Estimates the nonnormalized spectral density of a stationary time series using a spectral window given the time series data.

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

Let

for t = 1, …, N represent the centered and padded data where N = NOBS + NPAD,

and

is determined by

The modified periodogram of

for t = 1, …, N 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 over the discrete set of nonnegative frequencies

The routine PFFT is used to compute the modified periodogram of

The estimate of the nonnormalized spectral density hX (ω) is computed according to

where the spectral window Wn(θ) is specified by argument ISWVER. The following spectral windows Wn(θ) are available.

where FM(θ) corresponds to the Fejér kernel of order M.

where DM(θ) represents the Dirichlet kernel. The Tukey‑Hamming window is obtained when
a = 0.23 and the Tukey‑Hanning window is obtained when a = 0.25.

where M is even. If M is odd, then M + 1 is used instead of M in the above formula.

The argument NM specifies the number of window parameters M and corresponds to the number of spectral density estimates to be computed for a given spectral window. 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 SSWD to these data produces the following results: