Optimal sampling strategies for multiscale stochastic processes. (English) Zbl 1268.94018

Ribeiro, Vinay J.; Riedi, Rudolf H.; Baraniuk, Richard G.

Rojo, Javier (ed.), Optimality. The second Erich L. Lehmann symposium. Selected papers based on the presentations at the symposium, Houston, TX, USA, May 19–22, 2004. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 978-0-940600-66-9/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 49, 266-290 (2006).

Summary: In this paper, we determine which non-random sampling of fixed size gives the best linear predictor of the sum of a finite spatial population. We employ different multiscale superpopulation models and use the minimum mean-squared error as our optimality criterion. In multiscale superpopulation tree models, the leaves represent the units of the population, interior nodes represent partial sums of the population, and the root node represents the total sum of the population. We prove that the optimal sampling pattern varies dramatically with the correlation structure of the tree nodes. While uniform sampling is optimal for trees with “positive correlation progression”, it provides the worst possible sampling with “negative crrelation progression”. As an analysis tool, we introduce and study a class of independent innovations trees that are of interest in their own right. We derive a fast water-filling algorithm to determine the optimal sampling of the leaves to estimate the root of an independent innovations tree.
For the entire collection see [Zbl 1113.62002].

MSC:

94A20	Sampling theory in information and communication theory
62M30	Inference from spatial processes
60G18	Self-similar stochastic processes
62H11	Directional data; spatial statistics
62H12	Estimation in multivariate analysis
78M50	Optimization problems in optics and electromagnetic theory

Keywords:

multiscale stochastic processes; finite population; spatial data; networks; sampling; convex; concave; optimization; trees; sensor networks