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STAT 510  Applied Time Series Analysis
•
ANGEL
•
Department of Statistics
•
Eberly College of Science
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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 multiyear cycles like El Niño that are difficult to spot without using these
tools. Similarly business data has yearly cycles, but also multiyear 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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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.
We can generalize this model to include multiple frequencies with
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This note was uploaded on 09/10/2010 for the course STAT 510 at Pennsylvania State University, University Park.
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