Parameter Estimation for the Spatial Ornstein-Uhlenbeck Process with Missing Observations

Abstract

Suppose we are collecting a set of data on a rectangular sampling grid, it is reasonable to assume that observations (e.g. data that arise in weather forecasting, public health and agriculture) made on each sampling site are spatially correlated. Therefore, when building a model for this type of data, we often pair it with an underlying Gaussian process that contains different spatially dependent parameters. Here, we assume that the Gaussian process is characterized by the Ornstein-Uhlenbeck covariance function, which has the property of being both stationary and Markov under the assumption that no observations are missing. However, in reality, the full data assumption may not be a practical one. In this work, we consider two different scenarios where some observations are missing: 1) a block of observations is missing from the grid and 2) missing observations occur randomly throughout the sampling grid. In each case, we propose an approximate likelihood method to estimate the parameters for the covariance structure. We show that, either by an analytical or a numerical approach, the parameter estimates from the approximate method have similar properties to those obtained under the full data likelihood function. In particular, we show that the parameter estimators in the missing block case are strongly consistent and asymptotically normal under certain regularity condition, and conclude our work by comparing the results from implementing our methods with simulated data.