arXiv subjects

Ben D. Fulcher

Publications and source records attributed to Ben D. Fulcher.

At least 19 recordsLinked to original sources

arXiv2026

Parameter inference from a non-stationary unknown process using statistical feature-based slow feature analysis

Non-stationary phenomena are ubiquitous, with examples to be found in climatological measurements, brain activity, and the behavior of financial markets. Starting with a time series from a non-stationary process, a key challenge is to infer the time-varying parameters that underlie the non-stationarity in these systems, without requiring a generative model of the dynamics to be learned. This problem is referred to as Parameter Inference from a Non-stationary Unknown Process (PINUP). Here we introduce a PINUP method called feature-based Slow Feature Analysis (f -SFA) comprising the computation of time-series features across sliding windows, followed by dimension reduction using slow feature analysis (SFA). This allows us to detect slow variation in a potentially wide range of statistical properties of the measured dynamics on a timescale determined by the window length. Crucially, using a comprehensive time-series feature set avoids the subjectivity of feature selection, while the SFA slowness constraint overcomes the bias towards irrelevant correlated features seen with variance-based dimension reduction. The performance of f -SFA surpasses that of four benchmark PINUP methods across a diverse range of non-stationary chaotic processes, and we explore the impact of various parameters on performance, including observation noise, parameter timescales, parameter amplitudes, and unseen parameter values. Further, applying f -SFA to sleep polysomnography data, we show that it is able to infer a time-varying parameter underlying the non-stationary sleep recordings that closely tracks depth of sleep. To our knowledge, this work presents the first comparative study of PINUP methods, and we demonstrate that f -SFA is a simple, effective, and noise-robust approach for quantifying non-stationarity from time series, that can be applied in a range of fields.

physics.data-an↗

arXiv2026

Statistical comparisons of time-series feature sets on classification tasks

In recent years, numerous open-source software libraries have been developed for computing sets of features from univariate time series. The type and number of features vary across these feature sets, which have been constructed with varying disciplinary perspectives on quantifying structure in time-series data. To date, the relative strengths and weaknesses of these feature sets on time-series classification problems remains largely unexplored. Here we aimed to understand the relative performance of six open-source feature sets and three baseline feature sets (based on distributional and/or basic spectral structure) across 124 univariate time-series classification problems using a normalization-based approach to problem-level benchmarking that better indexes the relative strengths and weaknesses of different algorithms compared to prior rank-based approaches. Despite their dramatic differences in size, composition, and computation time, we found that feature sets performed relatively similarly overall (85.3% of pairwise comparisons resulted in ties), with the largest feature set, tsfresh, exhibiting the strongest overall performance (29.03% wins across all pairwise comparisons against other feature sets). We also highlighted specific problems on which the specific composition of a given feature set gave it a substantial performance advantage or disadvantage, and problems where simple baselines comprised of Fourier coefficients and quantiles were sufficient to achieve strong performance. Our results demonstrate the need to consider problem-level performance when benchmarking time-series feature sets, and highlight the importance of feature make-up in driving relative classification performance.

stat.ME↗

arXiv2026

Interpretable model-free inference of parametric variation across time-series data through large-scale feature extraction

Here we address the problem of estimating the dimensionality and nature of parametric variation in an unknown generative process directly from time-series data, without specifying or fitting a model. In particular we suppose that inter-instance variation in collections of time series is caused by parametric variation in the generating model. We hypothesize that, given a sufficiently large library of time-series features, low-dimensional parametric variation will manifest as low-dimensional structure in feature space, enabling interpretable estimators of the underlying degrees of freedom to be constructed. We test our hypothesis using a library of over 7000 diverse and interpretable time-series statistics and thirteen simulated systems with known parametric variation, spanning linear stochastic processes, nonlinear oscillators, and chaotic dynamics. Our unsupervised, data-driven approach often reconstructs the underlying parametric variation across this extensive range of simulated dynamical systems while also yielding interpretable estimators for each underlying dimension. Applied to the movement dynamics of 1143 fruit flies, we use this method to extract biologically meaningful components corresponding to sex and circadian rhythmicity. Our results pave the way for much-needed data-driven methods to bridge the gap between interpretable theoretical understanding of dynamics and the large and complex datasets that characterize modern scientific problems.

physics.data-an↗

arXiv2026

Estimating measures of information processing during cognitive tasks using functional magnetic resonance imaging

