Multivariate Statistik - Aufgabe 5 - Diskriminanz-/Hauptkomponentenanalyse

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Beispiel 1

The function to calculate the apriori probability is

p(k|x) = \frac{p(k) \cdot f(x|k)}{f(x)}

where f(x) = \sum p(k) \cdot f(x|k) \, 

library(mvtnorm)

aposteriori = function(x,apriori) {
	m=matrix(c(0,0,0,1,2,3,3,3,4),nrow=3)
	s=matrix(c(1,0.5,0.5,0.5,1,0.5,0.5,0.5,1),nrow=3,byrow=T)

	denum=0
	for(i in 1:ncol(m)) {
		denum=denum+dmvnorm(x,m[,i],s)*apriori[i]
	}

	ret = rep(0,ncol(m))

	for(i in 1:ncol(m)) {
		ret[i] = (dmvnorm(x,m[,i],s)*apriori[i]) / denum
	}
	ret
}

#


Error in loadNamespace(name) : there is no package called 'distrEx'
Calls: sys.load.image ... tryCatch -> tryCatchList -> tryCatchOne -> 
Execution halted

dmvnorm(...) returns f_X(x_1, \dots, x_N) = \frac {1} {(2\pi)^{N/2} \left|\Sigma\right|^{1/2}} \exp \left( -\frac{1}{2} ( x - \mu)^\top \Sigma^{-1} (x - \mu) \right) as found in en:Multivariate normal distribution


For the 3 observations x_1 = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix} \,, x_2 = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \, and x_3 = \begin{pmatrix} 3 \\ 3 \\ 4 \end{pmatrix} \, with aprioris p(1) = p(2) = p(3) = \frac{1}{3} \, the aposterios are given by 

library(mvtnorm)
aposteriori(c(0,0,0),rep(1/3,3))
aposteriori(c(1,2,3),rep(1/3,3))
aposteriori(c(0.5,1,1.5),rep(1/3,3))

#


Error in loadNamespace(name) : there is no package called 'distrEx'
Calls: sys.load.image ... tryCatch -> tryCatchList -> tryCatchOne -> 
Execution halted

Therefore x_1 \, is assigned to group 1, x_2 \, is assigned to group 2 and x_3 \, is just between group 1 and 2.


For aprioris p(1) = p(3) = 0.49 \, and p(2) = 0.02 \,. The aposterios are given by 

library(mvtnorm)
aposteriori(c(0,0,0),c(0.49,0.02,0.49))
aposteriori(c(1,2,3),c(0.49,0.02,0.49))
aposteriori(c(0.5,1,1.5),c(0.49,0.02,0.49)) 

#


Error in loadNamespace(name) : there is no package called 'distrEx'
Calls: sys.load.image ... tryCatch -> tryCatchList -> tryCatchOne -> 
Execution halted

Therefore x_1 \, and x_3 \, are assigned to group 1 and x_2 \, is assigned to group 3.

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