From 236a9561db118518a07cbc75824ebf5059a63a4f Mon Sep 17 00:00:00 2001 From: Cesar Taconeli <taconeli@ufpr.br> Date: Thu, 19 May 2016 20:25:47 -0300 Subject: [PATCH] Atualiza arquivos do minicurso --- data-raw/Baseauto2.csv | 16656 ++++++++++++++++ data-raw/Dadoscomp.csv | 79 + inst/slides/Cesar.tex | 579 +- inst/slides/images/{graf2.pdf => Graf_BN.pdf} | Bin 5655 -> 5655 bytes .../images/{graf1.pdf => Graf_Poisson.pdf} | Bin 5384 -> 5385 bytes inst/slides/images/Graf_Risco.pdf | Bin 0 -> 6069 bytes vignettes/Ovelhas.Rmd | 57 +- vignettes/Sinistros.Rmd | 112 +- 8 files changed, 16948 insertions(+), 535 deletions(-) create mode 100644 data-raw/Baseauto2.csv create mode 100644 data-raw/Dadoscomp.csv rename inst/slides/images/{graf2.pdf => Graf_BN.pdf} (69%) rename inst/slides/images/{graf1.pdf => Graf_Poisson.pdf} (69%) create mode 100644 inst/slides/images/Graf_Risco.pdf diff --git a/data-raw/Baseauto2.csv b/data-raw/Baseauto2.csv new file mode 100644 index 0000000..09d698e --- 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PESADA,SP,58,M,Casado,58781,"0,14",0 +2,SEDAN,AM,43,F,Casado,21854,"0,78",1 +9,HATCH,SP,54,F,Solteiro,22401,"0,77",0 +10,HATCH,SP,76,F,Casado,34451,"0,56",0 +9,SUV,SP,70,M,Casado,42174,"0,80",0 +8,HATCH,SP,71,F,Viuvo,19398,"0,80",0 +4,SEDAN,SP,33,F,Casado,36659,"0,05",0 +3,SEDAN,SP,44,M,Casado,30266,"0,01",0 +9,HATCH,SP,64,M,Casado,17060,"0,79",0 +10,MONOVOLUME,SP,55,M,Casado,13114,"0,80",0 +0,HATCH,RS,65,M,Casado,37106,"0,81",1 +10,MONOVOLUME,SP,34,M,Casado,61497,"0,77",0 +3,HATCH,SP,42,M,Casado,30664,"0,59",0 +10,HATCH,SP,35,M,Divorciado,50104,"0,79",0 +1,SUV,PR,52,F,Casado,48676,"0,83",1 +10,HATCH,SP,31,M,Solteiro,48122,"0,76",0 +10,MONOVOLUME,SP,49,M,Casado,20131,"0,78",0 +10,HATCH,CE,79,M,Casado,39585,"0,77",0 +10,PICAPE PESADA,MG,49,M,Casado,35940,"0,76",0 +5,SUV,SP,46,M,Casado,58998,"0,77",0 +10,SEDAN,SP,64,F,Casado,28616,"0,83",0 +2,HATCH,SP,39,F,Casado,21651,"0,76",0 +1,MONOVOLUME,SP,59,M,Casado,21588,"0,79",0 +10,HATCH,SP,55,M,Casado,27903,"0,16",0 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beta-binomial, binomial negativa; distribuições fundamentadas na modelagem do tempo entre eventos, como a Gamma-Count... + \item Algumas alternativas: Distribuição Binomial, Poisson e generalizações; distribuições geradas por misturas, como a beta-binomial, binomial negativa; distribuições fundamentadas na modelagem do tempo entre eventos, na razão de probabilidades sucessivas... \end{itemize} @@ -302,7 +302,7 @@ aquém, em quantidade e diversidade, em relação ao verificado para dados cont $$ - P\left ( Y=k \right )=\frac{e^{-\mu}\mu^{k}}{k!},\hspace{5mm}k=0,1,2,...; \mu>0. + P\left ( Y=k \right )=\frac{e^{-\lambda}\lambda^{k}}{k!},\hspace{5mm}k=0,1,2,...; \lambda>0. $$ \vspace{0,2cm} @@ -310,7 +310,7 @@ $$ \item Se os eventos sob contagem ocorrem independentemente e sujeitos a uma taxa constante $\lambda >0$, sob o modelo Poisson, para um intervalo de exposição de tamanho $t$ tem-se: $$ - P\left ( Y=k \right )=\frac{e^{-\lambda t}(\lambda t)^{k}}{k!},\hspace{5mm}k=0,1,2,.... + P\left ( Y_t=k \right )=\frac{e^{-\lambda t}(\lambda t)^{k}}{k!},\hspace{5mm}k=0,1,2,.... $$ \end{itemize} @@ -318,6 +318,54 @@ $$ %%%----------------------------------------------------------------------------------------- + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 11 + +\begin{frame}{Propriedades da distribuição de Poisson} + +Dentre as principais propriedades da distribuição de Poisson, tem-se: + +\vspace{0,3cm} + +\begin{itemize} + + \item Média: $E(Y)= \lambda$; + \vspace{0,5cm} + + \item Variância: $Var(Y)=\lambda$ (equidispersão); + \vspace{0,5cm} + + \item Razão de probabilidades sucessivas: $\frac{P\left ( X=k \right )}{P\left ( X=k-1 \right )}=\frac{\lambda}{k},$ gerando a relação de recorrência: + + $$ + P(Y=k)k=P(Y=k-1)\lambda; + $$ + + \item Se $Y_{1},Y_{2},...,Y_{n}$ são v.a.s independentes com $Y_{i}\sim Poisson(\lambda_{i})$, e $\sum\lambda_{i}<\infty$, então $\sum Y_{i}\sim Poisson(\sum\lambda_{i})$. + + \end{itemize} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + +%%% Slide 13 + +\begin{frame}{Distribuição Poisson para diferentes valores de $\lambda$} + + \begin{figure}[h] + \includegraphics[height=6cm,width=9cm]{images/Graf_Poisson.pdf} + \caption{Distribuição de Poisson para diferentes valores de $\lambda$} + \label{Fig1} + \centering + +\end{figure} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + %%% Slide 8 \begin{frame}{Motivações para a distribuição de Poisson} @@ -327,13 +375,15 @@ $$ \vspace{0,5cm} \begin{itemize} - \item Caso limite da distribuição binomial$(n, \pi)$ quando $n\rightarrow \infty$ e $\pi\rightarrow 0$, fixado $\mu=n\pi$, ou seja: + \item Caso limite da distribuição binomial$(n, \pi)$ quando $n\rightarrow \infty$ e $\pi\rightarrow 0$, fixado $\lambda=n\pi$, ou seja: + +\vspace{0.4cm} $$ - \underset{n \to \infty, \pi \to 0 }{lim} + \underset{n \to \infty \pi \to 0 }{lim} \left [ \begin{pmatrix} n\\k -\end{pmatrix} \left ( \frac{\mu}{n} \right )^{k}\left ( 1-\frac{\mu}{n} \right )^{n-k}\right ]=\frac{e^{-\mu}\mu^{k}}{k!