Principal Component Analysis

The Karhunen-Lo eve basis functions, more frequently referred to as principal components or empirical orthogonal functions (EOFs), of the noise response of the climate system are an important tool for geophysical studies. Many researchers have used this tool to examine the geophysical and climatological phenomena. Perhaps more frequent use of EOFs in recent studies is in conjunction with the development of the signal detection and estimation methods of the background uctuations of a detection variable serve as an orthogonal basis set and are used to design optimal techniques for detecting and estimating signals. A detection and prediction approach is to design a lter or optimal weights for the signal to be detected. It has been reported that weighted averaging of data over the surface of the Earth improved the detectability of climatic changes (Hasselmann 1979; Stefanick 1981; Bell 1982). Since the signal-to-noise ratio (SNR) varies geographically, there exists an optimal geographical weighting of the signal which maximizes the SNR. The design of an optimal weighting function may require detailed knowledge on the natural uctuation of the climate system. A conceptually similar approach is to employ a particular pattern (or patterns) of climatic change for detection and prediction (e.g., Barnett and Hasselmann 1979; Hasselmann 1979). The patterns of interest (also called the predictors) may include the principal components (empirical orthogonal functions) (e. von Storch 1990) among others. This approach also requires complete knowledge of the natural variability of the climate system. To test and improve the detection and prediction techniques addressed above, a complete cross-spectral covariance matrix, or similarly, a complete set of the principal components of natural uctuations of the climate system for each frequency band of the spectrum is necessary. In reality, a reliable spectrum of observational covariance matrix is not available because observations are not suuciently long and sampling errors contaminate the observational records (Preisendorfer and Barnett 1977; North et al. 1982). Further, inadequate spatial coverage of observations may introduce bias. Therefore, the covariance matrix of the noise response is often estimated from a simple stochastic model. Kim and North (1991, 1992) examined the covariance matrix in terms of various second-moment statistics earlier. Examined here are the principal components of the covariance matrix of the surface temperature uctuations in a simple coupled climate model in comparison with observations. The principal components not only are an