Speaker
Description
In modern data analysis, technological advancements frequently result in the collection of Functional Data (FD), where observations are naturally represented as smooth functions, curves, or surfaces over a continuum (e.g., time or space). Examples include daily stock prices, continuous temperature recordings, or spectroscopic measurements. Functional Data Analysis (FDA) offers a powerful framework for modeling such phenomena, addressing challenges like high dimensionality and irregular sampling better than traditional multivariate methods. On the other hand, a fundamental task in statistics is examining the relationship, or independence, between variables. While widely studied for classical multivariate data, testing independence in the functional setting remains a significant challenge. Existing methods, while valuable (e.g., Krzyśko et al., 2022, 2025), may not always be optimal, necessitating the development of alternative and more robust procedures.
This work, which is also a master's project, proposes a novel approach to test the independence of functional data by leveraging a basis expansion technique. Functional data is first approximated as a linear combination of basis functions (e.g., Fourier or B-spline) in a Hilbert space. The resulting finite set of basis coefficients effectively translates the functional problem into a standard multivariate one. We adapt and extend the recently proposed multivariate independence measures and tests (Wang et al., 2025) to this functional context.
We conduct a comprehensive simulation study to assess the new method's statistical properties, focusing on the control of type I error rate and the power. The proposed methodology has broad practical implications in fields where functional dependence is crucial: finance (determining the independence between two stock price trajectories); environmental science (assessing if the annual temperature curve in one region is independent of the annual rainfall curve in another); biometrics (testing the relationship between two continuous physiological signals, e.g., two types of brain activity traces). We illustrate the use of new methods in such or similar practical problems.
References:
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Krzyśko, M., Smaga, Ł., Kokoszka, P. (2022). Marginal distance and Hilbert-Schmidt covariances-based independence tests for multivariate functional data. Journal of Artificial Intelligence Research 73, 101613.
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Krzyśko, M., Smaga, Ł., Wydra, J. (2025). Distance of mean embedding for testing independence of functional data. Signal Processing 233, 109959.
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Wang, L., Zhou, H., Ma, W., Yang, Y. (2025). A conditional distribution function-based measure for independence and K-sample tests in multivariate data. Journal of Multivariate Analysis 205, 105378.
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