Bivariate Fourier (Cross Spectrum) Analysis Results

Click the Two series Fourier analysis (select dependent var.) button in the Fourier (Spectral) Analysis dialog box to display the Bivariate Fourier (Cross Spectrum) Analysis Results dialog box, which contains four tabs: Quick, Advanced, Periodogram & density plots, and Append.

Use the options on these tabs to compute cross-spectrum analysis results statistics. The interpretation of the different statistics is discussed in the Cross-spectrum Analysis Overview (see also, the Time Series Analysis Index).

When reviewing the results from long series, you can select the Examine subset of periodogram check box on the Advanced tab; when selected, the results computed by the Summary option and all graphs will be computed only for the range of cases (frequencies) specified in the edit boxes below the check box.

To append any of the computed statistics to the active work area, select the respective check boxes in the Append to work area on Exit group box on the Append tab. Note that if there are not enough unused or unlocked backups for the respective input variable available, STATISTICA will increase the Number of backups per variable parameter by as much as possible (see also the Fourier (Spectral) Analysis specifications dialog).

Summary box. The Summary box at the top of the dialog box displays the names of the series that were analyzed and the number of observations (before and after padding/truncating).

Copy button. Click the Copy button to copy either the selected text (if text has been selected) in the Summary box or all of the text (if no text has been selected) to the Clipboard. Note that the copied text retains formatting information (such as font, color, etc.).

Contract/Expand button. Click the Contract/Expand button to contract or expand the Summary box. When contracted, you can see only one line of the Summary box text and can scroll through the text using a scroll bar. Note that when contracted the text is scrolled so that the first non-blank line is at the top. When expanded (the default setting), the entire Summary box will be displayed on the Multiple Correspondence Analysis Results dialog.

Summary. Click this button to produce a summary spreadsheet containing the frequencies, periods, cosine and sine coefficients, periodogram values, spectral density estimates (computed according to the options selected in the Data windows for density estimates group box on the Advanced tab), cross-periodogram values (real and imaginary), cospectral density and quadrature spectrum values (computed according to the options selected in the Data windows for density estimates group box on the Advanced tab), the cross amplitude values, coherency values, gain values, phase spectrum values, and the weights used to produce the spectral density estimates. If the Highlight values larger than... check box is selected on the Advanced tab, then all values in the periodogram and spectral density columns that are larger than the specified value will be highlighted. The values in this spreadsheet are computed as follows:

Frequency. The frequency is defined as the number of cycles per unit time. In the Time Series module, since one observation is used as the time unit (i.e., frequency is expressed in terms of cycles per observation), the successive frequencies are computed as k/N (for k=0 to N/2) where N is the number observations in the series. Thus, for example, a frequency of .0833 would mean that each observation completes .0833 of the full cycle, or that 12 observations complete one full cycle (.0833*12=1). Thus, if the series contains monthly data collected over several years, the respective periodicity identifies an annual cycle.

Period. The period is computed as the inverse of the frequency. Thus it can be interpreted as the number of observations that is necessary in order to complete one cycle at the respective frequency.

Cosine coefficients. The cosine coefficients can be interpreted as regression coefficients; that is, they tell us the degree to which the respective cosine functions are correlated with the data at the respective frequencies. Cosine coefficients are computed for both the X and Y variable.

Sine coefficients. The sine coefficients can be interpreted analogous to the cosine coefficients (see previous paragraph). Sine coefficients are computed for both the X and Y variable.

Periodogram. The periodogram values are computed as the sum of the squared sine and cosine coefficients at each frequency (times N/2). The periodogram values can be interpreted in terms of variance (sums of squares) of the data at the respective frequency or period. Periodogram values are computed for both the X and Y variable.

Spectral density estimates. The spectral density estimates are computed by smoothing the periodogram values, using the options selected in the Data windows for density estimates group box on the Advanced tab. By smoothing the periodogram, you can identify the general frequency "regions" or (spectral densities) that significantly contribute to the cyclical behavior of the series. Note that the weights used for the smoothing will always be standardized so that they add to 1.0. Also, at the beginning and end of the series, the smoothing is done via reflection.

Cross-periodogram values. Analogous to the results for the single variables, the spreadsheet will also display periodogram values for the cross periodogram. However, the cross-spectrum consists of complex numbers that can be divided into a real and an imaginary part.

Cospectral density. The cospectral density is computed by smoothing the real part of the cross-periodogram values (using the options selected in the Data windows for density estimates group box on the Advanced tab).  

Quadrature spectrum. The quadrature spectrum values (quad density for short) are computed by smoothing the imaginary part of the cross-periodogram values (using the options selected in the Data windows for density estimates group box on the Advanced tab).  

Cross amplitude. The cross amplitude values are computed as the square root of the sum of the squared cross-density and quad-density values. The cross-amplitude can be interpreted as a measure of covariance between the respective frequency components in the two series.  

Coherency. You can standardize the cross-amplitude values by squaring them and dividing by the product of the spectrum density estimates for each series. The result is called the squared coherency, which can be interpreted similar to the squared correlation coefficient (see the Correlations Overview), that is, the coherency value is the squared correlation between the cyclical components in the two series at the respective frequency. However, the coherency values should not be interpreted by themselves; for example, when the spectral density estimates in both series are very small, large coherency values may result (the divisor in the computation of the coherency values will be very small), even though there are no strong cyclical components in either series at the respective frequencies.

Gain. The gain value is computed by dividing the cross-amplitude value by the spectrum density estimates for one of the two series in the analysis. Consequently, two gain values are computed, which can be interpreted as the standard least squares regression coefficients for the respective frequencies

Phase spectrum. The phase spectrum estimates are computed as tan-1 of the ratio of the quad density estimates over the cross-density estimate. The phase shift estimates (usually denoted by Greek letter y) are measures of the extent to which each frequency component of one series leads the other.

Weights. This column will report the actual weights used in the smoothing window to produce the spectral density estimates (see above). The different smoothing windows are described in the Advanced tab topic: Data windows for density estimates group box). Note that the weights are standardized so that they will always sum to 1.

Cancel. Click the Cancel button to close this dialog, ignoring any changes made, and return to the Fourier (Spectral) Analysis dialog box.

Options. Click the Options button to display the Options menu.