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Nonparametric two sample test of spectral densities

Ilaria Nadin, Tatyana Krivobokova, Farida Enikeeva

TL;DR

This work introduces a nonparametric two-sample test for equality of spectral densities of Gaussian stationary processes with potentially unequal lengths. It leverages a data-transformation pipeline based on the discrete cosine transform and binning to produce comparable inputs, and estimates log-spectral densities with a periodic smoothing spline, using their $L_2$ distance as the contrast. The authors establish minimax-optimal separation rates for Hölder-smooth spectral densities and prove the test achieves asymptotic level $\alpha$ while adapting to unknown smoothness up to $2q$ in the equal-length setting. Empirical validation via simulations and an EEG application demonstrates competitive power and practical utility, with an R implementation available in the sdf.test package.

Abstract

A novel nonparametric test for the equality of the covariance matrices of two Gaussian stationary processes, possibly of different lengths, is proposed. The test translates to testing the equality of two spectral densities and is shown to be minimax rate-optimal. Test performance is validated in a simulation study, and the practical utility is demonstrated in the analysis of real electroencephalography data. The test is implemented in the R-package sdf.test.

Nonparametric two sample test of spectral densities

TL;DR

This work introduces a nonparametric two-sample test for equality of spectral densities of Gaussian stationary processes with potentially unequal lengths. It leverages a data-transformation pipeline based on the discrete cosine transform and binning to produce comparable inputs, and estimates log-spectral densities with a periodic smoothing spline, using their distance as the contrast. The authors establish minimax-optimal separation rates for Hölder-smooth spectral densities and prove the test achieves asymptotic level while adapting to unknown smoothness up to in the equal-length setting. Empirical validation via simulations and an EEG application demonstrates competitive power and practical utility, with an R implementation available in the sdf.test package.

Abstract

A novel nonparametric test for the equality of the covariance matrices of two Gaussian stationary processes, possibly of different lengths, is proposed. The test translates to testing the equality of two spectral densities and is shown to be minimax rate-optimal. Test performance is validated in a simulation study, and the practical utility is demonstrated in the analysis of real electroencephalography data. The test is implemented in the R-package sdf.test.
Paper Structure (19 sections, 8 theorems, 132 equations, 5 figures, 2 tables)

This paper contains 19 sections, 8 theorems, 132 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Testing procedure
  4. Data transformation
  5. Test statistics
  6. Main result
  7. Simulation study
  8. Application to EEG data
  9. Discussion
  10. Appendix
  11. Proof of Theorem \ref{['th:MinimaxTest']}
  12. Proof of Corollary \ref{['cor:AdaptiveTest']}
  13. Supplementary Material
  14. Kernel matrix
  15. Proof of Lemma \ref{['lem:DataDecomposition']}
  16. ...and 4 more sections

Key Result

Lemma 1

Consider $\mathbf{X}\sim\mathcal{N}_n(0_n,\Sigma)$, with spectral density $f\in\mathcal{F}_{\eta}$, $\eta > 0$. If $h > 0$ is such that $h \to 0$ and $hT \to \infty$, with $T=\lfloor n^{\nu}\rfloor$ for $\nu \in (1-\min\{1,\eta\}/3,1)$, and $\mathbf{Y}^*$ is the vector of transformed data as in the where for $\left\lvert\epsilon_{t}\right\rvert \leq c' (n^{-1} + n^{-\eta} \log n)$ small determin

Figures (5)

  • Figure 1: In the first column, the spectral densities $f_1$ (black) and $\widetilde{f}$ (grey). In the second column, the powers of $\psi_{n_1,n_2}$ (solid), and the wavelet-based test (dashed) in scenario (A). In the third column, the powers of $\psi_{n_1,n_2}$ (solid), and the wavelet-based test (dashed) in scenario (B).
  • Figure 2: In the first column, difference of potential (in µV) registered by the channel E159-E160 along time windows (in s) of rest, first frontal tDCS, posterior tDCS, and second frontal tDCS. In the second column, the corresponding estimated spectral density in the interval $[0,1]$.
  • Figure 3: In the first column, the spectral densities $f_1$ (black) and $\widetilde{f}$ (grey). In the second column, the powers of $\psi_{n_1,n_2}$ (solid), and the wavelet-based test (dashed) in scenario (A). In the third column, the powers of $\psi_{n_1,n_2}$ (solid) with maximum likelihood method of smoothing parameter selection, and the wavelet-based test (dashed) in scenario (B).
  • Figure 4: Spectral densities $f$ (black) and $\widetilde{f}$ (grey) for each setting.
  • Figure 5: ROC curves of $\psi_{n_1,n_2}$ (solid yellow), the wavelet-based test (dashed red), the test by preuss_hildebrandt2013 (dotted teal), the test by caiado2012tests (dot-dash blue), and the test by dette_paparoditis2009 (long dash dark blue).

Theorems & Definitions (16)

  • Definition 1
  • Lemma 1: Data decomposition
  • Theorem 1
  • Corollary 1
  • proof : of Theorem \ref{['th:MinimaxTest']}
  • proof
  • Lemma 2: Properties of the kernel matrix
  • proof
  • Proposition 1
  • proof
  • ...and 6 more