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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
Smoothing
Submitted by gfj100 on Sun, 03/28/2010 - 16:37
Quick Review .
..
Let's quickly review what was done before. We looked at the discrete Fourier transform of the data
where ω
j
=
j
/
n
when
j
/
n
≤ 0.5 and ω
j
=
j
/
n-
1 when
j
/
n
> 0.5 for
j
= 0, .
.. ,
n
-1.
Now, these can be interpreted as the coefficients of a regression.
where
Now, for a particular frequency
This is hardly surprising since

This
** preview**
has intentionally

as an approximation of the spectral density.
This is definitely an approximation, but we can make this more concrete by saying that
Another interesting thing to note is that the d
c
(ω
j
) and d
c
(ω
k
) for different
j
and
k
are nearly independent
-- and are even less dependent for larger
n
.
If we use a slightly different notation -- for
n
data points, we perform the periodogram at the ω
j
. Let's
write this as ω
j
:
n
. Imagine that we get more and more data (
n
→ ∞) and that ω
j
:
n
→ ω. It's not hard to
imagine the that:
E
[
I
(ω
j
:
n
)] →
f
(ω)
That's great, but there is a problem. One important thing that we do in statistics is to find confidence
intervals for our estimators. However, for the periodogram, the problem is that the number of parameters
that we are fitting (the
f
(ω
j
:
n
)) is growing at the same rate as the data. More data means more frequencies
where we want to estimate the spectral density. Asymptotically,
This gives a confidence interval of
Note that the width of the confidence interval never shrinks -- regardless of the size of the sample.
This contrasts with a similar result for sample variance, where
Note, that the mean of a
is
n
- 1 and variance is 2(
n
- 1). If we divide by
n
- 1 we see the variance of
S
2
is shrinking. This does not occur above for the periodogram.
Another problem is this -- what we really want to do is to have a good representation of
f
(ω) which is
presumably a continuous function in ω
and we want to obtain this continuous function from discrete
observations{in this case the periodogram
I
(ω
j
) which is a function across the discrete values ω
j
.
We need to find a way to fix this.

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