Entropy-based correlated shrinkage of spatial random processes

Abstract

This paper proposes a two-stage correlated non-linear shrinkage estimation methodology for spatial random processes. A block hard thresholding design, based on Shannon’s entropy, is formulated in the first stage. The thresholding design is adaptive to each resolution level, because it depends on the empirical distribution function of the mutual information ratios between empirical wavelet blocks and the random variables of interest, at each scale. In the second stage, a global correlated (inter- and intra-scale) shrinkage is applied to approximate the values of interest of the underlying spatial process. Additionally, a simulation study is developed, in the Gaussian context, to analyze the sensitivity, measured by empirical stochastic ordering, of the entropy-based block hard thresholding stage in relation to the parameters characterizing local variability (fractality) and dependence range of the spatial process of interest, the noise level, and the design of the region of interest.

Appendix

It well known that an orthonormal basis of wavelet functions of ${L^{2}({\mathbb{R}}^{d})},$ the space of square integrable functions defined on ${{\mathbb{R}}^{d}, \,d\in \mathbb{N}},$ provides a multiresolution analysis of this space. Thus, it allows the description of a L ²-signal at different resolution levels, from the macro-scale to the micro-scale, that is, from its global characteristics and asymptotic properties to its local variability properties (see, for example, De Boor et al. 1993; Daubechies 1992; Mallat 1989; Meyer 1992).

A multiresolution approximation of ${L^{2}( {\mathbb{R}}^{d}) }$ is defined as an increasing sequence ${\left\{ V_{j}:j\in \mathbb{Z}\right\} }$ of closed linear subspaces of ${L^{2}({\mathbb{R}}^{d}) }$ satisfying the following properties:

• (i) $\bigcap_{-\infty }^{\infty }V_{j}=\left\{0\right\}, \bigcup_{-\infty }^{\infty }V_{j}$ is dense in ${L^{2}( {\mathbb{R}}^{d})};$

• (ii) $f\left({\mathbf x}\right) \in V_{0}$ iff $f\left( 2{\mathbf x}\right) \in V_{j+1},$ for all ${{\mathbf x}\in {\mathbb{R}}^{d}}$ and ${j\in \mathbb{Z}};$

• (iii) $f\left({\mathbf x}\right) \in V_{0}$ iff $f\left( {\mathbf x}-{\mathbf k}\right) \in V_{0},$ for all ${{\mathbf x}\in{\mathbb{R}}^{d}}$ and ${{\mathbf k}\in {\mathbb{Z}}^{d}},$ and

• (iv) there exists a function $g\left( \cdot \right) \in V_{0}$ such that the sequence ${\left\{ g\left( {\mathbf x}- {\mathbf k}\right): {\mathbf k}\in {\mathbb{Z}}^{d}\right\} }$ is a Riesz basis of V ₀.

Different approaches have been adopted in the construction of an orthonormal basis of wavelet functions associated with a multiresolution analysis of ${L^{2}({\mathbb{R}}^{d})}.$ In particular, from a Riesz basis¹ of V ₀, an orthonormal basis of wavelet functions can be constructed using the spectral theory of $L^{2}\left( [0,2\pi )^{d}\right)$-functions (see, for example, Meyer 1992). Formally, the following decomposition of ${L^{2}({\mathbb{R}}^{d})}$ is obtained from an orthonormal basis of wavelet functions:

$$ L^{2}({\mathbb{R}}^{d})=V_{0}\oplus \sum_{j=0}^{\infty} W_{j}. $$

Thus, for a function ${f\in L^{2}({\mathbb{R}}^{d})},$ its projection on an orthonormal scaling basis of V ₀ provides the draft of f, and its projections on the orthonormal wavelet bases of the spaces ${W_{j},\, j\in \mathbb{N}},$ generated by translations and contractions of the basic wavelet functions, reproduce local variability properties of f at different resolution levels ${j\in \mathbb{N}}.$ Note that, for every ${j \in \mathbb{N}},$

$$ V_{j}= V_{j-1}\oplus W_{j-1}. $$