From 6802f46df6abb89df8744197484af7a86a0adc1d Mon Sep 17 00:00:00 2001 From: Eduardo Junior <edujrrib@gmail.com> Date: Wed, 15 Jun 2016 16:50:43 -0300 Subject: [PATCH] =?UTF-8?q?Troca=20par=C3=A2metros=20na=20revis=C3=A3o=20t?= =?UTF-8?q?e=C3=B3rica=20para=20manter=20padr=C3=A3o=20adotado=20no=20cap?= =?UTF-8?q?=C3=ADtulo=20de=20resultados?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/01-tcc.pdf | Bin 1242779 -> 1242960 bytes docs/cap02_revisao-de-literatura.Rnw | 110 ++++++++++++++------------- 2 files changed, 56 insertions(+), 54 deletions(-) diff --git a/docs/01-tcc.pdf b/docs/01-tcc.pdf index dc5f079139ea8545515850d502cfbb43624399bb..13dda9e0a1e188e4fa37662c0dc15a459ae97275 100644 GIT binary patch delta 52381 zcmYhgV{qV2v^*SpV`F36wr!i6WMlgmTbpcb+qTV(ZQI8Gx%a*w-m0#isXA5jVX99} zpXrQC+1V>u|6tHyuwd{;h!eY%ybTx{U{hN*Zk+?U`&jctMF?t%CI{73J{^-jor4Oq z%f)*~M@KV+jK>Q*>f;U3uvvV}ZP_ZQ5jqG}`j(k>dy>->-If&H_ZhD3>G89oriYx2 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\ref{eqn:quase-verossimilhanca} que a função de +Na expressão \ref{eqn:quase-verossimilhanca} a função de quase-verossimilhança é definida a partir da especificação de $\mu_i$, -$V(\mu_i)$ e $\phi$. O processo de estimação via maximização dessa +$V(\mu_i)$ e $\sigma^2$. O processo de estimação via maximização dessa função compartilha as mesmas estimativas para $\mu_i$, porém a dispersão -de $y_i$, $V(y_i) = \phi V(\mu_i)$ é corrigida pelo parâmetro adicional -$\phi$. +de $y_i$, $V(y_i) = \theta V(\mu_i)$ é corrigida pelo parâmetro adicional +$\sigma^2$. Assim os problemas com a fuga da suposição de equidispersão podem ser superados quando a estimação por máxima quase-verossimilhança é @@ -215,23 +215,25 @@ adotado. Porém um resultado dessa abordagem é que \end{equation} \noindent -ou seja a informação a respeito de $\mu$ quando se conhece apenas $\phi$ +ou seja a informação a respeito de $\mu$ quando se conhece apenas $\sigma^2$ e $V(\mu)$, a relação entre média e variância, é menor do que a informação quando se conhece a distribuição da variável resposta, dada pela log-verossimilhança $\ell(\mu \mid y)$. Além disso ressalta-se que, de forma geral, não se recupera a distribuição de $Y$ somente com as -especificações de $\phi$ e $V(\mu)$. +especificações de $\sigma^2$ e $V(\mu)$. Em modelos de regressão, definimos $g(\mu_i) = X\beta$ e $V(\mu_i)$ que definem a função de quase-verossimilhança. Nessa abordagem são estimados -os parâmetros $\beta$ e $\phi$. A estimativa do vetor $\beta$ pode ser -obtidas pelo algoritmo \textit{IWLS}, usando as funções quase-escore e -matriz de quase-informação. Para o parâmetro $\phi$ um estimador usual é -o baseado na estatÃstica $\chi^2$ de Pearson. +os parâmetros $\beta$ e $\sigma^2$. A estimativa do vetor $\beta$ pode +ser obtidas pelo algoritmo \textit{IWLS}. Usando as funções quase-escore +e matriz de quase-informação chega-se ao mesmo algoritmo de estimação +dado no caso Poisson, que não depende de $\sigma^2$. O parâmetro +$\sigma^2$ é estimado separadamente, pós estimação dos $\beta$'s. Um +estimador usual é o baseado na estatÃstica $\chi^2$ de Pearson. \begin{equation} - \label{eqn:estimador-phi} - \hat{\phi} = \frac{1}{n-p} \sum_{i=1}^n + \label{eqn:estimador-theta} + \hat{\sigma^2} = \frac{1}{n-p} \sum_{i=1}^n \frac{(y_i - \hat{\mu_i})^2}{V(\hat{\mu_i})} \end{equation} @@ -248,14 +250,14 @@ que \label{eqn:proc-binomneg} \begin{split} Y \mid & b \sim Poisson(b) \\ - & b \sim Gama(\mu, \phi) + & b \sim Gama(\mu, \theta) \end{split} \end{equation} \noindent A função massa de probabilidade decorrente da estrutura descrita em -\ref{eqn:proc-binomneg} é deduzida integrando os efeitos aleatórios, -considere $f(y \mid b)$ como a função massa de probablidade da +\ref{eqn:proc-binomneg} é deduzida integrando os efeitos aleatórios. +Considere $f(y \mid b)$ como a função massa de probablidade da distribuição Poisson (vide expressão em \ref{eqn:pmf-poisson}) e $g(b \mid \mu, \phi)$ a função densidade da distribuição Gama \footnote{O desenvolvimento detalhado da integral pode ser visto em @@ -266,57 +268,57 @@ distribuição Poisson (vide expressão em \ref{eqn:pmf-poisson}) e $g(b \begin{equation} \label{eqn:proc-binomneg} \begin{split} - Pr(Y = y \mid \mu,\phi) &= \int_0^\infty f(y \mid b) - g(b \mid \mu,\phi) db\\ - &= \frac{\phi^\phi}{y!