The problem of detecting and localizing a moving source emitting a wide-band radio-frequency signal, starting from the noisy signals received from a pair of sensors, is addressed when the narrow-band condition is not satisfied. That is, when the product of signal bandwidth and data-record length is not much smaller than the ratio of the medium propagation speed and the relative radial speed. The proposed detection and localization techniques put together the advantages of wide-band correlation processing and the signal selectivity property of cyclostationarity-based signal processing.

Cyclostationary signals are widespread in many areas of research. Radar signals, GPS signals, vibromechanical signals all share the cyclostationary features. The aim of my talk is to show how recent advances in functional data analysis have impacted analysis of cyclostationary signals. The theoretical results will be illustrated with practical examples.

In this talk we present a survey of the recent results on the cyclic Fraction- Of-Time (FOT) analysis of a signal. The fraction-of-time probability analysis approach was introduced and developed by W.A. Gardner (1987). By applying the concept of relative measure used by Bochner, Bohr, Haviland, Jessen, Wiener, and Wintner and by Kac and Steinhaus, a probabilistic – but non-stochastic – model is built starting from a single function of time (the signal at hand). Several existing results are put in a common, rigorous, measure-theory based setup. By using the relative measure concept, a distribution function, and all the familiar probabilistic parameters (mean, variance, moments, cumulants) and concepts as FOT- independency can be constructed starting from a single function of time (W.A. Gardner 1987, A Napolitano & J. Le´skow 2006). The detection of cyclic components of a signal has induced to develop the cyclic FOT analysis introduced by W.A Gardner (1987) (W.A. Gardner & W.A. Brown 1991, A. Napolitano 2020). This has brought to consider the notions of cyclic FOT-distribution, cyclic FOT-density function and cyclic FOT-independency. Then the estimation problem of the cyclic FOT-density appeared. The consistency and the rate of convergence of kernel estimators has been stated. In particular, applying the notion of cyclic FOT-independence the case of a signal noise model was considered.

The report is focused on the issues of coexistence and prospects for the integration of signal processing technologies within the framework of the cyclostationary paradigm and modern technologies of artificial intelligence. Advantages and disadvantages of model-based and data-driving cyclic signal processing technologies are shown. Considered possible ways of integration (hybridization) of model-based technologies within the framework of the cyclostationary paradigm and machine learning technologies.

During this presentation, we will discuss the problem of identifying damage-related signatures in vibration signals characterized by non-Gaussian (impulsive) interference. The issue of non-Gaussian noise is increasingly addressed in the literature, with researchers frequently proposing new methods dedicated to such cases. These are most commonly robust methods that minimize the impact of large (outlying) observations. Our own research also explores this topic; our recent studies show that classical methods for identifying cyclostationary behavior in the presence of non-Gaussian noise may be insufficient, necessitating the use of more appropriate techniques. During the lecture, we will present results concerning the use of robust Cyclic Spectral Coherence (CSC) maps to identify cyclostationarity, which indicates local damage within the examined mechanical system. The procedure is based on a statistical testing methodology applied to CSC values. We will demonstrate that a similar approach based on classical CSC is insufficient for signals with impulsive background noise.

We explore periodic PARMA models with respect to periodicity in AR coefficients, MA coefficients and periodic variances of the white noise. We explore the need for periodic variances of the white noise by simulations, calculation of variances, autocorrelation and partial autocorrelation function of the process. We will further investigate estimation of the coefficients under normality in the presence of outliers by studying the spectrum of the process. Furthermore, we will expand our methods to processes with missing data under the assumption of missing at random. We study 5 different estimators of the spectrum: 1) classical periodogram, 2) Lomb-Scargle periodogram designed to handle unevenly spaced data, 3) M-periodogram version by Huber which is robust to outliers, 4) M-periodogram by Tukey which modifies the M-periodogram by Huber, 5) robust Whittle estimator. We use 3 different scenarios: 1) missing values but no outliers, 2) outliers but no missing values, and 3) outliers and missing data. Some preliminary simulations show the Robust Whittle Estimator has the smallest RMSE for missing data and outperforms the Lomb-Scargle periodogram designed for missing data. When we study outliers but no missing data we find that the Robust Whittle estimator outperforms the M-estimators and Lomb-Scargle estimators. Finally, when we have missing data as well as outliers M-estimators outperform the others but the Robust Whittle estimator performs better than the classical estimator and the Lomb-Scargle estimator. We conclude that the Robust Whittle is best overall in the presence of missing data when there may also be outliers.