Cognition is increasingly framed in terms of information processing, yet most fMRI analyses focus on activation or functional connectivity rather than quantifying how information is stored and transferred. To remedy this problem, we propose a framework for estimating measures of information processing: active information storage (AIS), transfer entropy (TE), and net synergy from task-based fMRI. AIS measures information maintained within a region, TE captures directed information flow, and net synergy contrasts higher-order synergistic to redundant interactions. Crucially, to enable this framework we utilised a recently developed approach for calculating information-theoretic measures: the cross mutual information. This approach combines resting-state and task data to address the challenges of limited sample size, non-stationarity and context in task-based fMRI. We applied this framework to the working memory (N-back) task from the Human Connectome Project (470 participants). Results show that AIS increases in fronto-parietal regions with working memory load, TE reveals enhanced directed information flows across control pathways, and net synergy indicates a global shift to redundancy. This work establishes a novel methodology for quantifying information processing in task-based fMRI.

q-bio.NC↗

arXiv2026

Unifying concepts in information-theoretic time-series analysis

Information theory is a powerful framework for quantifying complexity, uncertainty, and dynamical structure in time-series data, with widespread applicability across disciplines such as physics, finance, and neuroscience. However, the literature on these measures remains fragmented, with domain-specific terminologies, inconsistent mathematical notation, and disparate visualization conventions that hinder interdisciplinary integration. This work addresses these challenges by unifying key information-theoretic time-series measures through shared semantic definitions, standardized mathematical notation, and cohesive visual representations. We compare these measures in terms of their theoretical foundations, computational formulations, and practical interpretability -- mapping them onto a common conceptual space through an illustrative case study with functional magnetic resonance imaging time series in the brain. This case study exemplifies the complementary insights these measures offer in characterizing the dynamics of complex neural systems, such as signal complexity and information flow. By providing a structured synthesis, our work aims to enhance interdisciplinary dialogue and methodological adoption, which is particularly critical for reproducibility and interoperability in computational neuroscience. More broadly, our framework serves as a resource for researchers seeking to navigate and apply information-theoretic time-series measures to diverse complex systems.

cs.IT↗

arXiv2025

Identifying statistical indicators of temporal asymmetry using a data-driven approach

The dynamics of time-reversible systems are statistically indistinguishable when observed forward or backward in time. A rich literature of statistical methods to distinguish irreversible dynamics from the reversible dynamics of linear, Gaussian systems can provide insights into underlying mechanisms and aid modeling and statistical quantification of time-series data. But these existing time-reversibility metrics have been developed individually, forming a fragmented body of research that makes it challenging to identify the most effective approaches developed to date, and the most promising new directions for development. Here we address these issues by systematically evaluating over 6000 time-series summary statistics, derived from across the time-series analysis literature, on their ability to distinguish the time-irreversibility of data simulated from a diverse range of 35 systems. Our large-scale data-driven comparison highlights the effectiveness of several key families of statistics, including time-asymmetric forms of generalized autocorrelation functions, time-series symbolic sequences, and forecasting-related methods. All irreversible systems studied here could be accurately distinguished by a well-chosen time-series statistic, but no single statistic could accurately index the statistical form of irreversibility for all irreversible systems. This challenges the assumption that a given time-reversibility statistic will accurately capture time reversibility in general, and underscores the importance of tailoring statistical approaches to the time-reversal characteristics of a given system. Our results provide a unified understanding of the key algorithmic structures through which irreversibility can be effectively quantified from data, providing a foundation for connecting patterns in time series to the underlying mechanisms of the systems that generate them.

physics.data-an↗

arXiv2025

Cross Mutual Information

Mutual information (MI) is a useful information-theoretic measure to quantify the statistical dependence between two random variables: $X$ and $Y$. Often, we are interested in understanding how the dependence between $X$ and $Y$ in one set of samples compares to another. Although the dependence between $X$ and $Y$ in each set of samples can be measured separately using MI, these estimates cannot be compared directly if they are based on samples from a non-stationary distribution. Here, we propose an alternative measure for characterising how the dependence between $X$ and $Y$ as defined by one set of samples is expressed in another, \textit{cross mutual information}. We present a comprehensive set of simulation studies sampling data with $X$-$Y$ dependencies to explore this measure. Finally, we discuss how this relates to measures of model fit in linear regression, and some future applications in neuroimaging data analysis.

cs.IT↗

arXiv2025

Modeling the influences of non-local connectomic projections on geometrically constrained cortical dynamics

