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When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
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Dimension Reduction for Fréchet Regression.

Qi Zhang1, Lingzhou Xue1, Bing Li1

  • 1Department of Statistics, The Pennsylvania State University.

Journal of the American Statistical Association
|February 11, 2025
PubMed
Summary

This study introduces a new dimension reduction method for Fréchet regression, enabling analysis of complex data in non-Euclidean spaces. The approach effectively reduces dimensionality and aids visualization for metric space-valued responses.

Keywords:
Ensembled sufficient dimension reductionInverse regressionStatistical objectsUniversal kernelWasserstein space

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Area of Science:

  • Statistics
  • Machine Learning
  • Data Science

Background:

  • Increasingly complex data objects arise in statistical applications, often residing in non-Euclidean spaces.
  • The Fréchet regression model offers a framework for analyzing metric space-valued responses.
  • High-dimensional predictors can lead to the curse of dimensionality, complicating regression analysis.

Purpose of the Study:

  • To develop a flexible sufficient dimension reduction (SDR) method for Fréchet regression.
  • To mitigate the curse of dimensionality in Fréchet regression with high-dimensional predictors.
  • To provide a visual inspection tool for Fréchet regression models.

Main Methods:

  • Proposed a novel SDR method adaptable to existing Euclidean SDR techniques.
  • Mapped metric space-valued random objects to real-valued variables using a class of functions (ensemble).
  • Utilized a universal kernel (cc-universal kernel) to generate the function ensemble, ensuring coverage of the Fréchet SDR space.

Main Results:

  • Established the consistency and asymptotic convergence rates of the proposed SDR methods.
  • Demonstrated the method's effectiveness through simulations on various metric spaces (Wasserstein, SPD matrices, sphere).
  • Successfully illustrated the data visualization capabilities using human mortality data.

Conclusions:

  • The proposed SDR method effectively addresses dimensionality challenges in Fréchet regression.
  • The approach generalizes existing SDR methods for use with complex, non-Euclidean data.
  • The method offers valuable tools for both statistical analysis and data visualization in metric spaces.