Be the vector with ith element let be an diagonal

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Unformatted text preview: on- Raphson update is where the derivatives are evaluated at The iteration will terminate when is very close to . The iteration can be described in matrix form. wikicour senote.com/w/index.php?title= Stat841&pr intable= yes 30/74 10/09/2013 Stat841 - Wiki Cour se Notes Let Let Let be the column vector of . ( ) be the input matrix. be the vector with ith element Let be an . diagonal matrix with ith element then The Newton- Raphson step is This equation is sufficient for computation of the logistic regression model. However, we can simplify further to uncover an interesting feature of this equation. where This is a adjusted response and it is solved repeatedly when , , and changes. This algorithm is called iteratively reweighted least squares (http://en.wikipedia.org/wiki/Iteratively_reweighted_least_squares) because it solves the weighted least squares problem repeatedly. Recall that linear regression by least square finds the following minimum: we have Similarly, we can say that is the solution of a weighted least square problem: WLS Actually, the weighted least squares estimator minimizes the weighted sum of squared errors where . Hence the WLS estimator is given by A weighted linear regression of the iteratively computed response Therefore, we obtain note :Here we obtain , which is a regression, will be a vector, because we construct the model like . If we construct the model like , then similar to linear vector. Choosing seems to be a suitable starting value for the Newton- Raphson iteration procedure in this case. However, this does not guarantee convergence. The procedure will usually converge since the log- likelihood function is concave(or convex), but overshooting can occur. In the rare cases that the log- likelihood wikicour senote.com/w/index.php?title= Stat841&pr intable= yes 31/74 10/09/2013 Stat841 - Wiki Cour se Notes decreases, cut step size by half, then we can always have convergence. In the case that it does not, we can just prove the local con...
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This document was uploaded on 03/07/2014.

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