BUILT-IN MATHEMATICA SYMBOL
CorrelationFunction
CorrelationFunction[proc, hspec]
represents the correlation function at lags hspec for the random process proc.
CorrelationFunction[proc, s, t]
represents the correlation function at times s and t for the random process proc.
- CorrelationFunction is also known as autocorrelation or cross-correlation function (ACF or CCF).
- CorrelationFunction of the process proc is the CovarianceFunction c normalized by the outer product of the standard deviation function
at times s and t:
-
| c[s,t]/([s][t]) | for scalar-valued data or processes |
| c[s,t]/([s]  [t]) | for vector-valued data or processes |
- CorrelationFunction[proc, h] is defined only if proc is a weakly stationary process and is equivalent to CorrelationFunction[proc, 0, h].
- The process proc can be any random process such as ARMAProcess or WienerProcess.
- CorrelationFunction[{x1, ..., xn}, h] is equivalent to
with 
=Mean[{x1, ..., xn}].
- When data is TemporalData containing an ensemble of paths, the output represents the average across all paths.
- The following specifications can be given for hspec:
-
|  | at time or lag |
| {max} | unit spaced from 0 to  |
| {min,max} | unit spaced from  to  |
| {min,max,d} | from  to  in steps of d |
| {{1,2,...}} | use explicit  |
Estimate the correlation function at lag 2:
| Out[1]= |  |
The sample correlation function for a random sample from an autoregressive time series:
| Out[2]= |  |
The correlation function for a discrete-time process:
| Out[1]= |  |
| Out[2]= |  |
The correlation function for a continuous-time process:
| Out[1]= |  |
| Out[2]= |  |
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