On the inverse geostatistical problem of inference on missing locations

Emanuele Giorgi, Peter J. Diggle

Research output: Contribution to journalArticle

Abstract

The standard geostatistical problem is to predict the values of a spatially continuous phenomenon, S(x) say, at locations x using data (yi, xi):i=1, ..., n where yi is the realisation at location xi of S(xi), or of a random variable Yi that is stochastically related to S(xi). In this paper we address the inverse problem of predicting the locations of observed measurements y. We discuss how knowledge of the sampling mechanism can and should inform a prior specification, π(x) say, for the joint distribution of the measurement locations X={xi:i=1, ..., n}, and propose an efficient Metropolis-Hastings algorithm for drawing samples from the resulting predictive distribution of the missing elements of X. An important feature in many applied settings is that this predictive distribution is multi-modal, which severely limits the usefulness of simple summary measures such as the mean or median. We present three simulated examples to demonstrate the importance of the specification for π(x) and show how a one-by-one approach can lead to substantially incorrect inferences in the case of multiple unknown locations. We also analyse rainfall data from Paraná State, Brazil to show how, under additional assumptions, an empirical estimate of π(x) can be used when no prior information on the sampling design is available.

Original languageEnglish (US)
Pages (from-to)35-44
Number of pages10
JournalSpatial Statistics
Volume11
DOIs
StatePublished - Feb 1 2015
Externally publishedYes

Fingerprint

inverse problem
Inverse problems
Inverse Problem
Predictive Distribution
sampling
Specification
Sampling
Specifications
Metropolis-Hastings Algorithm
Sampling Design
Rainfall
Prior Information
rainfall
Random variables
Joint Distribution
Rain
Efficient Algorithms
Random variable
distribution
Predict

Keywords

  • Geostatistics
  • Kernel density estimation
  • Missing locations
  • Multi-modal distributions

ASJC Scopus subject areas

  • Computers in Earth Sciences
  • Statistics and Probability
  • Management, Monitoring, Policy and Law

Cite this

On the inverse geostatistical problem of inference on missing locations. / Giorgi, Emanuele; Diggle, Peter J.

In: Spatial Statistics, Vol. 11, 01.02.2015, p. 35-44.

Research output: Contribution to journalArticle

Giorgi, Emanuele ; Diggle, Peter J. / On the inverse geostatistical problem of inference on missing locations. In: Spatial Statistics. 2015 ; Vol. 11. pp. 35-44.
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