Measurement variance control for optimal forecasting

This paper addresses the question of optimal allocation of measurement resources, when [1] the overall cost and time duration of the measurement process are fixed, and [2] the cost of an individual measurement varies inversely with the measurement variance. The goal is to determine the time-distribution of the measurement variance which maximizes the accuracy of forecasting a stochastic process. The solution is found to depend upon the autocorrelation properties,of the underlying stochastic process (assumed known) and upon the forecast horizon. An explicit solution is presented for an integrated Wiener process. In this case, optimal prediction accuracy is achieved with two measurements of unequal variance. The time separation of the two measurements depends upon the process noise, the prediction time, and the total measurement cost.