Time Series Prediction Functions Based on Imprecise Observations

Abstract

Let $T \subseteq I$ be sets of real numbers. Let $\{Y(t): t\in I\}$ be a real time series whose covariance kernel is assumed known and positive definite. The mean is assumed either to be known or to be an unknown member of a known class of functions on $I$. For each fixed $s\in I, Y(s)$ is predicted by a minimum mean square error unbiased linear predictor $\hat{Y}(s)$ based on $\{Y(t): t\in T\}$. If $\hat{y}(s)$ is the evaluation of $\hat{Y}(s)$ given that the sample path for $\{Y(t): t\in T\}$ is an unknown element of a known collection of functions on $T$, then $\hat{y}(s)$ is a prediction for $Y(s)$ and the function $\hat{y}$ is called a prediction function. Mean-estimation functions are defined similarly. For certain prediction problems based on imprecise observations, characterizations are obtained for these functions in terms of the covariance structure of the process. For a particular prediction problem $\hat{y}$ is shown to be a spline function interpolating a convex set.