tim's journal

Logits and Classification

How to structure the problem is a big issue. Usually logisitic regression is used in situations where the potential factors are quite clear. For instance, weight, sex, family history as they relate to a disease like heart disease. In our case, it is not so evident what the factors to use in the model are. We could, for instance, use logistic regression as a sort of bootstrap for our existing classification scheme. We can determine how each channel contributes to correct classification and create a weight. Or, we could start at a more basic level and let each observation point be a factor to be analyzed. A consideration in the latter case is the great potential for overfitting.

Potential methods. In the current implementation, we are able to obtain, for each training trial, N channels of match or no match. We have this for each grid point. We can go further, by considering the class that each channel matches too (1 through 7). The next level would be to examine the least squares or correlation coefficient to each class. Finally, we can discard our measure all together and use each observation point (or Fourier component) as a factor in our analysis. Note: From now on I will only consider data downsampled using Marcos' (traditional) method so that there is no controversy regarding the base data.

Match - first cut. In the "match" analysis, rather than simply using results of the best channel, we can consider all other channels. But how can we weight the significance of other channels? We can do it with logistic regression. In this case, the response is binary - either 1 or 0, does match or does not match. The dependent variables are the results for each channel, either 1 or 0. In VA monopolar data there are 60 channels. However, in the bipolar data there are 1770 channels - clearly a recipe for overfitting. We can reduce the data set by considering only the best m channels, or, we can run all of them and take out channels that seem irrelevant after we do the logisitic regression.

To build the model, we would something like:

channel	train	total
Cz	43	193
Fz	12	193
Pz	54	193
C4	22	193

But does this really help us? I tried running this dummy data set suing SAS: Analysis and got the following results (under "Analysis of Maximum Likelihood"):

parameter	DF	estimate	std err
intercept	1	-1.7398	0.1106
C4	1	-0.3108	0.1946
Cz	1	0.4903	0.1649
Fz	1	-0.9738	0.2380

What does this mean? Now I'm thinking that I thought about this wrong. I found some notes on deciphering SAS output. I feel like somewhere along the line I needed to tell SAS that I was supposed to get 193/193, but I haven't done that... Basically, the analysis is says that the log of the odds is given by

g = -1.7398 - (0.3108 x <1>) + (0.4903 x <2>) - (0.9738 x <3>)

This is probably not what I wanted. The translation table for the channels is given by:

	1	2	3
C4	1	0	0
Cz	0	1	0
Fz	0	0	1
Pz	-1	-1	-1

So, in fact, what we have calculated here is what the expected odds of classifying using any given channel. This does not mix the channels like I wanted. Let's see how well it worked. Suppose my channel is Pz. Then

g_Pz = -1.7398 + 0.3108 - 0.4903 + 0.9738 = -0.9455

p/(1-p) = e^-0.9455 = 0.3885

p = 0.3855/1.3855 = .2782

Of course, this is right where we started, since

.2782 x 193 = 53.7

So, we are still not thinking about this correctly. Though it is nice to see that the results are not to wierd.

Posted by torque at April 3, 2003 10:25 AM