Spectral Plots

The Spectral Plots procedure is used to identify periodic behavior in time series. Instead of analyzing the variation from one time point to the next, it analyzes the variation of the series as a whole into periodic components of different frequencies. Smooth series have stronger periodic components at low frequencies; random variation ("white noise") spreads the component strength over all frequencies.

Series that include missing data cannot be analyzed with this procedure.

Example. The rate at which new houses are constructed is an important barometer of the state of the economy. Data for housing starts typically exhibit a strong seasonal component. But are there longer cycles present in the data that analysts need to be aware of when evaluating current figures?

Statistics. Sine and cosine transforms, periodogram value, and spectral density estimate for each frequency or period component. When bivariate analysis is selected: real and imaginary parts of cross-periodogram, cospectral density, quadrature spectrum, gain, squared coherency, and phase spectrum for each frequency or period component.

Plots. For univariate and bivariate analyses: periodogram and spectral density. For bivariate analyses: squared coherency, quadrature spectrum, cross amplitude, cospectral density, phase spectrum, and gain.

Spectral Plots Data Considerations

Data. The variables should be numeric.

Assumptions. The variables should not contain any embedded missing data. The time series to be analyzed should be stationary and any non-zero mean should be subtracted out from the series. For instructions on handling missing data, see the topic on replacing missing values. The most effective way to transform a nonstationary series into a stationary one is through a difference transformation. To learn how to perform a difference transformation, see the topic on creating time series.

Stationary. A condition that must be met by the time series to which you fit an ARIMA model. Pure MA series will be stationary; however, AR and ARMA series might not be. A stationary series has a constant mean and a constant variance over time.

Obtaining a Spectral Analysis

This feature requires the Forecasting option.

From the menus choose:
Analysis > Time Series > Spectral Analysis...
Select one or more variables from the available list and move them to the Variable(s) list. Note that the list includes only numeric variables.
Select one of the Spectral Window options to choose how to smooth the periodogram in order to obtain a spectral density estimate. Available smoothing options are Tukey-Hamming, Tukey, Parzen, Bartlett, Daniell (Unit), and None.

Tukey-Hamming. The weights are Wk = .54Dp(2 pi fk) + .23Dp (2 pi fk + pi/p) + .23Dp (2 pi fk - pi/p), for k = 0, ..., p, where p is the integer part of half the span and Dp is the Dirichlet kernel of order p.
Tukey. The weights are Wk = 0.5Dp(2 pi fk) + 0.25Dp (2 pi fk + pi/p) + 0.25Dp(2 pi fk - pi/p), for k = 0, ..., p, where p is the integer part of half the span and Dp is the Dirichlet kernel of order p.
Parzen. The weights are Wk = 1/p(2 + cos(2 pi fk)) (F[p/2] (2 pi fk))**2, for k= 0, ... p, where p is the integer part of half the span and F[p/2] is the Fejer kernel of order p/2.
Bartlett. The shape of a spectral window for which the weights of the upper half of the window are computed as Wk = Fp (2*pi*fk), for k = 0, ... p, where p is the integer part of half the span and Fp is the Fejer kernel of order p. The lower half is symmetric with the upper half.
Daniell (Unit). The shape of a spectral window for which the weights are all equal to 1.
None. No smoothing. If this option is chosen, the spectral density estimate is the same as the periodogram.

Span. The range of consecutive values across which the smoothing is carried out. Generally, an odd integer is used. Larger spans smooth the spectral density plot more than smaller spans.

Center variables. Adjusts the series to have a mean of 0 before calculating the spectrum and to remove the large term that may be associated with the series mean.

Bivariate analysis—first variable with each. If you have selected two or more variables, you can select this option to request bivariate spectral analyses.

The first variable in the Variable(s) list is treated as the independent variable, and all remaining variables are treated as dependent variables.
Each series after the first is analyzed with the first series independently of other series named. Univariate analyses of each series are also performed.

Plot. Periodogram and spectral density are available for both univariate and bivariate analyses. All other choices are available only for bivariate analyses.

Periodogram. Unsmoothed plot of spectral amplitude (plotted on a logarithmic scale) against either frequency or period. Low-frequency variation characterizes a smooth series. Variation spread evenly across all frequencies indicates "white noise."
Squared coherency. The product of the gains of the two series.
Quadrature spectrum. The imaginary part of the cross-periodogram, which is a measure of the correlation of the out-of-phase frequency components of two time series. The components are out of phase by pi/2 radians.
Cross amplitude. The square root of the sum of the squared cospectral density and the squared quadrature spectrum.
Spectral density. A periodogram that has been smoothed to remove irregular variation.
Cospectral density. The real part of the cross-periodogram, which is a measure of the correlation of the in-phase frequency components of two time series.
Phase spectrum. A measure of the extent to which each frequency component of one series leads or lags the other.
Gain. The quotient of dividing the cross amplitude by the spectral density for one of the series. Each of the two series has its own gain value.

By frequency. All plots are produced by frequency, ranging from frequency 0 (the constant or mean term) to frequency 0.5 (the term for a cycle of two observations).

By period. All plots are produced by period, ranging from 2 (the term for a cycle of two observations) to a period equal to the number of observations (the constant or mean term). Period is displayed on a logarithmic scale.

This procedure pastes SPECTRA command syntax.