14.5.3 - Spectral Analysis

Suppose we believe that a time series, \(Y_{t}\), contains a periodic (cyclic) component. Spectral analysis takes the approach of specifying a time series as a function of trigonometric components. A natural model of the periodic component would be

\(\begin{equation*} Y_{t}=R\cos(ft+d)+e_{t}, \end{equation*}\)

where R is the amplitude of the variation, f is the frequency of periodic variation1, and d is the phase. Using the trigonometric identity \(\cos(A+B)=\cos(A)cos(B)-sin(A)sin(B)\), we can rewrite the above model as

\(\begin{equation*} Y_{t}=a\cos(ft)+b\sin(ft)+e_{t}, \end{equation*}\)

where \(a=R\cos(d)\) and \(b=-R\sin(d)\). Thus, the above is a multiple regression model with two predictors.

Variation in time series may be modeled as the sum of several different individual waves occurring at different frequencies. The generalization of the above model as a sum of k frequencies is

\(\begin{equation*} Y_{t}=\sum_{j=1}^{k}R_{j}\cos(f_{j}t+d_{j})+e_{t}, \end{equation*}\)

which can be rewritten as

\(\begin{equation*} Y_{t}=\sum_{j=1}^{k}a_{j}\cos(f_{j}t)+\sum_{j=1}^{k}b_{j}\sin(f_{j}t)+e_{t}. \end{equation*}\)

Note that if the \(f_{j}\) values were known constants and we let \(X_{t,r}=\cos(f_{r}t)\) and \(Z_{t,r}=\sin(f_{r}t)\), then the above could be rewritten as the multiple regression model

\(\begin{equation*} Y_{t}=\sum_{j=1}^{k}a_{j}X_{t,j}+\sum_{j=1}^{k}b_{j}Z_{t,j}+e_{t}. \end{equation*}\)

The above is an example of a harmonic regression.

Fourier analysis is the study of approximating functions using the sum of sine and cosine terms. Such a sum is called the Fourier series representation of the function. Spectral analysis is identical to Fourier analysis, except that instead of approximating a function, the sum of sine and cosine terms approximates a time series that includes a random component. Moreover, the coefficients in the harmonic regression (i.e., the a's and b's) may be estimated using multiple regression techniques.

Figures can also be constructed to help in the spectral analysis of a time series. While we do not develop the details here, the basic methodology consists of partitioning the total sum of squares into quantities associated with each frequency (like an ANOVA). From these quantities, histograms of the frequency (or wavelength) can be constructed, which are called periodograms. A smoothed version of the periodogram, called a spectral density, can also be constructed and is generally preferred to the periodogram.