arXiv subjects

Agnieszka Wyłomańska

Publications and source records attributed to Agnieszka Wyłomańska.

At least 19 recordsLinked to original sources

arXiv2026

Real-time and adaptive anomaly detection algorithm for cyclostationary models

This article introduces PeriodicCALM, an effective real-time anomaly detection framework designed for cyclostationary data streams. While classical cyclostationary processes feature periodically time-varying statistical properties, real-world signals often contain recurring impulsive components that conceal abnormal behavior. Existing real-time methods for struggle with these dynamics, frequently misinterpreting phase-dependent variability as non-cyclic anomalies and causing excessive false alarms. To address this, PeriodicCALM incorporates cycle-dependent variability to systematically ignore regular cyclic impulses while accurately isolating genuine anomalies. Operating in real time with continuous retraining capabilities, the method adapts dynamically to evolving signal characteristics. Comparative evaluations against the baseline CALM framework using simulated data demonstrate significant improvements in detection accuracy and training efficiency, alongside a reduction in prediction latency. Furthermore, the practical utility of PeriodicCALM is validated on real-world vibration signals collected from a compressor monitoring system.

stat.ME↗

arXiv2026

A systematic framework for the identification and statistical quantification of impulsivity in condition monitoring signals

This article proposes a comprehensive framework for the identification and statistical quantification of impulsive behavior in signals, with a primary focus on condition monitoring. We concentrate on evaluating impulsivity, where such behavior results from normal operation or additional disturbances. Such an evaluation is crucial in the context of local damage detection, as the presence of impulsive disturbances significantly complicates the machine condition monitoring process. To address problem of impulsivity assessment we introduce a two-stage methodology to make processing workflow effective. First, we propose an objective selection criterion for "best-performing" impulsivity measures based on the Mann-Whitney statistic, allowing for the systematic comparison of various classical and advanced metrics across diverse signal scenarios. Second, we establish a formal procedure for assessing statistical significance using bootstrap-driven resampling and define a magnitude index to quantify the intensity of detected impulsivity. The framework is validated through extensive Monte Carlo simulations for three reference signal scenarios and applied to real-world vibration data from an industrial compressor. By systematizing existing measures and providing a statistically grounded pipeline, this research extends prior works, offering a scalable tool for distinguishing between diagnostically useful signals and those corrupted by anomalous interference.

stat.ME↗

arXiv2026

Turning angle analysis reveals hidden anisotropies in the anomalous diffusion of molecules in live cells

Molecular motion within living cells provides a window into the physical principles underlying intracellular organization and transport. Yet a fundamental limitation remains: measured trajectories are often short, noisy, and randomly oriented, rendering spatial anisotropies inaccessible to conventional analyses. We show that turning-angle statistics provide a robust approach for uncovering anisotropies in anomalous diffusion. Using a two-dimensional anisotropic fractional Brownian motion model with random orientations, we show theoretically and through simulations that turning-angle distributions preserve signatures of anisotropy. We apply this approach to single-particle tracking data, including quantum dots in the cytoplasm of HeLa cells and membrane proteins in hippocampal neurons. Turning angles reveal hidden anisotropies in the motion of quantum dots and Nav 1.6 channels, and in specific dynamical states of glycoprotein CD4. These results establish turning-angle analysis as a powerful strategy for detecting organization in complex environments and reveal that anisotropic anomalous diffusion is an overlooked feature of intracellular dynamics.

cond-mat.stat-mech↗

arXiv2026

Real-time anomaly detection in base station testbeds via scalable kernel density estimation framework

Large-scale testing infrastructures are critical for validating telecommunication systems, yet their growing complexity makes efficient resource utilization and anomaly detection increasingly challenging. In reservation-based testbed environments, errors in resource allocation or preparation often manifest as abrupt spikes or regime changes in time-based metrics. This paper proposes a scalable, unsupervised framework for real-time anomaly detection in such environments. We introduce CALM (Continuous Anomaly Localization for univariate and Multivariate data), a nonparametric method based on kernel density estimation and bootstrap-based thresholding, designed for anomaly detection at the individual testbed level. To address system-wide visibility, we further propose AggCALM, an aggregation framework that consolidates local anomaly signals across multiple testbeds to detect statistically significant global anomalies while mitigating alarm fatigue. The methodology is evaluated using simulated multivariate data and real-world data from a large-scale base station testing platform. Results demonstrate that the proposed framework enables timely, flexible, and accurate anomaly detection without requiring labeled data, supporting reliable operation of complex test environments. Although the proposed methodology is presented within the context of a telecommunication testing labs, it can be effectively used to other applications, such as condition monitoring, where anomaly detection serves as a pivotal pre-processing step for diagnostic signals.

