arXiv subjects

Michał Balcerek

Publications and source records attributed to Michał Balcerek.

11 recordsLinked to original sources

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↗

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

Machine learning models for predicting catastrophe bond coupons using climate data

In recent years, the growing frequency and severity of natural disasters have increased the need for effective tools to manage catastrophe risk. Catastrophe (CAT) bonds allow the transfer of part of this risk to investors, offering an alternative to traditional reinsurance. This paper examines the role of climate variability in CAT bond pricing and evaluates the predictive performance of various machine learning models in forecasting CAT bond coupons. We combine features typically used in the literature with a new set of climate indicators, including Oceanic Ni{ñ}o Index, Arctic Oscillation, North Atlantic Oscillation, Outgoing Longwave Radiation, Pacific-North American pattern, Pacific Decadal Oscillation, Southern Oscillation Index, and sea surface temperatures. We compare the performance of linear regression with several machine learning algorithms, such as random forest, gradient boosting, extremely randomized trees, and extreme gradient boosting. Our results show that including climate-related variables improves predictive accuracy across all models, with extremely randomized trees achieving the lowest root mean squared error (RMSE). These findings suggest that large-scale climate variability has a measurable influence on CAT bond pricing and that machine learning methods can effectively capture these complex relationships.

q-fin.PR↗

arXiv2025

CINNAMON: A hybrid approach to change point detection and parameter estimation in single-particle tracking data

Change point detection has become an important part of the analysis of the single-particle tracking data, as it allows one to identify moments, in which the motion patterns of observed particles undergo significant changes. The segmentation of diffusive trajectories based on those moments may provide insight into various phenomena in soft condensed matter and biological physics. In this paper, we propose CINNAMON, a hybrid approach to classifying single-particle tracking trajectories, detecting change points within them, and estimating diffusion parameters in the segments between the change points. Our method is based on a combination of neural networks, feature-based machine learning, and statistical techniques. It has been benchmarked in the second Anomalous Diffusion Challenge. The method offers a high level of interpretability due to its analytical and feature-based components. A potential use of features from topological data analysis is also discussed.

q-bio.QM↗

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

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

Two-dimensional Brownian motion with dependent components: turning angle analysis

Brownian motion in one or more dimensions is extensively used as a stochastic process to model natural and engineering signals, as well as financial data. Most works dealing with multidimensional Brownian motion consider the different dimensions as independent components. In this article, we investigate a model of correlated Brownian motion in $\mathbb{R}^2$, where the individual components are not necessarily independent. We explore various statistical properties of the process under consideration, going beyond the conventional analysis of the second moment. Our particular focus lies on investigating the distribution of turning angles. This distribution reveals particularly interesting characteristics for processes with dependent components that are relevant to applications in diverse physical systems. Theoretical considerations are supported by numerical simulations and analysis of two real-world datasets: the financial data of the Dow Jones Industrial Average and the Standard and Poor's 500, and trajectories of polystyrene beads in water. Finally, we show that the model can be readily extended to trajectories with correlations that change over time.

cond-mat.stat-mech↗

arXiv2024

Langevin equation in heterogeneous landscapes: how to choose the interpretation

The Langevin equation is a common tool to model diffusion at a single-particle level. In non-homogeneous environments, such as aqueous two-phase systems or biological condensates with different diffusion coefficients in different phases, the solution to a Langevin equation is not unique unless the interpretation of stochastic integrals involved is selected. We analyze the diffusion of particles in such systems and evaluate the mean, the mean square displacement, and the distribution of particles, as well as the variance of the time-averaged mean-squared displacements. Our analytical results provide a method to choose the interpretation parameter from single particle tracking experiments.

cond-mat.stat-mech↗

arXiv2023

Modelling intermittent anomalous diffusion with switching fractional Brownian motion

The stochastic trajectories of molecules in living cells, as well as the dynamics in many other complex systems, often exhibit memory in their path over long periods of time. In addition, these systems can show dynamic heterogeneities due to which the motion changes along the trajectories. Such effects manifest themselves as spatiotemporal correlations. Despite the broad occurrence of heterogeneous complex systems in nature, their analysis is still quite poorly understood and tools to model them are largely missing. We contribute to tackling this problem by employing an integral representation of Mandelbrot's fractional Brownian motion that is compliant with varying motion parameters while maintaining long memory. Two types of switching fractional Brownian motion are analysed, with transitions arising from a Markovian stochastic process and scale-free intermittent processes. We obtain simple formulas for classical statistics of the processes, namely the mean squared displacement and the power spectral density. Further, a method to identify switching fractional Brownian motion based on the distribution of displacements is described. A validation of the model is given for experimental measurements of the motion of quantum dots in the cytoplasm of live mammalian cells that were obtained by single-particle tracking.

cond-mat.stat-mech↗

arXiv2022

Fractional Brownian motion with random Hurst exponent: accelerating diffusion and persistence transitions

Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity, or the Hurst exponent. However, recent single particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that can not be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process which preserves the properties of fractional Brownian motion at a single trajectory level, however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical and statistical analysis of fractional Brownian motion with random Hurst exponent. The explicit formulas for probability density function, mean square displacement and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely two-point, uniform and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition which we demonstrate analytically and numerically.

cond-mat.stat-mech↗

arXiv2022

About subordinated generalizations of 3 classical models of option pricing

In this paper, we investigate the relation between Bachelier and Black-Scholes models driven by the infinitely divisible inverse subordinators. Such models, in contrast to their classical equivalents, can be used in markets where periods of stagnation are observed. We introduce the subordinated Cox-Ross-Rubinstein model and prove that the price of the underlying in that model converges in distribution and in Skorokhod space to the price of underlying in the subordinated Black-Scholes model defined in [31]. Motivated by this fact we price the selected option contracts using the binomial trees. The results are compared to other numerical methods.

math.NA↗