# 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

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## Basic Examples(4)

Estimate the correlation function at lag 2:

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

The correlation function for a discrete-time process:

The correlation function for a continuous-time process:

## Scope(13)

### Empirical Estimates(7)

Estimate the correlation function for some data at lag 9:

Obtain empirical estimates of the correlation function up to lag 9:

Compute the correlation function for lags 1 to 9 in steps of 2:

Compute the correlation function for a time series:

The correlation function of a time series for multiple lags is given as a time series:

Estimate the correlation function for an ensemble of paths:

Compare empirical and theoretical correlation functions:

Plot the cross-correlation for vector data:

### Random Processes(6)

The correlation function for a weakly stationary discrete-time process:

The correlation function only depends on the antidiagonal :

The correlation function for a weakly stationary continuous-time process:

The correlation function only depends on the antidiagonal :

The correlation function for a non-weakly stationary discrete-time process:

The correlation function depends on both time arguments:

The correlation function for a non-weakly stationary continuous-time process:

The correlation function depends on both time arguments:

The correlation function for some time series processes:

Cross-correlation plots for a vector ARProcess:

## Applications(2)

Determine whether the following data is best modeled with an MAProcess or an ARProcess:

It is difficult to determine the underlying process from sample paths:

The correlation function of the data decays slowly:

ARProcess is clearly a better candidate model than MAProcess:

Create an ACF plot with white-noise confidence bands:

Plot the correlation for lags 0 to 20 with 95% white-noise confidence bands:

Compare to uncorrelated white noise:

## Properties & Relations(12)

Sample correlation function is a biased estimator for the process correlation function:

Calculate the sample correlation function:

Correlation function for the process:

Plot both functions:

Correlation function for a process is the off-diagonal entry in the Correlation matrix:

Sample correlation at lag 0 is always 1:

Sample correlation function is related to CovarianceFunction:

Scaled sample covariance function:

Sample correlation function is related to AbsoluteCorrelationFunction:

Scale by the first element:

Compare to the sample correlation function:

Use Expectation to calculate correlation:

Define mean and standard deviation functions:

Correlation function for equal times reduces to 1:

Correlation function is related to the CovarianceFunction :

For , the standard deviation function is :

The correlation function is related to the Correlation:

It is the off-diagonal entry in the covariance matrix:

Correlation function is invariant for ToInvertibleTimeSeries:

Correlation function is invariant to centralizing:

The data has nonzero mean:

Centralize data:

Compare correlation functions:

Sum of the sample correlation function is constant:

The sample is random:

Calculate the sample correlation function from 1 to n-1:

Calculate the sum:

## Possible Issues(1)

CorrelationFunction output may contain DifferenceRoot:

Use FunctionExpand to recover explicit powers: