Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Efficient nonparametric estimation of Toeplitz covariance matrices

Karolina Klockmann, Tatyana Krivobokova

Abstract

A new nonparametric estimator for Toeplitz covariance matrices is proposed. This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression. The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast. Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices. These results are readily extended to estimation of inverses of Toeplitz covariance matrices. Also, an alternative version of the Whittle likelihood for the spectral density based on the Discrete Cosine Transform (DCT) is proposed. The method is implemented in the R package vstdct that accompanies the paper.

Efficient nonparametric estimation of Toeplitz covariance matrices

Abstract

A new nonparametric estimator for Toeplitz covariance matrices is proposed. This estimator is based on a data transformation that translates the problem of Toeplitz covariance matrix estimation to the problem of mean estimation in an approximate Gaussian regression. The resulting Toeplitz covariance matrix estimator is positive definite by construction, fully data-driven and computationally very fast. Moreover, this estimator is shown to be minimax optimal under the spectral norm for a large class of Toeplitz matrices. These results are readily extended to estimation of inverses of Toeplitz covariance matrices. Also, an alternative version of the Whittle likelihood for the spectral density based on the Discrete Cosine Transform (DCT) is proposed. The method is implemented in the R package vstdct that accompanies the paper.
Paper Structure (28 sections, 7 theorems, 112 equations, 6 figures, 9 tables)

This paper contains 28 sections, 7 theorems, 112 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Set up and Diagonalization of Toeplitz Matrices
  3. Methodology
  4. Theoretical Properties
  5. Practical Issues
  6. Simulation Study
  7. Application to Non-Gaussian Data
  8. Application to Protein Dynamics
  9. Appendix
  10. Proof of Lemma 1
  11. Periodic Smoothing Splines
  12. Proof of Theorem \ref{['theorem1']}
  13. An upper bound on $E \|\widehat{H(f)} - H(f)\|_\infty^2$
  14. An upper bound on $E \|\hat{f} - f\|_\infty^2$
  15. An upper bound on $E\|\widehat{\Sigma}-\Sigma\|^2$
  16. ...and 13 more sections

Key Result

Lemma 1

Let $\Sigma=\Sigma(f)$ with $f\in \mathcal{P}_\beta(M_0,M_1)$ and $x_j=(j-1)/(p-1)$ for $j=1,...,p$. Then where $\delta_{i,j}$ is the Kroneker delta, $\mathcal{O}(\cdot)$ terms are uniform over $i,j=1,\dots,p$ and is the discrete cosine transform I matrix.

Figures (6)

  • Figure 1: Spectral density functions (first row) and covariance functions (second row) for examples (1)--(3).
  • Figure 2: Distance between the first atom and the first center of mass of aquaporin (left) and the opening diameter $y_t$ over time $t$ (right).
  • Figure 3: On the left, the correlation function of $Y$ (grey) and of $\widehat{\Sigma}^{-1/2}Y$ (black), where $\widehat{\Sigma}$ is estimated with our method; On the right, correlation between the true values on the test data set and prediction based on partial least squares (grey) and corrected partial least squares (black).
  • Figure 4: Simulation study with Gaussian data; QQ-plots of binned and transformed data $Y_k^*\, (k=1,...,2T-2)$ with different bin numbers $T$ for example (1) with $p=5000,\, n=1$.
  • Figure 5: Simulation study with gamma data; QQ-plots of binned and transformed data $Y_k^*\, (k=1,...,2T-2)$ with $T=500$ for examples (1)--(3) with $p=5000,\, n=1$.
  • ...and 1 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 2 more