Modeling and Multivariate Methods - SAS

358 Performing Time Series Analysis Chapter 14
Time Series Commands
N 2 2
where f i
= i⁄
N are combined to form the periodogram If ( i
) = --- ( a , which represents the intensity
at frequency f i .
2 i
+ b i
)
The periodogram is smoothed and scaled by 1/(4π) to form the spectral density.
The Fisher’s Kappa statistic tests the null hypothesis that the values in the series are drawn from a normal
distribution with variance 1 against the alternative hypothesis that the series has some periodic component.
Kappa is the ratio of the maximum value of the periodogram, I(f i ), and its average value. The probability of
observing a larger Kappa if the null hypothesis is true is given by
q
Pr( k > κ) = 1–
(–
1) j q
max jk 1 – --- , 0
q – 1
j
q
j = 0
where q = N /2 if N is even, q =(N -1)/2 if N is odd, and κ is the observed value of Kappa. The null
hypothesis is rejected if this probability is less than the significance level α.
For q - 1 > 100, Bartlett’s Kolmogorov-Smirnov compares the normalized cumulative periodogram to the
cumulative distribution function of the uniform distribution on the interval (0, 1). The test statistic equals
the maximum absolute difference of the cumulative periodogram and the uniform CDF. If it exceeds
a⁄
( q)
, then reject the hypothesis that the series comes from a normal distribution. The values a = 1.36
and a = 1.63 correspond to significance levels 5% and 1% respectively.
Figure 14.5 Spectral Density Plots
Save Spectral Density
Save Spectral Density creates a new table containing the spectral density and periodogram where the
(i+1)th row corresponds to the frequency f i =i/N (that is, the ith harmonic of 1 / N).
The new data table has these columns:
Period is the period of the ith harmonic, 1 / f i .
Frequency is the frequency of the harmonic, f i .