}. +\end{pmatrix} \left ( \frac{\lambda}{n} \right )^{k}\left ( 1-\frac{\lambda}{n} \right )^{n-k}\right ]=\frac{e^{-\lambda}\lambda^{k}}{k!}. $$ \vspace{0,5cm} @@ -381,6 +431,23 @@ dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo %%%----------------------------------------------------------------------------------------- +%%% Slide 13 + +\begin{frame}{Diferentes comportamentos para $\lambda (t)$} + \vspace{-1.5cm} + \begin{figure}[h] + \includegraphics[height=8cm,width=10cm]{images/Graf_Risco.pdf} + \vspace{-0.8cm} + \caption{Diferentes comportamentos para $\lambda (t)$} + \label{Fig1} + \centering + +\end{figure} +\end{frame} + +%%%----------------------------------------------------------------------------------------- + + %%% Slide 10 \begin{frame}{O processo de Poisson} @@ -420,51 +487,7 @@ dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo \end{figure} \end{frame} -%%%----------------------------------------------------------------------------------------- - -%%% Slide 11 -\begin{frame}{Propriedades da distribuição de Poisson} - -Dentre as principais propriedades da distribuição de Poisson, tem-se: - -\vspace{0,3cm} - -\begin{itemize} - - \item Média: $E(Y)=\mu = \lambda t$; - \vspace{0,5cm} - - \item Variância: $Var(Y)=\mu$ (equidispersão); - \vspace{0,5cm} - - \item Razão de probabilidades sucessivas: $\frac{P\left ( X=k \right )}{P\left ( X=k-1 \right )}=\frac{\mu}{k},$ gerando a relação de recorrência: - - $$ - P(Y=k)k=P(Y=k-1)\mu; - $$ - - \item Se $Y_{1},Y_{2},...,Y_{n}$ são v.a.s independentes com $Y_{i}\sim Poisson(\mu_{i})$, e $\sum\mu_{i}<\infty$, então $\sum Y_{i}\sim Poisson(\sum\mu_{i})$, - - \end{itemize} -\end{frame} - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 13 - -\begin{frame}{Distribuição Poisson para diferentes valores de $\mu$} - - \begin{figure}[h] - \includegraphics[height=6cm,width=9cm]{images/graf1.pdf} - \caption{Distribuição de Poisson para diferentes valores de $\mu$} - \label{Fig1} - \centering - -\end{figure} -\end{frame} - -%%%----------------------------------------------------------------------------------------- %%% Slide 15 @@ -492,7 +515,7 @@ Dentre as principais propriedades da distribuição de Poisson, tem-se: \begin{frame}{Regressão Poisson - Especificação do modelo} \begin{itemize} - \item Sejam $Y_{1},Y_{2},...,Y_{n}$ variáveis aleatórias condicionalmente independentes, dado o vetor de covariáveis ${\vec{x_{i}}}'=(x_{i1},x_{i2},...,x_{ip}), i=1,2,...,n$. A regressão Poisson é definida pela distribuição de Poisson: + \item Sejam $Y_{1},Y_{2},...,Y_{n}$ variáveis aleatórias condicionalmente independentes, dado o vetor de covariáveis ${\boldsymbol{x_{i}}}'=(x_{i1},x_{i2},...,x_{ip}), i=1,2,...,n$. A regressão Poisson é definida pela distribuição de Poisson: $$ @@ -501,7 +524,7 @@ $$ \vspace{0,2cm} -sendo as covariáveis inseridas ao modelo da seguinte forma: +sendo as covariáveis inseridas ao modelo por: $$ \ln(\mu _{i})=\boldsymbol{x_{i}'} \boldsymbol{\beta}, @@ -528,7 +551,8 @@ em que ${\boldsymbol{\beta }}$ é o vetor de parâmetros de regressão. \begin{itemize} - \item $f(y_{i}|\boldsymbol{x_{i}})=\frac{\exp^{- \vec{{x_{i}}' \beta}}( \vec{{x_{i}}' \beta})^{y_{i}}}{y_{i}!},\hspace{5mm}y=0,1,2,...$ + \item $f(y_{i}|\boldsymbol{x_{i}})=\frac{e^{-exp(\boldsymbol{x'_i\beta})}{exp({\boldsymbol{x'_i\beta}})}^{y_i}}{y_i!}$ + \vspace{0,8cm} @@ -551,286 +575,11 @@ em que ${\boldsymbol{\beta }}$ é o vetor de parâmetros de regressão. -%%%----------------------------------------------------------------------------------------- - -%%% Slide 25 - -\begin{frame}{Modelos Lineares Generalizados(MLG)} - -\begin{itemize} - -\item A teoria dos modelos lineares generalizados se estende às variáveis aleatórias cujas funções (densidade) de probabilidades possa ser expressas na forma exponencial linear: - - \vspace{0,5cm} - -$$ - f\left ( y;\theta,\phi \right) =exp\left \{ \frac{y\theta-b\left ( \theta \right )}{\phi}+c\left ( y;\phi \right ) \right \}, -$$ - \vspace{0,5cm} -sendo $\theta=\theta(\mu)$ denominado \textit{parâmetro canônico} e $\phi$ o parâmetro de dispersão. - - \vspace{0,5cm} - -\item Casos particulares: Binomial, Poisson, Normal, Gama, Exponencial... - -\end{itemize} -\end{frame} - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 26 - -\begin{frame}{Modelos Lineares Generalizados} -\begin{itemize} - -\item Propriedades: - -$$ - E[Y]=b'(\theta)=\frac{\partial b(\theta)}{\partial \theta}; -$$ - -$$ - Var[Y]=\phi b''(\theta)=\phi\frac{\partial \mu}{\partial \theta}=\phi V(\mu) -$$ - - -\item Para a Poisson: - -$$\theta = \log(\mu); \hspace{0.2cm} b(\theta)=\exp(\theta); \hspace{0.2cm} \phi=1; \hspace{0.2cm} V(\mu)=\mu.$$ - -\end{itemize} -\end{frame} - - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 27 - -\begin{frame}{Modelo Linear Generalizado} - -\begin{itemize} - - - \item Definição do modelo: - - \vspace{0,5cm} - -$$ - g(\mu_{i})=\boldsymbol{x'_{i}\beta}=\beta_{0}+\beta_{1}x_{i1}+...