The function and dynamics of the cortex are fundamentally shaped by the specific wiring configurations of its constituent axonal fibers, also known as the connectome. However, many dynamical properties of macroscale cortical activity are well captured by instead describing the activity as propagating waves across the cortical surface, constrained only by the surface's two-dimensional geometry. It thus remains an open question why the local geometry of the cortex can successfully capture macroscale cortical dynamics, despite neglecting the specificity of Fast-conducting, Non-local Projections (FNPs) which are known to mediate the rapid and non-local propagation of activity between remote neural populations. Here we address this question by developing a novel mathematical model of macroscale cortical activity in which cortical populations interact both by a continuous sheet and by an additional set of FNPs wired independently of the sheet's geometry. By simulating the model across a range of connectome topologies, external inputs, and timescales, we demonstrate that the addition of FNPs strongly shape the model dynamics of rapid, stimulus-evoked responses on fine millisecond timescales ($\lessapprox 30~\text{ms}$), but contribute relatively little to slower, spontaneous fluctuations over longer timescales ($> 30~\text{ms}$), which increasingly resemble geometrically constrained dynamics without FNPs. Our results suggest that the discrepant views regarding the relative contributions of local (geometric) and non-local (connectomic) cortico-cortical interactions are context-dependent: While FNPs specified by the connectome are needed to capture rapid communication between specific distant populations (as per the rapid processing of sensory inputs), they play a relatively minor role in shaping slower spontaneous fluctuations (as per resting-state functional magnetic resonance imaging).

q-bio.NC↗

arXiv2025

Using matrix-product states for time-series machine learning

Matrix-product states (MPS) have proven to be a versatile ansatz for modeling quantum many-body physics. For many applications, and particularly in one-dimension, they capture relevant quantum correlations in many-body wavefunctions while remaining tractable to store and manipulate on a classical computer. This has motivated researchers to also apply the MPS ansatz to machine learning (ML) problems where capturing complex correlations in datasets is also a key requirement. Here, we develop and apply an MPS-based algorithm, MPSTime, for learning a joint probability distribution underlying an observed time-series dataset, and show how it can be used to tackle important time-series ML problems, including classification and imputation. MPSTime can efficiently learn complicated time-series probability distributions directly from data, requires only moderate maximum MPS bond dimension $χ_{\rm max}$, with values for our applications ranging between $χ_{\rm max} = 20-160$, and can be trained for both classification and imputation tasks under a single logarithmic loss function. Using synthetic and publicly available real-world datasets, spanning applications in medicine, energy, and astronomy, we demonstrate performance competitive with state-of-the-art ML approaches, but with the key advantage of encoding the full joint probability distribution learned from the data, which is useful for analyzing and interpreting its underlying structure. This manuscript is supplemented with the release of a publicly available code package MPSTime that implements our approach. The effectiveness of the MPS-based ansatz for capturing complex correlation structures in time-series data makes it a powerful foundation for tackling challenging time-series analysis problems across science, industry, and medicine.

stat.ML↗

arXiv2024

Analyzing the Brain's Dynamic Response to Targeted Stimulation using Generative Modeling

Generative models of brain activity have been instrumental in testing hypothesized mechanisms underlying brain dynamics against experimental datasets. Beyond capturing the key mechanisms underlying spontaneous brain dynamics, these models hold an exciting potential for understanding the mechanisms underlying the dynamics evoked by targeted brain-stimulation techniques. This paper delves into this emerging application, using concepts from dynamical systems theory to argue that the stimulus-evoked dynamics in such experiments may be shaped by new types of mechanisms distinct from those that dominate spontaneous dynamics. We review and discuss: (i) the targeted experimental techniques across spatial scales that can both perturb the brain to novel states and resolve its relaxation trajectory back to spontaneous dynamics; and (ii) how we can understand these dynamics in terms of mechanisms using physiological, phenomenological, and data-driven models. A tight integration of targeted stimulation experiments with generative quantitative modeling provides an important opportunity to uncover novel mechanisms of brain dynamics that are difficult to detect in spontaneous settings.

