From 74159442a0888affd9dd3a0009f35640d0cc21ba Mon Sep 17 00:00:00 2001 From: Eduardo Junior <edujrrib@gmail.com> Date: Tue, 7 Jun 2016 18:31:06 -0300 Subject: [PATCH] =?UTF-8?q?Corre=C3=A7=C3=B5es=20pontuais=20de=20express?= =?UTF-8?q?=C3=B5es=20matem=C3=A1ticas=20e=20retira=20espa=C3=A7os=20no=20?= =?UTF-8?q?final=20das=20linhas?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- inst/slides/modelo_misto.Rnw | 2 +- inst/slides/pois_bineg.Rnw | 232 +++++++++++++++++------------------ inst/slides/slides-mrdcr.pdf | Bin 487306 -> 491948 bytes 3 files changed, 117 insertions(+), 117 deletions(-) diff --git a/inst/slides/modelo_misto.Rnw b/inst/slides/modelo_misto.Rnw index 76a2b44..889d98d 100644 --- a/inst/slides/modelo_misto.Rnw +++ b/inst/slides/modelo_misto.Rnw @@ -67,7 +67,7 @@ set_parent("slides-mrdcr.Rnw") \item Pode ser em muitas dimensões ($q > 1$,intercepto, inclinação, etc) \item Efeitos podem ser múltiplos, aninhados ou cruzados - \item Para modelar subdispersão com efeito ao nÃvel de observação, + \item Para modelar superdispersão com efeito ao nÃvel de observação, tem-se que $b_{i}$ é $b_{ij}$, ou seja, na mesma dimensão dos dos dados. \end{itemize} diff --git a/inst/slides/pois_bineg.Rnw b/inst/slides/pois_bineg.Rnw index 12a73c8..8833c00 100644 --- a/inst/slides/pois_bineg.Rnw +++ b/inst/slides/pois_bineg.Rnw @@ -30,12 +30,12 @@ Alguns exemplos de problemas envolvendo contagens: \begin{itemize} \item Modelos probabilÃsticos para variáveis aleatórias discretas, com suporte no conjunto de números inteiros não-negativos, são potenciais candidatos para a análise de dados de contagens. - + \vspace{0.5cm} - + \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} \end{frame} @@ -66,22 +66,22 @@ aquém, em quantidade e diversidade, em relação ao verificado para dados cont %%% Slide 5 \begin{frame}{Regressão com erros normais na análise de dados de contagens} - + \vspace{0,5cm} - + \begin{itemize} \item O modelo linear com erros normais não considera a natureza discreta dos dados; - + \vspace{0,5cm} - + \item Associa probabilidade nula a qualquer possÃvel contagem; - + \vspace{0,5cm} - + \item Admite probabilidades não nulas a valores negativos da variável; - + \end{itemize} - + \end{frame} %%%----------------------------------------------------------------------------------------- @@ -90,25 +90,25 @@ aquém, em quantidade e diversidade, em relação ao verificado para dados cont \begin{frame}{Regressão com erros normais na análise de dados de contagens} - - + + \vspace{0,2cm} - + \begin{itemize} - + \item O uso de transformações dificulta a interpretação dos resultados; - + \vspace{0,5cm} - + \item O uso da transformação logarÃtmica apresenta problemas para contagens iguais a zero; - + \vspace{0,5cm} - + \item Não se contempla a relação não constante entre variância e média, caracterÃstica de dados de contagens. - - + + \end{itemize} - + \end{frame} %%%----------------------------------------------------------------------------------------- @@ -118,26 +118,26 @@ aquém, em quantidade e diversidade, em relação ao verificado para dados cont \label{sec-poisson} \begin{frame}{A distribuição de Poisson} - + \begin{itemize} \item A distribuição de Poisson é a principal referência para a análise de dados de contagens. - \vspace{0,3cm} + \vspace{0,3cm} \item Função de probabilidades: - + $$ P\left ( Y=k \right )=\frac{e^{-\lambda}\lambda^{k}}{k!},\hspace{5mm}k=0,1,2,...; \lambda>0. $$ - - \vspace{0,2cm} - - \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: - + + \vspace{0,2cm} + + \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: + $$ P\left ( Y_t=k \right )=\frac{e^{-\lambda t}(\lambda t)^{k}}{k!