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Distributional data analysis via quantile functions and its application to modeling digital biomarkers of gait in Alzheimer's Disease

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BIOSTATISTICS 2023年第3期24卷 539-561页

作者： Ghosal, Rahul Varma, Vijay R. Volfson, Dmitri Hillel, Inbar Urbanek, Jacek Hausdorff, Jeffrey M. Watts, Amber Zipunnikov, Vadim Johns Hopkins Bloomberg Sch Publ Hlth Dept Biostat Baltimore MD 21205 USA NIA NIH Baltimore MD USA Computat Biol Neurosci Analyt Cambridge MA USA Tel Aviv Sourasky Med Ctr Neurol Inst Ctr Study Movement Cognit & Mobil Tel Aviv Israel Johns Hopkins Univ Sch Med Dept Med Baltimore MD 21205 USA Tel Aviv Univ Dept Phys Therapy Sackler Fac Med Tel Aviv Israel Tel Aviv Univ Sagol Sch Neurosci Tel Aviv Israel Rush Univ Med Ctr Rush Alzheimers Dis Ctr Chicago IL USA Rush Univ Dept Orthoped Surg Med Ctr Chicago IL USA Univ Kansas Dept Psychol Lawrence KS 66045 USA

With the advent of continuous health monitoring with wearable devices, users now generate their unique streams of continuous data such as minute-level step counts or heartbeats. Summarizing these streams via scalar summaries often ignores the distributional nature of wearable data and almost unavoidably leads to the loss of critical information. We propose to capture the distributional nature of wearable data via user-specific quantile functions (QF) and use these QFs as predictors in scalar-on-quantile-function-regression (SOQFR). As an alternative approach, we also propose to represent QFs via user-specific L-moments, robust rank-based analogs of traditional moments, and use L-moments as predictors in SOQFR (SOQFR-L). These two approaches provide two mutually consistent interpretations: in terms of quantile levels by SOQFR and in terms of L-moments by SOQFR-L. We also demonstrate how to deal with multi-modal distributional data via Joint and Individual Variation Explained using L-moments. The proposed methods are illustrated in a study of association of digital gait biomarkers with cognitive function in Alzheimers disease. Our analysis shows that the proposed methods demonstrate higher predictive performance and attain much stronger associations with clinical cognitive scales compared to simple distributional summaries.

关键词： Alzheimer's disease Gait JIVE L-Moments quantile functions Scalar-on-quantile-function regression Wearable data

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Risk preferences on the space of quantile functions

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MATHEMATICAL PROGRAMMING 2014年第1-2期148卷 181-200页

作者： Dentcheva, Darinka Ruszczynski, Andrzej Stevens Inst Technol Dept Math Hoboken NJ 07030 USA Rutgers State Univ Dept Management Sci & Informat Syst Piscataway NJ 08854 USA

We propose a novel approach to quantification of risk preferences on the space of nondecreasing functions. When applied to law invariant risk preferences among random variables, it compares their quantile functions. The axioms on quantile functions impose relations among comonotonic random variables. We infer the existence of a numerical representation of the preference relation in the form of a quantile-based measure of risk. Using conjugate duality theory by pairing the Banach space of bounded functions with the space of finitely additive measures on a suitable algebra , we develop a variational representation of the quantile-based measures of risk. Furthermore, we introduce a notion of risk aversion based on quantile functions, which enables us to derive an analogue of Kusuoka representation of coherent law-invariant measures of risk.

关键词： Risk measures quantile functions Conjugate duality Kusuoka representation

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quantile-based overlap measures

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RICERCHE DI MATEMATICA 2024年第4期73卷 1919-1936页

作者： Mathew, Angel Joseph, Chinu Maharajas Coll Dept Stat Ernakulam Kerala India

The closeness or similarity between two populations is usually assessed by means of overlap measures based on their probability distributions. Matusita's measure, Morisita's measure and Weitzman's measure are three commonly used overlap measures. These measures are defined based on the probability density functions of the two distributions. There are several probability distributions that do not have a tractable probability density function, though the corresponding quantile density function has an explicit form. In such cases, the traditional approach fails and the overlap measures need to be defined in terms of quantile functions. The present paper considers a quantile-based study of the overlap measures. We also illustrate the usefulness of the proposed quantile-based overlap measures using several examples.

关键词： Overlap measures quantile functions Matusita's measure Morisita's measure Weitzman's measure

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HOW ARE PRELAUNCH ONLINE MOVIE REVIEWS RELATED TO BOX OFFICE REVENUES?

