Spectrum

Definition: Spectrum summarizes the periodicity properties of a time series or time series sample x_t. Often represented in a graph with frequency, or period, (often denoted little omega) on the horizontal axis, and S_x(omega), which is defined below, on the vertical axis. S_x is zero for frequencies that are not found in the time series or sample, and is increasingly positive for frequencies that are more important in the data.

S_x(omega) = (2pi)^-1(sum for j from -infinity to +infinity of) gamma_je^-ijomega

where gamma_j is the jth autocovariance, omega is in the range [-pi, pi], and i is the square root of -1.

Example 1: If x_t is white noise, the spectrum is flat. All cycles are equally important. If they were not, the series would be forecastable.

Example 2: If x_t is an AR(1) process, with coefficient in (0, 1), the spectrum has a peak at frequency zero and declines monotonically with distance from zero. This process does not have an observable cycle.

(Econterms)

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