q-bio.NC↗

arXiv2024

Parameter inference from a non-stationary unknown process

Non-stationary systems are found throughout the world, from climate patterns under the influence of variation in carbon dioxide concentration, to brain dynamics driven by ascending neuromodulation. Accordingly, there is a need for methods to analyze non-stationary processes, and yet most time-series analysis methods that are used in practice, on important problems across science and industry, make the simplifying assumption of stationarity. One important problem in the analysis of non-stationary systems is the problem class that we refer to as Parameter Inference from a Non-stationary Unknown Process (PINUP). Given an observed time series, this involves inferring the parameters that drive non-stationarity of the time series, without requiring knowledge or inference of a mathematical model of the underlying system. Here we review and unify a diverse literature of algorithms for PINUP. We formulate the problem, and categorize the various algorithmic contributions. This synthesis will allow researchers to identify gaps in the literature and will enable systematic comparisons of different methods. We also demonstrate that the most common systems that existing methods are tested on - notably the non-stationary Lorenz process and logistic map - are surprisingly easy to perform well on using simple statistical features like windowed mean and variance, undermining the practice of using good performance on these systems as evidence of algorithmic performance. We then identify more challenging problems that many existing methods perform poorly on and which can be used to drive methodological advances in the field. Our results unify disjoint scientific contributions to analyzing non-stationary systems and suggest new directions for progress on the PINUP problem and the broader study of non-stationary phenomena.

physics.data-an↗

arXiv2024

Tracking the distance to criticality in systems with unknown noise

Many real-world systems undergo abrupt changes in dynamics as they move across critical points, often with dramatic consequences. Much existing theory on identifying the time-series signatures of nearby critical points -- such as increased variance and slower timescales -- is derived for the case of fixed, low-amplitude noise. However, real-world systems are often corrupted by unknown levels of noise that can distort these temporal signatures. Here we aimed to develop noise-robust indicators of the distance to criticality (DTC) for systems affected by dynamical noise in two cases: when the noise amplitude is fixed, or is unknown and variable across recordings. To approach this problem, we compare the ability of over 7000 candidate time-series features to track the DTC in the vicinity of a supercritical Hopf bifurcation. We recover existing theory in the fixed-noise case, highlighting conventional time-series features that accurately track the DTC. But in the variable-noise setting, where these conventional indicators perform poorly, we highlight new types of high-performing time-series features and show that their success is accomplished by capturing the shape of the invariant density (which depends on both the DTC and the noise amplitude) relative to the spread of fast fluctuations (which depends on the noise amplitude). We introduce a new high-performing time-series statistic, the Rescaled Auto-Density (RAD), that combines these two algorithmic components. We then use RAD to provide new evidence that brain regions higher in the visual hierarchy are positioned closer to criticality, supporting existing hypotheses about patterns of brain organization that are not detected using conventional metrics of the DTC. Our results demonstrate how large-scale algorithmic comparison can yield theoretical insights that can motivate new theory and interpretable algorithms for real-world problems.

physics.data-an↗

arXiv2024

A feature-based information-theoretic approach for detecting interpretable, long-timescale pairwise interactions from time series

Quantifying relationships between components of a complex system is critical to understanding the rich network of interactions that characterize the behavior of the system. Traditional methods for detecting pairwise dependence of time series, such as Pearson correlation, Granger causality, and mutual information, are computed directly in the space of measured time-series values. But for systems in which interactions are mediated by statistical properties of the time series (`time-series features') over longer timescales, this approach can fail to capture the underlying dependence from limited and noisy time-series data, and can be challenging to interpret. Addressing these issues, here we introduce an information-theoretic method for detecting dependence between time series mediated by time-series features that provides interpretable insights into the nature of the interactions. Our method extracts a candidate set of time-series features from sliding windows of the source time series and assesses their role in mediating a relationship to values of the target process. Across simulations of three different generative processes, we demonstrate that our feature-based approach can outperform a traditional inference approach based on raw time-series values, especially in challenging scenarios characterized by short time-series lengths, high noise levels, and long interaction timescales. Our work introduces a new tool for inferring and interpreting feature-mediated interactions from time-series data, contributing to the broader landscape of quantitative analysis in complex systems research, with potential applications in various domains including but not limited to neuroscience, finance, climate science, and engineering.

physics.data-an↗

arXiv2023

On the information-theoretic formulation of network participation

The participation coefficient is a widely used metric of the diversity of a node's connections with respect to a modular partition of a network. An information-theoretic formulation of this concept of connection diversity, referred to here as participation entropy, has been introduced as the Shannon entropy of the distribution of module labels across a node's connected neighbors. While diversity metrics have been studied theoretically in other literatures, including to index species diversity in ecology, many of these results have not previously been applied to networks. Here we show that the participation coefficient is a first-order approximation to participation entropy and use the desirable additive properties of entropy to develop new metrics of connection diversity with respect to multiple labelings of nodes in a network, as joint and conditional participation entropies. The information-theoretic formalism developed here allows new and more subtle types of nodal connection patterns in complex networks to be studied.