stat.AP↗

arXiv2026

Scaling limits of Lévy walks with random velocities

This paper investigates Lévy walks with random velocities, extending classical models beyond constant speed assumptions. We derive scaling limits, demonstrating that diffusion depends on interplay between heavy-tailed duration and velocity distributions. Three distinct scaling regimes are identified, including a critical case with logarithmic corrections, offering a precise framework for modeling anomalous transport in heterogeneous systems.

math.PR↗

arXiv2026

Testing and estimation of the index of stability of univariate and bivariate symmetric $α-$stable distributions via modified Greenwood statistic

We propose a testing and estimation methodology for univariate and bivariate symmatric $α$-stable distributions using a modified version of the Greenwood statistic. Originally designed for positive-valued random variables, the Greenwood statistic, and its modified version tailored for symmetric distributions, have been predominantly applied to univariate random samples. In this paper, we extend the modified Greenwood statistic to a bivariate setting and examine its probabilistic properties within the class of $α$-stable distributions, with a focus on the sub-Gaussian case. Additionally, we introduce a novel testing approach that considers two variations of the modified Greenwood statistic as test statistics for the bivariate case. In the univariate setting, we adapt the proposed testing methodology for estimating the stability index. The simulation studies presented demonstrate that our proposed methodology outperforms classical approaches previously used in this context and serves as an effective tool for distinguishing between Gaussian and $α$-stable distributions with a stability index close to 2. The theoretical and simulation results are further illustrated with practical data examples.

stat.ME↗

arXiv2026

Fractional lower-order covariance-based measures for cyclostationary time series with heavy-tailed distributions: application to dependence testing and model order identification

This article introduces new methods for the analysis of cyclostationary time series with infinite variance. Traditional cyclostationary analysis, based on periodically correlated (PC) processes, relies on the autocovariance function (ACVF). However, the ACVF is not suitable for data exhibiting a heavy-tailed distribution, particularly with infinite variance. Thus, we propose a novel framework for the analysis of cyclostationary time series with heavy-tailed distribution, utilizing the fractional lower-order covariance (FLOC) as an alternative to covariance. This leads to the introduction of two new autodependence measures: the periodic fractional lower-order autocorrelation function (peFLOACF) and the periodic fractional lower-order partial autocorrelation function (peFLOPACF). These measures generalize the classical periodic autocorrelation function (peACF) and periodic partial autocorrelation function (pePACF), offering robust tools for analyzing infinite-variance processes. Two practical applications of the proposed measures are explored: a portmanteau test for testing dependence in cyclostationary series and a method for order identification in periodic autoregressive (PAR) and periodic moving average (PMA) models with infinite variance. Both applications demonstrate the potential of new tools, with simulations validating their efficiency. The methodology is further illustrated through the analysis of real-world air pollution data, which showcases its practical utility. The results indicate that the proposed measures based on FLOC provide reliable and efficient techniques for analyzing cyclostationary processes with heavy-tailed distributions.

stat.ME↗

arXiv2025

Two-dimensional fractional Brownian motion: Analysis in time and frequency domains

This article introduces a novel construction of the two-dimensional fractional Brownian motion (2D fBm) with dependent components. Unlike similar models discussed in the literature, our approach uniquely accommodates the full range of model parameters and explicitly incorporates cross-dependencies and anisotropic scaling through a matrix-valued Hurst operator. We thoroughly analyze the theoretical properties of the proposed causal and well-balanced 2D fBm versions, deriving their auto- and cross-covariance structures in both time and frequency domains. In particular, we present the power spectral density of these processes and their increments. Our analytical findings are validated with numerical simulations. This work provides a comprehensive framework for modeling anomalous diffusion phenomena in multidimensional systems where component interdependencies are crucial.

math.PR↗

arXiv2025

Super-resolved anomalous diffusion: deciphering the joint distribution of anomalous exponent and diffusion coefficient

The molecular motion in heterogeneous media displays anomalous diffusion by the mean-squared displacement $\langle X^2(t) \rangle = 2 D t^α$. Motivated by experiments reporting populations of the anomalous diffusion parameters $α$ and $D$, we aim to disentangle their respective contributions to the observed variability when this last is due to a true population of these parameters and when it arises due to finite-duration recordings. We introduce estimators of the anomalous diffusion parameters on the basis of the time-averaged mean squared displacement and study their statistical properties. By using a copula approach, we derive a formula for the joint density function of their estimations conditioned on their actual values. The methodology introduced is indeed universal, it is valid for any Gaussian process and can be applied to any quadratic time-averaged statistics. We also explain the experimentally reported relation $D\propto\exp(αc_1+c_2)$ for which we provide the exact expression. We finally compare our findings to numerical simulations of the fractional Brownian motion and quantify their accuracy by using the Hellinger distance.

