node32 Tapering STAT 510 - Applied Time Series Analysis

# node32 Tapering STAT 510 - Applied Time Series Analysis -...

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This is Google's cache of http://onlinecourses.science.psu.edu/stat510/node/32 . It is a snapshot of the page as it appeared on 20 Jul 2010 12:35:21 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 Tapering Submitted by gfj100 on Sun, 03/28/2010 - 16:38 To review, the spectral density is the Fourier Transform of the autocovariance function and the periodogram a truncated version of the Fourier Transform of the sample autocovariance function (ACF). What happens if I hand you as much data as you want. Will the periodogram estimate the spectral density? It looks like it should, but what is the difference between these two? The problem is that we are increasing the number of parameters that we are trying to estimate. So, what is the solution for getting the periodogram to estimate the spectral density? Smoothing. We know that with smoothing, increasing the bandwidth affects both bias and variance. The more smoothing, the larger the bandwidth, there is more bias and smaller variance. In the end, if you make the bandwidth very wide to encompass all of your data, we will get a straight line. This representation is not helpful at all. We are trying to have the lowest bias and variances together. This is why we begin with a small bandwidth and move out from there until we can balance these two elements. As we increase the bandwidth, you are increasing bias, but decreasing variance. And, as you shrink bandwidth, we are decreasing bias but increasing variance. In regular smoothing, we take the data as it is with a lot of variance but very small bias. If we take a traditional approach to smoothing, we are trying to estimate a curve which is perhaps something like: x t = μ t + y t , where y t has a zero mean. If we take as an estimate of , is this biased? This is unbiased.

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If we take the expectation of this: The So, if we just take each single time point it is unbiased. The problem is that we have a lot of variance and have only one data point. This is not the best estimator. We can improve this estimator to reduce the variance (and reduce mean squared error) by averaging over some of its neighbors. We will be introducing bias, but decreasing MSE. If we try to do this with a periodogram, what happens? Is
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## This note was uploaded on 09/10/2010 for the course STAT 510 at Penn State.

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node32 Tapering STAT 510 - Applied Time Series Analysis -...

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