Time Series Analysis - Single Series Fourier (Spectral) Analysis Summary Spreadsheet

Both the Quick tab and the Advanced tab of the Single Series Fourier (Spectral) Analysis Results dialog have an option for producing a summary spreadsheet with the frequencies, periods, cosine and sine coefficients, periodogram values, spectral density estimates (computed according to the selection in the Data windows for spectral density estimates group box on the Single Series Fourier (Spectral) Analysis Results dialog - Advanced tab), and the weights used to produce the spectral density estimates. Note that as the default graph for this spreadsheet, you may select to plot the sine/cosine coefficients, the periodogram values (or log-periodogram), or the spectral density estimates (or the log-densities) against the frequencies or period.

Given a series (see Shumway, 1988, page 49):

x_i = a₀/2 + (Σ_k{a_k*cos[2πf_k(t-1)]} + {b_k*sin[2πf_k(t-1)]})

where the values are computed as follows:

Frequency (fk). The frequency (f_k) is defined as the number of cycles per unit time. Since in the Time Series module, 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 (1/fk). The period (1/f_k) 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 (ak). The cosine coefficients (a_k) 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.

Sine coefficients (bk). The sine coefficients (b_k) can be interpreted analogous to the cosine coefficients (see previous paragraph).