+\beta_{p}x_{ip}, -$$ - - \vspace{0,5cm} - -sendo $g(\cdot)$ uma função real, monótona e diferenciável. - - \vspace{0,5cm} - -\item Para a distribuição Poisson é usual considerar: - -$$ - g(\mu_{i})= \log(\mu_i) = \boldsymbol{x'_{i}\beta}=\beta_{0}+\beta_{1}x_{i1}+...+\beta_{p}x_{ip}. -$$ - -\end{itemize} - -\end{frame} - - -%%%----------------------------------------------------------------------------------------- - - -\begin{frame}{Modelo Linear Generalizado} - -\begin{itemize} - - -\item Sejam $Y_{1},Y_{2},...,Y_{n}$ independentes, com distribuição expressa na forma exponencial linear. Temos a seguinte log-verossimilhança: - -$$ - l(\boldsymbol{\beta })=\phi^{-1}\sum_{i=1}^{n}(y_{i}\theta_{i}-b(\theta_{i}))+\sum_{i=1}^{n}c(y_{i},\phi), -$$ - -sendo $\theta_i=q(\mu_i) \text{ e } g(\mu_i)=\eta_i=\beta_{0}+\beta_{1}x_{i1}+...+\beta_{p}x_{ip}.$ - - -\item Função escore: - -$$ - S(\beta_j)=\phi^{-1}\sum_{i=1}^{n}\frac{y_{i}-\mu_{i}}{V(\mu_{i})} \frac{\partial \mu_{i}}{\partial \eta_{i}}x_{ij}, \hspace{0,3cm} j=0,1,2,...,p. -$$ - - - - -\end{itemize} - -\end{frame} - - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 29 - -\begin{frame}{Modelo Linear Generalizado} - -\begin{itemize} - - -\item A estimação via máxima verossimilhança de $\boldsymbol{\beta}$ se dá pela solução do sistema de equações $S_j=S(\beta_j)=0$, $j=0,1,2,...,p$. - - \vspace{0,5cm} - - -\item Matriz de informação esperada (informação de Fisher): - -$$ -I(\boldsymbol{\beta})=-E -\begin{bmatrix} -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{0}^{2} } & -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{0} \partial \beta_{1} } & -\cdots & -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{0} \partial \beta_{p} }\\ -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{0} \partial \beta_{1} } & -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{1}^{2} } & -\cdots & -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{1} \partial \beta_{p} } - \\ -\vdots & \vdots & \ddots & \vdots \\ -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{0} \partial \beta_{p} } & -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{1} \partial \beta_{p} } & -\cdots & -\frac{\partial l^{2}(\boldsymbol{\beta , y})}{\partial \beta_{p}^{2} } -\end{bmatrix} -$$ - -\end{itemize} -\end{frame} - - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 30 - -\begin{frame}{Modelo Linear Generalizado} - -\begin{itemize} - - \item O termo característico da matriz informação esperada, $I_{jk}$, é dado por: - - $$ - I_{jk}=\phi^{-1} \sum_{i=1}^{n}\omega_{i}x_{ij}x_{ik}, - $$ - -sendo - -$$ - \omega_{i}=V(\mu_{i})\left ( \frac{\partial \mu_{i}}{\partial \eta_{i}} \right )^{2}. -$$ - - \item Matricialmente: - -$$ - \boldsymbol{I}= \boldsymbol{I}(\boldsymbol{\beta})=\phi^{-1}\boldsymbol{X'WX}, -$$ -\end{itemize} - -sendo $\boldsymbol{X}$ a matriz do modelo e $\boldsymbol{W}$ a matriz diagonal com elementos $\{\omega_{1}, \omega_{2}, ..., \omega_{n}\}$. - -\end{frame} - - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 31 - -\begin{frame}{Modelo Linear Generalizado} - -\begin{itemize} - -\item Estimação dos parâmetros: algoritmo escore de Fiher (processo iterativo): - - \vspace{0,5cm} - -$$ - \boldsymbol{\beta}^{(m+1)}=\boldsymbol{\beta}^{(m)}+(\boldsymbol{I}^{(m)})^{-1}\boldsymbol{S^{(m)}}, -$$ - - \vspace{0,5cm} - -sendo $\boldsymbol{S^{(m)}}$ e $\boldsymbol{I}^{(m)}$ o vetor escore e a matriz informação avaliados no passo $m$. - - \vspace{0,5cm} - -\item Substituindo $\boldsymbol{S^{(m)}}$ e $\boldsymbol{I}^{(m)}$ pelas correspondentes expressões para um modelo linear generalizado, temos: - -$$ - \boldsymbol{\beta}^{(m+1)}=(\boldsymbol{X'WX})^{-1}\boldsymbol{X'W^{(m)}z^{(m)}} -$$ - - -\end{itemize} -\end{frame} - - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 32 - -\begin{frame}{Modelo Linear Generalizado} - -\begin{itemize} - -\item Como resultado temos um processo iterativo de mínimos quadrados ponderados, em que $\boldsymbol{W}^{(m)}$ é a matriz de pesos, atualizados a cada passo do processo, e - -$$ - \boldsymbol{z}^{(m+1)}=\boldsymbol{\eta}^{(m)}+\boldsymbol{G}^{(m)}\boldsymbol{(y-\mu^{(m)})}, -$$ - -é a variável dependente ajustada, sendo $\boldsymbol{G}$ a matriz diagonal com elementos $\{d\eta_{1}/d\mu_{1}, -d\eta_{2}/d\mu_{2},...,d\eta_{n}/d\mu_{n}\}.