# Eigenvector Analysis for Prediction of Time Series

Glenn W. Brier Department of Atmospheric Science, Colorado State University, Fort Collins 80523

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Gayle T. Meltesen Department of Atmospheric Science, Colorado State University, Fort Collins 80523

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## Abstract

The theorem of singular value decomposition is used to represent a data matrix X as the product of a system with a response R to a forcing function F. Algebraically, R is the matrix of principal components and F the transpose of the matrix of eigenvectors of X′X. If the data are such that the eigenvectors are orthogonal functions of time and they have some recognizable non-random structure permitting predictability in time, then the observed response at time t can be used with the extrapolated forcing function to predict some physical quantity (e.g., temperature, pressure). This method is called the time extrapolated eigenvector prediction (TEEP). An example is given to illustrate the method with a known forcing function, the annual solar heating cycle. We have access to efficient computer routines which will facilitate an extension to much larger data sets.

## Abstract

The theorem of singular value decomposition is used to represent a data matrix X as the product of a system with a response R to a forcing function F. Algebraically, R is the matrix of principal components and F the transpose of the matrix of eigenvectors of X′X. If the data are such that the eigenvectors are orthogonal functions of time and they have some recognizable non-random structure permitting predictability in time, then the observed response at time t can be used with the extrapolated forcing function to predict some physical quantity (e.g., temperature, pressure). This method is called the time extrapolated eigenvector prediction (TEEP). An example is given to illustrate the method with a known forcing function, the annual solar heating cycle. We have access to efficient computer routines which will facilitate an extension to much larger data sets.

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