},\hspace{5mm}k=0,1,2,.... $$ \end{itemize} - + \end{frame} %%%----------------------------------------------------------------------------------------- @@ -157,18 +157,18 @@ Dentre as principais propriedades da distribuição de Poisson, tem-se: \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} @@ -177,13 +177,13 @@ Dentre as principais propriedades da distribuição de Poisson, tem-se: %%% 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} @@ -193,11 +193,11 @@ Dentre as principais propriedades da distribuição de Poisson, tem-se: %%% Slide 8 \begin{frame}{Motivações para a distribuição de Poisson} - + \vspace{0,5cm} - + \begin{itemize} \item Caso limite da distribuição binomial$(n, \pi)$ quando $n\rightarrow \infty$ e $\pi\rightarrow 0$, fixado $\lambda=n\pi$, ou seja: @@ -206,7 +206,7 @@ Dentre as principais propriedades da distribuição de Poisson, tem-se: $$ \underset{n \to \infty \pi \to 0 }{lim} \left [ \begin{pmatrix} - n\\k + n\\k \end{pmatrix} \left ( \frac{\lambda}{n} \right )^{k}\left ( 1-\frac{\lambda}{n} \right )^{n-k}\right ]=\frac{e^{-\lambda}\lambda^{k}}{k!}. $$ @@ -215,7 +215,7 @@ $$ \item Resultado do processo estocástico de Poisson, em que os eventos contados ocorrem \textbf{aleatoriamente} ao longo do tempo, espaço,... \end{itemize} - + \end{frame} %%%----------------------------------------------------------------------------------------- @@ -225,32 +225,32 @@ $$ \begin{frame}{Motivações para a distribuição de Poisson} \begin{itemize} - + \item Se o tempo decorrido entre sucessivos eventos é uma variável aleatória com distribuição exponencial de média $\mu=1/\lambda$, então o número de eventos ocorridos em um intervalo t de tempo tem distribuição de Poisson com média $\lambda t$. - + \vspace{0,3cm} - -\begin{itemize} - + +\begin{itemize} + \item A dualidade entre as distribuições Poisson e exponencial implica que a taxa de ocorrência do evento, definida por: - - + + \end{itemize} - + $$ \lambda (t) =\lim_{\Delta t\rightarrow 0}\frac{P\left \{ \text{evento ocorrer em} \left ( t,t+\Delta t \right ) \right \}}{\Delta t}, -$$ +$$ \vspace{0,3cm} - + dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo $t>0$. - + \end{itemize} - -\end{frame} + +\end{frame} %%%----------------------------------------------------------------------------------------- @@ -265,7 +265,7 @@ dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo \caption{Diferentes comportamentos para $\lambda (t)$} %% \label{Fig1} \centering - + \end{figure} \end{frame} @@ -275,21 +275,21 @@ dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo %%% Slide 10 \begin{frame}{O processo de Poisson} - + O Processo de Poisson configura um processo de contagem em que $Y(t),t\geqslant 0$, representa o número de eventos que ocorrem até $t$, satisfazendo: - + \vspace{0,5cm} - + \begin{enumerate} \item $Y(t)$ é inteiro e não negativo; \item Para $s<t$, $Y(s)\leq Y(t)$; \item $Y(t)-Y(s)$ é o número de eventos que ocorrem no intervalo $(s,t]$; \item O processo é estacionário: - + $$ Y(t_{2}+s)-Y(t_{1}+s) \overset{i.d. }{\sim}Y(t_{2})-Y(t_{1}), \forall s>0 $$ - + \item O processo tem incrementos independentes, ou seja, os números de eventos verificados em intervalos disjuntos são independentes. \end{enumerate} @@ -301,13 +301,13 @@ dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo %%% Slide 14 \begin{frame}{Diferentes padrões em processos de contagens} - + \begin{figure}[h] \includegraphics[scale=0.6]{images/processos14.pdf} \caption{Ilustração de diferentes tipos de processos pontuais} \label{Fig2} \centering - + \end{figure} \end{frame} @@ -315,19 +315,19 @@ dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo %%% Slide 15 -\begin{frame}{Regressão Poisson} +\begin{frame}{Regressão Poisson} \begin{itemize} \item O modelo de regressão Poisson (ou modelo log linear de Poisson) é o mais usado para a análise de dados de contagens. - + \vspace{0,5cm} - + \item A regressão Poisson baseia-se nos pressupostos inerentes ao processo e à distribuição de Poisson. - + \vspace{0,5cm} - + \item Caso tais pressupostos não sejam atendidos, a regressão Poisson ainda pode ser uma alternativa apropriada, desde que usada com os cuidados necessários. - + \end{itemize} \end{frame} @@ -336,7 +336,7 @@ dado que o evento não ocorreu até o tempo $t$, \textbf{é constante} para todo %%% Slide 16 -\begin{frame}{Regressão Poisson - Especificação do modelo} +\begin{frame}{Regressão Poisson - Especificação do modelo} \begin{itemize} \item Sejam $Y_{1},Y_{2},...,Y_{n}$ variáveis aleatórias condicionalmente independentes, dado o vetor de covariáveis ${\boldsymbol{x_{i}}}'=(x_{i1},x_{i2},...,x_{ip}), i=1,2,...,n$. A regressão Poisson é definida pela distribuição de Poisson: @@ -347,7 +347,7 @@ $$ $$ \vspace{0,2cm} - + sendo as covariáveis inseridas ao modelo por: $$ @@ -355,9 +355,9 @@ $$ $$ \vspace{0,2cm} - + em que ${\boldsymbol{\beta }}$ é o vetor de parâmetros de regressão. - + \end{itemize} @@ -368,29 +368,29 @@ em que ${\boldsymbol{\beta }}$ é o vetor de parâmetros de regressão. %%% Slide 17 -\begin{frame}{Regressão Poisson - Propriedades} +\begin{frame}{Regressão Poisson - Propriedades} -\vspace{0,8cm} - +\vspace{0,8cm} + \begin{itemize} - - \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} + \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} + + + \item $E\left [ y_{i}|\boldsymbol{x_{i}} \right ]= \mu_{i}=\exp\left ( \boldsymbol{x'_{i}\beta} \right );$ + +\vspace{0,8cm} + + \item $Var\left [ y_{i}|\mathbf{x_{i}} \right ]= \mu_{i}=\exp\left ( \boldsymbol{x'_{i}\beta} \right ).$ + - - \item $E\left [ y_{i}|\boldsymbol{x_{i}} \right ]= \mu_{i}=exp\left ( \boldsymbol{x'_{i}\beta} \right );$ -\vspace{0,8cm} - - \item $Var\left [ y_{i}|\mathbf{x_{i}} \right ]= \mu_{i}=exp\left ( \boldsymbol{x'_{i}\beta} \right ).$ - - - \end{itemize} \end{frame} @@ -403,32 +403,32 @@ em que ${\boldsymbol{\beta }}$ é o vetor de parâmetros de regressão. %%% Slide 18 -\begin{frame}{Regressão Poisson - Estimação por máxima verossimilhança} +\begin{frame}{Regressão Poisson - Estimação por máxima verossimilhança} Para a regressão Poisson: \vspace{0.5cm} -\begin{itemize} +\begin{itemize} -\item Log-verossimilhança: $l(\boldsymbol{\beta})=\sum_{i=1}^{n} \{ y_{i}\boldsymbol{x_{i}'\beta}-\exp{(\boldsymbol{x_{i}'\beta}}\}-\ln{y_{i}!});$ +\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 Vetir escore: $\boldsymbol{S}(\boldsymbol{\beta})=\frac{\partial l(\boldsymbol{\beta};\boldsymbol{y})}{\partial \boldsymbol{\beta}}= - \sum_{i=1}^{n}(y_{i}-\exp(\boldsymbol{x_{i}'\beta}))\boldsymbol{x_{i}};$ + \sum_{i=1}^{n}(y_{i}-\exp(\boldsymbol{x_{i}'\beta}))\boldsymbol{x_{i}};$ \vspace{0.5cm} - + \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}}; +\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 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 ). -$ - +\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} \end{frame} @@ -439,7 +439,7 @@ $ %%% Slide 34 -\begin{frame}{Regressão Poisson - Modelo Linear Generalizado} +\begin{frame}{Regressão Poisson - Modelo Linear Generalizado} A Regressão Poisson é um caso particular dos Modelos Lineares Generalizados (MLG). Algumas propriedades dessa classe de modelos: @@ -473,7 +473,7 @@ A Regressão Poisson é