Time: Wednesdays, Fridays, 9:00 AM
Room: Munk Conference Room, 303, IGPP
My segments concern spectral analysis of stochastic processes. Here is the syllabus.
The final segment (Chapter 4) will concern somewhat more advanced topics, including cross spectra, spectral analysis on the plane and the sphere, and wavelets.
The first class in my portion will be on Wednesday, 5/05/10. Notes will be posted on this site. Please check here before class for notes, homework assignments and other information.
I have fixed the link to German's site and put the real
solution to set 4 this afternoon.
6/03/10:
I have graded the homework, and posted
my solution.
It is a mystery to me why so many of you decided,
rather than just executing plotxy on the
scripts generated by PSD, decided to replot the output and
obscure the information with inappropriate axes
and sizing (sigh).
For the most up-to-date code for spectral and cross spectral
estimation see this
link to the site of German Prieto, a former IGPP student.
5/27/10:
Chapter 4 of the notes is up on the
site.
5/26/10:
As promised I am posting the
last homework set.
You will also need this
data set.
You are asked to download the spectral estimation program PSD
from the same site where you found plotxy. PSD is a Fortran
program which takes all the hard work out of spectral
estimation. Here are some
notes on installation.
Your solutions will be collected on Wednesday 6/02/10; late
solutions will not be accepted.
5/25/10:
Much to my embarrassment I have discovered a critical typo in
Chapter 2 in Section 5. I missed a
factor of &Deltat in equation (5.5), which carried
through to (5.8). Corrected pages are
here. The units of the continuous and
the discrete PDF must be the same, and they weren't in my
version. Answers with or without the factor are considered
correct in the homework. My apologies.
5/24/10:
The homework required you to find the Fourier transform of a
fairly complicated function, which might perhaps best be
evaluated numerically. No one succeeded in this. Doing
numerical FTs is something you may find yourselves doing quite
often if you do any kind of theory. Here is a
brief essay on the subject, which you
may find helpful.
You are encouraged to evaluate this class and its instructors at
this site.
5/21/10:
My
solution to the second homework is now up on
the site. There will be no new homework until next Wednesday, so
you can take a brief rest.
5/20/10:
I understand many of you are having problems with the
second homework. Here are some hints. I expect your solutions
tomorrow.
3(a). Rather than deriving the expression from first principles, it is much simpler to build up from results already in the Notes. Break the problem into two parts. Consider the averaging process as a continuous activity, generating a continuous-time signal: what is that signal? How is its PSD related to the original? Now sample that signal at t=n &Deltat. How does the sampling affect the PSD?
3(b). This question is best answered not using part (a), although it's certainly possible that way. Just consider consecutive terms in the discrete process: are they correlated? In general, are any two terms correlated? From the answers to these question you can construct the form of the autocovariance.
4. The Planck spectrum characterizes sunlight (approximately); it
gives the energy flux through a unit area normal to the direction
of the radiation. The Note shows how energy flux is related to
magnetic field: the total flux is simply proportional to the
B field variance, and so we can assume that the energy in
a particular wavenumber interval is proportional (with the same
scaling factor) to the field variance in the same interval.
Variance in a wavenumebr band is the intuitive way to define PSD.
5/13/10:
I have graded the first set of homework. You may wish to inspect
my solution; the
plotxy scripts are posted
separately.
5/12/10:
First, here are the
notes on the PSD of the derivative that
I derived in class today.
Next, Chapter 3 on spectral estimation is available, and we get to these notes on Friday.
Finally, due next Wednesday, 5/19/10, the
second homework set.
5/07/10:
An item arising from today's class: the
wkipedia article on the Kolmogorov-Smirnov test is perfectly
adequate for our purposes. Because you will soon have a working
version of plotxy and some familiarity with compiling Fortran,
I give you
qqplot, a program that draws a QQ-plot and
histogram, and performs the KS test for normality. Given a diskfile
x.dat with one value per line, run
qqplot < x.dat
and the program outputs various statistics to the terminal and writes
the file qq.p with plotxy instructions for the graphs. Matlab does
not appear to have the KS test.
5/05/10:
We will finish the short Chapter 1 on Friday. Here is the
Chapter 2.
5/03/10:
Here is the
first set of notes for the lecture on
Wednesday.
My
first assignment requires you to
download and compile plotxy, for which Fortran source and documentation
can be found
here.
You will also need a
data set for the homework.
Your solutions are due on Wednesday, 5/12/10.