Estimate Spectral Density of a Time Series by a Smoothed Periodogram

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Description

spec.pgram calculates the periodogram using a fast Fourier transform, and optionally smooths the result with a series of modified Daniell smoothers (moving averages giving half weight to the end values).

Usage

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spec.pgram(x, spans = NULL, kernel, taper = 0.1,
           pad = 0, fast = TRUE, demean = FALSE, detrend = TRUE,
           plot = TRUE, na.action = na.fail, ...)

Arguments

x

univariate or multivariate time series.

spans

vector of odd integers giving the widths of modified Daniell smoothers to be used to smooth the periodogram.

kernel

alternatively, a kernel smoother of class "tskernel".

taper

specifies the proportion of data to taper. A split cosine bell taper is applied to this proportion of the data at the beginning and end of the series.

pad

proportion of data to pad. Zeros are added to the end of the series to increase its length by the proportion pad.

fast

logical; if TRUE, pad the series to a highly composite length.

demean

logical. If TRUE, subtract the mean of the series.

detrend

logical. If TRUE, remove a linear trend from the series. This will also remove the mean.

plot

plot the periodogram?

na.action

NA action function.

...

graphical arguments passed to plot.spec.

Details

The raw periodogram is not a consistent estimator of the spectral density, but adjacent values are asymptotically independent. Hence a consistent estimator can be derived by smoothing the raw periodogram, assuming that the spectral density is smooth.

The series will be automatically padded with zeros until the series length is a highly composite number in order to help the Fast Fourier Transform. This is controlled by the fast and not the pad argument.

The periodogram at zero is in theory zero as the mean of the series is removed (but this may be affected by tapering): it is replaced by an interpolation of adjacent values during smoothing, and no value is returned for that frequency.

Value

A list object of class "spec" (see spectrum) with the following additional components:

kernel

The kernel argument, or the kernel constructed from spans.

df

The distribution of the spectral density estimate can be approximated by a (scaled) chi square distribution with df degrees of freedom.

bandwidth

The equivalent bandwidth of the kernel smoother as defined by Bloomfield (1976, page 201).

taper

The value of the taper argument.

pad

The value of the pad argument.

detrend

The value of the detrend argument.

demean

The value of the demean argument.

The result is returned invisibly if plot is true.

Author(s)

Originally Martyn Plummer; kernel smoothing by Adrian Trapletti, synthesis by B.D. Ripley

References

Bloomfield, P. (1976) Fourier Analysis of Time Series: An Introduction. Wiley.

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

Venables, W.N. and Ripley, B.D. (2002) Modern Applied Statistics with S. Fourth edition. Springer. (Especially pp. 392–7.)

See Also

spectrum, spec.taper, plot.spec, fft

Examples

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require(graphics)

## Examples from Venables & Ripley
spectrum(ldeaths)
spectrum(ldeaths, spans = c(3,5))
spectrum(ldeaths, spans = c(5,7))
spectrum(mdeaths, spans = c(3,3))
spectrum(fdeaths, spans = c(3,3))

## bivariate example
mfdeaths.spc <- spec.pgram(ts.union(mdeaths, fdeaths), spans = c(3,3))
# plots marginal spectra: now plot coherency and phase
plot(mfdeaths.spc, plot.type = "coherency")
plot(mfdeaths.spc, plot.type = "phase")

## now impose a lack of alignment
mfdeaths.spc <- spec.pgram(ts.intersect(mdeaths, lag(fdeaths, 4)),
   spans = c(3,3), plot = FALSE)
plot(mfdeaths.spc, plot.type = "coherency")
plot(mfdeaths.spc, plot.type = "phase")

stocks.spc <- spectrum(EuStockMarkets, kernel("daniell", c(30,50)),
                       plot = FALSE)
plot(stocks.spc, plot.type = "marginal") # the default type
plot(stocks.spc, plot.type = "coherency")
plot(stocks.spc, plot.type = "phase")

sales.spc <- spectrum(ts.union(BJsales, BJsales.lead),
                      kernel("modified.daniell", c(5,7)))
plot(sales.spc, plot.type = "coherency")
plot(sales.spc, plot.type = "phase")

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