\mu^\phi\Gamma(\phi)} - \int_0^\infty e^{-b(1 + \phi/\mu)} b^{y+\phi-1}db \\ - &= \frac{\Gamma(\phi + y)}{\Gamma(y+1)\Gamma(\phi)} - \left ( \frac{\mu}{\mu + \phi} \right )^y - \left ( \frac{\phi}{\mu + \phi} \right )^\phi + Pr(Y = y \mid \mu,\theta) &= \int_0^\infty f(y \mid b) + g(b \mid \mu,\theta) db\\ + &= \frac{\theta^\theta}{y!\mu^\theta\Gamma(\theta)} + \int_0^\infty e^{-b(1 + \theta/\mu)} b^{y+\theta-1}db \\ + &= \frac{\Gamma(\theta + y)}{\Gamma(y+1)\Gamma(\theta)} + \left ( \frac{\mu}{\mu + \theta} \right )^y + \left ( \frac{\theta}{\mu + \theta} \right )^\theta \qquad y = 0, 1, 2, \cdots \end{split} \end{equation} \noindent -com $\mu >0$ e $\phi > 0$. Ressaltamos que esse é um caso particular de +com $\mu >0$ e $\theta > 0$. Ressaltamos que esse é um caso particular de um modelo de efeito aleatório cuja a integral tem solução analÃtica e por consequência o modelo marginal tem forma fechada. Outro caso que se baseia no mesmo princÃpio é o modelo \textit{Inverse Gaussian Poisson}, que como o nome sugere adota a distribuição Inversa Gaussiana para os efeitos aleatórios. Na figura \ref{fig:distr-binomneg} são apresentadas -as distribuições Binomial Negativa para diferentes parâmetros $\phi$ em +as distribuições Binomial Negativa para diferentes parâmetros $\theta$ em comparação com a distribuição Poisson equivalente em locação. Note que -quanto menor o parâmetro $\phi$, maior a dispersão da distribuição. Isso -introduz uma propriedade importante desse modelo, para $\phi \rightarrow +quanto menor o parâmetro $\theta$, maior a dispersão da distribuição. Isso +introduz uma propriedade importante desse modelo, para $\theta \rightarrow \infty$ a distribuição reduz-se a Poisson. -<<distr-binomneg, fig.cap="Probabilidades pela distribuição Binomial Negativa para diferentes valores de $\\phi$ com $\\mu = 5$", fig.height=3.5, fig.width=7>>= +<<distr-binomneg, fig.cap="Probabilidades pela distribuição Binomial Negativa para diferentes nÃveis de dispersão, fixando a média em 5.", fig.height=3.5, fig.width=7>>= ##------------------------------------------- ## Parametros da distribuição mu <- 5 -phis <- c("p1" = 1, "p2" = 5, "p3" = 30) -vars <- mu + (1/phis) * mu^2 +thetas <- c("p1" = 1, "p2" = 5, "p3" = 30) +vars <- mu + (1/thetas) * mu^2 ##------------------------------------------- ## Calculando as probabilidades y <- 0:15 ## Binomial Negativa -py.bn <- sapply(phis, function(p) dnbinom(y, size = p, mu = mu)) +py.bn <- sapply(thetas, function(p) dnbinom(y, size = p, mu = mu)) da.bn <- as.data.frame(py.bn) da.bn <- cbind(y, stack(da.bn)) ## Poisson -py.po <- sapply(phis, function(p) dpois(y, lambda = mu)) +py.po <- sapply(thetas, function(p) dpois(y, lambda = mu)) da.po <- as.data.frame(py.po) da.po <- cbind(y, stack(da.po)) ##------------------------------------------- ## Objetos para grafico da lattice fl <- substitute( - expression(phi == p1, phi == p2, phi == p3), - list(p1 = phis[1], p2 = phis[2], p3 = phis[3])) + expression(theta == p1, theta == p2, theta == p3), + list(p1 = thetas[1], p2 = thetas[2], p3 = thetas[3])) cols <- trellis.par.get("superpose.line")$col[1:2] yaxis <- pretty(da.po$values, n = 2) ylim <- c(-0.08, max(da.po$values)*1.2) @@ -359,27 +361,27 @@ trellis.unfocus() @ Os momentos média e variância da distribuição Binomial Negativa são -expressos como $E(Y) = \mu$ e $V(Y) = \mu + \mu^2/\phi$. Note que pelas -expressões fica evidente a caracterÃstica da Binomial Negativa de +expressos como $E(Y) = \mu$ e $V(Y) = \mu + \mu^2/\sigma^2$. Note que +pelas expressões fica evidente a caracterÃstica da Binomial Negativa de acomodar somente superdispersão, pois $E(Y)$ é menor que $V(Y)$ para -qualquer $\phi$. Percebemos também quanto maior o parâmetro $\phi$ mais -$E(Y)$ se aproxima de $V(Y)$, e no limite $\phi \rightarrow \infty$, -$E(Y) = V(Y)$ fazendo com que a distribuição Binomial Negativa se reduza -a Poisson. +qualquer $\sigma^2$. Percebemos também quanto maior o parâmetro +$\sigma^2$ mais $E(Y)$ se aproxima de $V(Y)$, e no limite, quando +$\sigma^2 \rightarrow \infty$, $E(Y) = V(Y)$ fazendo com que a +distribuição Binomial Negativa se reduza a Poisson. <<mv-binomneg, fig.cap="Relação Média e Variância na distribuição Binomial Negativa", fig.height=4, fig.width=4, fig.show="hide", results="asis">>= ##------------------------------------------- ## Parâmetros considerados -phi <- seq(0.5, 50, length.out = 50) +theta <- seq(0.5, 50, length.out = 50) col <- rev(brewer.pal(n = 8, name = "RdBu")) -col <- colorRampPalette(colors = col)(length(phi)) +col <- colorRampPalette(colors = col)(length(theta)) ##------------------------------------------- ## Etiquetas da legenda labels <- substitute( - expression(phi == p1, phi == p2, phi == p3), - list(p1 = min(phi), p2 = median(phi), p3 = max(phi))) + expression(theta == p1, theta == p2, theta == p3), + list(p1 = min(theta), p2 = median(theta), p3 = max(theta))) ##------------------------------------------- ## Gráfico @@ -388,20 +390,20 @@ labels <- substitute( par(mar = c(5.5, 4.2, 3, 3), las = 1) curve(mu + 1*0, from = 0, to = 10, xname = "mu", - ylab = expression(V(Y) == mu + mu^2~"/"~phi), + ylab = expression(V(Y) == mu + mu^2~"/"~theta), xlab = expression(E(Y) == mu)) grid() ## Curvas da relação média e variância da Binomial Negativa -for (a in seq_along(phi)) { - curve(mu + (mu^2)/phi[a], +for (a in seq_along(theta)) { + curve(mu + (mu^2)/theta[a], add = TRUE, xname = "mu", col = col[a], lwd = 2) } plotrix::color.legend( xl = 11, yb = 2.5, xr = 12, yt = 6.5, gradient = "y", align = "rb", - legend = round(fivenum(phi)[c(1, 3, 5)]), + legend = round(fivenum(theta)[c(1, 3, 5)]), rect.col = col) -mtext(text = expression(phi), side = 3, cex = 1.5, +mtext(text = expression(theta), side = 3, cex = 1.5, line = -4, at = 11.5) fonte("Fonte: Elaborado pelo autor.") @@ -410,25 +412,25 @@ wrapfigure() A relação funcional entre média e variância é ilustrada na figura \ref{fig:mv-binomneg} onde apesentamos as médias e variâncias -para $\mu$ entre 0 e 10 e $\phi$ entre 0 e 50. O comportamento dessa +para $\mu$ entre 0 e 10 e $\theta$ entre 0 e 50. O comportamento dessa relação proporciona um mairo flexibilidade à distribuição em acomodar superdispersão, uma caracterÃstica importante exibida nesta figura é que para a Binomial Negativa se aproximar a Poisson em contagens altas o -$\phi$ deve ser extremamente grande. +$\theta$ deve ser extremamente grande. O emprego do modelo Binomial Negativo em problemas se regressão ocorre de maneira similar aos MLG's, com excessão de que a distribuição só -pertence a famÃlia exponencial de distribuições se o parâmetro $\phi$ +pertence a famÃlia exponencial de distribuições se o parâmetro $\theta$ for conhecido e assim o processo sofre algumas alterações. Primeiramente, assim como na Poisson, definimos $g(\mu_i) = X\beta$, comumente utiliza-se a função $g(\mu_i) = \log(\mu_i)$. Desenvolvendo a log-verossimilhança e suas funções derivadas, função escore e matriz de informação de Fisher chegamos que a matriz de informação é bloco diagonal caracterizando a ortogonalidade -dos parâmetros $\beta$ de locação e $\phi$ de dispersão. Deste fato +dos parâmetros $\beta$ de locação e $\theta$ de dispersão. Deste fato decorre que a estimação dos parâmetros pode ser realizada em paralelo, ou seja, estima-se o vetor $beta$ pelo método de \textit{IWLS} e -posteriormente o parâmetro $\phi$ pelo método de Newton-Raphson, faz-se +posteriormente o parâmetro $\theta$ pelo método de Newton-Raphson, faz-se os dois procedimentos simultaneamente até a convengência dos parâmetros. \section{Modelo COM-Poisson} -- GitLab