Comments on: Making Hockey Sticks the Jones Way

By: John Creighton

John Creighton — Sun, 02 Jul 2006 07:09:41 +0000

The signal processing is coming back to me. So recall that the noise decreases with 1/sqrt(n) the number of independent measurements. Since a sign wave takes a whole period to repeat, each period constitutes an independent measurement. Thus the amplitude (stand deviation) of the noise should fall with one over the square root of the number of periods. That is 1/sqrt(f*T) where f is the frequency and T is the data length.

The power spectrum is the magnitude squared of the frequency distribution. That is:
Ps=Ys Ys*

Where Ps is the power spectrum, Ys is the frequency spectrum and Ys* is the complex conjugate of the frequency spectrum. Taking the inverse Fourier transform of the power spectrum we get the autocorrelation function.

Rs(Tau)=F^-1{Ps} where Rs is the auto correlation function Tau is the time delay
F^-1 is the inverse Fourier transform

The correlation matrix can be written from the auto correlation function as follows:

RR=
[Rs(t1) Rs(t2) ... Rs(Tn)]
[Rs(t2) Rs(t1) ... Rs(Tn-1)]
…
[[Rs(tn) Rs(t1) ... Rs(T1})]

Now that we have the correlation matrix we can whiten the linear equation

Y=Ax

By multiplying both sides of the equation by the square root of the inverse of the correlation matrix. The inverse of the correlation matrix is known as the information matrix.

By: John Creighton

John Creighton — Sun, 02 Jul 2006 06:29:48 +0000

#45 It is too bad that the hockey team doesn’t realize that a moving average filter is exactly what you don’t want. If anything you want a high pass filter rather then a low pas filter as there is simply not enough data to identify low frequency signals especially since the number of parameters you are trying to fit is large.

Perhaps we should weight our information in the frequency components as:

1/sqrt(f*T) Where T is the data length in time and f is the frequency. From this frequency weighting we should be able to devise a whitening filter.

By: mark

mark — Wed, 28 Jun 2006 04:43:06 +0000

Ah, gotcha. Not sure, actually, though on inspection, I would guesstimate a moving average filter which seems to get discussed often around here.

Mark

By: John Creighton

John Creighton — Wed, 28 Jun 2006 03:57:49 +0000

#43
“Figure 2. Top – archived version. Middle – 3 series version pre-stabilization. Bottom – 3-series version, post-stabilization.”

By: mark

mark — Wed, 28 Jun 2006 03:45:59 +0000

? Stabilizing method?

Mark

By: John Creighton

John Creighton — Wed, 28 Jun 2006 02:49:00 +0000

I would of liked to here more about this stabilizing method. A link, a justification some mathematics something.

By: Mark

Mark — Tue, 06 Jun 2006 23:59:59 +0000

C&A is EXTREMELY technical. The only background they give, btw, is methods for solving linear systems of equations with unknown sources (and known mixing matrix and observation vector). I’ve had plenty of instruction in that realm, but I went through it just to have a good read on the methods they prefer since those methods will appear in their later solutions (I’m guessing).

I saw the Stone book listed on Amazon, and may buy it at some point, maybe even the HKO book. I have access to both of those papers via IEEE as well. Thank you for the pointers. Odd that the new Karhunen would be unrelated, yet doing nearly identical work with the same name…

In the end, this all has to result in a dissertation topic, and, preferably, an actual dissertation!

Mark

By: Jean S

Jean S — Tue, 06 Jun 2006 22:06:32 +0000

re #37: The approach in those two books is rather different, and which one is better for you depends from your background/preferences. In general, IMO, C&A is more "technical" (and contains more information) but HKO is much easier to read. Addition to these two books there is a newer book on ICA by James Stone. I have not seen it, but judging from the Table of Contents (and Amazon reviews) seems to be a rather good book on the topic. BTW, if you really want to learn the statistical fundamentals of ICA/BSS, I suggest you to read/master the following two papers: 1) P. Comon, Independent Component Analysis, a new concept ?, Signal Processing, 36(3):287--314, April 1994. 2) J-F. Cardoso, Blind signal separation: statistical principles, Proceedings of the IEEE, 90(8):2009--2026, October 1998. re #38: Juha Karhunen has no relation to Karhunen-Loeve, and to my knowledge is not an offspring either.

By: Reid

Reid — Tue, 06 Jun 2006 20:38:56 +0000

Re #36: “Is there any basis on which a private citizen could bring suit against a state government – say, someone whose business has been damaged by CO2 emission restraints – and summons Mann ?”

A number of lawsuits have been filed by proponents of AGW to force the Federal Government to impose CO2 emission controls. The City of Boulder has filed suit. Calling Mann or any other scientist, both believer or skeptic, would be an option.

I don’t know the status of these suits but the AGW community should be careful what they wish for. The skeptics will be able to present their entire case under oath and on the record. To quote President Bush, “Bring it on!”.

By: Mark

Mark — Tue, 06 Jun 2006 19:59:56 +0000

Oh, curiously, I just looked at your reference for Hyvarinen’s book and I see it is co-authored by Juha Karhunen… any relation to the Karhunen of Karhunen-Loeve fame? My advisor seems to think the latter is probably deceased, but maybe an offspring?

mark