BUILT-IN MATHEMATICA SYMBOL
CovarianceFunction
CovarianceFunction[data, hspec]
estimates the covariance function at lags hspec from data.
CovarianceFunction[proc, hspec]
represents the covariance function at lags hspec for the random process proc.
CovarianceFunction[proc, s, t]
represents the covariance function at times s and t for the random process proc.
- CovarianceFunction is also known as autocovariance function.
- CovarianceFunction for a process proc with mean function
and value 
at time t is given by:
-
| Expectation[(x[s]- [s])(x[t]-[t])] | for a scalar-valued process |
| Expectation[(x[s]-[s] (x[t]-[t])] | for a vector-valued process |
- CovarianceFunction[proc, h] is defined only if proc is a weakly stationary process and is equivalent to CovarianceFunction[proc, 0, h].
- The process proc can be any random process such as ARMAProcess and WienerProcess.
- CovarianceFunction[{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 covariance function at lag 2:
| Out[1]= |  |
The sample covariance function for a random sample from an autoregressive time series:
| Out[2]= |  |
Calculate the covariance function for a discrete-time process:
| Out[1]= |  |
| Out[2]= |  |
Calculate the covariance function for a continuous-time process:
| Out[1]= |  |
| Out[2]= |  |
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