Changes between Version 2 and Version 3 of GLM

Timestamp:

Aug 7, 2014, 10:52:54 AM (10 years ago)

Author:

Pedro Gea

Comment:

--

GLM

@@ -49 +49 @@
 
 {{{
-Grad = d(LogL(B))/dB = - Sum_i( Y_i*Exp(-B'X)*(-X_i)/(1+Exp(-B'X)) + (1-Y_i)*Exp(B'X)*X_i/(1+Exp(B'X)) )
+G(B) = d(LogL(B))/dB =
 
+     = - Sum_i( Y_i*Exp(-B'X_i)*(-X_i)/(1+Exp(-B'X_i)) + (1-Y_i)*Exp(B'X_i)*X_i/(1+Exp(B'X_i)) ) =
+
+     = Sum_i( Y_i*X_i/(1+Exp(B'X_i)) - (1-Y_i)*X_i/(1+Exp(-B'X_i)) ) =
+
+     = Sum_i( ( Y_i/(1+Exp(B'X_i)) - (1-Y_i)/(1+Exp(-B'X_i)) ) * X_i )
 }}}
+
+==== Hessian ====
+
+La segunda derivada respecto a la matriz de parámetros ({{{B}}}) es la hessiana del logaritmo de la verosimilitud:
+
+{{{
+H(B) = d^2(LogL(B))/(dB dB') =
+
+     = Sum_i( ( Y_i*Exp(B'X_i)/(1+Exp(B'X_i))^2 - (1-Y_i)*Exp(-B'X_i)/(1+Exp(-B'X_i))^2 ) * X_i * X'_i ) = 
+
+     = Sum_i( Exp(B'X_i)/(1+Exp(B'X_i))^2 * X_i * X'_i )
+}}}