### Abstract

The standard geostatistical problem is to predict the values of a spatially continuous phenomenon, S(x) say, at locations x using data (y_{i}, x_{i}):i=1, ..., n where y_{i} is the realisation at location x_{i} of S(x_{i}), or of a random variable Y_{i} that is stochastically related to S(x_{i}). 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={x_{i}: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 language | English (US) |
---|---|

Pages (from-to) | 35-44 |

Number of pages | 10 |

Journal | Spatial Statistics |

Volume | 11 |

DOIs | |

State | Published - Feb 1 2015 |

Externally published | Yes |

### Fingerprint

### 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

*Spatial Statistics*,

*11*, 35-44. https://doi.org/10.1016/j.spasta.2014.11.002

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

Research output: Contribution to journal › Article

*Spatial Statistics*, vol. 11, pp. 35-44. https://doi.org/10.1016/j.spasta.2014.11.002

}

TY - JOUR

T1 - On the inverse geostatistical problem of inference on missing locations

AU - Giorgi, Emanuele

AU - Diggle, Peter J.

PY - 2015/2/1

Y1 - 2015/2/1

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

UR - http://www.scopus.com/inward/record.url?scp=84920099457&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84920099457&partnerID=8YFLogxK

U2 - 10.1016/j.spasta.2014.11.002

DO - 10.1016/j.spasta.2014.11.002

M3 - Article

VL - 11

SP - 35

EP - 44

JO - Spatial Statistics

JF - Spatial Statistics

SN - 2211-6753

ER -