Estimations of the nonnormalized spectral density of a stationary time series based on specified periodogram weights given the time series data.

Routine SWED estimates the nonnormalized spectral density function of a stationary time series using a fixed sequence of weights, given a sample of n = NOBS observations {Xt}, for
t = 1, 2, …, n.

Let

represent the centered and padded data where N = NOBS + NPAD,

and

is determined by

The modified periodogram of

is estimated by

where

and

represent the

cosine and sine transforms, respectively, and K is the scale factor equal to 1/(2πn). Since the periodogram is an even function of the frequency, it is sufficient to estimate the periodogram at the discrete set of nonnegative frequencies

(Here, ⌊a⌋ means the greatest integer less than or equal to a.) The routine PFFT is used to compute the modified periodogram of

Consider the sequence of m = NWT weights

where

These weights are fixed in the sense that they do not depend on the frequency ω at which to estimate the nonnormalized spectral density hX(ω). The estimate of the nonnormalized spectral density is computed according to

where

and k(ω) is the integer such that ωk,0 is closest to ω. The weights specified by argument WT may be relative since they are normalized to sum to one in the actual computation of

Usually, m is odd with the weights symmetric about the middle weight w0. If m is even, the weight to the right of the middle is considered w0. Note that periodogram ordinate

is replaced by

and the sum reflects at each end.

The nonnormalized spectral density is estimated over the set of frequencies

where nƒ = NF. These frequencies are in the scale specified by the argument IFSCAL but are transformed to the scale of radians per unit time for computational purposes.

The above formula for

assumes the data {Xt} correspond to a realization of a discrete‑parameter stationary process observed consecutively in time. In this case, the observations are equally spaced in time with interval Δt = TINT equivalent to one. However, if the data correspond to a realization of a continuous‑parameter stationary process recorded at equal time intervals, then the estimate of the nonnormalized spectral density must be adjusted for the effect of aliasing. In general, the estimate of hX(ω) is given by

Note that the frequency ω of the desired spectral estimate is assumed to be input in a form already adjusted for the time interval Δt.

Approximate confidence intervals for h(ω) can be computed using formulas given in the introduction.

Consider the Wölfer Sunspot Data (Anderson 1971, page 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. Application of routine SWED to these data produces the following results: