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POLocally weighted logistic regression
 

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    <p>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.</p> <p>My main resource has been the <a href="http://www.cs.cmu.edu/~kdeng/thesis/logistic.pdf" rel="noreferrer">dissertation of Dr. Kan Deng</a>, in which he described the algorithm in what I feel is pretty light detail. My work so far on the library is <a href="https://github.com/orthogonal/USCF-Stats/blob/master/lib/LocalLogisticRegression.rb" rel="noreferrer">here</a>.</p> <p>I've run into trouble when trying to calculate <code>B</code> (beta). From what I understand, <code>B</code> is a <code>(1+d x 1)</code> vector that represents the local weighting for a particular point. After that, <code>pi</code> (the probability of a positive output) for that point is the sigmoid function based on the <code>B</code> for that point. To get <code>B</code>, use the Newton-Raphson algorithm recursively a certain number of times, probably no more than ten.</p> <p>Equation 4-4 on page 66, the Newton-Raphson algorithm itself, doesn't make sense to me. Based on my understanding of what <code>X</code> and W are, <code>(x.transpose * w * x).inverse * x.transpose * w</code> should be a <code>(1+d x N)</code> matrix, which doesn't match up with <code>B</code>, which is <code>(1+d x 1)</code>. The only way that would work, then, is if e were a <code>(N x 1)</code> vector.</p> <p>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.</p> <p>The use of <code>pi'</code> 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, <code>pi'</code> is just supposed to be a number, to be multiplied by w and form the diagonal of the weight algorithm W.</p> <p>So, my two main questions here are, what is <code>e</code> on page 67 and is that the <code>1xN</code> matrix I need, and how does <code>pi'</code> in equation 4-5 end up a number?</p> <p>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.</p> <p>If anyone has any experience with this algorithm or knows of any other resources, any help would be much appreciated!</p>
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    1. USAndrew Latham
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    1. COShould you only implement Dr.Deng's equation? Have you considered other variants of logistic regression?
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    2. COI am looking, in particular, for locally weighted logistic regression. Ordinary logistic regression is not able to handle multiple intervals, and there are a couple inelegant solutions but the locally weighted algorithm is, I think, smoother and particularly appropriate in my specific application.
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      1. USAndrew Latham
    3. COTo see a rough example of the kind of chart I'm looking to fit, go to http://www.uscfstats.com/deltas and enter 12842311. I'd like to figure out a curve to roughly divide green/blue dots and red dots.
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