the most often cases occurs when signal is a mixture of different 
processes that reveal different statistical properties. Common way to deal with is the data 
segmentation. In the following paper we propose an automatic segmentation procedure based on 
gamma distribution approach. In the algorithm we estimate the parameters of gamma distribution 
for subsequent batches of distance values between consecutive impulses (waiting times). Then we 
use Expectation-Maximization algorithm to classify estimated parameters. Obtained classes refer 
to particular signal segments. Procedure has been applied to real vibration signal from roadheader 
working in underground mining industry. 
Keywords: segmentation, gamma distribution, expectation-maximization algorithm, roadheader. 
1. Introduction 
Key aspect of modeling and analysis of real-life signals is a possibility of obtaining the 
information about the considered process. Unfortunately, it is common that observed time series 
is characterized with high level of non-stationarity, which comes from the fact that it consists of 
different processes occurring subsequently. Typically, those processes cannot be described by the 
same model, because they reveal very different statistical properties. On the other hand, it is often 
seen that they can be described with the same type of model, but its parameters vary. Because of 
that, segmentation of the signal should be performed before any further analysis or modeling, and 
the aim is to extract parts of the signal with homogenous statistical properties. In the same time, 
extracted segments define points in time when properties change and signal switches to the other 
regime [1-4]. According to presented methodology, those properties of the signal can be estimated 
and then used multidimensionally as classification parameters.  
Signal segmentation methods very often are dedicated to the applications they are used. We 
mention only methods related to speech analysis [5], condition monitoring for shocks isolation [6], 
seismic signal analysis [7], physics [3], machine performance analysis for operation modes 
separation [4] or biomedical signals [8]. 
In this paper we propose the segmentation algorithm based on modeling of empirical 
distribution function of waiting times between consecutive impulses by gamma distribution and 
clustering obtained parameters with Expectation-maximization (EM) algorithm. The gamma 
distribution has been chosen because of its closest similarity to empirical distributions derived 
from the data, as well as its theoretical predisposition to be used for modeling of waiting times. 
2. Methodology 
This section describes segmentation procedure presented in this paper. Firstly, impulses 
observed in the signal are identified and localized in time and distances between them are 
measured. Then parameters of gamma distribution related to shape and scale are estimated for 
those waiting times within a sliding window. This outputs two vectors of parameters, that are 
A NEW SEGMENTATION METHOD OF ROADHEADER SIGNAL BASED ON THE STATISTICAL ANALYSIS OF WAITING TIMES.  
JACEK WODECKI, AGNIESZKA WYŁOMAŃSKA, RAFAL POŁOCZAŃSKI, RADOSŁAW ZIMROZ 
60 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
provided to the EM algorithm for clustering. Finally, post-processing translates clustering results 
into time segments containing signal parts that reveal homogenous statistical behavior. General 
schematic of the procedure is presented in Fig. 1. Key aspects are discussed in Sections 2.1-2.4. 
 
