stats (version 3.3.1)

# acf: Auto- and Cross- Covariance and -Correlation Function Estimation

## Description

The function `acf` computes (and by default plots) estimates of the autocovariance or autocorrelation function. Function `pacf` is the function used for the partial autocorrelations. Function `ccf` computes the cross-correlation or cross-covariance of two univariate series.

## Usage

```acf(x, lag.max = NULL, type = c("correlation", "covariance", "partial"), plot = TRUE, na.action = na.fail, demean = TRUE, ...)
pacf(x, lag.max, plot, na.action, ...)
"pacf"(x, lag.max = NULL, plot = TRUE, na.action = na.fail, ...)
ccf(x, y, lag.max = NULL, type = c("correlation", "covariance"), plot = TRUE, na.action = na.fail, ...)
"["(x, i, j)```

## Arguments

x, y
a univariate or multivariate (not `ccf`) numeric time series object or a numeric vector or matrix, or an `"acf"` object.
lag.max
maximum lag at which to calculate the acf. Default is \$10*log10(N/m)\$ where \$N\$ is the number of observations and \$m\$ the number of series. Will be automatically limited to one less than the number of observations in the series.
type
character string giving the type of acf to be computed. Allowed values are `"correlation"` (the default), `"covariance"` or `"partial"`. Will be partially matched.
plot
logical. If `TRUE` (the default) the acf is plotted.
na.action
function to be called to handle missing values. `na.pass` can be used.
demean
logical. Should the covariances be about the sample means?
...
further arguments to be passed to `plot.acf`.
i
a set of lags (time differences) to retain.
j
a set of series (names or numbers) to retain.

## Value

An object of class `"acf"`, which is a list with the following elements:
lag
A three dimensional array containing the lags at which the acf is estimated.
acf
An array with the same dimensions as `lag` containing the estimated acf.
type
The type of correlation (same as the `type` argument).
n.used
The number of observations in the time series.
series
The name of the series `x`.
snames
The series names for a multivariate time series.
The lag `k` value returned by `ccf(x, y)` estimates the correlation between `x[t+k]` and `y[t]`.The result is returned invisibly if `plot` is `TRUE`.

## Details

For `type` = `"correlation"` and `"covariance"`, the estimates are based on the sample covariance. (The lag 0 autocorrelation is fixed at 1 by convention.)

By default, no missing values are allowed. If the `na.action` function passes through missing values (as `na.pass` does), the covariances are computed from the complete cases. This means that the estimate computed may well not be a valid autocorrelation sequence, and may contain missing values. Missing values are not allowed when computing the PACF of a multivariate time series.

The partial correlation coefficient is estimated by fitting autoregressive models of successively higher orders up to `lag.max`.

The generic function `plot` has a method for objects of class `"acf"`.

The lag is returned and plotted in units of time, and not numbers of observations.

There are `print` and subsetting methods for objects of class `"acf"`.

## References

Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth Edition. Springer-Verlag.

(This contains the exact definitions used.)

`plot.acf`, `ARMAacf` for the exact autocorrelations of a given ARMA process.

## Examples

Run this code
```require(graphics)

## Examples from Venables & Ripley
acf(lh)
acf(lh, type = "covariance")
pacf(lh)

acf(ldeaths)
acf(ldeaths, ci.type = "ma")
acf(ts.union(mdeaths, fdeaths))
ccf(mdeaths, fdeaths, ylab = "cross-correlation")
# (just the cross-correlations)

presidents # contains missing values
acf(presidents, na.action = na.pass)
pacf(presidents, na.action = na.pass)
```

Run the code above in your browser using DataCamp Workspace