It is considered properties of Hilbert transform of nonstationary signal with the wide-band high frequency modulation, which is represented by a superposition of stochastically modulated in amplitude and phase harmonics with multiple frequencies. It is shown that the condition for the periodic nonstationarity of an analytical signal is the correlation of the quadratures of different components. It is shown that the periodically non-stationary random process and its Hilbert transform are related, their zero-order auto covariation components are the same, and the zero-order cross-covariation components differ only in sign.

This work is devoted to a Web-based software platform for analyzing and modeling complex signals with a cyclic structure. It is an effective tool for identifying the characteristics of cyclic signals, providing visual representations and building individual workflows adapted to the individual needs of users. The main advantage of the mathematical support of the software platform is its significantly greater functionality compared to known analogues, which is due to the use of the modern theory of cyclic functional relations, which covers a wide range of mathematical models and methods for processing cyclic signals within the framework of deterministic, stochastic, fuzzy and interval paradigms of mathematical modeling.

Modern neurointerface, cardiodiagnostic, and biometric systems rely increasingly on the analysis of cyclic biomedical signals such as electroencephalograms (EEG) and various types of cardiosignals—including electrocardiograms, seismocardiograms, photoplethysmograms, etc. In real-world conditions, these signals exhibit variable rhythm and morphology, which limits the applicability of traditional stationary or quasi-periodic models and reduces the stability of feature extraction, classification, and decision-making. This work presents a unified rhythm-adaptive framework for modeling and processing cyclic biomedical signals based on the concept of cyclic random processes with an explicit rhythm function. The approach allows both regular and irregular rhythms to be represented within a single mathematical model and supports the joint processing of vector signals with shared rhythmic structures. Within this framework, rhythm-adaptive estimates of statistical moments and their harmonic representations form compact, informative features applicable across different signal modalities. For neurointerfaces, the model enhances the extraction of control signals from EEG during voluntary mental activity, while for cardiosignals it enables robust identification and authentication of individuals under varying physiological conditions. The proposed rhythm-adaptive methodology improves both accuracy and interpretability of biomedical signal analysis and provides a general mathematical foundation for a wide class of non-invasive systems—from brain–computer interfaces and wearable sensors to secure biometric and cardiodiagnostic technologies.

Analysis of cyclical data is a key component of statistical analysis of random processes. However, many real-world signals exhibit irregular cyclic patterns, which pose challenges for standard modeling approaches. To address this, we introduce a semiparametric continuous-time model for signals with irregular cyclicities. This model is based on stochastic modulation of the amplitude and phase of a deterministic signal (an almost periodic function). We investigate its theoretical properties, focusing on the behavior of the mean and autocovariance functions, and we demonstrate that the process gradually loses its cyclic structure over time due to random disturbances. Estimators of the asymptotic mean and autocovariance functions are introduced. The performance of the autocovariance function estimator is examined in a simulation study. This is joint work with Anna Dudek and Łukasz Lenart.

A continuous-time conditional linear random process (CLRP) is defined as a stochastic integral of a random function (often called the stochastic kernel) driven by a process with independent increments. It is a useful mathematical model for stochastic signals generated as a sum of a large quantity of stochastically dependent random impulses occurring at Poisson times. Using the method of characteristic functions, the properties of CLRP are analyzed in the context of mathematical modelling of cyclostationary signals. In particular, the conditions required for the kernel and the driving process to yield a cyclostationary CLRP are investigated. Similar results are obtained for discrete-time conditional linear cyclostationary random processes. This discrete-time class includes the random coefficient periodic autoregressive model, which is suitable for estimation, forecasting, and computer simulation of cyclostationary signals with conditional heteroskedasticity. Finally, applications related to signal modelling in the fields of energy informatics and medicine are discussed.

This research presents a novel framework of rhythm-adaptive stochastic technologies designed for the statistical processing and forecasting of cyclic economic processes. We model a cyclical economic time‐series as the sum of a deterministic trend (polynomial) component and a cyclic random process component which explicitly incorporates variability in the rhythm of oscillations (i.e., the length and timing of cycles). By introducing a rhythm function, we enable adaptive estimation of probabilistic characteristics (expectation, variance, autocorrelation) of the cyclic component, thus improving accuracy relative to fixed‐period models. The methodology is demonstrated through computational experiments using real macroeconomic time‐series data. Results indicate that the rhythm-adaptive approach reduces mean absolute forecasting error by more than 50 % compared to comparable non-adaptive methods. We discuss implementation via a software system (Python ecosystem), outline implications for economic decision support systems, and suggest future directions including multi-process generalisations and integration with machine learning-based forecasting.

