Nonstationary Systems: Theory and Applications

The anomalous diffusive processes are widely discussed in the research papers. In contrast to the diffusive processes with the linear second moment, they are characterized by the nonlinear variance. The anomalous diffusive processes exhibit many interesting properties which are not adequate to the diffusive systems and thus they have found various applications including, among others, biology, physics and environmental engineering. Very useful in the testing of the anomalous diffusive behavior are the time-averaged statistics which are based on the sample trajectory of the given process. Similar as the empirical second moment, they exhibit different behavior for anomalous diffusive and diffusive processes. Thus, they can be very effective tools for the estimation and statistical testing of the anomalous diffusive behavior. One can find different theoretical anomalous diffusive processes. One of the classical examples is the fractional Brownian motion. In this chapter, we demonstrate how the selected time-averaged statistics behave for the fractional Brownian motion and show how they can be applied in order to estimate the Hurst exponent (responsible for the anomalous diffusive behavior). By using Monte Carlo simulations, we compare the effectiveness of the presented estimation methods for the considered stochastic process. The described methodology can be applied to any other anomalous diffusive processes.

To deal with time series process with excess of zeros, we extend the INAR(1) process by considering that the innovations follow different zero-inflated models, called the ZI-INAR(1) model. We present some of its theoretical properties, develop an efficient EM algorithm for parameter estimation and propose several bootstrap techniques to construct confidence intervals for the parameters. Finally, we present the relevance and applicability the of proposed ZI-INAR(1) model through simulation studies and an application to a real dataset.

In this paper, we concentrate on the two-dimensional \(\alpha \)-stable autoregressive model of order 1. For this model, we analyze the auto-dependence functions applied to describe the interdependence of the time series components considered separately as the one-dimensional processes. Since the classical second moment-based dependence measures are not defined in the \(\alpha -\)stable case, we examine here the auto-codifference and the auto-covariation functions. The main result of this paper is the derivation of the formulas which describe the asymptotics of the considered measures of interdependence. The theoretical results are supported by a simulation study.

Many real datasets exhibit non-Gaussian distribution, which is mainly manifested by impulsive behavior that is rather not visible in the Gaussian-based models. One can recognize such behavior in both one-dimensional and multi-dimensional datasets. In this paper, we study the problem of the linear dependence for two-dimensional non-Gaussian (infinite-variance) random variables. In the Gaussian case, the perfect measure is the correlation which clearly indicates the linear dependence between considered random variables. However, when the random variables are infinite-variance distributed, the alternative dependence measures need to be considered. We remind in this paper the most known alternative dependence measures which are adequate for infinite-variance systems and for two example two-dimensional random variables which are linear dependent, we calculate the selected measures. Moreover, by using the Monte Carlo simulations, we check how the empirical dependence measures tend to their theoretical values. In the simulation study, we check how the parameters of the considered random variables influence the behavior of the empirical correlation and empirical scov measure. The results presented in the simulation study clearly indicate that the empirical correlation should not be considered as the measure of dependence for the infinite-variance random vectors.

We explore Granger causality and cointegration between main stock indices, macroeconomic indicators (PMI) and central banks monetary expansion for US data in presence of extreme market movements: bubbles and crashes. Two stock indices are caused in Granger sense either by economic fundamentals or by money supply provided by Federal Reserve’s monetary policy. The causation is found to be dynamic: vanishing during moderate expansions and recurring around long–term market peaks followed by market crashes. Cointegration between the time series dynamics, here considered within Vector Autoregressive framework, has been empirically shown to be a time–variant, recurrent phenomenon, too.

Regime-switching models have been recently becoming increasingly important as they allow structural changes to be taken into account in modelling financial data. Application of such models to describe prices behaviour is especially valuable in commodities markets, where China’s industrialisation in the past three decades, created a huge demand for metals and energy, changing fundamentals of the market and affecting commodity prices. Moreover, in the financial time series very often we observe the non-Gaussian behaviour, which is manifested by large observations related to the market conditions. Considering the mentioned features of the financial data (especially commodity prices) in this paper we propose a stochastic model which takes under consideration the possible regime changes, the non-Gaussian character of the data and finally, the non-constant in time characteristics (like the mean function). We describe the main properties of the considered model and introduce a novel estimation procedure for the estimation of its parameters. Finally, we apply the proposed model to the real data describing the copper prices, one of the main risk factor for the KGHM mining company.

