arXiv subjects

Piotr Kokoszka

Publications and source records attributed to Piotr Kokoszka.

At least 19 recordsLinked to original sources

arXiv2026

Dimension-invariant uniform consistency of the empirical spatial distribution function and its associated spatial depth estimator

We provide a proof that the empirical spatial distribution estimator in $\mathbb R^d$ as well as the corresponding plug-in estimator of the spatial depth are uniformly $L^1$-consistent. The consistency rate only depends on the sample size $n$, not on the dimension $d$ or any tuning or regularization parameters. This is a rare property. The result of this note originates from a conversation with ChatGPT 5.4 Pro as part of some of our own earlier experiments on its mathematical reasoning capabilities.

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arXiv2026

Does PCA Work for Rough Functional Data?

Functional data analysis is concerned with the analysis of infinite-dimensional data functions. Functional principal component analysis (FPCA) is a key method to obtain finite-dimensional summaries. Consistency of FPCA has been theoretically established for sufficiently regular data functions. However, empirical evidence shows that FPCA can become severely inconsistent when the underlying functions are too rough. This paper provides the first theoretical explanation for this phenomenon. We propose a model that explicitly captures the roughness of functional data and allows us to quantify the resulting bias of FPCA, depending on the functional roughness. The model undergoes a phase transition marking the point at which FPCA becomes entirely uninformative. Based on these probabilistic results, we discuss diagnostic tests for informative principal components. As an additional contribution, we derive results on spectral statistics that may serve as a foundation for goodness-of-fit tests for rough functional data. Mathematically, our approach combines recent advances in random matrix theory and generic chaining with tools from FDA. We illustrate the effects of roughness on FPCA using simulations, as well as climate and environmental datasets.

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arXiv2026

Deep learning estimation of the spectral density of functional time series on large domains

We derive an estimator of the spectral density of a functional time series that is the output of a multilayer perceptron neural network. The estimator is motivated by difficulties with the computation of existing spectral density estimators for time series of functions defined on very large grids that arise, for example, in climate compute models and medical scans. Existing estimators use autocovariance kernels represented as large $G \times G$ matrices, where $G$ is the number of grid points on which the functions are evaluated. In many recent applications, functions are defined on 2D and 3D domains, and $G$ can be of the order $G \sim 10^5$, making the evaluation of the autocovariance kernels computationally intensive or even impossible. We use the theory of spectral functional principal components to derive our deep learning estimator and prove that it is a universal approximator to the spectral density under general assumptions. Our estimator can be trained without computing the autocovariance kernels and it can be parallelized to provide the estimates much faster than existing approaches. We validate its performance by simulations and an application to fMRI images.

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arXiv2025

Two approaches to multiple canonical correlation analysis for repeated measures data

In classical canonical correlation analysis (CCA), the goal is to determine the linear transformations of two random vectors into two new random variables that are most strongly correlated. Canonical variables are pairs of these new random variables, while canonical correlations are correlations between these pairs. In this paper, we propose and study two generalizations of this classical method: (1) Instead of two random vectors we study more complex data structures that appear in important applications. In these structures, there are $L$ features, each described by $p_l$ scalars, $1 \le l \le L$. We observe $n$ such objects over $T$ time points. We derive a suitable analog of the CCA for such data. Our approach relies on embeddings into Reproducing Kernel Hilbert Spaces, and covers several related data structures as well. (2) We develop an analogous approach for multidimensional random processes. In this case, the experimental units are multivariate continuous, square-integrable functions over a given interval. These functions are modeled as elements of a Hilbert space, so in this case, we define the multiple functional canonical correlation analysis, MFCCA. We justify our approaches by their application to two data sets and suitable large sample theory. We derive consistency rates for the related transformation and correlation estimators, and show that it is possible to relax two common assumptions on the compactness of the underlying cross-covariance operators and the independence of the data.

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arXiv2025

Extremal correlation coefficient for functional data

We propose a coefficient that measures dependence in paired samples of functions. It has properties similar to the Pearson correlation, but differs in significant ways: (i) it is designed to measure dependence between curves, (ii) it focuses only on extreme curves. The new coefficient is derived within the framework of regular variation in Banach spaces. A consistent estimator is proposed and justified by an asymptotic analysis and a simulation study. The usefulness of the new coefficient is illustrated on financial and and climate functional data.

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arXiv2025

Monitoring for a Phase Transition in a Time Series of Wigner Matrices

We develop methodology and theory for the detection of a phase transition in a time-series of high-dimensional random matrices. In the model we study, at each time point $ t = 1,2,\ldots $, we observe a deformed Wigner matrix $ \mathbf{M}_t $, where the unobservable deformation represents a latent signal. This signal is detectable only in the supercritical regime, and our objective is to detect the transition to this regime in real time, as new matrix--valued observations arrive. Our approach is based on a partial sum process of extremal eigenvalues of $\mathbf{M}_t$, and its theoretical analysis combines state-of-the-art tools from random-matrix-theory and Gaussian approximations. The resulting detector is self-normalized, which ensures appropriate scaling for convergence and a pivotal limit, without any additional parameter estimation. Simulations show excellent performance for varying dimensions. Applications to pollution monitoring and social interactions in primates illustrate the usefulness of our approach.

