Final published version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - On the inverse geostatistical problem of inference on missing locations
AU - Giorgi, Emanuele
AU - Diggle, Peter John
PY - 2015/2
Y1 - 2015/2
N2 - 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.
AB - 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.
KW - Geostatistics
KW - Kernel density estimation
KW - Missing locations
KW - Multi-modal distributions
U2 - 10.1016/j.spasta.2014.11.002
DO - 10.1016/j.spasta.2014.11.002
M3 - Journal article
VL - 11
SP - 35
EP - 44
JO - Spatial Statistics
JF - Spatial Statistics
SN - 2211-6753
ER -