Changes between Version 23 and Version 24 of OfficialTolArchiveNetworkGrzLinModel

Timestamp:

Feb 21, 2012, 2:38:59 PM (13 years ago)

Author:

Víctor de Buen Remiro

Comment:

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OfficialTolArchiveNetworkGrzLinModel

@@ -94 +94 @@
    [[LatexEquation( f\left(y;\mu\right)=\frac{1}{y!}e^{-\mu}\mu^{y} )]] [[BR]] [[BR]] 
  * and its logarithm will be [[BR]] [[BR]] 
-   [[LatexEquation( \ln f\left(y;\mu\right)=-\ln\left(\Gamma\left(y+1\right)\right)+y\ln\left(\mu\right)-\mu = -\ln\left(\Gamma\left(y+1\right)\right)+y\eta-e^{\eta} )]] [[BR]] [[BR]] 
+   [[LatexEquation( \ln f\left(y;\mu\right)=-\ln\left(y!\right)+y\ln\left(\mu\right)-\mu = -\ln\left(y!\right)+y\eta-e^{\eta} )]] [[BR]] [[BR]] 
  * the partial derivatives of log-density function respect to the linear prediction is [[BR]] [[BR]]
    [[LatexEquation( \frac{\partial\ln f}{\partial\eta}=y-e^{\eta} )]] [[BR]] [[BR]]
@@ -153 +153 @@
    [[LatexEquation( f\left(y;\mu,\lambda\right)=\begin{cases}\lambda+\left(1-\lambda\right)e^{-\mu} & \forall y=0\\\left(1-\lambda\right)\frac{1}{y!}e^{-\mu}\mu^{y} & \forall y>0\end{cases} )]] [[BR]] [[BR]] 
  * and its logarithm will be [[BR]] [[BR]] 
-   [[LatexEquation( \ln f\left(y;\mu,\lambda\right)=\begin{cases}\ln\left(\lambda+\left(1-\lambda\right)e^{-\mu}\right) & \forall y=0\\\ln\left(1-\lambda\right)-\ln\left(\Gamma\left(y+1\right)\right)+y\ln\mu-\mu & \forall y>0\end{cases} )]] [[BR]] [[BR]] 
- * the partial derivatives of log-density function respect to the linear prediction are [[BR]] [[BR]]
-   [[LatexEquation( \frac{\partial\ln f}{\partial\eta}=\frac{\partial\ln f}{\partial\mu}\frac{\partial\mu}{\partial\eta}=\begin{cases}\frac{1-\lambda}{\lambda+\left(1-\lambda\right)e^{-\mu}}\mu & \forall y=0\\ \left(\frac{y}{\mu}-1\right)\mu & \forall y>0\end{cases}=\begin{cases} \frac{\left(1-\lambda\right)e^{\eta}}{\lambda+\left(1-\lambda\right)e^{-e^{\eta}}} & \forall y=0\\ y-e^{\eta} & \forall y>0\end{cases} )]] [[BR]] [[BR]]
-   [[LatexEquation( \frac{\partial^{2}\ln f}{\partial\eta^{2}}=-e^{\eta} )]] [[BR]] [[BR]]
+   [[LatexEquation( \ln f\left(y;\mu,\lambda\right)=\begin{cases}\ln\left(\lambda+\left(1-\lambda\right)e^{-\mu}\right) & \forall y=0\\\ln\left(1-\lambda\right)-\ln\left(y!\right)+y\ln\mu-\mu & \forall y>0\end{cases} )]] [[BR]] [[BR]] 
+ * the first and second partial derivatives of log-density function respect to the linear prediction are [[BR]] [[BR]]
+   [[LatexEquation( \frac{\partial^{2}\ln f}{\partial\eta^{2}}=\begin{cases}-\frac{\left(1-{e}^{\eta}\right)\,{e}^{\eta-{e}^{\eta}}\,\left(1-\lambda\right)}{\lambda+{e}^{-{e}^{\eta}}\,\left(1-\lambda\right)}-\frac{{e}^{2\,\eta-2\,{e}^{\eta}}\,{\left(1-\lambda\right)}^{2}}{{\left(\lambda+{e}^{-{e}^{\eta}}\,\left(1-\lambda\right)\right)}^{2}} & \forall y=0\\-e^{\eta} & \forall y>0\end{cases} )]] [[BR]] [[BR]]
+ * the first partial derivative of log-density function respect to the zero-inflation parameter is [[BR]] [[BR]]
+   [[LatexEquation( \frac{\partial\ln f}{\partial\lambda}=\begin{cases}\frac{1-{e}^{-{e}^{\eta}}}{\lambda+{e}^{-{e}^{\eta}}\,\left(1-\lambda\right)} & \forall y=0\\-\frac{1}{1-\lambda} & \forall y>0\end{cases} )]]