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Local Fréchet regression with toroidal predictors

Chang Jun Im, Jeong Min Jeon

Abstract

We provide the first regression framework that simultaneously accommodates responses taking values in a general metric space and predictors lying on a general torus. We propose intrinsic local constant and local linear estimators that respect the underlying geometries of both the response and predictor spaces. Our local linear estimator is novel even in the case of scalar responses. We further establish their asymptotic properties, including consistency and convergence rates. Simulation studies, together with an application to real data, illustrate the superior performance of the proposed methodology.

Local Fréchet regression with toroidal predictors

Abstract

We provide the first regression framework that simultaneously accommodates responses taking values in a general metric space and predictors lying on a general torus. We propose intrinsic local constant and local linear estimators that respect the underlying geometries of both the response and predictor spaces. Our local linear estimator is novel even in the case of scalar responses. We further establish their asymptotic properties, including consistency and convergence rates. Simulation studies, together with an application to real data, illustrate the superior performance of the proposed methodology.
Paper Structure (20 sections, 27 theorems, 237 equations, 1 figure, 1 table)

This paper contains 20 sections, 27 theorems, 237 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem setting and estimators
  3. Problem setting
  4. Local constant estimator
  5. Local linear estimator
  6. Asymptotic theory
  7. Consistency
  8. Rate of convergence
  9. Simulation study
  10. Real data analysis
  11. Proofs of \ref{['thm:consistency']} and \ref{['thm:main4.3']} for $\hat{m}_{\mathbf{h}, 0}(\mathbf{x})$
  12. Lemmas for proof of \ref{['thm:consistency']}
  13. Proof of \ref{['thm:consistency']} for $\hat{m}_{\mathbf{h}, 0}(\mathbf{x})$
  14. Lemmas for proof of \ref{['thm:main4.3']}
  15. Proof of Theorem \ref{['thm:main4.3']} for $\hat{m}_{\mathbf{h}, 0}(\mathbf{x})$
  16. ...and 5 more sections

Key Result

Theorem 3.1

Assume that the conditions con:L1, con:B1, con:D1--con:D3 and con:M1 hold. For each $s\in\{0,1\}$, it holds that

Figures (1)

  • Figure 1: Fitted yellow taxi networks in Manhattan at selected hours and calendar dates. Edge thickness is proportional to absolute ridership, while color indicates relative ridership compared with the overall data average.

Theorems & Definitions (56)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof : Proof of \ref{['lemma:1.4']}
  • ...and 46 more