# CorrelationFunction

CorrelationFunction[data,hspec]

estimates the correlation function at lags hspec from data.

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.

# Details

• CorrelationFunction is also known as autocorrelation or cross-correlation function (ACF or CCF).
• The following specifications can be given for hspec:
•  τ at time or lag τ {τmax} unit spaced from 0 to τmax {τmin,τmax} unit spaced from τmin to τmax {τmin,τmax,dτ} from τmin to τmax in steps of dτ {{τ1,τ2,…}} use explicit {τ1,τ2,…}
• 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.
• CorrelationFunction of the process proc is the CovarianceFunction c normalized by the outer product of the standard deviation function σ at times s and t:
•  c[s,t]/(σ[s]σ[t]) for scalar-valued data or processes c[s,t]/(σ[s] ⊗ σ[t]) for vector-valued data or processes
• The symbol represents KroneckerProduct.
• CorrelationFunction[proc,h] is defined only if proc is a weakly stationary process and is equivalent to CorrelationFunction[proc,h,0].
• The process proc can be any random process, such as ARMAProcess and WienerProcess.

# Examples

open allclose all

## Basic Examples(4)

Estimate the correlation function at lag 2:

 In[1]:=
 Out[1]=

The sample correlation function for a random sample from an autoregressive time series:

 In[1]:=
 In[2]:=
 Out[2]=

The correlation function for a discrete-time process:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=

The correlation function for a continuous-time process:

 In[1]:=
 Out[1]=
 In[2]:=
 Out[2]=