Chapter 6: Time Series Analysis

ATM 552 Notes: Time Series Analysis - Section 6a Page 114or, alternativelyN2 1yt ( ) = y + C k cos 2kk =1T( )t t kC k2 = Ak2 + Bk2 and tk = + A NtN 2 cos T T2k1Btan k A k (6.23)Of course, normally these formulas would be obtained analytically using the a prioriinformation that equally spaced data on a finite interval can be used to exactly calculate theFourier representation on that interval (assuming cyclic continuation ad infinitum in bothdirections).The fraction of the variance explained by a particular function is given byr 2 ( y, x k ) = x' k y'2x'2 k y'2= A k 2 + B 2 k2y' 2 ,for k =1,2,... N 2 12A N 2y' 2 for k = N 2(6.24)The variance explained by a particular k isC2 k2 for k =1,2, ... N 2 1; A 2N 2 for k = N 2 (6.25)6.2.2 The Power SpectrumThe plot of C k2 vs. k is called the power spectrum of y(t) - the frequency spectrumif t represents time and the wavenumber spectrum if t represents distance. Strictly speakingC k2 represents a line spectrum since it is defined only for integral values of k, whichcorrespond to particular frequencies or wavenumbers. If we are sampling a finite datarecord from a larger time series, then this line spectrum has serious drawbacks.1. Integral values of k do not have any special significance, but are simplydetermined by the length of the data record T, which is usually chosen on thebasis of what is available.k j = j tT , j = 0,1,2,... N 22. The individual spectral lines each contain only about 2 degrees of freedom,since N data points were used to determine a mean, N/2 amplitudes and N/2 - 1phases (a mean and N/2 variances). Hence, assuming a reasonable amount ofCopyright 2005 Dennis L. Hartmann 1/25/05 5:04 PM 114