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Test of the Latent Dimension of a Spatial Blind Source Separation Model

Christoph Muehlmann, François Bachoc, Klaus Nordhausen, Mengxi Yi

Abstract

We assume a spatial blind source separation model in which the observed multivariate spatial data is a linear mixture of latent spatially uncorrelated Gaussian random fields containing a number of pure white noise components. We propose a test on the number of white noise components and obtain the asymptotic distribution of its statistic for a general domain. We also demonstrate how computations can be facilitated in the case of gridded observation locations. Based on this test, we obtain a consistent estimator of the true dimension. Simulation studies and an environmental application demonstrate that our test is at least comparable to and often outperforms bootstrap-based techniques, which are also introduced in this paper.

Test of the Latent Dimension of a Spatial Blind Source Separation Model

Abstract

We assume a spatial blind source separation model in which the observed multivariate spatial data is a linear mixture of latent spatially uncorrelated Gaussian random fields containing a number of pure white noise components. We propose a test on the number of white noise components and obtain the asymptotic distribution of its statistic for a general domain. We also demonstrate how computations can be facilitated in the case of gridded observation locations. Based on this test, we obtain a consistent estimator of the true dimension. Simulation studies and an environmental application demonstrate that our test is at least comparable to and often outperforms bootstrap-based techniques, which are also introduced in this paper.

Paper Structure

This paper contains 17 sections, 7 theorems, 79 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Setup and Model
  3. Theory and Methodology
  4. Asymptotic Tests for Dimension
  5. Regular Domain as a Special Example
  6. Estimation of the Number of Signal Components
  7. General Mean
  8. Bootstrap Tests for Dimension
  9. Simulation
  10. Simulation Study 1: Hypothesis Testing
  11. Simulation Study 2: Computation Time Comparison
  12. Simulation Study 3: Estimation of the Signal Dimension
  13. Data Example
  14. Concluding Remarks
  15. Acknowledgments
  16. ...and 2 more sections

Key Result

Proposition 3.1

Assume that Conditions c:1-c:8 hold. Then, as $n\to\infty$,

Figures (8)

  • Figure 1: Left: Matérn correlation functions for model setting 1, which consists of the signal random field $(z_1,z_2,z_{3,1})$ with parameters $(\nu, \phi) \in \{ (3,2), (2,1.5), (1,1) \}$ and model setting 2 formed by the signal random field $(z_1,z_2,z_{3,2})$ with parameters $(\nu, \phi) \in \{(3,2), (2,1.5), (0.6,0.6)\}$. Middle and right: uniform (middle) and skewed (right) coordinate sample pattern for a spatial domain of size $30 \times 30$ with three circles of radii $(2,4,6)$ representing ring kernel functions.
  • Figure 2: Median running times of the five different test methods for different domain sizes with regular sampling sites based on five simulation repetitions. Computations were carried out with code designed for regular and irregular sampling sites.
  • Figure 3: Frequencies of the estimated signal dimension for model setting 1.
  • Figure 4: Frequencies of the estimated signal dimension for model setting 2.
  • Figure 5: Latent field components of the change-point between estimated signal and noise components for kernel function setting 1. Map tiles by Stamen Design, under CC BY 3.0. Data by OpenStreetMap, under ODbL.
  • ...and 3 more figures

Theorems & Definitions (8)

  • Remark 1
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.1
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Lemma A.1