The aim of this paper is to provide readers with basic concepts and techniques for analysis of strongly periodically correlated fields (SCF) over \(Z^2\). We show that every SCF over \(Z^2\) can be transformed into a coordinate-wise SCF (Fact 3.1) studied in [13]. The main result of the paper however is a specific decomposition of a strongly periodically correlated field (Theorem 4.1) which was not available for coordinate-wise SCFs. As consequences of the latter we obtain a description and an easy proof of existence of the spectral measures of an SCF (Theorem 5.1) as well as a functional description of an absolutely continuous SCF (Theorem 6.1). Most of the facts are explained in details and proved, with an exception of the proof of Theorem 6.1, which was too long for this publication and is left for a forthcoming paper.

The component and the least square (LS) estimators of mean and covariance functions of biperiodically correlated random processes (BPCRPs) as the model of the signal with binary stochastic recurrence are analyzed. The formulae for biases and the variances for estimators are obtained and the sufficient condition for the mean square consistency of mean function and Gaussian BPCRP covariance function are given. It is shown that the leakage errors are absent for the LS estimators in contrast to the component ones. The comparison of the bias and variance of the component and the LS estimators is carried out for BPCRP particular case.

This paper addresses the problem of deterministic/random separation in vibration signals when the machine is operating under nonstationary regime. The solution to this problem is well established in the stationary regime case, where the deterministic component is simply periodic. In this regard, the synchronous average provides an optimal way to separate a deterministic synchronous source from other interferences. However, synchronous averaging theoretically requires the machine to operate under stationary regime (i.e. the related vibration signals are cyclostationary) and is otherwise jeopardized by the presence of amplitude and phase modulations. The local synchronous fitting presents a powerful generalization of the synchronous average to the non-stationary regime case (i.e. the related vibration signals are cyclo-non-stationary). The idea is to replace the (cyclic) empirical average operation by a (cyclic) local curve fitting using the Savitsky-Golay algorithm. This paper studies the temporal and spectral properties of this filter, and demonstrates its potentiality on real-world helicopter data recorded under varying operating speed.

Phonocardiogram is a concept that is used for recording heart sound signals and murmurs. This acoustic recording helps to reveal important information that human ear cannot recognize easily. A phonocardiogram signal, in the healthy case, consists of two fundamental sounds \(s_1\) and \(s_2\) which are derived from the mechanical functioning of the heart. Actually any change, even small, in the heart sounds might indicate heart valve problems, and hence the need of correctly analyzing and characterizing phonocardiogram signals. Recently, the analysis of phonocardiogram signals becomes an interesting field of research. There are several tools that have been studied and presented in the literature review. The majority of these studies are based on time-frequency and partially exploiting the periodic character of phonocardiogram signal due to the heart functioning. The objective of this research is to propose a coherent mathematical model and an analytical framework based on cyclostationarity. This allows the use of cyclostationary tools for the characterization and the analysis of phonocardiogram signals which are analyzed and discussed in details over synthetic and experimental datasets. The simulation shows promising results that can help with the early detection of some heart diseases.

The problem of calculation of parameters based on spectra might seem trivial, but it is so only for MATLAB^® (or other advanced software) users. Commercial condition monitoring systems work on pure platforms and they use basic mathematical functions. For this reason, even such a simple thing as evaluation of scalar indicators from spectral signal representation might raise several problems and possible errors during industrial implementations, especially for untypical signal parameters. The paper presents selected guidelines how to cope with this problem in practical implementations. After reading this paper, one will learn some practical recipes, which enable conversion of spectral data to trend data.

This research aims to develop and validate a method for tacho-less automatic detection of faults in rolling bearings for mechanical systems comprised by gears. The proposed method was based on the application of some mode decomposition technique in order to extract monocomponent signals from the vibration and to calculate an indicator of the modulation produced by the rolling element bearing fault. The computation of this indicator was performed by means of Lock-in Amplifiers, which are used in order to extract, through a synchronous approach, spectral components from non-stationary signals. A novel algorithm, previously applied on gear fault detection, was adapted and used in order to estimate the rotational speed. The effectiveness of the method was assessed through experiments with real signals. Besides, the capability of the indicator for serving as a relative measure of the fault severity was verified.

