Assignment 5: Spectral Analysis
Due
Mo Dec 11.
Consider the
sunspot data.
The data set is available in
R ,
and possibly also in S-plus.
Feel free to take a subset of the data if you prefer to work with a
smaller data set, for example ,the annual data.
If you have troubles getting the data, you can download here the
annual data or the
monthly data .
- Find the raw periodogram.
If you use R (or Splus) you could use the function
spectrum.
(If you use Matlab, please let me know (for my curiosity) which functions
you could use).
-
Using a harmonic regression model
y[t] = a[t]*cos(w t) + b[t]*sin(w t) + eps[t], t=1,..,n
find the posterior distribution p(w|y) for the unknown frequency w.
Complete the model by
- ... any reasonable prior distribution, and
- ... any reasonable assumption about the distribution of the
measurement errors eps[t].
Please state the specific prior distribution and
assumption for eps[t] which you use.
-
Now consider a harmonic regression model with two unknown
frequencies:
y[t] = a1[t]*cos(w1 t) + b1[t]*sin(w1 t) +
a2[t]*cos(w2 t) + b2[t]*sin(w2 t) + eps[t], t=1,..,n
Find the posterior distribution p(w1,w2|y).
See hints..
-
Fourier form DLM.
Analyze the annual sunspot data for
seasonal a patterns of period p=11 years.
Use a full seasonal effects Fourier form DLM with a linear
trend.
- Find the five harmonic frequencies 'omega[j]', j=1,..,(p-1)/2.
- Find the design matrix 'G' for a full seasonal effects DLM with these
five frequencies
and a linear trend.
- Find the corresponding design vector 'F'.
- Implement inference and plot
- the data with
- the filtered posterior mean 'E(mu[t] | D[t])' and
- smoothed means 'E(mu[t] | D[T])', given all data 'D[T]'.
See hints..
Here 'mu[t] = F'*theta[t]' is the mean function at time t.