Prediction of random fields with incomplete quarter-plane past

Abstract

Let $\{X(n,m);(n,m)\in {\mathbf{Z}}^2\}$ be a zero mean weakly stationary random field and consider the problem of predicting X(0, 0) based on the incomplete quarter-plane past. An explicit formula is obtained for the prediction error of a future value of weakly stationary random fields when the quarter-plane $\mathcal{Q}=\{(n,m)\in {\mathbf{Z}}^2;n\le 0,m\le 0,(n,m)\ne (0,0)\}$ is altered by some missing observations. Some properties for autoregressive (AR) random fields and for moving average (MA) are established.

Keywords:

• Random fields
• moving average representation
• autoregressive representation
• missing value
• prediction

Acknowledgement

We would like to thank the anonymous reviewers and associate editor for their valuable suggestions and helpful remarks that improved the quality and the presentation of the paper.