| OCR Text |
Show This implies that (/) (T) and G (w) both contain the same information about the statistical properties of x (t). Spectrum analysis is aimed at determining the smoothed spectrum, which is the same for different portions or samples of an infinitely long stationary time series. For the long population series a smooth spectrum can be defined by a suitable limiting process. The p roblem of spectrum analysis is to estimate this smooth spectrum on the basis of the . given short time series. If the samples are drawn at random and the distribution of the variate is normal, the sample spectrum estimates at a given frequency are distributed about the corresponding population spectrum approxi~ I mately according to the distribution of chi square divided by degrees of freedom. The number of degrees of freedom is: 2N - (M/2) M · where M is the number of lags used and N is the original number of observations. In general, the theory shows that the sampling fluctuations are large and that peaks and troughs are significant only if .they are extreme, or if several adjacent points in the spectral estimates are either high or low. · A typical example for snow rate analysis would be a twelve hour storm period. For this time series N would be 288 (2. 5 min. samples), and if this were analyzed at 2 0 lags ( M = 2 0), the degrees of freedom k = 2 X 288 - (20/2) = 20 2 8.~3 The limiting value of chi-square at the 5o/~ le vel is 4 1. .69, so th~t ' x 2 / k = 1. 475. Therefore , if th e population is the natural or unseeded time series, the seeded period spectral estimates m_Jlst exceed 1. 475 times the unseeded spectral density va lue for that frequency, in order to be significant at the 5% level. When two time series appear to be correlated, it is desired to know whether the correlation is due to high frequency components or low frequency components. Also , it is poss ible that two time series may appear to be uncorrelated because hi gh frequency components are positively correlated and low frequ en cy components are n egatively co rrelated. Cross spectrum analysis allows for examination of the relationship between frequency components of the two series. A particularly useful statistic is the coherence, which gives • a measure of the relationship between the two variables or time series for various periods. It is defined by 268 |
