On high-dimensional wavelet eigenanalysis

Abstract

In this paper, we mathematically construct wavelet eigenanalysis in high dimensions Didier (2018a, 2018b)) by characterizing the scaling behavior of the eigenvalues of large wavelet random matrices. We assume that possibly non-Gaussian, finite-variance p-variate measurements are made of a low-dimensional r-variate (r ≪ p) fractional stochastic process with non-canonical scaling coordinates and in the presence of additive high-dimensional noise. We show that the r largest eigenvalues of the wavelet random matrices, when appropriately rescaled, converge to scale invariant functions in the high-dimensional limit. By contrast, the remaining p − r eigenvalues remain bounded. In addition, we show that, up to a log transformation, the r largest eigenvalues of wavelet random matrices exhibit asymptotically Gaussian distributions. We further show how the asymptotic and large-scale behavior of wavelet eigenvalues can be used to construct statistical inference methodology for a highdimensional signal-plus-noise system.

Hurst multimodality detection based on large wavelet random matrices

“…In the context of multivariate Hurst exponent estimation using wavelet-based linear regressions, as in ( 5), when both n, p → +∞, thorough analysis of scale invariant systems further implies that the range of scales where linear regressions are performed should also grow towards infinity: (j 1 , j 2 ) → +∞. Technically, in such settings, the high-dimensional behavior analyzed by means of large random matrices actually leads us to consider the three-way limit [9], [30] n, p, j → +∞,…”

“…Mathematical results in [9], [30] imply that, in the threeway limit (6) and under (2), the eigenvalues of the sample wavelet spectrum behave as…”

“…, p, for some constant C ℓ , where o P (1) vanishes in probability. In fact, eigenvalue scaling relations of the type ( 7) are shown to hold for high-dimensional signal-plus-noise instances, broader than the noise-free model (2) [9], [30]. Further, assume measurements given by the highdimensional model (2).…”