Locally weighted logistic regression

Question

I have been trying to implement a locally-weighted logistic regression algorithm in Ruby. As far as I know, no library currently exists for this algorithm, and there is very little information available, so it's been difficult.

My main resource has been the dissertation of Dr. Kan Deng, in which he described the algorithm in what I feel is pretty light detail. My work so far on the library is here.

I've run into trouble when trying to calculate B (beta). From what I understand, B is a (1+d x 1) vector that represents the local weighting for a particular point. After that, pi (the probability of a positive output) for that point is the sigmoid function based on the B for that point. To get B, use the Newton-Raphson algorithm recursively a certain number of times, probably no more than ten.

Equation 4-4 on page 66, the Newton-Raphson algorithm itself, doesn't make sense to me. Based on my understanding of what X and W are, (x.transpose * w * x).inverse * x.transpose * w should be a (1+d x N) matrix, which doesn't match up with B, which is (1+d x 1). The only way that would work, then, is if e were a (N x 1) vector.

At the top of page 67, under the picture, though, Dr. Deng just says that e is a ratio, which doesn't make sense to me. Is e Euler's Constant, and it just so happens that that ratio is always 2.718:1, or is it something else? Either way, the explanation doesn't seem to suggest, to me, that it's a vector, which leaves me confused.

The use of pi' is also confusing to me. Equation 4-5, the derivative of the sigmoid function w.r.t. B, gives a constant multiplied by a vector, or a vector. From my understanding, though, pi' is just supposed to be a number, to be multiplied by w and form the diagonal of the weight algorithm W.

So, my two main questions here are, what is e on page 67 and is that the 1xN matrix I need, and how does pi' in equation 4-5 end up a number?

I realize that this is a difficult question to answer, so if there is a good answer then I will come back in a few days and give it a fifty point bounty. I would send an e-mail to Dr. Deng, but I haven't been able to find out what happened to him after 1997.

If anyone has any experience with this algorithm or knows of any other resources, any help would be much appreciated!

Answer 1

As far as I can see, this is just a version of Logistic regression in which the terms in the log-likelihood function have a multiplicative weight depending on their distance from the point you are trying to classify. I would start by getting familiar with an explanation of logistic regression, such as http://czep.net/stat/mlelr.pdf. The "e" you mention seems to be totally unconnected with Euler's constant - I think he is using e for error.

If you can call Java from Ruby, you may be able to make use of the logistic classifier in Weka described at http://weka.sourceforge.net/doc.stable/weka/classifiers/functions/Logistic.html - this says "Although original Logistic Regression does not deal with instance weights, we modify the algorithm a little bit to handle the instance weights." If nothing else, you could download it and look at its source code. If you do this, note that it is a fairly sophisticated approach - for instance, they check beforehand to see if all the points actually lie pretty much in some subspace of the input space, and project down a few dimensions if they do.