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ANNALS OF APPLIED STATISTICS 2024年第2期18卷 1686-1708页

作者： Guan, Tianyu Ho, Jason Krider, Robert Cao, Jiguo Fogg, Andrew York Univ Dept Math & Stat Toronto ON Canada Simon Fraser Univ Beedie Sch Business Burnaby BC Canada Simon Fraser Univ Dept Stat & Actuarial Sci Burnaby BC Canada Roku Inc San Jose CA USA

This paper studies the dynamic patterns of the prelaunch online movie reviews, or movie electronic word-of-mouth (eWOM), over time and investigates their relations to the subsequent box office revenues. The volume and valence of prelaunch eWOM have been shown to be early indicators of strong or weak box office. The time patterns of prelaunch eWOM evolution, which are essentially functional data, on the other hand, tend to be overlooked. We apply the functional principal component analysis, a dimension reduction technique in functional data analysis, to analyze the dynamic patterns of various quantile trajectories of the movie eWOM, instead of directly studying the whole eWOM functional data. The functional principal component (FPC) scores of quantile trajectories at various quantile levels are used to predict the box office revenues. We use the sparse group lasso method to select the quantile levels and individual FPC scores that make significant contributions to the prediction of box office revenues. The results show that compared with other measures, such as valence and variance, the top-end quantiles would be a better measure in capturing the relations between the prelaunch product ratings time pattern and launch sales.

关键词： Electronic word of mouth functional data analysis functional principal component analysis quantile functions variable selection

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Bayesian meta-elliptical multivariate regression models with fixed marginals on unit intervals

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 2023年第3期52卷 918-938页

作者： Rodrigues, Josemar Benites, Yury R. Cancho, Vicente G. Balakrishnan, N. Suzuki, Adriano K. Univ Sao Paulo Inst Math & Comp Sci Dept Appl Math & Stat Ave Trabalhador Sao Carlense400 Ctr BR-13566590 Sao Carlos SP Brazil MacMaster Univ Dept Math & Stat Hamilton ON Canada

In this paper, we make use of meta-elliptical copula functions to build a new multivariate distribution with fixed marginal distributions and dependence structure to analyze bounded data. Specifically, we present a flexible p-elliptical multivariate probability distribution in the hypercube (0,1)p p with fixed marginal GF-quantile distributions. We then present some illustrative examples and a meta-elliptical multivariate regression model as a flexible alternative to the multivariate normal regression model on unit intervals. A simulation study and real-life data analysis using a Bayesian framework with the extreme-value quantile functions show the flexibility of the proposed meta-multivariate normal regression model for modeling the observed proportion response variables.

关键词： Elliptical copulas quantile functions Bayesian method marginal GF quantile distributions

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Distributional Synthetic Controls

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ECONOMETRICA 2023年第3期91卷 1105-1117页

作者： Gunsilius, F. F. Univ Michigan Dept Econ Ann Arbor MI 48109 USA

The method of synthetic controls is a fundamental tool for evaluating causal effects of policy changes in settings with observational data. In many settings where it is applicable, researchers want to identify causal effects of policy changes on a treated unit at an aggregate level while having access to data at a finer granularity. This article proposes an extension of the synthetic controls estimator that takes advantage of this additional structure and provides nonparametric estimates of the heterogeneity within the aggregate unit. The idea is to replicate the quantile function associated with the treated unit by a weighted average of quantile functions of the control units. This estimator relies on the same mathematical theory as the changes-in-changes estimator and can be applied in both repeated cross-sections and panel data with as little as a single pre-treatment period. It also provides a unique counterfactual quantile function for any type of distribution.

关键词： Causal inference comparative case studies heterogeneous treatment effects quantile functions synthetic controls Wasserstein distance

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quantile-based residual Matusita's measure

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JAPANESE JOURNAL OF STATISTICS AND DATA SCIENCE 2025年 1-19页

作者： Joseph, Chinu Mathew, Angel Maharajas Coll Dept Stat Ernakulam Kerala India

In reliability and survival studies, assessing the similarity between the probability distributions of the residual lifetimes of two populations is of great importance. Matusita's overlapping measure is a widely used similarity measure based on the concept of distance between the probability distributions. In this paper, we consider the quantile-based Matusita's measure for residual lifetimes. The quantile-based approach is particularly useful when one of the probability distributions being studied lacks an analytically tractable form, but its corresponding quantile function is explicitly defined. The properties of the proposed measure, including bounds and the effect of monotonic transformations, are discussed. A nonparametric estimator for the quantile-based residual Matusita's measure is also proposed. A simulation study is carried out to assess the performance of the proposed estimator.