physics.soc-ph↗

arXiv2023

Feature-Based Time-Series Analysis in R using the theft Package

Time series are measured and analyzed across the sciences. One method of quantifying the structure of time series is by calculating a set of summary statistics or `features', and then representing a time series in terms of its properties as a feature vector. The resulting feature space is interpretable and informative, and enables conventional statistical learning approaches, including clustering, regression, and classification, to be applied to time-series datasets. Many open-source software packages for computing sets of time-series features exist across multiple programming languages, including catch22 (22 features: Matlab, R, Python, Julia), feasts (42 features: R), tsfeatures (63 features: R), Kats (40 features: Python), tsfresh (779 features: Python), and TSFEL (390 features: Python). However, there are several issues: (i) a singular access point to these packages is not currently available; (ii) to access all feature sets, users must be fluent in multiple languages; and (iii) these feature-extraction packages lack extensive accompanying methodological pipelines for performing feature-based time-series analysis, such as applications to time-series classification. Here we introduce a solution to these issues in an R software package called theft: Tools for Handling Extraction of Features from Time series. theft is a unified and extendable framework for computing features from the six open-source time-series feature sets listed above. It also includes a suite of functions for processing and interpreting the performance of extracted features, including extensive data-visualization templates, low-dimensional projections, and time-series classification operations. With an increasing volume and complexity of time-series datasets in the sciences and industry, theft provides a standardized framework for comprehensively quantifying and interpreting informative structure in time series.

stat.ML↗

arXiv2023

Unifying Pairwise Interactions in Complex Dynamics

Scientists have developed hundreds of techniques to measure the interactions between pairs of processes in complex systems. But these computational methods, from correlation coefficients to causal inference, rely on distinct quantitative theories that remain largely disconnected. Here we introduce a library of 237 statistics of pairwise interactions and assess their behavior on 1053 multivariate time series from a wide range of real-world and model-generated systems. Our analysis highlights new commonalities between different mathematical formulations, providing a unified picture of a rich interdisciplinary literature. Using three real-world case studies, we then show that simultaneously leveraging diverse methods from across science can uncover those most suitable for addressing a given problem, yielding interpretable understanding of the conceptual formulations of pairwise dependence that drive successful performance. Our framework is provided in extendable open software, enabling comprehensive data-driven analysis by integrating decades of methodological advances.

physics.data-an↗

arXiv2023

Never a Dull Moment: Distributional Properties as a Baseline for Time-Series Classification

The variety of complex algorithmic approaches for tackling time-series classification problems has grown considerably over the past decades, including the development of sophisticated but challenging-to-interpret deep-learning-based methods. But without comparison to simpler methods it can be difficult to determine when such complexity is required to obtain strong performance on a given problem. Here we evaluate the performance of an extremely simple classification approach -- a linear classifier in the space of two simple features that ignore the sequential ordering of the data: the mean and standard deviation of time-series values. Across a large repository of 128 univariate time-series classification problems, this simple distributional moment-based approach outperformed chance on 69 problems, and reached 100% accuracy on two problems. With a neuroimaging time-series case study, we find that a simple linear model based on the mean and standard deviation performs better at classifying individuals with schizophrenia than a model that additionally includes features of the time-series dynamics. Comparing the performance of simple distributional features of a time series provides important context for interpreting the performance of complex time-series classification models, which may not always be required to obtain high accuracy.

stat.ME↗

arXiv2022

Classifying Kepler light curves for 12,000 A and F stars using supervised feature-based machine learning

With the availability of large-scale surveys like Kepler and TESS, there is a pressing need for automated methods to classify light curves according to known classes of variable stars. We introduce a new algorithm for classifying light curves that compares 7000 time-series features to find those which most effectively classify a given set of light curves. We apply our method to Kepler light curves for stars with effective temperatures in the range 6500--10,000K. We show that the sample can be meaningfully represented in an interpretable five-dimensional feature space that separates seven major classes of light curves (delta Scuti stars, gamma Doradus stars, RR Lyrae stars, rotational variables, contact eclipsing binaries, detached eclipsing binaries, and non-variables). We achieve a balanced classification accuracy of 82% on an independent test set of Kepler stars using a Gaussian mixture model classifier. We use our method to classify 12,000 Kepler light curves from Quarter 9 and provide a catalogue of the results. We further outline a confidence heuristic based on probability density with which to search our catalogue, and extract candidate lists of correctly-classified variable stars.

astro-ph.SR↗