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arXiv2025

Prokhorov Metric Convergence of the Partial Sum Process for Reconstructed Functional Data

Motivated by applications in functional data analysis, we study the partial sum process of sparsely observed, random functions. A key novelty of our analysis are bounds for the distributional distance between the limit Brownian motion and the entire partial sum process in the function space. To measure the distance between distributions, we employ the Prokhorov and Wasserstein metrics. We show that these bounds have important probabilistic implications, including strong invariance principles and new couplings between the partial sums and their Gaussian limits. Our results are formulated for weakly dependent, nonstationary time series in the Banach space of d-dimensional, continuous functions. Mathematically, our approach rests on a new, two-step proof strategy: First, using entropy bounds from empirical process theory, we replace the function-valued partial sum process by a high-dimensional discretization. Second, using Gaussian approximations for weakly dependent, high-dimensional vectors, we obtain bounds on the distance. As a statistical application of our coupling results, we validate an open-ended monitoring scheme for sparse functional data. Existing probabilistic tools were not appropriate for this task.

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arXiv2025

Estimation of the long-run variance of nonlinear time series with an application to change point analysis

For a broad class of nonlinear time series known as Bernoulli shifts, we establish the asymptotic normality of the smoothed periodogram estimator of the long-run variance. This estimator uses only a narrow band of Fourier frequencies around the origin and so has been extensively used in local Whittle estimation. Existing asymptotic normality results apply only to linear time series, so our work substantially extends the scope of the applicability of the smoothed periodogram estimator. As an illustration, we apply it to a test of changes in mean against long-range dependence. A simulation study is also conducted to illustrate the performance of the test for nonlinear time series.

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arXiv2024

Detection of a structural break in intraday volatility pattern

We develop theory leading to testing procedures for the presence of a change point in the intraday volatility pattern. The new theory is developed in the framework of Functional Data Analysis. It is based on a model akin to the stochastic volatility model for scalar point-to-point returns. In our context, we study intraday curves, one curve per trading day. After postulating a suitable model for such functional data, we present three tests focusing, respectively, on changes in the shape, the magnitude and arbitrary changes in the sequences of the curves of interest. We justify the respective procedures by showing that they have asymptotically correct size and by deriving consistency rates for all tests. These rates involve the sample size (the number of trading days) and the grid size (the number of observations per day). We also derive the corresponding change point estimators and their consistency rates. All procedures are additionally validated by a simulation study and an application to US stocks.

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arXiv2023

Functional diffusion driven stochastic volatility model

We propose a stochastic volatility model for time series of curves. It is motivated by dynamics of intraday price curves that exhibit both between days dependence and intraday price evolution. The curves are suitably normalized to stationary in a function space and are functional analogs of point-to-point daily returns. The between curves dependence is modeled by a latent autoregression. The within curves behavior is modeled by a diffusion process. We establish the properties of the model and propose several approaches to its estimation. These approaches are justified by asymptotic arguments that involve an interplay between between the latent autoregression and the intraday diffusions. The asymptotic framework combines the increasing number of daily curves and the refinement of the discrete grid on which each daily curve is observed. Consistency rates for the estimators of the intraday volatility curves are derived as well as the asymptotic normality of the estimators of the latent autoregression. The estimation approaches are further explored and compared by an application to intraday price curves of over seven thousand U.S. stocks and an informative simulation study.

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arXiv2021

Testing normality of spatially indexed functional data

We develop a test of normality for spatially indexed functions. The assumption of normality is common in spatial statistics, yet no significance tests, or other means of assessment, have been available for functional data. This paper aims at filling this gap in the case of functional observations on a spatial grid. Our test compares the moments of the spatial (frequency domain) principal component scores to those of a suitable Gaussian distribution. Critical values can be readily obtained from a chi-squared distribution. We provide rigorous theoretical justification for a broad class of weakly stationary functional random fields. We perform simulation studies to assess the the power of the test against various alternatives. An application to Surface Incoming Shortwave Radiation illustrates the practical value of this procedure.

