node30 Variability, Spectral Density and the Periodogram STAT 510 - Applied Time Series Analysis

# Node30 Variability, Spectral Density and the Periodogram STAT 510 - Applied Time Series Analysis

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This is Google's cache of http://onlinecourses.science.psu.edu/stat510/node/30 . It is a snapshot of the page as it appeared on 18 Jul 2010 01:58:10 GMT. The current page could have changed in the meantime. Learn more Text-only version STAT 510 - Applied Time Series Analysis ANGEL Department of Statistics Eberly College of Science Home // Section 3: Spectral Domain Models Variability, Spectral Density and the Periodogram Submitted by gfj100 on Sun, 03/28/2010 - 16:35 Our goal is to estimate the spectral density. We know that if we are looking at something stationary, it should have a spectral density. The spectral density tells us about the fluctuation of the autocovariance function, in terms of sines and cosines. What will we use to estimate this? In general, we need to find an estimator. Although we will have to modify it somewhat, the estimator is the periodogram. We have already talked about the periodogram but not how to calculate it. The way we have used it, and the way we have interpreted it (and we will see why we have interpreted it this way), is as a regression model. It is a huge regression model involving all of the frequencies that we can include -- a regression on the sines and cosines. We get coefficients, and the periodogram is the square of the coefficients. But, this was when we were interpreting the time series as a non-stationary time series - in fact, as a nonstationary time series that has a periodic trend in it. Now, we want to interpret this periodogram when what we are looking at is a stationary time series and this has no trend in it. First, we need to define the periodogram, and to do this we will begin by defining the Discrete Fourier Transform of x t , our data: (Recall ω j = j -1/ n where j = 1, 2, ... , N .) This may look similar to what we used before, the regular Fourier Transform we used with the autocovariance, the only difference here is that we are summing over a finite number. What we did before was to take this over an infinite number.

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The x t is our data. The periodogram, which and in some ways will be our primitive estimator for spectral density, will be: The modulus squared of the Fourier Transform of the data. Remember, when we take the modulus it moves things into the reals. Let's review how this works: If we took the modulus of a + bi : The modulus squared is The nice thing about the Fourier Transform is that computers can calculate them very quickly. There is a special alogorithmin that allows this to happen.
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