5.7 … para dispersão e precisão
A precisão é o inverso da variância no caso univariado, e o inverso da matriz de covariâncias no caso multivariado.
Univariado
\(P=\frac{1}{\sigma^2}\)
Multivariado
\(\Omega = \Sigma^{-1}\)
5.7.2 Gama inversa
## [1] 0.03893829
## [1] 0.8737863 0.2984410 0.2912499 0.5844254 0.2694825 0.5206008 0.2133415 0.5389982 0.4858066 1.5705256
# Plot Probability Functions
x <- seq(from=0.1, to=20, by=0.1)
plot(x, dinvgamma(x,1,1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dinvgamma(x,1,0.6), type="l", col="green")
lines(x, dinvgamma(x,0.6,1), type="l", col="blue")
legend(2, 0.9, expression(paste(alpha==1, ", ", beta==1),
paste(alpha==1, ", ", beta==0.6), paste(alpha==0.6, ", ", beta==1)),
lty=c(1,1,1), col=c("red","green","blue"))
5.7.3 Qui-quadrado inversa padronizada
## [1] 0.2419707
## [1] 7.6275257 2.3461109 1.8776043 2.3980416 1.5938337 5.1493120 0.2201109 0.2321013 0.1950130 8.0955823
#Plot Probability Functions
x <- seq(from=0.1, to=5, by=0.01)
plot(x, dinvchisq(x,0.5,1), ylim=c(0,1), type="l", main="Probability Function",
ylab="density", col="red")
lines(x, dinvchisq(x,1,1), type="l", col="green")
lines(x, dinvchisq(x,5,1), type="l", col="blue")
legend(3, 0.9, expression(paste(nu==0.5, ", ", lambda==1),
paste(nu==1, ", ", lambda==1), paste(nu==5, ", ", lambda==1)),
lty=c(1,1,1), col=c("red","green","blue"))
5.7.5 Normal-Inversa-Gama
5.7.6 Wishart
Occasionally, in the statistical distribution literature, Wishart distributions have been referred to as “multivariate gamma distributions”. We, however, restrict this term to those distributions for which the marginal distribution are of gamma form. (Kotz, Balakrishnan, and Johnson 2000, 431)
De acordo com (Box and Tiao 1973, 427) e [Statisticat and LLC. (2021)]6, a distribuição Wishart (Wishart 1928) é uma generalização multivariada da distribuição qui-quadrado. Não é chamada de distribuição qui-quadrado multivariada, no entanto, pelo fato de a distribuição marginal dos elementos fora da diagonal não é qui-quadrado.
library(LaplacesDemon)
Omega <- matrix(c(2,-.3,-.3,4),2,2)
dwishart(Omega, 3, matrix(c(1,.1,.1,1),2,2))
## [1] 0.003785585
## [,1] [,2]
## [1,] 4.5672769 -0.7975769
## [2,] -0.7975769 0.1724443
Com a parametrização de Cholesky, considerando o fator triangular superior \(U\) da matriz de precisão \(\Omega\).
library(LaplacesDemon)
Omega <- matrix(c(2,-.3,-.3,4),2,2)
U <- chol(Omega)
dwishartc(U, 3, matrix(c(1,.1,.1,1),2,2))
## [1] 0.003785585
## [,1] [,2]
## [1,] 2.137119 -0.3732020
## [2,] 0.000000 0.1821115
5.7.7 Wishart inversa
library(LaplacesDemon)
Sigma <- matrix(c(2,-.3,-.3,4),2,2)
dinvwishart(Sigma, 3, matrix(c(1,.1,.1,1),2,2))
## [1] 0.0001079824
## [,1] [,2]
## [1,] 1.434238 -0.3196810
## [2,] -0.319681 0.2656181
Com a parametrização de Cholesky, considerando o fator triangular superior \(U\) da matriz de covariâncias \(\Sigma\).
library(LaplacesDemon)
Sigma <- matrix(c(2,-.3,-.3,4),2,2)
U <- chol(Sigma)
dinvwishartc(U, 3, matrix(c(1,.1,.1,1),2,2))
## [1] 0.0001079824
## [,1] [,2]
## [1,] 1.197597 -0.2669354
## [2,] 0.000000 0.4408669
5.7.8 Normal-Wishart
library(LaplacesDemon)
K <- 3
set.seed(1); mu <- rnorm(K)
set.seed(2); mu0 <- rnorm(K)
nu <- K + 1
S <- diag(K)
set.seed(3); lambda <- runif(1) #Real scalar
set.seed(4); Omega <- as.positive.definite(matrix(rnorm(K^2),K,K))
x <- dnormwishart(mu, mu0, lambda, Omega, S, nu, log=TRUE)
set.seed(5); out <- rnormwishart(n=10, mu0, lambda, S, nu)
joint.density.plot(out$mu[,1], out$mu[,2], color=TRUE)
5.7.9 Normal-Inversa-Wishart
library(LaplacesDemon)
K <- 3
set.seed(1); mu <- rnorm(K)
set.seed(2); mu0 <- rnorm(K)
nu <- K + 1
S <- diag(K)
set.seed(3); lambda <- runif(1) # Real scalar
set.seed(4); Sigma <- as.positive.definite(matrix(rnorm(K^2),K,K))
x <- dnorminvwishart(mu, mu0, lambda, Sigma, S, nu, log=TRUE)
set.seed(5); out <- rnorminvwishart(n=10, mu0, lambda, S, nu)
joint.density.plot(out$mu[,1], out$mu[,2], color=TRUE)
Referências
Documentação da função
LaplacesDemon::dist.Wishart
.↩︎