Time Series: The Frequency Domain

In the previous chapter, we studied how a series of observations evolves over time. Another approach is to study how the series varies in frequency: the periods of cyclic phenomena. To do this, we require an estimate of the spectral density function which is a complementary function to that expressed in the correlogram. Spectral analysis involves decomposing a function into a sum of sines and cosines, an adaptation of Fourier analysis to stochastic variables. Suppose that our time series can be represented by an unknown function of time, E(y_t = g(t;ω). Then we decompose this function:7.1$$ g(t;\omega ){{ = }_{{\frac{{{{\user2{a}}_{o}}}}{2}}}} + \sum\limits_{{\user2{r} = 1}}^{\user2{M}} {({{\user2{a}}_{\user2{r}}}\cos {{\user2{\omega }}_{\user2{r}}}\user2{t}} + {{\user2{b}}_{\user2{r}}}\sin {{\user2{\omega }}_{\user2{r}}}\user2{t)} $$7.2$$ _{{\frac{{{{\user2{a}}_{o}}}}{2}}} + \sum\limits_{{\user2{r} = 1}}^{\user2{M}} {{{\user2{R}}_{\user2{r}}}} \cos ({{\user2{\omega }}_{\user2{r}}}\user2{t + |}{{\phi }_{\user2{r}}}) $$is the amplitude of the rth harmonic of the series,ω_r = 2π/P_r = 2πr/N is the frequency of the rth cycle, where p_r is its period or length, φ_r = tan-1 (-b_r/a_r) is the phase, and r (1 ≤ r ≤ M = [N/2]) are the fundamental or Fourier frequencies. Then, 7.3$$ \user2{I(}{{\omega }_{r}}) = NR_{1}^{2}/4\pi $$ is known as the periodogram, and is plotted against ω_r. In this way, the total area under the periodogram equals the variance of the time series. (R_r2/2 plotted against r would give the same result.)