Statistics of spatial averages and optimal averaging in the presence of missing data

TL;DR

This paper analyzes how missing data affects the bias and variance of spatial averages, proposing methods to optimize weighting strategies to minimize errors, with applications to rainfall data over India.

Contribution

It extends existing optimal averaging strategies to account for missing data, providing a framework to reduce bias and variance in spatial averages.

Findings

Missing data increases variance and bias in spatial averages.

Optimal weighting depends on local variance, covariance, and proximity to the true average.

The model effectively estimates standard error in all-India rainfall measurements.

Abstract

We consider statistics of spatial averages estimated by weighting observations over an arbitrary spatial domain using identical and independent measuring devices, and derive an account of bias and variance in the presence of missing observations. We test the model relative to simulations, and the approximations for bias and variance with missing data are shown to compare well even when the probability of missing data is large. Previous authors have examined optimal averaging strategies for minimizing bias, variance and mean squared error of the spatial average, and we extend the analysis to the case of missing observations. Minimizing variance mainly requires higher weights where local variance and covariance is small, whereas minimizing bias requires higher weights where the field is closer to the true spatial average. Missing data increases variance and contributes to bias, and reducing both effects involves emphasizing locations with mean value nearer to the spatial average. The framework is applied to study spatially averaged rainfall over India. We use our model to estimate standard error in all-India rainfall as the combined effect of measurement uncertainty and bias, when weights are chosen so as to yield minimum mean squared error.