Fig. 1. Functional schematic of presented procedure 
2.1. Impulses localization 
To localize the impulses in time, firstly one needs to choose positive or negative side of the 
signal in terms of values. To choose positive side, absolute value of signal is performed. After that 
signal is smoothed with moving average with the window length equal to 160 samples to extract 
the general shape, and then the result is smoothed again with moving average with the window 
length equal to 70 samples, just to get rid of the smallest ripples that would interfere during the 
derivative analysis. Finally, localization of the impulses in time is performed as a simple 
derivative-based local maxima detection. When location of individual impulses is known, we can 
proceed to modeling their distances with gamma distribution. 
2.2. Gamma distribution 
There are many parameterizations of gamma distribution. In this paper we assume the random 
variable ܺ has gamma distribution with shape parameter ݇ and scale parameter ߠ if it has the 
following probability density function [9]: 
݂ሺݔሻ = ݔ
௞ିଵ݁ି௫/ఏ
ߠ௞Γሺ݇ሻ ,					ݔ ൐ 0. (1)
This distribution has found many practical applications. It is especially important in the 
insurance theory, where individual claims are modeled by using gamma distribution [9]. In many 
research papers the gamma distribution is also considered as an appropriate one for modeling 
so-called waiting times in the continuous time random walk model (CTRW), [10]. 
2.3. Parameters estimation 
When the real independent data are modeled by using gamma distribution, the most classical 
method that can be used to parameter’s estimation is the method of moments [9]. However, if the 
considered vector of observations is related to the waiting times, then the method of moments 
cannot be directly applied. The most commonly used method to identify the correct distribution 
in this case is visualization of empirical cumulative distribution function (CDF) on a plot with 
log-log scale and comparison with fitted parametric distribution [10]. However, a testing 
procedures which would allow to identify the correct class of distribution for observations 
A NEW SEGMENTATION METHOD OF ROADHEADER SIGNAL BASED ON THE STATISTICAL ANALYSIS OF WAITING TIMES.  
JACEK WODECKI, AGNIESZKA WYŁOMAŃSKA, RAFAL POŁOCZAŃSKI, RADOSŁAW ZIMROZ 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 61 
corresponding to waiting times are very rarely considered in the literature. The difficulty arises 
from the discretization of the time intervals which is a natural consequence of time-discretization 
of measuring devices. Furthermore, standard estimation procedures like method of moments might 
lead to under- or overestimation of parameters.  
In this paper we use an estimation procedure of gamma distribution parameters for waiting 
times proposed in [11]. This is a new method which is based on the distance between theoretical 
CDF and so-called rescaled modified CDF. The main issue during estimation of waiting times 
parameters comes from the fact that the exact waiting time is unknown and usually comes from 
continuous distribution. For example, if we observe that a character of the process has changed 
after 3 units of time, it is not known at which point of time the change actually happened, the 
correct value lies in the interval (2, 4). In other words, a constant time period equal to 2.5 might 
be classified as 2 or 3 with equal probability. 
The estimators of gamma distribution parameters are calculated on the basis of the 
Kolmogorov-Smirnov distance for CDFs (theoretical and empirical one). More precisely, for 
vector of observations corresponding to waiting times we calculate first the empirical cumulative 
distribution function: 
ܨ෠௡ሺݐሻ =
1
݊෍1௫೔ஸ௧
௡
௜ୀଵ
, (2)
where 1஺ denotes indicator of the set ܣ and ݔଵ, ݔଶ,..., ݔ௡ is a vector of observations of waiting 
times. In the next step we calculate the Kolmogorov-Smirnov statistic as follows: 
ܭܵ൫ܨ௡෡ , ܩ൯ = sup௫หܩሺݔሻ − ܨ෠௡ሺݔሻห, (3)
where ܩ  is rescaled modified CDF corresponding to gamma distribution. The estimators of 
parameters ݇ and ߠ are calculated such that they minimize the distance ܭܵ. More details one can 
find in [11].  
2.4. Expectation-maximization algorithm 
Expectation-maximization is an iterative optimization method for estimation of unknown 
parameters, given measured data and latent variables representing missing data. EM is particularly 
useful for separating mixtures of Gaussian distributions over the considered feature space. It 
consists of two main steps: Expectation (E-step) and Maximization (M-step), which are iterated 
until convergence [12-15].  
In the first iteration algorithm has to be provided with some initial values of parameters. It can 
be done by picking random means, covariances and distribution weights, but it is a good practice 
to pre-estimate means ߤԦ௟ using some simpler algorithm like ݇-means or hierarchical clustering, 
then compute covariance matrices Σ௟ basing on results of this pre-clustering, and set weights ߙ௟ to 
normalized amount of points in each pre-cluster. 
It is important to remember about limitations of EM methodology. EM only tries to find the 
maximum likelihood estimate, and not finds it with 100 % confidence, because EM estimate only 
guarantees not to get worse in the process. If the likelihood function has multiple peaks 
(non-concavity case) EM will not necessarily find the global optimum of the likelihood. In  
practice, one can never trust one single run. It is very common to start EM multiple times with 
multiple random initial guesses, and choose the one with the largest likelihood as the final estimate 
for parameters.  
EM is widely used for data clustering in machine learning and computer vision techniques. In 
natural language processing, two prominent instances of the algorithm are the Baum-Welch 
algorithm and the inside-outside algorithm for unsupervised induction of probabilistic context-free 
A NEW SEGMENTATION METHOD OF ROADHEADER SIGNAL BASED ON THE STATISTICAL ANALYSIS OF WAITING TIMES.  
JACEK WODECKI, AGNIESZKA WYŁOMAŃSKA, RAFAL POŁOCZAŃSKI, RADOSŁAW ZIMROZ 
62 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
grammars. In our method we also propose to estimate optimal amount of clusters with Silhouette 
criterion [16, 17] for limited range of number of clusters ݇  (in our application ݇ =	1:5) with 
Euclidean measure of distance. In our application we use Expectation – Maximization algorithm 
under the assumption that point clouds in feature space will form clusters distributed normally. 
3. Application to real data 
The analyzed real-life data set is a vibration signal of a roadheader. The sampling rate was 
equal to ܨݏ = 25 kHz. Duration of measurement was 8 seconds, and it was performed by using a 
linear accelerometer placed vertically on the mining head’s arm. The analyzed data set is presented 
in Fig. 2.  
At first, impulses have been localized in time according to the description in Section 2.1 (see 
Fig. 3), and waiting times between them have been calculated as a numerical derivative of 
impulses’ timestamps.  
 