um caso particular dos Modelos Lineares Generalizados (M \section{Estimação via Quase-Verossimilhança} \label{sec-quase-verossimilhanca} -\begin{frame}{Regressão Poisson - Quase-Verossimilhança} +\begin{frame}{Regressão Poisson - Quase-Verossimilhança} \begin{itemize} \item Para o ajuste de um modelo alternativo via Quase-Verossimilhança, definimos: @@ -505,7 +505,7 @@ $$ Q\left ( \mu \right )=\int _y^\mu \frac{y-t}{\phi V(t)}dt $$. %%% Slide 39 -\begin{frame}{Estimação via Quase-Verossimilhança} +\begin{frame}{Estimação via Quase-Verossimilhança} \begin{itemize} @@ -535,7 +535,7 @@ $$\ln(E(y_i|\boldsymbol{x_i}))=\boldsymbol{x'_i \beta};$$ %%% Slide45 -\begin{frame}{Estimação via Quase Verossimilhança} +\begin{frame}{Estimação via Quase Verossimilhança} \begin{itemize} @@ -555,7 +555,7 @@ com $\mu_{i}=exp({\boldsymbol{x'_{i}\beta}})$ e $\omega_{i}=Var(y_{i}|\boldsymbo \item Um estimador para $\phi$: -$$ \hat{\phi} = \frac{1}{n-p}\sum_{i=1}^n \frac{y_i-\hat{\mu}_i}{V(\hat{\mu}_i)}. $$ +$$ \hat{\phi} = \frac{1}{n-p}\sum_{i=1}^n \frac{(y_i-\hat{\mu}_i)^2}{V(\hat{\mu}_i)}. $$ \end{itemize} @@ -576,7 +576,7 @@ $$ \hat{\phi} = \frac{1}{n-p}\sum_{i=1}^n \frac{y_i-\hat{\mu}_i}{V(\hat{\mu}_i)} \section{Modelo Binomial Negativa} \label{sec-binomial-negativa} -\begin{frame}{Distribuição binomial negativa} +\begin{frame}{Distribuição binomial negativa} \begin{itemize} @@ -606,7 +606,7 @@ $$ %%% Slide 48 -\begin{frame}{Distribuição binomial negativa} +\begin{frame}{Distribuição binomial negativa} \begin{itemize} @@ -616,14 +616,14 @@ $$ $$ P(Y=k) = \left ( \begin{matrix} -r+k-1\\ +r+k-1\\ r-1 \end{matrix} \right ) (1-p)^rp^k, \hspace{0,5cm} k=0,1,2,..., $$ \vspace{0.5cm} -sendo $r=\alpha$ e $p=\lambda/(\lambda+\alpha),$ 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 $r=1$). @@ -642,7 +642,7 @@ sendo $r=\alpha$ e $p=\lambda/(\lambda+\alpha),$ com $0<p<1$ e $r>0$. %%% Slide 48 -\begin{frame}{Distribuição binomial negativa} +\begin{frame}{Distribuição binomial negativa} \begin{itemize} @@ -669,13 +669,13 @@ com $E(\theta)=\theta=\alpha /\beta$ e variância $Var(\theta)=\alpha /\beta^2.$ %%% Slide 50 \begin{frame}{Distribuição binomial negativa para diferentes parâmetros} - + \begin{figure}[h] \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 - + \end{figure} \end{frame} @@ -685,11 +685,11 @@ com $E(\theta)=\theta=\alpha /\beta$ e variância $Var(\theta)=\alpha /\beta^2.$ %%% Slide 51 -\begin{frame}{Distribuição binomial negativa} +\begin{frame}{Distribuição binomial negativa} \begin{itemize} -\item O modelo de regressão com resposta binomial negativa pode ser especificado fazendo $E(y|\boldsymbol{x})=exp(\boldsymbol{x'\beta}).$ +\item O modelo de regressão com resposta binomial negativa pode ser especificado fazendo $E(y|\boldsymbol{x})=\exp(\boldsymbol{x'\beta}).$ \vspace{0.5cm} @@ -708,4 +708,4 @@ com $E(\theta)=\theta=\alpha /\beta$ e variância $Var(\theta)=\alpha /\beta^2.$ {\it Vignette} \href{run:../doc/Sinistros.html}{\tt Sinistros.html} -\end{frame} \ No newline at end of file +\end{frame} diff --git a/inst/slides/slides-mrdcr.pdf b/inst/slides/slides-mrdcr.pdf index 50ae15d3033807611334a24f3999be7e5155c561..e69c4ea36af37033ac0c61176d2b57a9a023a378 100644 GIT binary patch delta 37318 zcmYhgb95(Nu)iBm>`ZK%6Wg|piEZZ_+n(6AZQHhOV{$X^Ip?=-|I@v?S9Nt&?Y-;S zpAO%~tq8!2B~7sBql%Nm1lUj&B;&zu$>$FQ%jATVeRb9&a52X43wny1m~?X8U$VuJ zo<JixH3;E}L&y7f1flr{9;ib{rzWP(!D<AMab@G<uBjVN7R+~QH`5Ej2Rmy8q<yhf zs~3$QmK@U?_L!>&XK&ZC+*e-DO7!dkuCz@L&bu}QK{7p`r-j2Q0kbC#M~<WF;Yo(# zMj#P7WZ5yZ^r6c+XExY#v7=&3$a*lxDirT{%_4V8W!^<9$+YuZGDew1X$lpE7W8E7 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