Prediction Intervals, Factor Analysis Models, and High-Dimensional Empirical Linear Prediction

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Abstract
We discuss a technique that provides prediction intervals based on a model called an empirical linear model. The technique, high-dimensional empirical linear prediction (HELP), involves principal component analysis, factor analysis and model selection. In fact, a special case of the empirical model is the factor analysis model. A factor analysis model does not generally aim at prediction, however. Therefore, HELP can be viewed as a technique that provides prediction (and confidence) intervals based on a factor analysis model or a more generalized model, possibly with unknown dimension to be estimated. Although factor analysis models do not typically have justifiable theory due to nonidentifiability, we show that our intervals are justifiable asymptotically. An interval for a future response is called a prediction interval; an interval for the mean of the future response is called a confidence interval. These intervals were compared to the intervals of Hwang and Liu, which were derived using standard asymptotic theory where the relevant covariance matrix has a fixed dimension. In contrast, our intervals are derived asymptotically with the dimension of the covariance matrix approaching infinity, a result much more difficult to obtain. However, the numerical results show that the intervals of this article are much more satisfactory in many cases, including the motivating application. The application that motivated us arises from the work of a group of electrical engineers led by Souders and Stenbakken at the National Institute of Standards and Technology (NIST). Their aim is to reduce the number of measurements of a high-dimensional variable of dimension, 213 = 8,192, called the future “measurements,” by using the past measurements of similar electric components, such as A/D converters. They claim that only 64 out of 8,192 measurements need to be measured to predict the rest of unobserved measurements well. In this article, we construct our intervals using only 64 measurements of the “future observations” and show that the intervals seem narrow enough to justify their claim.

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