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arXiv2020

Wasserstein Autoregressive Models for Density Time Series

Data consisting of time-indexed distributions of cross-sectional or intraday returns have been extensively studied in finance, and provide one example in which the data atoms consist of serially dependent probability distributions. Motivated by such data, we propose an autoregressive model for density time series by exploiting the tangent space structure on the space of distributions that is induced by the Wasserstein metric. The densities themselves are not assumed to have any specific parametric form, leading to flexible forecasting of future unobserved densities. The main estimation targets in the order-$p$ Wasserstein autoregressive model are Wasserstein autocorrelations and the vector-valued autoregressive parameter. We propose suitable estimators and establish their asymptotic normality, which is verified in a simulation study. The new order-$p$ Wasserstein autoregressive model leads to a prediction algorithm, which includes a data driven order selection procedure. Its performance is compared to existing prediction procedures via application to four financial return data sets, where a variety of metrics are used to quantify forecasting accuracy. For most metrics, the proposed model outperforms existing methods in two of the data sets, while the best empirical performance in the other two data sets is attained by existing methods based on functional transformations of the densities.

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arXiv2018

Principal components analysis of regularly varying functions

The paper is concerned with asymptotic properties of the principal components analysis of functional data. The currently available results assume the existence of the fourth moment. We develop analogous results in a setting which does not require this assumption. Instead, we assume that the observed functions are regularly varying. We derive the asymptotic distribution of the sample covariance operator and of the sample functional principal components. We obtain a number of results on the convergence of moments and almost sure convergence. We apply the new theory to establish the consistency of the regression operator in a functional linear model.

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arXiv2018

Testing Separability of Functional Time Series

We derive and study a significance test for determining if a panel of functional time series is separable. In the context of this paper, separability means that the covariance structure factors into the product of two functions, one depending only on time and the other depending only on the coordinates of the panel. Separability is a property which can dramatically improve computational efficiency by substantially reducing model complexity. It is especially useful for functional data as it implies that the functional principal components are the same for each member of the panel. However such an assumption must be verified before proceeding with further inference. Our approach is based on functional norm differences and provides a test with well controlled size and high power. We establish our procedure quite generally, allowing one to test separability of autocovariances as well. In addition to an asymptotic justification, our methodology is validated by a simulation study. It is applied to functional panels of particulate pollution and stock market data.

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arXiv2016

Change point detection in heteroscedastic time series

Many time series exhibit changes both in level and in variability. Generally, it is more important to detect a change in the level, and changing or smoothly evolving variability can confound existing tests. This paper develops a framework for testing for shifts in the level of a series which accommodates the possibility of changing variability. The resulting tests are robust both to heteroskedasticity and serial dependence. They rely on a new functional central limit theorem for dependent random variables whose variance can change or trend in a substantial way. This new result is of independent interest as it can be applied in many inferential contexts applicable to time series. Its application to change point tests relies on a new approach which utilizes Karhunen--Lo{é}ve expansions of the limit Gaussian processes. After presenting the theory in the most commonly encountered setting of the detection of a change point in the mean, we show how it can be extended to linear and nonlinear regression. Finite sample performance is examined by means of a simulation study and an application to yields on US treasury bonds.

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arXiv2016

Principal component analysis of periodically correlated functional time series

Within the framework of functional data analysis, we develop principal component analysis for periodically correlated time series of functions. We define the components of the above analysis including periodic, operator-valued filters, score processes and the inversion formulas. We show that these objects are defined via convergent series under a simple condition requiring summability of the Hilbert-Schmidt norms of the filter coefficients, and that they poses optimality properties. We explain how the Hilbert space theory reduces to an approximate finite-dimensional setting which is implemented in a custom build R package. A data example and a simulation study show that the new methodology is superior to existing tools if the functional time series exhibit periodic characteristics.

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arXiv2016

A randomness test for functional panels

Functional panels are collections of functional time series, and arise often in the study of high frequency multivariate data. We develop a portmanteau style test to determine if the cross-sections of such a panel are independent and identically distributed. Our framework allows the number of functional projections and/or the number of time series to grow with the sample size. A large sample justification is based on a new central limit theorem for random vectors of increasing dimension. With a proper normalization, the limit is standard normal, potentially making this result easily applicable in other FDA context in which projections on a subspace of increasing dimension are used. The test is shown to have correct size and excellent power using simulated panels whose random structure mimics the realistic dependence encountered in real panel data. It is expected to find application in climatology, finance, ecology, economics, and geophysics. We apply it to Southern Pacific sea surface temperature data, precipitation patterns in the South-West United States, and temperature curves in Germany.

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arXiv2016

Detection of periodicity in functional time series

We derive several tests for the presence of a periodic component in a time series of functions. We consider both the traditional setting in which the periodic functional signal is contaminated by functional white noise, and a more general setting of a contaminating process which is weakly dependent. Several forms of the periodic component are considered. Our tests are motivated by the likelihood principle and fall into two broad categories, which we term multivariate and fully functional. Overall, for the functional series that motivate this research, the fully functional tests exhibit a superior balance of size and power. Asymptotic null distributions of all tests are derived and their consistency is established. Their finite sample performance is examined and compared by numerical studies and application to pollution data.

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