Fig. 2. Raw vibration data 
 
a) Raw vibration data 
 
b) Impulse localization 
Fig. 3. Raw vibration data and localized impulses (zoomed for better visibility) 
After that two vectors of shape and scale parameters have been estimated by modeling waiting 
times data (see Section 2.3) within the window of the two lengths.  
Statistics presented in Fig. 4 represent set of ܰ-points vectors of statistics for window length 
equal to 20 and 30 respectively. Dataset is then centered and variance over each dimension is set 
to 1. This operation allows to more reliable and efficient clustering and also estimation of cluster 
amount. 
After calculation of parameters we estimate the amount of clusters ݇ using Silhouette criterion 
for limited range (݇ =	1:5). When optimal amount of clusters ݇ is established, EM algorithm 
clusters the dataset dividing it into ݇ classes. For the analyzed dataset we discovered that the 
optimal number of clusters is equal to two.  
A NEW SEGMENTATION METHOD OF ROADHEADER SIGNAL BASED ON THE STATISTICAL ANALYSIS OF WAITING TIMES.  
JACEK WODECKI, AGNIESZKA WYŁOMAŃSKA, RAFAL POŁOCZAŃSKI, RADOSŁAW ZIMROZ 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533 63 
a) 
 
b) 
Fig. 4. Statistics used for classification: a) window of 20 impulses, b) window of 30 impulses 
We present the division of the raw signal for window length equal to 20 samples (Fig. 5) and 
equal to 30 (Fig. 6) into two clusters after application of the mentioned procedure. As a result of 
the presented procedure we obtain parts of signal belonging to certain clusters. Investigation of 
indices of data points revealed that both window lengths resulted in identical classification results. 
Detection and further analysis of such time map can help improving machine operating efficiency, 
e.g. to minimize amount and duration of idle modes. 
 
Fig. 5. Result of classification (window 20) 
 
Fig. 6. Result of classification (window 30) 
A NEW SEGMENTATION METHOD OF ROADHEADER SIGNAL BASED ON THE STATISTICAL ANALYSIS OF WAITING TIMES.  
JACEK WODECKI, AGNIESZKA WYŁOMAŃSKA, RAFAL POŁOCZAŃSKI, RADOSŁAW ZIMROZ 
64 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. AUG 2016, VOL. 7. ISSN 2345-0533  
4. Conclusions 
In this paper we have proposed a segmentation technique which can be applied to the vibration 
signal from roadheader working in the mining industry. Different operation modes, difficult 
working conditions and some degree of uncertainty due to external interferences cause the analysis 
to be difficult from diagnostic point of view. Therefore, it is needed to design new data-driven 
method for signal segmentation for operational modes classification. Proposed technique works 
automatically and is based on Silhouette criterion, gamma distribution approach and 
Expectation-maximization algorithm for parameters classification. The method outputs time 
segments within which given signal reveals homogenous character. Obtained time periods 
correspond to operation modes of the machine.  
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A new segmentation method of roadheader signal based on the statistical analysis of waiting times

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A new segmentation method of roadheader signal based on the statistical analysis of waiting times

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