$ - -\item Assintoticamente, - -$$ - \boldsymbol{\hat{\beta}}\overset{a}{\sim} N_{p+1}\left ( \boldsymbol{\beta}, \boldsymbol{I^{-1}\left ( \boldsymbol{\beta} \right )} \right ), -$$ - -com $\boldsymbol{I^{-1}(\boldsymbol{\beta})}=Var(\boldsymbol{\hat{\beta}})=\phi (\boldsymbol{X'WX})^{-1}$. - - -\end{itemize} - -\end{frame} - - - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 32 - -\begin{frame}{Modelo Linear Generalizado} - -\begin{itemize} - - -\item O parâmetro de dispersão, $\phi$, pode ser estimado usando o método de momentos, resultando em: - -\vspace{0.5cm} - -$$ \hat{\phi}=\frac{\sum_{i=1}^{n} (y_{i}-\hat{\mu}_{i})^{2}}{n-p}. $$ - -\end{itemize} - -\end{frame} - -%%%----------------------------------------------------------------------------------------- - - %%% Slide 18 -\begin{frame}{Regressão Poisson} +\begin{frame}{Regressão Poisson - Estimação por máxima verossimilhança} Para a regressão Poisson: @@ -838,22 +587,22 @@ Para a regressão Poisson: \begin{itemize} -\item $l(\boldsymbol{\beta})=\sum_{i=1}^{n} \{ y_{i}\boldsymbol{x_{i}'\beta}-\exp{(\boldsymbol{x_{i}'\beta}}\}-\ln{y_{i}!});$ +\item Log-verossimilhança: $l(\boldsymbol{\beta})=\sum_{i=1}^{n} \{ y_{i}\boldsymbol{x_{i}'\beta}-\exp{(\boldsymbol{x_{i}'\beta}}\}-\ln{y_{i}!});$ \vspace{0.5cm} -\item $\boldsymbol{S}(\boldsymbol{\beta})=\frac{\partial l(\boldsymbol{\beta};\boldsymbol{y})}{\partial \boldsymbol{\beta}}= +\item Vetir escore: $\boldsymbol{S}(\boldsymbol{\beta})=\frac{\partial l(\boldsymbol{\beta};\boldsymbol{y})}{\partial \boldsymbol{\beta}}= \sum_{i=1}^{n}(y_{i}-\exp(\boldsymbol{x_{i}'\beta}))\boldsymbol{x_{i}};$ \vspace{0.5cm} -\item $ +\item Matriz Informação: $ \boldsymbol{I({\beta})} = \sum_{i=1}^n \mu_i \boldsymbol{x_i x'_i} = \exp{(\boldsymbol{x'_i \beta})\boldsymbol{x_i x'_i}}; $ \vspace{0.5cm} -\item $ -\boldsymbol{\hat{\beta}} \sim N \left ( \boldsymbol{\beta}, \left [ \sum_{i=1}^n \mu_i \boldsymbol{x_i x'_i} \right ]^{-1} \right ). +\item Distribuição assintótica: $ +\boldsymbol{\hat{\beta}} \overset{a}{\sim} N \left ( \boldsymbol{\beta}, \left [ \sum_{i=1}^n \mu_i \boldsymbol{x_i x'_i} \right ]^{-1} \right ). $ \end{itemize} @@ -866,22 +615,23 @@ $ %%% Slide 34 -\begin{frame}{Modelo Linear Generalizado} +\begin{frame}{Regressão Poisson - Modelo Linear Generalizado} + -Propriedades adicionais dos modelos lineares generalizados: +A Regressão Poisson é um caso particular dos Modelos Lineares Generalizados (MLG). Algumas propriedades dessa classe de modelos: \begin{itemize} -\item Os EMV's são consistentes ainda que a distribuição especificada seja incorreta, mas desde que a média condicional de $Y$ seja declarada corretamente; +\item Os estimadores são consistentes ainda que a distribuição especificada seja incorreta, mas desde que a média condicional de $Y$ seja declarada corretamente; \item Os erros padrões, intervalos de confiança e testes de hipóteses, no entanto, ficam comprometidos; -\item O ajuste de um MLG, no entanto, requer apenas a especificação: +\item O ajuste de um MLG requer apenas a especificação: \begin{itemize} \item Da esperança de $Y$ condicional às covariáveis, mediante especificação do preditor linear e da função de ligação; -\item Da variância condicional, mediante especificação da função de variância $V(\mu)$, possível inclusão do parâmetro de dispersão $(\phi)$, ou sua estimação por métodos robustos. +\item Da variância condicional, mediante especificação da função de variância $V(\mu)$, possível inclusão do parâmetro de dispersão $(\phi)$, ou sua estimação por métodos robustos (abordagem de Quase-Verossimilhança). \end{itemize} @@ -894,151 +644,57 @@ Propriedades adicionais dos modelos lineares generalizados: %%%----------------------------------------------------------------------------------------- - - -%%%----------------------------------------------------------------------------------------- - -%%% Slide 39 - -\begin{frame}{Modelo Linear Generalizado - Quase-Verossimilhança} +\begin{frame}{Regressão Poisson - Quase-Verossimilhança} \begin{itemize} + \item Para o ajuste de um modelo alternativo via Quase-Verossimilhança, definimos: -\item Considere a variável aleatória $Y$ com média $\mu$ e variância $\phi V(\mu)$. O método de quase-verossimilhança baseia-se na função quase-escore: +\vspace{0.4cm} -$$ U(\mu) = \frac{y-\mu}{\phi V(\mu)}$$. +$$g(E(y_i|\boldsymbol{x_i}))=\boldsymbol{x'_i \beta};$$ -\item Função de quase-verossimilhança: - -$$ Q\left ( \mu \right )=\int _y^\mu \frac{y-t}{\phi V(t)}dt $$. - -\end{itemize} - -\end{frame} +$$Var(y_i|\boldsymbol{x_i})=\phi V(\mu_i).$$ +\item A obtenção dos estimadores se dá pela maximização da função de quase-verossimilhança: -%%%----------------------------------------------------------------------------------------- - -%%% Slide 40 - -\begin{frame}{Modelo Linear Generalizado - Quase-Verossimilhança} - -\begin{itemize} - -\item Propriedades: - -\vspace{0.5cm} - -\begin{itemize} - -\item $ E\left ( \frac{\partial Q(\mu)}{\partial \mu} \right ) = 0 $ +$$ Q\left ( \mu \right )=\int _y^\mu \frac{y-t}{\phi V(t)}dt $$. -\vspace{0.5cm} -\item $ E\left [ \left ( \frac{\partial Q(\mu)}{\partial \mu} \right )^2 \right ]= --E\left [ \frac{\partial^2Q(\mu)}{\partial \mu^2} \right ] $ +\item As funções de quase-verossimilhança, quase-escore e quase-informação compartilham propriedades comuns às correspondentes funções no caso paramétrico, para MLGs. \end{itemize} -\vspace{0.5cm} - -\item A depender da especificação