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Pj(Xi) = exp(XiBj)/((sum\[j=1..(k-1)\]exp(Xi*Bj))+1) The last class has probability 1-(sum\[j=1..(k-1)\]Pj(Xi)) = 1/((sum\[j=1..(k-1)\]exp(Xi*Bj))+1) The (negative) multinomial log-likelihood is thus: L = -sum\[i=1..n\]\{ sum\[j=1..(k-1)\](Yij \* ln(Pj(Xi))) +(1 - (sum\[j=1..(k-1)\]Yij)) \* ln(1 - sum\[j=1..(k-1)\]Pj(Xi)) \} + ridge * (B^2) |
In order to find the matrix B for which L is minimised, a Quasi-Newton Method is used to search for the optimized values of the m*(k-1) variables. Note that before we use the optimization procedure, we 'squeeze' the matrix B into a m*(k-1) vector. For details of the optimization procedure, please check weka.core.Optimization class.
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