From 1751d1dfeab3465d0994256f40b66a8e4937fe81 Mon Sep 17 00:00:00 2001 From: Walmes Zeviani <walmes@ufpr.br> Date: Sun, 15 May 2016 20:27:40 -0300 Subject: [PATCH] Adiciona slides da gamma count. --- inst/slides/gamma_count.Rnw | 218 ++++++++++++++++++++++++++++ inst/slides/images/winkelman95.jpeg | Bin 0 -> 64514 bytes 2 files changed, 218 insertions(+) create mode 100644 inst/slides/gamma_count.Rnw create mode 100755 inst/slides/images/winkelman95.jpeg diff --git a/inst/slides/gamma_count.Rnw b/inst/slides/gamma_count.Rnw new file mode 100644 index 0000000..90bc40d --- /dev/null +++ b/inst/slides/gamma_count.Rnw @@ -0,0 +1,218 @@ +<<setup-child, include=FALSE>>= +set_parent("slides-mrdcr.Rnw") +@ + +\begin{frame} + \begin{center} + \includegraphics[width=11cm]{images/winkelman95.jpeg} + \end{center} + \begin{thebibliography}{99} + \bibitem{Winkelmann1995} + \MakeUppercase{Winkelmann, R.} + \newblock{Duration Dependence and Dispersion in Count-Data Models}. + \textbf{Journal of Business \& Economic Statistics}, v.13, n.4, + p.467--474, 1995. + \end{thebibliography} +\end{frame} + +\begin{frame}[allowframebreaks] + \frametitle{Duração dependência} + \begin{itemize} + \item Considere um processo estocástico definido pela sequência da + v.a. $\tau_i$, intervalo de tempo entre eventos. + \item Se $\{\tau_1, \tau_2,\ldots\}$ são independentes e identicamente + distribuídos, todos com densidade $f(\tau)$, esse processo é chamado + de \emph{renewal process}. + \item Defina a variável de contagem $N_T$ como o número de eventos no + intervalo $[0,T)$. + \item Defina $\vartheta_n = \sum_{i=1}^{n} \tau_i$ o tempo até o + $n$-ésimo evento. + \item A distribuição de $\vartheta_n$ determina a distribuição de + $N_T$, mas é baseada em covolução. + \item São distribuições fechadas para covolução: normal, Poisson, + binomial e gama. + \item Destas, apenas a gama é contínua e positiva. + + \framebreak + + \item Denote $\text{E}(\tau) = \mu$, $\text{V}(\tau) = \sigma^2$ e + $\text{CV}(\tau) = \sigma/\mu$. + \item Defina $\lambda(\tau) = \frac{f(\tau)}{1-F(\tau)}$ como a função + de risco e assuma que é monótona. + \item Existe relação entre o tipo de duração dependência e o + coeficiente de variância + \begin{equation} + \frac{\text{d}\lambda(t)}{\text{d}t} \left.\begin{matrix} + < \\ + = \\ + > + \end{matrix}\right\} 0 \Rightarrow + \upsilon \left.\begin{matrix} + < \\ + = \\ + > + \end{matrix}\right\} 1 + \end{equation} + \end{itemize} +\end{frame} + +\begin{frame}[allowframebreaks] + \frametitle{Relação entre número de eventos e intervalo entre eventos} + \begin{itemize} + \item Intervalos entre tempo $\tau \sim \text{Gama}(\alpha,\beta)$, + $$f(\tau, \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} + \cdot \tau^{\alpha-1}\cdot \exp\{-\beta\tau\},$$ + $$ \text{E}(\tau) = \frac{\alpha}{\beta}, \quad + \text{V}(\tau) = \frac{\alpha}{\beta^2}.$$ + \item Tempo até o $n$-ésimo evento + $$\vartheta_n = \tau_1+\cdots+\tau_n ~ \sim + \text{Gama}(n\alpha, \beta),$$ + $$f_n(\vartheta, \alpha, \beta) = + \frac{\beta^{n\alpha}}{\Gamma(n\alpha)}\cdot + \vartheta^{n\alpha-1}\cdot \exp\{-\beta\vartheta\},$$ + $$ \text{E}(\vartheta) = \frac{n\alpha}{\beta}, \quad + \text{V}(\vartheta) = \frac{n\alpha}{\beta^2}.$$ + + \framebreak + + \item A distribuição acumulada do tempo até $\vartheta_{n}$ é + \item $$F_n(T) = \Pr(\vartheta_n \leq T) = \int_{0}^{T} + \frac{\beta^{n\alpha}}{\Gamma(n\alpha)}\cdot t^{n\alpha-1}\cdot + \exp\{-\beta t\}\,\text{d}t.$$ + \item Seja $[0,T)$ um intervalo e $N_{T}$ a v.a. número de eventos + neste intervalo. + \item Segue que $N_T < n$ se e somente se $\vartheta_n \geq + T$. Assim + $$\Pr(N_T<n) = \Pr(\vartheta_n \geq T) = 1-F_n(T);$$ + \item Já que $\Pr(N_T = n) = \Pr(N_T < n+1) - \Pr(N_T < n)$, então + $$\Pr(N_T = n) = F_n(T) - F_{n+1}(T).$$ + + \framebreak + + \item Portanto, distribuição de $N_T$ é resultado da diferença de + acumuladas da distribuição Gama, pois + \begin{equation} + F_n(T) = G(n\alpha, \beta T) = + \int_{0}^{T} \frac{\beta^{n\alpha}}{\Gamma(n\alpha)} + t^{n\alpha-1}\cdot\exp\{-\beta t\}\, \text{d}t. + \end{equation} + \item Assim + \begin{align*} + \Pr(N_T=n) &= G(n\alpha, \beta T) - G((n+1)\alpha, \beta T) \\ + &= \left[ \int_{0}^{T} + \frac{\beta^{n\alpha}}{\Gamma(n\alpha)} + t^{n\alpha-1}\cdot + \exp\{-\beta t\}\, \text{d}t \right] - + \left[ \int_{0}^{T} + \frac{\beta^{(n+1)\alpha}}{\Gamma((n+1)\alpha)} + t^{(n+1)\alpha-1}\cdot + \exp\{-\beta t\}\, \text{d}t \right] + \end{align*} + \end{itemize} +\end{frame} + +\begin{frame}[allowframebreaks] + \frametitle{Parametrização para modelo de regressão} + \begin{itemize} + \item A média da variável aleatória $N_T$ é resultado de + \begin{align*} + E(N) &= \sum_{i=0}^{\infty} i\cdot \Pr(i) \\ + &= \sum_{i=1}^{\infty} i\cdot \Pr(i)\\ + &= \sum_{i=1}^{\infty} G(i\alpha, \beta T).\\ + \end{align*} + \item Para um $T$ cada vez maior, tem-se que + \begin{equation*} + N(T)\; \dot{\sim}\; \text{Normal}\left( + \frac{\beta}{\alpha}, + \frac{\beta}{\alpha^2}\right). + \end{equation*} + \item Considere que + $$\frac{\beta}{\alpha} = \exp\{x^{\top}\theta\} \Rightarrow + \beta = \alpha \exp\{x^{\top}\theta\}.$$ + Essa parametrização produz um modelo de regressão para a média + do tempo entre eventos definida por + $$\text{E}(\tau|x) = \frac{\alpha}{\beta} = + \exp\{-x^{\top}\theta\}.$$ + \item O modelo de regressão é para o tempo entre eventos ($\tau$) + e não diretamente para contagem porque, a menos que + $\alpha = 1$, não é certo que + $\text{E}(N_i|x_i) = [\text{E}(\tau_i|x_i)]^{-1}$. + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{Função de log-verossimilhança} + Considerando uma amostra aleatória $y_i, i=1,2,\ldots,n$, a + verossimilhança é + \begin{equation} + L(y; \alpha, \beta) = + \prod_{i=1}^{n} \left( + G(y_i\alpha, \beta) - + G((y_i+1)\alpha, \beta) \right) + \end{equation} + e a função de log-verossimilhança é + \begin{equation} + \ell(y; \alpha, \beta) = + \sum_{i=1}^{n} \ln\left( + G(y_i\alpha, \beta) - + G((y_i+1)\alpha, \beta) \right) + \end{equation} +\end{frame} + +\begin{frame}[fragile] + \frametitle{Implementação da log-verossimilhança} +<<echo = TRUE>>= +library(MRDCr) +llgcnt +@ +\end{frame} + +\begin{frame}[fragile] +<<fig.width = 9, fig.height = 4.5, out.width = "0.95\\textwidth">>= +grid <- expand.grid(lambda = c(2, 8, 15), + alpha = c(0.5, 1, 2.5)) +y <- 0:30 +py <- mapply(FUN = dgcnt, + lambda = grid$lambda, + alpha = grid$alpha, + MoreArgs = list(y = y), SIMPLIFY = FALSE) +grid <- cbind(grid[rep(1:nrow(grid), each = length(y)), ], + y = y, + py = unlist(py)) + +useOuterStrips(xyplot(py ~ y | factor(lambda) + factor(alpha), + ylab = expression(f(y)), + xlab = expression(y), + data = grid, type = "h", + panel = function(x, y, ...) { + m <- sum(x * y) + panel.xyplot(x, y, ...) + panel.abline(v = m, lty = 2) + }), + strip = strip.custom( + strip.names = TRUE, + var.name = expression(lambda == ""), + sep = ""), + strip.left = strip.custom( + strip.names = TRUE, + var.name = expression(alpha == ""), + sep = "")) +@ +\end{frame} + +\begin{frame}{Estudos de caso} + {\it Vignette} \href{run:../doc/v06_gamma_count.html}{\tt + gamma\_count.html} + \begin{description} + \item[\tt soja]: Número de vagens, de grãos e de grãos por vagem. + \item[\tt capdesfo]: Número de capulhos produzidos em algodão. + \item[\tt nematoide]: Número de nematoides em raízes de linhagens de + feijoeiro. + \item[\tt cambras]: Gols do Campeonato Brasileiro de 2010. + \end{description} +\end{frame} + +\begin{frame} + \frametitle{Conclusões} + TODO +\end{frame} diff --git a/inst/slides/images/winkelman95.jpeg 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