dos momentos, as equações de estimação são idênticas às obtidas mediante especificação completa da verossimilhança. -\end{itemize} - \end{frame} -%%%----------------------------------------------------------------------------------------- - - - -%%% Slide 41 - -\begin{frame}{Modelo Linear Generalizado - Quase-Verossimilhança} - -\begin{itemize} - - -\item Os estimadores dos parâmetros de um MLG via quase-verossimilhança, $(\boldsymbol{\hat{\beta}_{QL}})$ são obtidos pela solução das equações quase-escore: -$$U(\boldsymbol{\boldsymbol{\beta}})=U(\boldsymbol{\mu}({\boldsymbol{\beta}}))=\frac{\partial{Q(\boldsymbol{\beta)}}}{\partial{\boldsymbol{\beta}}}= \frac{1}{\phi}\boldsymbol{D'V^{-1}(y-\mu)},$$ - -em que $\boldsymbol{D}=\partial{\boldsymbol{\mu}}/\partial{\boldsymbol{\beta}}=\boldsymbol{W^{1/2}V^{1/2}X}$, $\boldsymbol{V}=diag(V_1,V_2,...,V_n)$, $\boldsymbol{W}=diag(W_1,W_2,...,W_n)$, com $\omega_i=(\partial{\mu_i}/\partial{\eta_i})^2 /V_i$. - - -\item Matriz de quase-informação: - -$$-E\left [ \frac{\partial^2Q(\mu)}{\partial \mu^2} \right ] = \frac{1}{\phi}\boldsymbol{D'V^{-1}D}$$ - - -\end{itemize} - -\end{frame} - %%%----------------------------------------------------------------------------------------- +%%% Slide 39 - -%%% Slide 41 - -\begin{frame}{Modelo Linear Generalizado - Quase-Verossimilhança} +\begin{frame}{Estimação via Quase-Verossimilhança} \begin{itemize} - -\item O algoritmo de estimação baseia-se na solução das equações quase-escore. $U(\boldsymbol{\boldsymbol{\beta}})=\boldsymbol{0}$, resultando no algoritmo: - -\vspace{0.5cm} - -$$\boldsymbol{\beta} ^{(m+1)} = \boldsymbol{\beta} ^{(m)} + -\boldsymbol{(D'^{(m)}V^{-1(m)}D^{(m)})^{-1}} -\boldsymbol{D'^{(m)}V^{-1(m)}(y-\mu^{(m)}}) -$$ - -\vspace{0.5cm} - \item Distribuição assintótica: \vspace{0.3cm} -$$ \boldsymbol{\hat{\beta}_{QL}} \overset{a}{\sim} N(\boldsymbol{\beta}, \phi \boldsymbol{(D'V^{-1}D)}^{-1} )$$ +$$ \boldsymbol{\hat{\beta}_{QL}} \overset{a}{\sim} N(\boldsymbol{\beta}, Var(\boldsymbol{\hat{\beta}_{QL}} ))$$ -\end{itemize} +\vspace{0.8cm} -\end{frame} +\item Para o modelo Quase-Poisson, assume-se: +$$\ln(E(y_i|\boldsymbol{x_i}))=\boldsymbol{x'_i \beta};$$ -%%%----------------------------------------------------------------------------------------- -%%% Slide 41 - -\begin{frame}{Modelo Linear Generalizado - Quase-Verossimilhança} - -\begin{itemize} - - - - -\item Adotando-se $V(\mu)=\mu$ temos as seguintes equações de estimação: - - - -$$U(\boldsymbol{\beta}) =\sum_{i=1}^{n}(y_{i}-exp(\boldsymbol{x'_{i}\beta}))\boldsymbol{x_{i}=0}.$$ - - -\vspace{0.5cm} - -\item As equações de estimação via quase-verossimilhança, neste caso, são idênticas às obtidas com base na verossimilhança da regressão Poisson. - - -\item Como principais alternativas para definição de $V(\mu)$, podemos considerar proporcionalidade em relação a alguma potência de $\mu$, como $V(\mu) \text{ } \propto \text{ } \mu$ e $V(\mu)\text{ } \propto \text{ } \mu^2$. - \end{itemize} @@ -1046,16 +702,15 @@ $$U(\boldsymbol{\beta}) =\sum_{i=1}^{n}(y_{i}-exp(\boldsymbol{x'_{i}\beta}))\bol - %%%----------------------------------------------------------------------------------------- %%% Slide45 -\begin{frame}{Modelo Linear Generalizado - Quase Verossimilhança} +\begin{frame}{Estimação via Quase Verossimilhança} \begin{itemize} -\item Para a regressão Poisson: +\item A matriz de covariâncias assintótica para $\boldsymbol{\hat{\beta}_{QL}}$ fica dada por: $$ Var(\boldsymbol{\hat{\beta}_{QL}})=\left [ \sum_{i=1}^{n} \boldsymbol{x_{i}x'_{i}}\mu_{i} \right ]^{-1} @@ -1066,7 +721,13 @@ com $\mu_{i}=exp({\boldsymbol{x'_{i}\beta}})$ e $\omega_{i}=Var(y_{i}|\boldsymbo \vspace{0.5cm} -\item Podemos considerar $\omega_i=V(\mu_i;\phi)$, como $\omega_i \propto \mu_i$, $\omega_i \propto \mu_i^2$ ou, simplesmente, o estimador robusto, baseado em $\omega_i=(y_i-\mu_i)^2$. + +\item Podemos considerar $\omega_i=V(\mu_i;\phi)$, como $\omega_i =\phi \mu_i$, $\omega_i =\phi \mu_i^2$ ou, simplesmente, o estimador robusto, baseado em $\omega_i=(y_i-\mu_i)^2$. + +\item Um estimador para $\phi$: + +$$ \hat{\phi} = \frac{1}{n-p}\sum_{i=1}^n \frac{y_i-\hat{\mu}_i}{V(\hat{\mu}_i)}. $$ + \end{itemize} @@ -1085,16 +746,16 @@ com $\mu_{i}=exp({\boldsymbol{x'_{i}\beta}})$ e $\omega_{i}=Var(y_{i}|\boldsymbo \item Função de probabilidades: $$ -P(Y=k)=\frac{\Gamma(\alpha+k)}{\Gamma(k+1)\Gamma(\alpha)}\left ( \frac{\mu}{\mu+\alpha} \right )^{k} \left( \frac{\alpha}{\mu+\alpha} \right )^{\alpha}, k=0,1,2,...; \alpha > 0, \mu>0 +P(Y=k)=\frac{\Gamma(\alpha+k)}{\Gamma(k+1)\Gamma(\alpha)}\left ( \frac{\lambda}{\lambda+\alpha} \right )^{k} \left( \frac{\alpha}{\lambda+\alpha} \right )^{\alpha}, k=0,1,2,...; \alpha > 0, \lambda>0 $$ \item Propriedades: $$ -E(Y)=\mu ; \quad Var(Y)= \mu+ \alpha^{-1} \mu^2 +E(Y)=\lambda ; \quad Var(Y)= \lambda+ \alpha^{-1} \lambda^2 $$ -\item Assim, para qualquer $\alpha>0$, temos $Var(Y)>\mu$. +\item Assim, para qualquer $\alpha>0$, temos $Var(Y)>\lambda$. \item A distribuição binomial negativa tem como caso limite distribuição Poisson, quando $\alpha \rightarrow \infty$. @@ -1125,7 +786,7 @@ $$ \vspace{0.5cm} -sendo $r=\alpha$ e $p=\mu/(\mu+\alpha),$ com $0<p<1$ e $r>0$. +sendo $r=\alpha$ e $p=\lambda/(\lambda+\alpha),$ com $0<p<1$ e $r>0$. \vspace{0.5cm} \item Modelagem do número de "sucessos" até o r-ésimo "fracasso" ($r = 1,2,3,...