Chapter 15
Multivariate Statistical
Modeling: POP-Model as
a First Order
Approximation
by Jin-Song von Storch
15.1 Introduction
The study of a time series is a standard exercise in statistical analysis. A
time series, which is an ordered set of random variables, and its associated
probability distribution are called a stochastic process. This mathematical
construct can be applied to time series of climate variables. Strictly speaking,
a climate variable is generated by deterministic processes. However since a
myriad of processes contribute to the behavior of a climate variable, a climate
time series behaves like one generated by a stochastic process. More detailed
discussion of this problem is given by H. von Storch and Zwiers (1995).
The most important assumption made about a time series is that the corresponding stochastic process is stationary, and that a stationary stochastic
process may be adequately described by the lower moments of its probability distribution. The lower moments include the mean, variance, covariance
function and the Fourier transform of the covariance function, the power
spectrum. For multivariate time series, one has additionally cross-covariance
functions between components of the multivariate time series and the Fourier