# Correlation Analysis of Signals

The package provides functions to perform correlation analysis of sequential signals.

## Autocovariance and Autocorrelation

`StatsBase.autocov`

— Function`autocov(x, [lags]; demean=true)`

Compute the autocovariance of a vector or matrix `x`

, optionally specifying the `lags`

at which to compute the autocovariance. `demean`

denotes whether the mean of `x`

should be subtracted from `x`

before computing the autocovariance.

If `x`

is a vector, return a vector of the same length as `lags`

. If `x`

is a matrix, return a matrix of size `(length(lags), size(x,2))`

, where each column in the result corresponds to a column in `x`

.

When left unspecified, the lags used are the integers from 0 to `min(size(x,1)-1, 10*log10(size(x,1)))`

.

The output is not normalized. See `autocor`

for a function with normalization.

`StatsBase.autocov!`

— Function`autocov!(r, x, lags; demean=true)`

Compute the autocovariance of a vector or matrix `x`

at `lags`

and store the result in `r`

. `demean`

denotes whether the mean of `x`

should be subtracted from `x`

before computing the autocovariance.

If `x`

is a vector, `r`

must be a vector of the same length as `lags`

. If `x`

is a matrix, `r`

must be a matrix of size `(length(lags), size(x,2))`

, and where each column in the result will correspond to a column in `x`

.

The output is not normalized. See `autocor!`

for a method with normalization.

`StatsBase.autocor`

— Function`autocor(x, [lags]; demean=true)`

Compute the autocorrelation function (ACF) of a vector or matrix `x`

, optionally specifying the `lags`

. `demean`

denotes whether the mean of `x`

should be subtracted from `x`

before computing the ACF.

If `x`

is a vector, return a vector of the same length as `lags`

. If `x`

is a matrix, return a matrix of size `(length(lags), size(x,2))`

, where each column in the result corresponds to a column in `x`

.

When left unspecified, the lags used are the integers from 0 to `min(size(x,1)-1, 10*log10(size(x,1)))`

.

The output is normalized by the variance of `x`

, i.e. so that the lag 0 autocorrelation is 1. See `autocov`

for the unnormalized form.

`StatsBase.autocor!`

— Function`autocor!(r, x, lags; demean=true)`

Compute the autocorrelation function (ACF) of a vector or matrix `x`

at `lags`

and store the result in `r`

. `demean`

denotes whether the mean of `x`

should be subtracted from `x`

before computing the ACF.

If `x`

is a vector, `r`

must be a vector of the same length as `lags`

. If `x`

is a matrix, `r`

must be a matrix of size `(length(lags), size(x,2))`

, and where each column in the result will correspond to a column in `x`

.

The output is normalized by the variance of `x`

, i.e. so that the lag 0 autocorrelation is 1. See `autocov!`

for the unnormalized form.

## Cross-covariance and Cross-correlation

`StatsBase.crosscov`

— Function`crosscov(x, y, [lags]; demean=true)`

Compute the cross covariance function (CCF) between real-valued vectors or matrices `x`

and `y`

, optionally specifying the `lags`

. `demean`

specifies whether the respective means of `x`

and `y`

should be subtracted from them before computing their CCF.

If both `x`

and `y`

are vectors, return a vector of the same length as `lags`

. Otherwise, compute cross covariances between each pairs of columns in `x`

and `y`

.

When left unspecified, the lags used are the integers from `-min(size(x,1)-1, 10*log10(size(x,1)))`

to `min(size(x,1), 10*log10(size(x,1)))`

.

The output is not normalized. See `crosscor`

for a function with normalization.

`StatsBase.crosscov!`

— Function`crosscov!(r, x, y, lags; demean=true)`

Compute the cross covariance function (CCF) between real-valued vectors or matrices `x`

and `y`

at `lags`

and store the result in `r`

. `demean`

specifies whether the respective means of `x`

and `y`

should be subtracted from them before computing their CCF.

If both `x`

and `y`

are vectors, `r`

must be a vector of the same length as `lags`

. If either `x`

is a matrix and `y`

is a vector, `r`

must be a matrix of size `(length(lags), size(x, 2))`

; if `x`

is a vector and `y`

is a matrix, `r`

must be a matrix of size `(length(lags), size(y, 2))`

. If both `x`

and `y`

are matrices, `r`

must be a three-dimensional array of size `(length(lags), size(x, 2), size(y, 2))`

.

The output is not normalized. See `crosscor!`

for a function with normalization.

`StatsBase.crosscor`

— Function`crosscor(x, y, [lags]; demean=true)`

Compute the cross correlation between real-valued vectors or matrices `x`

and `y`

, optionally specifying the `lags`

. `demean`

specifies whether the respective means of `x`

and `y`

should be subtracted from them before computing their cross correlation.

If both `x`

and `y`

are vectors, return a vector of the same length as `lags`

. Otherwise, compute cross covariances between each pairs of columns in `x`

and `y`

.

When left unspecified, the lags used are the integers from `-min(size(x,1)-1, 10*log10(size(x,1)))`

to `min(size(x,1), 10*log10(size(x,1)))`

.

The output is normalized by `sqrt(var(x)*var(y))`

. See `crosscov`

for the unnormalized form.

`StatsBase.crosscor!`

— Function`crosscor!(r, x, y, lags; demean=true)`

Compute the cross correlation between real-valued vectors or matrices `x`

and `y`

at `lags`

and store the result in `r`

. `demean`

specifies whether the respective means of `x`

and `y`

should be subtracted from them before computing their cross correlation.

If both `x`

and `y`

are vectors, `r`

must be a vector of the same length as `lags`

. If either `x`

is a matrix and `y`

is a vector, `r`

must be a matrix of size `(length(lags), size(x, 2))`

; if `x`

is a vector and `y`

is a matrix, `r`

must be a matrix of size `(length(lags), size(y, 2))`

. If both `x`

and `y`

are matrices, `r`

must be a three-dimensional array of size `(length(lags), size(x, 2), size(y, 2))`

.

The output is normalized by `sqrt(var(x)*var(y))`

. See `crosscov!`

for the unnormalized form.

## Partial Autocorrelation Function

`StatsBase.pacf`

— Function`pacf(X, lags; method=:regression)`

Compute the partial autocorrelation function (PACF) of a real-valued vector or matrix `X`

at `lags`

. `method`

designates the estimation method. Recognized values are `:regression`

, which computes the partial autocorrelations via successive regression models, and `:yulewalker`

, which computes the partial autocorrelations using the Yule-Walker equations.

If `x`

is a vector, return a vector of the same length as `lags`

. If `x`

is a matrix, return a matrix of size `(length(lags), size(x, 2))`

, where each column in the result corresponds to a column in `x`

.

`StatsBase.pacf!`

— Function`pacf!(r, X, lags; method=:regression)`

Compute the partial autocorrelation function (PACF) of a matrix `X`

at `lags`

and store the result in `r`

. `method`

designates the estimation method. Recognized values are `:regression`

, which computes the partial autocorrelations via successive regression models, and `:yulewalker`

, which computes the partial autocorrelations using the Yule-Walker equations.

`r`

must be a matrix of size `(length(lags), size(x, 2))`

.