\HeaderA{spectrum}{Spectral Density Estimation}{spectrum}
\aliasA{spec}{spectrum}{spec}
\keyword{ts}{spectrum}
\begin{Description}\relax
The \code{spectrum} function estimates the spectral density of a
time series.
\end{Description}
\begin{Usage}
\begin{verbatim}
spectrum(x, ..., method = c("pgram", "ar"))
\end{verbatim}
\end{Usage}
\begin{Arguments}
\begin{ldescription}
\item[\code{x}] A univariate or multivariate time series.
\item[\code{method}] String specifying the method used to estimate the
spectral density.  Allowed methods are \code{"pgram"} (the default)
and \code{"ar"}.
\item[\code{...}] Further arguments to specific spec methods or
\code{plot.spec}.
\end{ldescription}
\end{Arguments}
\begin{Details}\relax
\code{spectrum} is a wrapper function which calls the methods
\code{\LinkA{spec.pgram}{spec.pgram}} and \code{\LinkA{spec.ar}{spec.ar}}.

The spectrum here is defined with scaling \code{1/\LinkA{frequency}{frequency}(x)},
following S-PLUS.  This makes the spectral density a density over the
range \code{(-frequency(x)/2, +frequency(x)/2]}, whereas a more common
scaling is \eqn{2\pi}{} and range \eqn{(-0.5, 0.5]}{} (e.g., Bloomfield)
or 1 and range \eqn{(-\pi, \pi]}{}.

If available, a confidence interval will be plotted by
\code{plot.spec}: this is asymmetric, and the width of the centre
mark indicates the equivalent bandwidth.
\end{Details}
\begin{Value}
An object of class \code{"spec"}, which is a list containing at
least the following components:
\begin{ldescription}
\item[\code{freq}] vector of frequencies at which the spectral
density is estimated. (Possibly approximate Fourier frequencies.)
The units are the reciprocal of cycles per unit time (and not per
observation spacing): see \sQuote{Details} below.
\item[\code{spec}] Vector (for univariate series) or matrix (for multivariate
series) of estimates of the spectral density at frequencies
corresponding to \code{freq}.
\item[\code{coh}] \code{NULL} for univariate series. For multivariate time
series, a matrix containing the \emph{squared} coherency
between different
series. Column \eqn{ i + (j - 1) * (j - 2)/2}{} of \code{coh}
contains the squared coherency between columns \eqn{i}{} and \eqn{j}{}
of \code{x}, where \eqn{i < j}{}.
\item[\code{phase}] \code{NULL} for univariate series. For multivariate
time series a matrix containing the cross-spectrum phase between
different series. The format is the same as \code{coh}.
\item[\code{series}] The name of the time series.
\item[\code{snames}] For multivariate input, the names of the component series.
\item[\code{method}] The method used to calculate the spectrum.
\end{ldescription}


The result is returned invisibly if \code{plot} is true.
\end{Value}
\begin{Note}\relax
The default plot for objects of class \code{"spec"} is quite complex,
including an error bar and default title, subtitle and axis
labels.  The defaults can all be overridden by supplying the
appropriate graphical parameters.
\end{Note}
\begin{Author}\relax
Martyn Plummer, B.D. Ripley
\end{Author}
\begin{References}\relax
Bloomfield, P. (1976) \emph{Fourier Analysis of Time Series: An
Introduction.} Wiley.

Brockwell, P. J. and Davis, R. A. (1991) \emph{Time Series: Theory and
Methods.} Second edition. Springer.

Venables, W. N. and Ripley, B. D. (2002) \emph{Modern Applied
Statistics with S-PLUS.} Fourth edition. Springer. (Especially
pages 392--7.)
\end{References}
\begin{SeeAlso}\relax
\code{\LinkA{spec.ar}{spec.ar}},
\code{\LinkA{spec.pgram}{spec.pgram}};
\code{\LinkA{plot.spec}{plot.spec}}.
\end{SeeAlso}
\begin{Examples}
\begin{ExampleCode}
require(graphics)

## Examples from Venables & Ripley
## spec.pgram
par(mfrow=c(2,2))
spectrum(lh)
spectrum(lh, spans=3)
spectrum(lh, spans=c(3,3))
spectrum(lh, spans=c(3,5))

spectrum(ldeaths)
spectrum(ldeaths, spans=c(3,3))
spectrum(ldeaths, spans=c(3,5))
spectrum(ldeaths, spans=c(5,7))
spectrum(ldeaths, spans=c(5,7), log="dB", ci=0.8)

# for multivariate examples see the help for spec.pgram

## spec.ar
spectrum(lh, method="ar")
spectrum(ldeaths, method="ar")
\end{ExampleCode}
\end{Examples}

