In
time series analysis
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
, the cross-spectrum is used as part of a
frequency domain
In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency, rather than time. Put simply, a time-domain graph shows how a s ...
analysis of the
cross-correlation or
cross-covariance between two time series.
Definition
Let
represent a pair of
stochastic processes that are jointly
wide sense stationary
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Con ...
with
autocovariance
In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process ...
functions
and
and
cross-covariance function
. Then the cross-spectrum
is defined as the
Fourier transform of
:
where
: