Difference between revisions of "The individual approach"

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p_Y(y_1, y_2, \ldots y_n ; \psi) &=& \prod_{j=1}^n p_{Y_j}(y_j ; \psi) \\
 
p_Y(y_1, y_2, \ldots y_n ; \psi) &=& \prod_{j=1}^n p_{Y_j}(y_j ; \psi) \\
 
&&  \frac{e^{-\frac{1}{2} \sum_{j=1}^n \left( \frac{y_j - f(t_j ; \psi)}{g(t_j ; \psi)} \right)^2}}{\prod_{j=1}^n \sqrt{2\pi g(t_j ; \psi)}}
 
&&  \frac{e^{-\frac{1}{2} \sum_{j=1}^n \left( \frac{y_j - f(t_j ; \psi)}{g(t_j ; \psi)} \right)^2}}{\prod_{j=1}^n \sqrt{2\pi g(t_j ; \psi)}}
\end{eqnarray*}
+
\end{eqnarray*}
  
 
Maximizing the likelihood is equivalent to minimizing the deviance (-2 $\times$ log-likelihood) which plays here the role of the objective function:
 
Maximizing the likelihood is equivalent to minimizing the deviance (-2 $\times$ log-likelihood) which plays here the role of the objective function:

Revision as of 17:25, 28 January 2013

An example of continuous data from a single individual

Graf1.png


A model for continuous data: \begin{eqnarray*} y_{j} &=& f(t_j ; \psi) + \varepsilon_j \quad ; \quad 1\leq j \leq n \\ &=& f(t_j ; \psi) + g(t_j ; \psi) \bar{\varepsilon_j} \end{eqnarray*}


  • $f$ : { structural model}
  • $\psi=(\psi_1, \psi_2, \ldots, \psi_d)$ : vector of parameters
  • $(t_1,t_2,\ldots , t_n)$ : observation times
  • $(\varepsilon_j, \varepsilon_2, \ldots, \varepsilon_n)$ : residual errors ($\Epsilon({\varepsilon_j}) =0$)
  • $g$ : { residual error model}
  • $(\bar{\varepsilon_1}, \bar{\varepsilon_2}, \ldots, \bar{\varepsilon_n})$ : normalized residual errors ($Var({\bar{\varepsilon_j}}) =1$)


Some tasks in the context of modelling, {\i.e.} when a vector of observations $(y_j)$ is available:

  • Simulate a vector of observations $(y_j)$ for a given model and a given parameter $\psi$,
  • Estimate the vector of parameters $\psi$ for a given model,
  • Select the structural model $f$
  • Select the residual error model $g$
  • Assess/validate the selected model


Maximum likelihood estimation of the parameters: \begin{equation} '"`UNIQ-MathJax18-QINU`"' maximizes '"`UNIQ-MathJax19-QINU`"' \bar{\varepsilon_i} \sim_{i.i.d} {\cal N}(0,1)'"`UNIQ-MathJax20-QINU`"'y_i'"`UNIQ-MathJax21-QINU`"'(y_1, y_2, \ldots y_n)'"`UNIQ-MathJax22-QINU`"'\times'"`UNIQ-MathJax23-QINU`"'y_{j} = f(t_j ; \phi) + a \, \teps_j'"`UNIQ-MathJax24-QINU`"'g=a+b*f'"`UNIQ-MathJax25-QINU`"'g=a'"`UNIQ-MathJax26-QINU`"'g=b\, f'"`UNIQ-MathJax27-QINU`"'g=a+b f'"`UNIQ-MathJax28-QINU`"'u(y_j)'"`UNIQ-MathJax29-QINU`"'y_j'"`UNIQ-MathJax30-QINU`"'g=a+b*f'"`UNIQ-MathJax31-QINU`"'\log(y)=\log(f) + a\, \teps$ \end{tabbing}