physics.bio-ph↗

arXiv2025

Multifractional Brownian motion with telegraphic, stochastically varying exponent

The diversity of diffusive systems exhibiting long-range correlations characterized by a stochastically varying Hurst exponent calls for a generic multifractional model. We present a simple, analytically tractable model which fills the gap between mathematical formulations of multifractional Brownian motion and empirical studies. In our model, called telegraphic multifractional Brownian motion, the Hurst exponent is modelled by a smoothed telegraph process which results in a stationary beta distribution of exponents as observed in biological experiments. We also provide a methodology to identify our model in experimental data and present concrete examples from biology, climate and finance to demonstrate the efficacy of our approach.

cond-mat.stat-mech↗

arXiv2025

Robust Suboptimal Local Basis Function Algorithms for Identification of Nonstationary FIR Systems in Impulsive Noise Environments

While local basis function (LBF) estimation algorithms, commonly used for identifying/tracking systems with time-varying parameters, demonstrate good performance under the assumption of normally distributed measurement noise, the estimation results may significantly deviate from satisfactory when the noise distribution is impulsive in nature, for example, corrupted by outliers. This paper introduces a computationally efficient method to make the LBF estimator robust, enhancing its resistance to impulsive noise. First, the choice of basis functions is optimized based on the knowledge of parameter variation statistics. Then, the parameter tracking algorithm is made robust using the sequential data trimming technique. Finally, it is demonstrated that the proposed algorithm can undergo online tuning through parallel estimation and leave-one-out cross-validation.

eess.SP↗

arXiv2025

Statistical applications of the 20/60/20 rule in risk management and portfolio optimization

This paper explores the applications of the 20/60/20 rule-a heuristic method that segments data into top-performing, average-performing, and underperforming groups-in mathematical finance. We review the statistical foundations of this rule and demonstrate its usefulness in risk management and portfolio optimization. Our study highlights three key applications. First, we apply the rule to stock market data, showing that it enables effective population clustering. Second, we introduce a novel, easy-to-implement method for extracting heavy-tail characteristics in risk management. Third, we integrate spatial reasoning based on the 20/60/20 rule into portfolio optimization, enhancing robustness and improving performance. To support our findings, we develop a new measure for quantifying tail heaviness and employ conditional statistics to reconstruct the unconditional distribution from the core data segment. This reconstructed distribution is tested on real financial data to evaluate whether the 20/60/20 segmentation effectively balances capturing extreme risks with maintaining the stability of central returns. Our results offer insights into financial data behavior under heavy-tailed conditions and demonstrate the potential of the 20/60/20 rule as a complementary tool for decision-making in finance.

q-fin.PM↗

arXiv2025

Evaluating Gaussianity of heterogeneous fractional Brownian motion

Heterogeneous diffusion processes are prevalent in various fields, including the motion of proteins in living cells, the migratory movement of birds and mammals, and finance. These processes are often characterized by time-varying dynamics, where interactions with the environment evolve, and the system undergoes fluctuations in diffusivity. Moreover, in many complex systems anomalous diffusion is observed, where the mean square displacement (MSD) exhibits non-linear scaling with time. Among the models used to describe this phenomenon, fractional Brownian motion (FBM) is a widely applied stochastic process, particularly for systems exhibiting long-range temporal correlations. Although FBM is characterized by Gaussian increments, heterogeneous processes with FBM-like characteristics may deviate from Gaussianity. In this article, we study the non-Gaussian behavior of switching fractional Brownian motion (SFBM), a model in which the diffusivity of the FBM process varies while temporal correlations are maintained. To characterize non-Gaussianity, we evaluate the kurtosis, a common tool used to quantify deviations from the normal distribution. We derive exact expressions for the kurtosis of the considered heterogeneous anomalous diffusion process and investigate how it can identify non-Gaussian behavior. We also compare the kurtosis results with those obtained using the Hellinger distance, a classical measure of divergence between probability density functions. Through both analytical and numerical methods, we demonstrate the potential of kurtosis as a metric for detecting non-Gaussianity in heterogeneous anomalous diffusion processes.

cond-mat.stat-mech↗

arXiv2025

Robust variance estimators in application to segmentation of measurement data distorted by impulsive and non-Gaussian noise

