Institute of Mathematical Statistics Lecture Notes - Monograph Series

Optimal sampling strategies for multiscale stochastic processes

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

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Abstract

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 correlation 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.

Chapter information

Source
Javier Rojo, ed., Optimality: The Second Erich L. Lehmann Symposium (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006), 266-290

Dates
First available in Project Euclid: 28 November 2007

Permanent link to this document
https://projecteuclid.org/euclid.lnms/1196283966

Digital Object Identifier
doi:10.1214/074921706000000509

Mathematical Reviews number (MathSciNet)
MR2338548

Zentralblatt MATH identifier
1268.94018

Subjects
Primary: 94A20: Sampling theory 62M30: Spatial processes 60G18: Self-similar processes
Secondary: 62H11: Directional data; spatial statistics 62H12: Estimation 78M50: Optimization

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

Rights
Copyright © 2006, Institute of Mathematical Statistics

Citation

Ribeiro, Vinay J.; Riedi, Rudolf H.; Baraniuk, Richard G. Optimal sampling strategies for multiscale stochastic processes. Optimality, 266--290, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000509. https://projecteuclid.org/euclid.lnms/1196283966


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