Distribution of Distances based Object Matching: Asymptotic Inference

A time series is an extremely abundant data type arising in many areas of scientific research, including the biological sciences. Any method that compares time series data relies on a pairwise distance between trajectories, and the choice of distance measure determines the accuracy and speed of the time series comparison. This paper introduces an optimal transport type distance for comparing time series trajectories that are allowed to lie in spaces of different dimensions and/or with differing numbers of points possibly unequally spaced along each trajectory. The construction is based on a modified Gromov–Wasserstein distance optimization program, reducing the problem to a Wasserstein distance on the real line. The resulting program has a closed-form solution and can be computed quickly due to the scalability of the one-dimensional Wasserstein distance. We discuss theoretical properties of this distance measure, and empirically demonstrate the performance of the proposed distance on several datasets with a range of characteristics commonly found in biologically relevant data. We also use our proposed distance to demonstrate that averaging oscillatory time series trajectories using the recently proposed Fused Gromov–Wasserstein barycenter retains more characteristics in the averaged trajectory when compared to traditional averaging, which demonstrates the applicability of Fused Gromov–Wasserstein barycenters for biological time series. Fast and user friendly software for computing the proposed distance and related applications is provided. The proposed distance allows fast and meaningful comparison of biological time series and can be efficiently used in a wide range of applications.

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Scalable Gromov–Wasserstein Based Comparison of Biological Time Series

Kravtsova

McGee

Dawes

2023

Bull Math Biol

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Empirical optimal transport under estimated costs: Distributional limits and statistical applications

Hundrieser,

Mordant,

Weitkamp

et al. 2024

Stochastic Processes and their Applications

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A new class of nonparametric tests for second‐order stochastic dominance based on the Lorenz P–P plot

Lando,

Legramanti

2024

Scandinavian J Statistics

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Given samples from two non‐negative random variables, we propose a family of tests for the null hypothesis that one random variable stochastically dominates the other at the second order. Test statistics are obtained as functionals of the difference between the identity and the Lorenz P–P plot, defined as the composition between the inverse unscaled Lorenz curve of one distribution and the unscaled Lorenz curve of the other. We determine upper bounds for such test statistics under the null hypothesis and derive their limit distribution, to be approximated via bootstrap procedures. We then establish the asymptotic validity of the tests under relatively mild conditions and investigate finite‐sample properties through simulations. The results show that our testing approach can be a valid alternative to classic methods based on the difference in the integrals of the cumulative distribution functions, which require bounded support and struggle to detect departures from the null in some cases. The same approach can be extended to a family of fractional‐degree stochastic orders, including the first order as a limiting case.

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Distribution of Distances based Object Matching: Asymptotic Inference

Cited by 5 publications

References 71 publications

Scalable Gromov–Wasserstein Based Comparison of Biological Time Series

Scalable Gromov–Wasserstein Based Comparison of Biological Time Series

Empirical optimal transport under estimated costs: Distributional limits and statistical applications

A new class of nonparametric tests for second‐order stochastic dominance based on the Lorenz P–P plot

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