# PartialCorrelationFunction

PartialCorrelationFunction[data,hspec]

estimates the partial correlation function at lags hspec from data.

PartialCorrelationFunction[tproc,hspec]

represents the partial correlation function at lags hspec for the time series process tproc.

# Details • PartialCorrelationFunction is also known as the partial autocorrelation function (PACF).
• PartialCorrelationFunction represents the correlation between x(t) and x(t+h), conditioned on x(u) for t<u<t+h, and x(t) representing tproc at time t.
• PartialCorrelationFunction[tproc,hspec] is defined only if tproc is a weakly stationary process.
• The process tproc can be any process such that WeakStationarity[tproc] gives True.
• 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,…}

# Examples

open allclose all

## Basic Examples(3)

Estimate the partial correlation function at lag 2:

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

Partial correlation function for an ARProcess:

## Scope(9)

### Empirical Estimates(6)

Estimate the partial correlation function for some data at lag 9:

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

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

Compute the partial correlation function for a time series:

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

Estimate the partial correlation function for an ensemble of paths:

Compare empirical and theoretical correlation functions:

### Random Processes(3)

Partial correlation function for a MAProcess has infinite support:

Partial correlation function for an ARProcess has finite support:

Partial correlation function for an ARMAProcess has infinite support:

## 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 partial correlation function of the data decays slowly:

MAProcess is clearly a better candidate model than ARProcess:

Create a PACF plot with white-noise confidence bands:

Plot the partial correlation to lag 20 with 95% white-noise confidence bands:

Compare to uncorrelated white noise:

## Properties & Relations(3)

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

Calculate the sample partial correlation function:

Partial correlation function for the process:

Plot both functions:

Use CorrelationFunction to directly calculate PartialCorrelationFunction:

Define a ToeplitzMatrix using the first components of the correlation vector:

Replace the last column in the matrix with the last components:

Calculate ratio of determinants:

Compare to the value of PartialCorrelationFunction:

Partial correlation function and correlation function agree for lag of 1:

For an ARProcess:

## Possible Issues(2)

Partial correlation function does not exist for non-weakly stationary processes:  Partial correlation function is not defined for zero time difference: