Estimate Spectral Density of a Time Series by a Smoothed Periodogram
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).
spec.pgram(x, spans = NULL, kernel, taper = 0.1, pad = 0, fast = TRUE, demean = FALSE, detrend = TRUE, plot = TRUE, na.action = na.fail, ...)
- univariate or multivariate time series.
- vector of odd integers giving the widths of modified Daniell smoothers to be used to smooth the periodogram.
- alternatively, a kernel smoother of class
- 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.
- proportion of data to pad. Zeros are added to the end of
the series to increase its length by the proportion
- logical; if
TRUE, pad the series to a highly composite length.
- logical. If
TRUE, subtract the mean of the series.
- logical. If
TRUE, remove a linear trend from the series. This will also remove the mean.
- plot the periodogram?
- graphical arguments passed to
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
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.
- A list object of class
spectrum) with the following additional components:
kernelargument, or the kernel constructed from
df The distribution of the spectral density estimate can be approximated by a (scaled) chi square distribution with
dfdegrees of freedom.
bandwidth The equivalent bandwidth of the kernel smoother as defined by Bloomfield (1976, page 201). taper The value of the
pad The value of the
detrend The value of the
demean The value of the
- The result is returned invisibly if
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.
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")