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Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first, find the median or second quartile. The first quartile, Q1, is the middle value of the lower half of the data, and the third quartile, Q3, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:
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High-Dimensional Spatial Quantile Function-on-Scalar Regression.

Zhengwu Zhang1, Xiao Wang2, Linglong Kong3

  • 1Department of Statistics and Operations Research, University of North Carolina, Chapel Hill, NC.

Journal of the American Statistical Association
|April 3, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a new spatial quantile regression model to understand how scalar predictors affect functional responses across entire spatial domains. The method aids in characterizing conditional distributions and generating new images from data.

Keywords:
CopulaFunction-on-scalar regressionImage analysisMinimax rate of convergenceQuantile regressionReproducing kernel Hilbert space

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Area of Science:

  • Statistics
  • Spatial Analysis
  • Machine Learning

Background:

  • Understanding the spatial distribution of functional responses is crucial in various scientific fields.
  • Existing methods often struggle to fully characterize complex conditional distributions, especially with high-dimensional data.

Purpose of the Study:

  • To develop a novel spatial quantile function-on-scalar regression model.
  • To explicitly characterize the conditional distribution of high-dimensional functional responses given scalar predictors.
  • To provide a comprehensive understanding of covariate effects across different quantile levels.

Main Methods:

  • Integration of quantile regression and copula modeling.
  • Development of a spatial quantile function-on-scalar regression framework.
  • Establishment of minimax rates of convergence for coefficient function estimation.
  • Implementation of an efficient primal-dual algorithm for high-dimensional image data.

Main Results:

  • The proposed model successfully characterizes the conditional distribution of functional or image responses over spatial domains.
  • The method offers insights into how scalar covariates influence functional responses at various quantile levels.
  • Theoretical guarantees for estimation convergence rates are established.
  • An efficient algorithm is developed for practical application with high-dimensional data.

Conclusions:

  • The novel spatial quantile regression model provides a powerful tool for analyzing functional data with spatial dependencies.
  • The approach enhances understanding of covariate effects and enables practical image generation.
  • The developed methodology is theoretically sound and computationally efficient for high-dimensional functional data analysis.