Nonparametric Spectral Analysis and Periodogram Analysis ?

The spectrum of a time series can be estimated by a variety of methods. In lesson 4 we looked at the Blackman-Tukey method, which uses a Fourier transform of the smoothed, truncated autocovariance function. In contrast, the smoothed periodogram method uses a Fourier transform of the time series itself. The transform yields a quantity called the raw periodogram, which in essence is a highly resolved spectral estimate. The raw periodogram was introduced in the late1800s for study of periodicity in time series. Unfortunately, the raw periodogram is a crude spectral estimate with high variance (great uncertainty). Smoothing the raw periodogram produces a better estimate of the spectrum – one with a smaller variance (tighter confidence interval). But smoothing reduces the ability of the spectrum to resolve periodic features that may be very close in wavelength. The smoothness, resolution and variance of spectral estimates is controlled by the choice of filters in smoothed periodogram analysis. Extremely broad smoothing produces an underlying smoothly varying spectrum, or null continuum, against which spectral peaks in less-severely smoothed periodogram can be tested for significance. This approach is an alternative to the specification of a functional form of the null continuum (e.g., red noise). 

Reduction to stationarity. The first step, which is taken before embarking on the cycle, is to examine the time plot of the data and to judge whether or not it could be the outcome of a stationary process. If a trend is evident in the data, then it must be removed. A variety of techniques of trend removal, which include the fitting of parametric curves and of spline functions, have been discussed in previous lectures. When such a function is fitted, it is to the sequence of residuals that the ARMA model is applied. However, Box and Jenkins were inclined to believe that many empirical series can be modelled adequately by supposing that some suitable difference of the process is stationary. Thus the process generating the observed series y(t) might be modelled by the ARIMA(p, d, q) equation (3) α(L)∇dy(t) = µ(L)ε(t), wherein ∇d = (I − L)d is the dth power of the difference operator. In that case, the differenced series z(t) = ∇dy(t) will be described by a stationary ARMA(p, q) model. The inverse operator ∇−1 is the summing or integrating operator, which accounts for the fact that the model depicted by equation (3) is described an autoregressive integrated moving-average model.