This chapter presents development of a novel structural health monitoring methodology used for incipient damage detection in moving structures. This method is built upon implementation of low-cost deposition of piezoceramic transducers on crucial substrates together with electromechanical impedance (EMI) to provide a practical solution to damage/fault detection of moving structures that suffer from mechanical fatigue, thermal fatigue or corrosion, before any catastrophic failure. Important steps towards application of such technology are: i) Chemo-physical fabrication of piezoceramic transducers, i.e., deposition of precursor solutions or piezoelectric materials, on geometrically irregular structures; ii) Development of a portable impedance-based transceiver and signal processing algorithm capable of transmitting actuating signals as well as receiving and analyzing response signals to/from piezoelectric transducers, respectively; iii) Development of a monitoring algorithm, and iv) Methodology verification through bunch of semi-filed tests by building the prototype of a moving structure. Multiple characterizations are performed to assure the micro and macro-structural functionality, accuracy, and precision of fabricated transducers. To verify the functionality of the custom-built system, electromechanical response of the transducers and the results obtained from the transceiver are compared with commercially available piezoelectric wafers and standard impedance analyzers. Frequency response of transducers measured in a wide bandwidth shows obvious frequency shift and change in the admittance/ impedance amplitude corresponding to the resonance/anti-resonance peaks at pristine vs. damaged conditions. Furthermore, for rotary structures, rotational speed and temperature play important role in this method. Application of this method could be extended to moving structures such as airplane engine blades, fuselage frames, wing ribs, helicopter main rotor assembly, critical parts of the exploration rovers, satellite loaded modules and thrusters, moving links/joints on the international space station, rotors in hydroelectric or nuclear power plants, autonomous underwater vehicles, submarine propellers, hydro/diving planes, wind turbines, hot/cold rollers in steel production lines, drones, mobile robots, etc.

In this paper, a newly developed signal decomposition technique called Ensemble Patch Transformation (EPT) is used for the first time to identify mechanical faults of a hammer type coal crusher machine. Rub-impact faults of a rotary machine result in amplitude modulation of the vibration signal. In this scenario, mode decomposition of a complex signal is essential when the signal is a combination (either convolutive or additive) of many simpler signals. EPT is a newly developed multi-resolution signal analysis method inspired by multi-scale concept of scale-space theory. As Empirical Mode Decomposition (EMD) method can decompose a complex signal into a number of simpler signals called intrinsic mode functions (IMF); the EPT process can also decompose a complex signal into a main frequency component (FC) and a residual signal. The residual signal again can be decomposed into a FC component and residual component and this process can be repeated for a number of times. Initially, the performance of the EPT is investigated on a simulated signal. Then, the same method is applied to find out rub-impact fault of a hammer type coal crusher machine in a steel plant. Finally, the superiority of the EPT based method is demonstrated during extracting multiple signals with comparison to the EMD based method.

Fault detection in non-stationary processes is a timely research topic in industrial process monitoring. The core objective of this research is to tackle anomaly detection in non-stationary industrial processes with manipulated set-point changes and uncertainties in the prior knowledge about the statistical nature of the measurements. In this research, the fault detection problem is investigated from an unsupervised perspective and a modified PCA approach is proposed. This method utilizes the base-line loading matrices and an upper bound to be determined for the variation range of time-series to relax the assumption on stationary characteristics. Hence, the mean used for normalizing the time-series are adaptively updated (using soft-calculation) without any need for a high-complexity recalibration procedure as needed in other existing adaptive/recursive PCA methods. Moreover, the first- and second-order error indices are introduced to monitor the statistical behaviour of process measurements. To develop a more reliable system condition indicator, an overall health index is given based on the proposed features using a non-parametric kernel density estimator (KDE). The proposed approach does not require a heavy online calculation in comparison with the existing adaptive solutions and it can successfully detect faults from healthy measurements’ mean changes. Finally, an alarm generator algorithm is presented which generates two alarm types, caution and actual fault for processes operators, utilizing the proposed overall health index. The effectiveness of the modified PCA approach is validated by both numerical examples and industrial case studies.

Power converters’ usage is expanding ever in industrial applications as they provide flexibility, high level of performances and new functionalities. However, with increased complexity come new constraints with respect to reliability. This chapter covers a study on reliability of a lab-scale power electronic module taken here as a vehicle. The downsizing of converters and new application-related operating constraints are accompanied by an increase in current density. The use of Silicon Carbide wide-gap technology in power modules is therefore attracting but remains a challenge because this technology is not yet mature and does not benefit from the deep knowledge established about Silicon counterpart. Therefore, health monitoring has naturally emerged as an effective way to implement a reliability assessment. After a brief description of the expected failure modes, an experimental failure monitoring bench will be presented. The choice and implementation of failure indicators through a classification using a neural network will be discussed and presented.

Robustness and operational reliability are some advantages that present rotating machine components as inevitable parts of most mechanical engineering systems. To keep rotating machines function at optimal conditions, control and maintenance of machine components must well be applied. Improper analysis of high modulated non-stationary data, acquired from machine components (e.g. gears and bearings) may lead to complete machine break. Despite the body of research work, available through the literature, most of existing condition monitoring (CM) methods proved to be inefficient for real world applications. Hence, this would still suggest an obvious need for fundamental modifications and/or development of new CM techniques. Here we aim to tackle this issue by proposing Signal Intensity Estimator (SIE) as an alternative technique, tailored to the task for monitoring of high modulated data. Our main interest lies with the introduction of the idea behind the SIE method and its previous successful applications for monitoring Suzlon and Repower (Wind Energy Companies) wind machines.

