SPECTRUM Procedure

@SPECTRUM computes and optionally graphs the estimated spectrum of a single series. Use @CROSSPEC to compute and graph cross-spectral statistics such as coherence and phase.

@SPECTRUM( options ) series start end

Parameters

series	series to analyze
start, end	range of series to use. By default, the defined range of series.

Options

ORDINATES=Number of ordinates [depends upon length,seasonal]

WINDOW=[FLAT]/TENT/QUADRATIC

WIDTH=window width [depends upon ordinates]

WEIGHT=VECTOR of customized relative weights for a half-window.

For, instance WEIGHT=||3,2,1|| would give a centered window with weights in the pattern 1,2,3,2,1 normalized to sum to one.

PERIODOGRAM/[NOPERIODOGRAM]

Compute the unsmoothed periodogram rather than a smoothed estimate.

TAPER=TRAPEZOIDAL/COSINE/[NONE]

WTAPER=Taper width [.25 of length]

[GRAPH]/NOGRAPH

HEADER="Header for graph"

FOOTER="Footer for graph"

[LOGSCALE]/NOLOGSCALE

Graph spectrum in log scale or not.

SPECTRUM=(output) series for $[0,\pi]$ spectrum (0 frequency at entry 1)

Variables Defined

%EDF	Equivalent Degrees of Freedom for spectral window.

Example

* Brockwell & Davis, "Time Series Analysis: Theory and Methods"

* Example 10.4.3 on page 354

open data sunspots.dat

calendar 1770

data(format=free,org=columns) 1770:1 1869:1 sunspots

diff(center) sunspots / cspots

@spectrum(footer="Figure 10.7 Periodogram of Wolfer Sunspot Numbers",$

periodogram,nologscale) cspots

@spectrum(footer="Figure 10.8 Spectral Estimate of Wolfer Sunspot Numbers",$

weights=||3.0,3.0,2.0,1.0||,nologscale,spectrum=smoothed,ordinates=128) cspots

* Confidence bands (text only does these for the artificial data). %EDF

* is the Equivalent Degrees of Freedom of the window used by @spectrum.

set lower = smoothed*%edf/%invchisqr(.025,%edf)

set upper = smoothed*%edf/%invchisqr(.975,%edf)

graph(footer="Spectral Estimate of Wolfer Sunspot Numbers with 95% Confidence Interval") 3

# smoothed

# lower / 2

# upper / 2

* Same thing done on log scale - the confidence bands are equal width

graph(footer="Log Spectral Estimate of Wolfer Sunspot Numbers with 95% Confidence Interval",log=10) 3

# smoothed

# lower / 2

# upper / 2