The generalized almost-cyclostationary processes (GACS) is a class of continuous sto- chastic processes, for which the statistical moments are almost-periodic functions of time, for every fixed delay. For such processes, the autocovariance function can be expressed as a trigonometric polynomial, with functions of delay used as frequencies. These func- tions are called lag-dependent cycle frequencies. The problem of estimating autocovariance function (or more precisely the generalised cycle autocovariance function), with the unk- nown lag-dependent cycle frequency, will be addressed. The mean-square consistency of the cyclic correlogram, with estimated lag-dependent cycle frequency (as estimator of the generalized cycle autocovariance function), will be considered. A method for estimating lag-dependent cycle frequencies will be presented. Moreover, the performence of the cyc- lic correlogram with estimated lag-depentend cycle frequencies will be evaluated through numerical simulation.

Fraction of Time and Robust Periodic ARMA Methods in Modeling Implied Volatility Time Series Financial time series of implied volatility—the market's forecast of price fluctuation—often exhibit irregularities like sudden spikes, data gaps, and seasonal cycles that challenge standard modeling techniques. We propose a novel framework combining the Fraction-of-Time (FOT) probability approach with robust Periodic ARMA (PARMA) models to address these issues. We analyze time series representing the statistical moments of risk (variance, skewness, and kurtosis) alongside underlying asset indices. By prioritizing time-averaged distributions over theoretical ensemble assumptions, the FOT perspective naturally aligns with the rhythmic, cyclostationary nature of the data. Simultaneously, our robust PARMA component employs estimators resilient to anomalies, handling outliers and missing values without distorting the model. Preliminary results indicate this integrated approach yields higher accuracy than traditional methods, effectively capturing periodicities while mitigating the impact of extreme observations. This methodology offers a resilient tool for analyzing complex, non-stationary time series beyond standard financial models.

The accuracy of electricity consumption forecasting is a critical factor for the efficiency of modern power systems; however, traditional approaches rely on the assumption of deterministic seasonality, neglecting the stochastic variability of cyclic parameters. The objective of this study is to validate a mathematical model of electricity consumption as a cyclical random process to adequately describe the stochastic nature of cyclic fluctuations. The paper employs the mathematical framework of cyclical random processes and conducts a statistical analysis of real-world hourly electricity consumption data acquired from residential smart meters. A model is proposed that enables the structural modeling of the randomness inherent in the cycle parameters themselves (amplitude and phase), in contrast to existing approaches that only model fluctuations around a fixed cycle. Empirical analysis confirmed periodic variations in mathematical expectation, variance, and higher-order moments, thereby demonstrating the cyclical nature of the process and the inadequacy of traditional stationary models. The correlation analysis revealed distinctive peaks in the autocovariance function, indicating the system's "memory" of its diurnal cycle. The results hold practical significance for enhancing forecasting accuracy in power systems and can be adapted for the analysis of other cyclic processes across various domains

The report considers some methods of diagnosing the technical condition of rotating units of power equipment. It is proposed to use linear of autoregressive processes (AR) as mathematical models of vibrations of power equipment units. Such processes belong to linear random processes with discrete time, which have infinitely divisible distribution laws. The peculiarities of such processes are that the autoregressive coefficients are directly related to the kernel of linear random processes with discrete time. This enables the construction of recurrent algorithms for estimating the kernels of linear random processes with discrete time. As an example of the use of the proposed approach, the vibration signal of the rotating unit of the rolling bearing of power equipment is considered. Vibration signals were registered, and estimates of the kernels of linear random processes were obtained. Some parameters of the kernels of linear AR processes are shown, which can be used as diagnostic parameters of the technical condition of the units of power equipment. Some methods for the development of decision rules when using such processes as mathematical models of the information signals are considered.

A model of the vibration signal of a friction pair as a periodically non-stationary random process is presented. The results of experimental studies of vibrations of “dry” and “wet” friction pairs under different friction modes using the proposed model are explained and discussed. The effectiveness of the proposed approach for obtaining estimates of friction modes based on vibration data is shown.

In this work, we present the modified Greenwood statistic and its applications for testing procedures for different distributions, both in univariate and multivariate case. We utilize the stochastic monotonicity property of the modified Greenwood statistic for the wide class of the so-called star-shaped ordered distributions. Firstly, we show the validity of the proposed methodology via simulations study for various classes of star-shaped distributions, i.e. Pareto, alpha-stable and Student’s t distributions, both for univariate and multivariate data. Moreover, we compare the proposed testing procedures with methods known in the literature. Finally, we demonstrate the efficacy of the proposed methodology for real data cases, namely for condition monitoring and financial applications.