关键词： Similarity measure Matusita's measure quantile functions Residual lifetimes Nonparametric estimation

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quantile Policy Effects: An Application to U.S. Macroprudential Policy

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Journal of Business and Economic Statistics 2025年第1期43卷 81-97页

作者： Lin, Hsin-Yi Hsiao, Yu-Hsiang Hsu, Yu-Chin Department of Economics National Chengchi University Taipei Taiwan Central Bank of the Republic of China (Taiwan) Taipei Taiwan Institute of Economics Academia Sinica Taipei Taiwan Department of Finance National Central University Taoyuan Taiwan CRETA National Taiwan University Taipei Taiwan

To assess the dynamic distributional impacts of macroeconomic policy, we propose quantile policy effects to quantify disparities between the quantiles of potential outcomes under different policies. We first identify quantile policy effects under the unconfoundedness assumption and propose an inverse probability weighting estimator. We then examine the asymptotic behavior of the proposed estimator in a time series framework and suggest a blockwise bootstrap method for inference. Applying this method, we investigate the effectiveness of U.S. macroprudential actions on bank credit growth from 1948 to 2019. Empirically, we find that the effects of macroprudential policy on credit growth are asymmetric and depend on the quantiles of credit growth. The tightening of macroprudential actions fails to rein in high credit growth, whereas easing policies do not effectively stimulate bank credit growth during low-growth periods. These findings suggest that U.S. macroprudential policies might not sufficiently address the challenges of soaring bank credit or ensure overarching financial stability. © 2024 American Statistical Association.

关键词： Causal effects Impulse response functions Macroprudential policy Propensity score quantile functions

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The Statistics of Superdirective Beam Patterns

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2022年 70卷 1959-1975页

作者： Trucco, Andrea Univ Genoa Dept Elect Elect Telecommun Engn & Naval Architec I-16145 Genoa Italy Ist Italian Tecnol ITT Pattern Anal & Comp Vis Dept I-16152 Genoa Italy

Superdirective arrays have been extensively studied because of their considerable potential accompanied, unfortunately, by a high sensitivity to random errors that affect the responses and positions of array elements. However, the statistics of their actual beam pattern (BP) has never been systematically investigated. This paper shows that the Rician probability density function (PDF), sometimes adopted to study the impact of errors in conventional arrays, is a valid approximation for superdirective BP statistics only where some mathematical terms are negligible. The paper also shows that this is the case for all linear end-fire arrays considered. A similar study is proposed concerning the correlation between BP lobes, showing that for the superdirective arrays considered the lobes, especially non-adjacent ones, are almost independent. Furthermore, knowledge of the PDF of the actual BP allows one to define quantile BP functions, whose probability of being exceeded, at any point, is fixed. Combining the lobes' independence with quantile BP functions, an empirical equation for the probability that the entire actual BP will not exceed a quantile function over an interval larger than a given size is obtained. This new knowledge and these tools make it possible to devise new methods to design robust superdirective arrays via optimization goals with clearer and more relevant statistical meaning.

关键词： Rician channels Probability density function Probability Array signal processing Optimization Mathematical models Robustness Beamforming superdirective arrays robust superdirectivity beam pattern statistics random errors correlation quantile functions Rician PDF

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Stochastic evolution of distributions and functional Bollinger bands

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APPLIED STOCHASTIC MODELS IN BUSINESS AND INDUSTRY 2022年第2期38卷 370-390页

作者： Bernis, Guillaume Brunel, Nicolas Kornprobst, Antoine Scotti, Simone Natixis Assurances Paris France Univ Paris Saclay Lab Math & Modelisat Evry UMR CNRS 8071 ENSIIE Gif Sur Yvette France Univ Western Ontario London ON Canada Univ Paris LPSM Paris France

We use mixture of percentile functions to model credit spread evolution, which allows to obtain a flexible description of indices and their components at the same time. We show regularity results in order to extend mixture percentile to the dynamic case. We characterize the stochastic differential equation of the flow of cumulative distribution function and we link it with the ordered list of the components of the credit index. The main financial goal is to introduce a functional version of Bollinger bands. The crossing of bands by the spread is associated with a trading signal. Finally, we show the richness of the signals produced by functional Bollinger bands compared with standard one with a practical example in credit asset.

关键词： Bollinger bands credit risk quantile functions stochastic models

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