The paper algorithmizes the problem of regime change point identification for data measured in a system exhibiting impulsive behaviors. This is a fundamental challenge for annotation of measurement data relevant, e.g., for designing data-driven autonomous systems. The contribution consists in the formulation of an offline robust methodology based on the classical approach for structural break detection. The problem of data segmentation is considered in the context of scale change, which physically can be translated into the occurrence of a critical event that reorganizes the system structure. The main advantage of our approach is that it does not require the existence of a variance of the data distribution. The efficiency has been evaluated for simulated data from two distributions and for real-world datasets measured in financial, mechanical, and medical systems. Simulation studies show that in the most challenging case, the error in estimating regime change is 20 times smaller for robust approach compared to the classical one.

stat.ME↗

arXiv2025

A procedure for assessing of machine health index data prediction quality

The paper discusses the challenge of evaluating the prognosis quality of machine health index (HI) data. Many existing solutions in machine health forecasting involve visually assessing the quality of predictions to roughly gauge the similarity between predicted and actual samples, lacking precise measures or decisions. In this paper, we introduce a universal procedure with multiple variants and criteria. The overarching concept involves comparing predicted data with true HI time series, but each procedure variant has a specific pattern determined through statistical analysis. Additionally, a statistically established threshold is employed to classify the result as either a reliable or non-reliable prognosis. The criteria include both simple measures (MSE, MAPE) and more advanced ones (Space quantiles-inclusion factor, Kupiec's POF, and TUFF statistics). Depending on the criterion chosen, the pattern and decision-making process vary. To illustrate effectiveness, we apply the proposed procedure to HI data sourced from the literature, covering both warning (linear degradation) and critical (exponential degradation) stages. While the method yields a binary output, there is potential for extension to a multi-class classification. Furthermore, experienced users can use the quality measure expressed in percentage for more in-depth analysis.

eess.SP↗

arXiv2025

Applications of robust statistics for cyclostationarity detection in non-Gaussian signals for local damage detection in bearings

Signals with periodic characteristics are ubiquitous in real-world applications. One of these areas is condition monitoring, where the vibration signals from rotating machines naturally display periodic behavior. Thus, the cyclostationary analysis has evolved into the investigation of such signals. For the traditional cyclostationary approaches, the autocovariance function (ACVF) and its bi-frequency representation, spectral coherence (SC), are regarded as the base. However, recent research has revealed that real vibration signals increasingly exhibit impulsive behavior in addition to periodicity. As a result, there was a need for new methods to identify periodic behavior that take into account the impulsiveness of the data. In this article, we provide a way to improve the SC method by using its robust variants in place of the classical ACVF estimator (sample ACVF). The suggested concept is intuitive and relatively simple. We create robust versions of the SC algorithm that more accurately detect periodic behavior in signals with significant disruptions in contrast to the classical techniques. The efficiency of the proposed approach is demonstrated for simulated signals with three different types of non-Gaussian noise distribution and different levels of periodic impulses imitating a local damage. The introduced approach is also validated on real vibration signal from the rolling element bearings operating in a crushing machine.

stat.AP↗

arXiv2024

Identification of fault frequency variation in the envelope spectrum in the vibration-based local damage detection in possible changing load/speed conditions

The problem of local damage diagnosis (based on the detection of impulsive and periodic signals) is discussed. Both features should be checked, as fault frequency must be linked to the true value calculated for a given machine and speed. The precise estimation of the fault frequency is hard due to several factors. If a speed fluctuation exists, it is solved by order analysis. A wider perspective is proposed here, namely, an automatic statistical approach to analyze the distribution of estimated fault frequencies. We propose a procedure to evaluate whether the fault frequency is constant or not. The algorithm uses frequency estimation based on peak detection in the envelope spectrum and statistical testing. We present simulation studies and industrial examples. We have found that if the fault frequency is not constant and its distribution does not follow Gaussian shape with minor variance, then one should use more advanced techniques, e.g. order analysis.

eess.SP↗

arXiv2024

Conditional correlation estimation and serial dependence identification

It has been recently shown in Jaworski, P., Jelito, D. and Pitera, M. (2024), 'A note on the equivalence between the conditional uncorrelation and the independence of random variables', Electronic Journal of Statistics 18(1), that one can characterise the independence of random variables via the family of conditional correlations on quantile-induced sets. This effectively shows that the localized linear measure of dependence is able to detect any form of nonlinear dependence for appropriately chosen conditioning sets. In this paper, we expand this concept, focusing on the statistical properties of conditional correlation estimators and their potential usage in serial dependence identification. In particular, we show how to estimate conditional correlations in generic and serial dependence setups, discuss key properties of the related estimators, define the conditional equivalent of the autocorrelation function, and provide a series of examples which prove that the proposed framework could be efficiently used in many practical econometric applications.

stat.ME↗