We present several results concerning two types of fractional Brownian motions with a random Hurst exponent, which are inspired by recent biological experiments in single particle tracking. In both cases, we introduce basic probabilistic properties such as the q-th moment of the absolute value of the process, the autocovariance function, and the expectation of the time-averaged mean squared displacement. Furthermore, we analyze the ergodic properties of both processes. Alongside the theoretical analysis, we also provide a numerical study of the results. The takk is based on the results of [1]. Bibliography [1] Hubert Woszczek, Agnieszka Wyłomańska, Aleksei Chechkin. "Riemann–Liouville Fractional Brownian Motion with Random Hurst Exponent." Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 35, no. 2, 2025, pp. 0243975.

Cyclostationary models are widely used to describe signals with periodically varying statistical properties. In practical applications such as machine condition monitoring, observed signals often exhibit both cyclostationarity and pronounced non-Gaussian, impulsive behavior whose characteristics may evolve over time. Classical cyclostationary theory, mainly based on Gaussian assumptions, becomes inadequate in such scenarios. This research addresses the problem of segmenting non-Gaussian cyclostationary signals with evolving statistics. Change points in the data patterns can be an indicator of machine damage. Hence, change point identification can be useful in early-stage defect detection. In the simulation study we use a periodic autoregressive model with a dedicated background noise distribution, defined as a mixture of Gaussian noise and random impulses. In our analysis, we assume that the distribution of the background noise varies over time with p (probability of the occurrence of large impulses). In our research, we compare two methods for identifying state (/regime) changes in data: Hidden Markov Model (HMM) and modified MIDAST [1] procedure. The methods are evaluated in terms of change detection accuracy and time complexity.

Cyclostationary analysis is one of the most widely used techniques for extracting diagnostic parameters. The bi-frequency representation of a signal, known as the cyclic spectral coherence (CSC) map, is an effective tool for identifying cyclic components associated with faults. However, the classical method for computing the CSC map becomes inadequate when the signal is corrupted by strongly impulsive, non-Gaussian disturbances. For this purpose, several methods have been proposed, including removing outliers and signal filtering to increase the signal-to-noise ratio. Another approach is to derive robust variants of the CSC map, where the classical autocovariance estimator based on the Pearson correlation coefficient is replaced with robust ones such as Spearman’s or Kendall’s correlation coefficients. Despite the effectiveness of these approaches, they are computationally complex. We propose a simplified framework that reduces complexity through a preliminary signal transformation. The raw signal is first linearly scaled to modify the impact of the subsequent nonlinear transformation (by e.g. arctangent, sigmoid, or logarithmic-based functions). This preprocessing technique improves the classical CSC map by reducing large non-informative observations. The effectiveness of the proposed methods is demonstrated using simulated signals with several levels of non-Gaussianity and validated via CSC map-based diagnostic parameters.

Anomalous diffusion describes processes where the mean squared displacement scales non-linearly with time, with the parameter beta called the anomalous exponent. This behavior, seen in complex systems like biological cell, often defies standard diffusion models. Classical models such as fractional Brownian motion (FBM) and scaled Brownian motion (SBM) assume constant exponents, failing to capture dynamics with varying anomalous parameters. To address this limitations, models like FBM with random exponents (FBMRE) and SBM with random exponents (SBMRE) were introduced. We propose the universal procedure based on statistical testing framework that distinguishes between anomalous diffusion models with constant and random anomalous exponents using time-averaged statistics and their ratio-based counterparts. A novel procedure for optimizing time lag selection via divergence measure (here the Hellinger distance) is also proposed. The introduced methodology applies broadly to constant vs. random anomalous diffusion scenarios, with effectiveness depending on statistic selection, time lags, and process properties, as shown in simulations (with the two-point distribution of anomalous exponent) and real-world data analysis.

This work presents an approach for detecting stress states based on the analysis of nonstationary biomedical signals, specifically electrocardiogram (ECG) and photoplethysmogram (PPG). The study was conducted at the Department of Biotechnical Systems of Ternopil National Technical University (TNTU) within the framework of developing intelligent functional diagnostic systems. The methodology combines time–frequency analysis techniques, including continuous wavelet transform (CWT) and short-time Fourier transform (STFT), with the stochastic energy theory of signals proposed by Prof. Yaroslav P. Dragan, which interprets biomedical signals as realizations of energy processes in the time–frequency domain. This approach is aligned with the scientific school of Prof. Bohdan I. Yavorsky and incorporates findings from Dr. Evhenia B. Yavorska in modeling and optimizing biosensor systems. Results demonstrate that incorporating energy-based descriptors of nonstationary signals allows for more accurate detection of stress-related physiological changes, improving classification accuracy by approximately 12% compared to conventional heart rate variability metrics. The Random Forest classifier achieved the best performance with 91% accuracy and an F1-score of 0.89. The proposed framework can serve as a foundation for intelligent wearable biosensor systems for continuous monitoring of human functional states, contributing to the development of personalized and preventive medicine.