$), configurando uma generalização da distribuição geométrica (para $r=1$). @@ -1171,11 +832,11 @@ com $E(\theta)=\theta=\alpha /\beta$ e variância $Var(\theta)=\alpha /\beta^2.$ %%% Slide 50 -\begin{frame}{Ilustração} +\begin{frame}{Distribuição binomial negativa para diferentes parâmetros} \begin{figure}[h] - \includegraphics[height=5.5cm,width=9cm]{images/graf2.pdf} - \caption{Distribuição binomial para $\mu=2$ e diferentes valores de $\alpha.} + \includegraphics[height=5.5cm,width=9cm]{images/Graf_BN.pdf} + \caption{Distribuição binomial negativa para $\lambda=2$ e diferentes valores de $\alpha.} \label{Fig1} \centering @@ -1192,7 +853,7 @@ com $E(\theta)=\theta=\alpha /\beta$ e variância $Var(\theta)=\alpha /\beta^2.$ \begin{itemize} -\item O modelo de regressão com resposta binomial negativa pode ser especificado fazendo $\boldsymbol{\mu}=E(y|\boldsymbol{x})=exp(\boldsymbol{x'\beta}).$ +\item O modelo de regressão com resposta binomial negativa pode ser especificado fazendo $E(y|\boldsymbol{x})=exp(\boldsymbol{x'\beta}).$ \vspace{0.5cm} diff --git a/inst/slides/images/graf2.pdf b/inst/slides/images/Graf_BN.pdf similarity index 69% rename from inst/slides/images/graf2.pdf rename to inst/slides/images/Graf_BN.pdf index 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zq4+cLXv;a5&&s#mkE_+H>)XfusQBY~ZSu_JNA0!$$NWC?<YxK)ia$mMqSmJWV!r&j zync54iv2hKUwW7NzU0Q2zu}VW)8eM@39K)Ep!+)a|Lz&(!p1w_Z+x$9Kcnu&Pl+!d z-!1R)pI^uPz2)<7j8?ZLd{mMPI*jR>mz<xHnU}7h4{Ej|c;T6Oj(NqIFaa)om&~-Z z)S}cp(BNdWf{CeqZf0I-u|h0puo2p(2e+wVtz5_`DY(V0A6%ML0v<F42^uQs`z7Y4 zDu7zyU`rutJW_MYQcE(E6S?$ZmMVap3vQUhEC99CNwB~S;wy*+!Iim5`8i0=ttd)O z<1$b%G~zOV0|hfvQ)5$wGzGYrk)Z-uRv`~AW@u)NE@otEVu2xMW{jcN$lTNfL!G&W zIl7pEk)aW$m?egosfh)qn5nrLx?WQ=b0jyH6eVWnq!w|3dLzM^RjCSwMhg0#d1?6y mpa2e1&=1PbFHr!Ed4og0xFoTt1RPqX=B9>Rs;aL3Zd?GqzyG%Y literal 0 HcmV?d00001 diff --git a/vignettes/Ovelhas.Rmd b/vignettes/Ovelhas.Rmd index 19ed0ba..dc2d6bf 100644 --- a/vignettes/Ovelhas.Rmd +++ b/vignettes/Ovelhas.Rmd @@ -14,16 +14,16 @@ vignette: > source("_setup.R") ``` -Dados referentes a um experimento delineado em blocos, com o objetivo de +Dados referentes a um experimento realizado com o objetivo de investigar o efeito de uma intervenção, por parte do cuidador, no comportamento de ovelhas. -Para isso, foram considradas ovelhas de duas +Para isso, foram consideradas ovelhas de duas linhagens distintas (pouco ou muito reativas), submetidas a dois tipos diferentes de intervenção (observação ou observação + intervenção). A variável resposta aqui considerada é o número de mudanças na postura -corporal do animal ao longo do período de observação. +corporal do animal ao longo do período de observação (3 minutos). ```{r, echo = FALSE, include=FALSE} @@ -59,6 +59,7 @@ require('car') require('RColorBrewer') require('sandwich') require('hnp') +require('knitr') ``` @@ -103,8 +104,12 @@ mdp ajusteps <- glm(Nposturas ~ tratamento + linhagem, data=datapost4, family=poisson) summary(ajusteps) +### Avaliando possível efeito de interação. +ajustepsint <- glm(Nposturas ~ tratamento * linhagem, data=datapost4, family=poisson) +anova(ajusteps,ajustepsint, test = 'Chisq') + exp(coef(ajusteps)[2]) -### Estima-se que a frequência média de variação de postura no grupo com intervenção seja aproximadamente +### Estima-se que a taxa de variação de postura no grupo com intervenção seja aproximadamente ### a metade em relação ao grupo sem intervenção. ##### Estimação do parâmetro de dispersão. @@ -164,11 +169,11 @@ envelope(ajusteps) ### Modelo quasi poisson (V(mu) = mu). ajuste12 <- glm(r4 ~ tratamento+linhagem, data=datapost4, family = 'quasipoisson') -summary(ajuste12) +# summary(ajuste12) ### Forma alternativa de declarar o Modelo quase-poisson (V(mu) = mu). ajuste13 <- glm(r4 ~ tratamento+linhagem, data=datapost4, family = quasi(variance = 'mu', link='log')) -summary(ajuste13) +# summary(ajuste13) ### Modelo de quase-verossimilhança (V(mu) = mu^2). ajuste14 <- glm(r4 ~ tratamento+linhagem, data=datapost4, family = quasi(variance = 'mu^2', link='log')) @@ -192,12 +197,11 @@ qqline(resid(ajuste14,type='deviance')) ```{r} estrb <- coeftest(ajusteps, vcov=sandwich) ### Estimador sanduíche -estrb +estrb ### Estimação robusta. ### Usando bootstrap (R=999 simulações) ajusteboot <- Boot(ajusteps) -plot(ajusteboot, index = 2) ### Distribuição bootstrap para o efeito de intervenção. -summary(ajusteboot) +summary(ajusteboot) ### Resultados obtidos via bootstrap. ``` @@ -218,24 +222,43 @@ quadres <- cbind(erroz, ics) rownames(quadres) <- c('Poisson', 'Quasi(mu)', 'Quasi(mu^2)', 'Robusto (sanduiche)', 'Bootstrap') -### Quadro resumo para as estimativas produzidas pelos quatro modelos. +### Quadro resumo para as estimativas produzidas pelos quatro modelos para o efeito de intervenção. kable(quadres, format = "markdown", caption = "Comparativo dos modelos ajustados") -### Vamos avaliar o efeito das observações com maiores resíduos nos achados do modelo +``` + +### Verificando efeito das observações com maiores resíduos na análise. + +```{r} dadosexclud <- datapost4[-c(8,18,28),] -ajusteexclud <- glm(Nposturas ~ tratamento+linhagem, data=dadosexclud, family = quasi(variance = 'mu', link='log')) + +### Ajustando o modelo Poisson sem as três observações. +ajusteexcludpoi <- glm(Nposturas ~ tratamento+linhagem, data=dadosexclud, family = poisson) + +### Estimativa do parâmetro de dispersão. +sum(resid(ajusteexcludpoi, type = 'pearson')**2)/ajusteexcludpoi$df.residual + +### Gráficos de diagnóstico. +par(mfrow=c(2,2)) +plot(ajusteexcludpoi) + +### Agora, o modelo quase-poisson +ajusteexcludquasi <- glm(Nposturas ~ tratamento+linhagem, data=dadosexclud, family = quasi(variance = 'mu', link='log')) ### Estimativas produzidas pelo modelo quasipoisson com e sem as três observações. -c1 <- compareCoefs(ajuste13, ajusteexclud, print = FALSE) -colnames(c1) <- c('Est. com outliers','E.P. com outliers','Est. sem outliers','E.P. sem outliers') -kable(c1) +c1 <- compareCoefs(ajusteps, ajuste13, ajusteexcludpoi, ajusteexcludquasi, print = FALSE) +colnames(c1) <- c('Est. Poisson c/out','E.P. Poisson c/out', + 'Est. Quasi c/out','E.P. Quasi c/out', + 'Est. Poisson s/out','E.P. Poisson s/out', + 'Est. Quasi s/out','E.P. Quasi s/out') +kable(round(c1,3), align = 'c') ### Efeito da intervenção desconsiderando as três observações. -exp(coef(ajusteexclud)[2]) +exp(coef(ajusteexcludpoi)[2]) ### O efeito de intervenção aumenta, e torna-se mais significativo, mediante exclusão dos outliers. -```` +``` diff --git a/vignettes/Sinistros.Rmd b/vignettes/Sinistros.Rmd index cb5d195..c112b89 100644 --- a/vignettes/Sinistros.Rmd +++ b/vignettes/Sinistros.Rmd @@ -32,11 +32,10 @@ seguintes variáveis: ```{r, echo = FALSE, include=FALSE} require(lattice) -require(hnp) require(MASS) require(effects) +require(knitr) -setwd("~/Desktop") dados <- read.csv('Baseauto2.csv', header=TRUE, sep=',', dec = ',') options(scipen = 3) ### Era zero. @@ -69,29 +68,29 @@ taxageral <- sum(dados$Sinistros)/sum(dados$Exposicao); taxageral ### Taxa de si tab <- aggregate(cbind(Exposicao, Sinistros) ~ Sexo, data = dados, sum) taxa <- with(tab, Sinistros/Exposicao) -tab <- cbind(tab, taxa); tab ### Distribuição do número de sinistros por sexo. +tab <- cbind(tab, taxa) +### Distribuição do número de sinistros por sexo. +kable(tab, align = 'c', caption = '**Taxa de sinistros segundo Sexo**') dados$idadecat <- cut(dados$Idade, breaks=c(18,30,60, 92), include.lowest = T) tab <- aggregate(cbind(Exposicao, Sinistros) ~ idadecat, data = dados, sum) taxa <- with(tab, Sinistros/Exposicao) -tab <- cbind(tab, taxa); tab ### Distribuição do número de sinistros por sexo. +tab <- cbind(tab, taxa) +### Distribuição do número de sinistros por sexo. +kable(tab, align = 'c', caption = '**Taxa de sinistros segundo Idade**') tabidsex <- aggregate(cbind(Exposicao, Sinistros) ~ Sexo + idadecat, data = dados, sum) -Taxaidsex <- with(tabidsex, Sinistros/Exposicao) -tabidsex <- cbind(tabidsex, Taxaidsex); tabidsex ### Distribuição do número de sinistros por idade e sexo. +taxa <- with(tabidsex, Sinistros/Exposicao) +tabidsex <- cbind(tabidsex, taxa) +### Distribuição do número de sinistros por idade e sexo. +kable(tabidsex, align = 'c', caption = '**Taxa de sinistros segundo Sexo e Idade**') tab <- aggregate(cbind(Exposicao, Sinistros) ~ Tipo, data = dados, sum) taxa <- with(tab, Sinistros/Exposicao) -tab <- cbind(tab, taxa); tab ### Distribuição do número de sinistros por tipo de veículo. +tab <- cbind(tab, taxa) +### Distribuição do número de sinistros por tipo de veículo. +kable(tab, align = 'c', caption = '**Taxa de sinistros segundo Tipo de Veículo**') -tab <- aggregate(cbind(Exposicao, Sinistros) ~ Civil, data = dados, sum) -taxa <- with(tab, Sinistros/Exposicao) -tab <- cbind(tab, taxa); tab ### Distribuição do número de sinistros por estado civil. - -dados$valorcat <- cut(dados$Valor, breaks=quantile(dados$Valor), include.lowest = T) -tab <- aggregate(cbind(Exposicao, Sinistros) ~ valorcat, data = dados, sum) -taxa <- with(tab, Sinistros/Exposicao) -tab <- cbind(tab, taxa); tab ### Distribuição do número de sinistros por valor do veículo. ``` ## Regressão usando o modelo log-linear poisson. @@ -101,24 +100,19 @@ dados <- na.omit(dados) ajusteps <- glm(Sinistros ~ Tipo+Sexo+Idade+I(Idade**2)+Valor+Civil+offset(log(Exposicao)), data = dados, family=poisson) summary(ajusteps) -##### Estimação do parâmetro de dispersão. +### Modelo sem as variáveis não significativas. + +ajusteps2 <- glm(Sinistros ~ Sexo+Idade+I(Idade**2)+Valor+offset(log(Exposicao)), data = dados, family=poisson) +anova(ajusteps2,ajusteps,test='Chisq') -exp(coef(ajusteps)) +### Estimação do parâmetro de dispersão. -X2 <- sum(resid(ajusteps,type='pearson')**2) -phichap <- X2/ajusteps$df.residual +X2 <- sum(resid(ajusteps2,type='pearson')**2) +phichap <- X2/ajusteps2$df.residual phichap ### Indicador de superdispersão. ``` -### Diagnóstico do ajuste (gráficos). - -```{r} -##### Diagnóstico do modelo - gráficos. -par(mfrow=c(2,2)) -plot(ajusteps) - - -par(mfrow=c(1,1)) +```{r, echo = FALSE} envelope=function(modelo){ dados=na.omit(modelo$data) nsim=100 @@ -145,35 +139,36 @@ envelope=function(modelo){ lines(quantis,ls,type='l') points(quantis,r1,pch=16,cex=0.75) } +``` -envelope(ajusteps) +### Diagnóstico do ajuste (gráficos). + +```{r} +##### Diagnóstico do modelo - gráficos. +par(mfrow=c(2,2)) +plot(ajusteps2) + +par(mfrow=c(1,1)) +envelope(ajusteps2) ``` -### Re-ajuste do modelo associando um parâmetro ao termo offset (log-exposicao) +### Ajuste do modelo associando um parâmetro ao termo offset (log-exposicao) ```{r} -ajusteps2 <- glm(Sinistros ~ Tipo + Sexo + Idade +I(Idade**2) + Valor + Civil+log(Exposicao), data = dados, family=poisson) -summary(ajusteps2) -anova(ajusteps, ajusteps2, test = 'Chisq') +ajusteps3 <- glm(Sinistros ~ Sexo + Idade +I(Idade**2) + Valor + log(Exposicao), data = dados, family=poisson) +summary(ajusteps3) +anova(ajusteps2, ajusteps3, test = 'Chisq') ``` ## Regressão usando a distribuição binomial negativa. ```{r} -ajustenb2 <- glm.nb(Sinistros ~ Tipo+Sexo+Idade+I(Idade**2)+Valor+Civil+log(Exposicao),data= dados) +ajustenb2 <- glm.nb(Sinistros ~ Sexo+Idade+I(Idade**2)+Valor+log(Exposicao),data= dados) summary(ajustenb2) -### Verificando possibilidade de excluir estado civil e tipo de veículo do modelo. -ajustenb3 <- update(ajustenb2, ~.-(Civil+Tipo)) -anova(ajustenb2, ajustenb3) - -## Estimação do parâmetro de dispersão -X2 <- sum(resid(ajustenb3,type='pearson')**2) -phichap <- X2/ajustenb3$df.residual -phichap ``` @@ -182,14 +177,14 @@ phichap ```{r} ##### Diagnóstico do modelo - gráficos. par(mfrow=c(2,2)) -plot(ajustenb3) +plot(ajustenb2) ``` ```{r, echo = FALSE} dadosnb3 <- dados[,c('Sexo','Idade','Valor','Exposicao','Sinistros')] - -par(mfrow=c(1,1)) +dadosnb3$lexpo <- log(dados$Exposicao) +ajustenb2 <- glm.nb(Sinistros ~ Sexo+Idade+I(Idade**2)+Valor+lexpo,data= dadosnb3) envelope=function(modelo){ dados=na.omit(dadosnb3) nsim=100 @@ -221,32 +216,31 @@ envelope=function(modelo){ ```{r} par(mfrow=c(1,1)) -envelope(ajustenb3) +envelope(ajustenb2) ``` - ### Explorando os efeitos das covariáveis. Estimativas pontuais e ICs (95%) ```{r} -intervalos <- confint(ajustenb3) -estimat <- cbind(ajustenb3$coefficients, intervalos) +intervalos <- confint(ajustenb2) +estimat <- cbind(ajustenb2$coefficients, intervalos) colnames(estimat)[1] <- 'Estimativa pontual' ### Quadro de estimativas -round(estimat, 5) +kable(round(estimat, 5), align = 'c', caption = '**Estimativas pontuais e intervalos de confiança - Modelo Binomial Negativo**') ``` ### Gráficos de efeitos -```{r} +```{r, echo = FALSE} -dados$lexpo <- log(dados$Exposicao) -ajustenb3 <- glm.nb(Sinistros ~ Sexo+Idade+I(Idade**2)+Valor+lexpo,data= dados) +``` -efeitos <- allEffects(ajustenb3, given.values=c(lexpo=0)) +```{r} +efeitos <- allEffects(ajustenb2, given.values=c(lexpo=0)) trellis.par.set(list(axis.text = list(cex = 1.2))) plot(efeitos[[2]], type='response',main=list( @@ -284,7 +278,7 @@ plot(efeitos[[4]], type='response',main=list( -```{r, echo = FALSE} +```{r, echo = FALSE, include=F} ## Frequências ajustadas pelas duas distribuições, com e sem covariaveis. @@ -312,18 +306,18 @@ ajustenb <- glm.nb(Sinistros ~ 1,data=dados) media <- exp(coefficients(ajustenb)) shape <- ajustenb$theta -freqbn <- round(n*dnbinom(0:10, mu = media, size = shape)); freqbn +freqbn <- round(n*dnbinom(0:10, mu = media, size = shape)) ### Binomial negativa com covariaveis -pred2 <- predict(ajustenb3,type='response') +pred2 <- predict(ajustenb2,type='response') intervalo <- 0:10 matprob <- matrix(0,nrow=nrow(dados),ncol=length(intervalo)) probnb <- function(interv, media, shape) dnbinom(intervalo, mu = media, size = shape) for(i in 1:nrow(dados)) - matprob[i,] <- probnb(interv = intervalo, media = pred2[i], shape = ajustenb3$theta) + matprob[i,] <- probnb(interv = intervalo, media = pred2[i], shape = ajustenb2$theta) pbarra <- colMeans(matprob) frebnaj <- round(n*pbarra) ams <- c(table(dados$Sinistros), rep(0,5)) @@ -337,7 +331,7 @@ rownames(matfreq) <- c('Amostra', 'Poisson não ajustada por covariáveis', 'Poi ## Comparação dos ajustes ```{r, results = 'markup'} -kable(matfreq, format = "markdown", caption = "Frequências amostrais e freuências ajustadas para o número de sinistros") +kable(matfreq, format = "markdown", caption = "Frequências amostrais e frequências ajustadas para o número de sinistros") ``` -- GitLab