The development and implementation are described of the model-based decision support system for selecting and correcting the possible charging programs of a blast furnace, working in non-stationary regimes. The verification of the adequacy of the calculated charging parameters obtained by the modelling system is performed using the information of the modern control facilities. The model-based decision support system allows technical personnel to make the right decisions about blast furnace charging modes and burden materials volumes. Among other useful functions is also the training technological personnel in conditions of mastering a new top charging apparatus. Using the system during the operation of the blast furnace since 2012, charging programs were developed and implemented: when using a various composition of charge materials as well as using natural gas and pulverized coal in the air blast. The developed charging programs have provided increased economic effect and high technical parameters of blast furnace operation.

Resonant and out-of-resonance vibration machines are investigated operating under conditions of non-stationary loads and the influence of variable parameters of treated bulk material. The design scheme, which is based on a beam system with intermediate elastic supports and concentrated in the middle of the span mass, which performs transverse oscillations, is considered. This scheme allows changing the natural frequency of oscillations of the single-mass system due to the shift of the intermediate supports along the beam. Nonlinear dependences of the natural frequency of mass oscillations in the case of elastic intermediate supports are determined and analysed. The optimization of the vibration system is proposed to minimize the mass and provide the given frequency characteristics. A scheme of the frequency characteristics adjusting in the process of the vibrating machine operation has been introduced. The control scheme is based on feedback by the mass vibration signal.

The paper considers the use of mathematical and computer modelling methods for the analysis of the technical state of complex rotary machines with controlled electronic components. These are active magnetic bearings (AMBs) with control systems. Assessment of the vibrational state of any rotor system is one of the most important tasks in their design and synthesis. Stationary and non-stationary processes take place in such systems. The main focus of the research is the use of a specially designed mathematical apparatus and software for implementation of proposed techniques for assessing the dynamic behaviour of rotors of power engineering turbomachines with active magnetic bearings in the entire range of excitation frequencies. Mathematical modeling is based on analytical representation using Lagrange-Maxwell magnetomechanical equations. The Runge-Kutta procedure is applied for numerical simulation. These methods and computer tools have several advantages over existing ones since they take into account completeness of a nonlinear relationship between mechanical, magnetic, and electrical processes occurring in the system, including continuous or discrete control actions. Created analytical and numerical approaches allowed to perform the modelling of rotor dynamics occurring in an energy gas turbine installation taking into account AMBs. Design calculation studies and analysis of the rotor dynamics of turbocompressor and turbogenerator of this installation show advantages of the proposed method for systems including AMBs. All the results of numerical studies are verified by comparison with numerical and experimental data known from open sources of information.

The study of the material microstructure allows obtaining information about the state of the part without additional tests and full-scale experiments. The paper offers computer methods for constructing parametric, statistically equivalent models of cast iron microstructure with the inclusion of spheroidal graphite. The studied material has a transient microstructure that exhibits variability at various material points. To analyze the unsteadiness of deformations, the Monte Carlo method is used. A finite element model is constructed to find the elastic characteristics of the material. The stress state is considered based on plane models. Numerical experiments are carried out for various concentrations of inclusions. The results obtained for the elastic constants are statistically averaged, and the dependences of the Poisson’s ratio, the moduli of elasticity, and the shear moduli on the concentration of the inclusions are established. For veracity assessment, the values obtained are compared with those obtained using the mixture rule. The results of the application of the rule confirm the correctness of the built models. The yield surfaces are found, going beyond the surface indicates the appearance of plastic strains in the material.

The global energy crisis has allowed marine currents to enter the field of vision of all countries. Marine current turbine (MCT) is a kind of deep-sea equipment that converts marine current energy into electric power, and its safe and reliable operation is very important. In order to facilitate monitoring of the blade status of MCTs in multiple scenarios, the chapter deals with the diagnosis method of MCT blade followed by different methods. First, a review of the MCT blade fault diagnosis method has been presented. Then two different methods are discussed in this chapter. One is the signal processing method based on the stator current, which includes VMD denoising and proposed novel LDA classifier; the another one is the image semantic segmentation technique based on the MCT image, which includes semantic segmentation and adaptive recognition technical. These two methods can be organically combined under different biofouling cases. When the biofouling is low to affect the output torque of the turbine, the method based on image processing can be useful. On the contrary, the method based on current signal is more convenient and effective. The experimental results show that the two methods proposed in this chapter can effectively detect MCT biofouling in different scenarios. It also proposes several trends for a handle with biofouling problem in conclusion.