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node29 Spectral Density Estimation Spectral Analysis STAT 510 - Applied Time Series Analysis

Node29 Spectral Density Estimation Spectral Analysis STAT 510 - Applied Time Series Analysis

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This is Google's cache of http://onlinecourses.science.psu.edu/stat510/node/29 . It is a snapshot of the page as it appeared on 20 Jul 2010 08:32:24 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 Spectral Density Estimation / Spectral Analysis Submitted by gfj100 on Sun, 03/28/2010 - 16:34 Introduction to Spectral Analysis -- Spectral Density So far, we've been looking at time domain methods. Specifically, we've discussed models which give an explicit formula for the current observation in terms of past observations and past white noise terms. (As Shumway and Stoffer say, this is like a regression of the present on the past.) This is a dynamic definition, a rule on how to move through time. Another way we can look at stationary process is to look at the current observation as a combination of waves -- of a mixture of objects that are defined for all time. (As Shumway and Stoffer say, a regression of the current time on sines and cosines of various frequencies.) We would like to look at the spectrum to give us information about the underlying system. We want to identify dominant frequencies within the data. For example, in weather data, the yearly cycle is very important, but there are also multi-year cycles like El Niño that are difficult to spot without using these tools. Similarly business data has yearly cycles, but also multi-year business cycles. One thing that we've already discussed is the periodogram, which gives the total variation at a given sequence of frequencies (which are natural to the sampling rate). We will later discuss the spectral density which is the population version of the periodogram. The periodogram (sample) estimates the spectral density (population). Recall that period and frequency are inversely related. If we have quarterly data, then there are four data points per year (cycle). This corresponds to 0.25 cycles per data point. The notation we've used would then yield In chapter 2, we discussed periodic functions on the integers that have the following form
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for t = 0, ±1, ±1 where ω is the frequency, A is the amplitude, and φ is the phase. As we discussed before having the φ inside the cosine function can be problematic. If we want to do a regression it would be non- linear. We solved this by using a trig identity to rewrite the above as which we may rewrite as Here's where things are very different from before; we're going to assume that U 1 and U 2 are iid Gaussian with ZERO mean and fixed variance, σ 2 . This is different than a periodic trend. For example, in temperature data (in the northern hemisphere), January would be cold, and July would be hot. But if these are zero mean random variables, the the opposite is just as likely.
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