selectiongain (version 2.0.1)

multistagevariance: Expected gain for k-stages selection

Description

This function uses the algorithm described by Tallis(1964) to calculate the variance after multi-stage selection. The variance among candidates of y in the selected area $\textbf{S}_{Q}$ is defined as the second central moment, $\psi_n(y)=E(Y^2|\textbf{S}_{Q}) - \left[E(Y|\textbf{S}_{Q})\right]^2$, where $E(Y^2|\textbf{S}_{Q}) = \alpha^{-1} \int_{-\infty} ^\infty \int_{q_{1}}^\infty...\int_{q_{n}}^\infty y^2\, \phi_{n+1}(\textbf{x}^{*}; \bm{\Sigma}^{*}) \, d \textbf{x}^*$

Usage

multistagevariance(Q, corr, alg, lim.y)

Arguments

Q

(length n) refers to the coordinates of the truncation points Q, which is the output of the next function (multistagetp) that we are going to introduced.

corr

(n+1-dimensional matrix) is the correlation matrix of y and X. The correlation matrix must be symmetric and positive-definite. Before starting the calculations, the user is recommended to check the correlation matrix, which is usually obtained by analysis

alg

is used to switch between two algorithms. If alg = GenzBretz(), which is by default, the quasi-Monte Carlo algorithm from Genz(1999) will be used. If alg = Miwa(), the program will use the Miwa algorithm (Mi2009), which an analytical solution of the MVN i

lim.y

is the lower limit of y and is set to -200 as default, which is on the safe side.

Value

• The output is the value of $\psi_n(y|\textbf{S}_{Q})=E(y^2|\textbf{S}_{Q})-[E(y|\textbf{S}_{Q})]^2$.

Details

More details are in the JSS paper section 3.4.

References

G.M. Tallis. Moment generating function of truncated multi-normal distribution. Journal of the Royal Statistical Society, Series B, 23(1):223-229, 1961. X. Mi, T. Miwa and T. Hothorn. Implement of Miwa's analytical algorithm of multi-normal distribution, R Journal, 1:37-39, 2009. X. Mi, H.F. Utz. and A.E. Melchinger. R package selectiongain: A tool for efficient calculation and optimization of the expected gain from multi-stage selection. J Stat Softw. (submitted)

See Also

No link

Examples

Q =c(0.4308,0.9804,1.8603)

corr=matrix( c(1,       0.3508,0.3508,0.4979,
               0.3508  ,1,     0.3016,0.5630,
               0.3508,  0.3016,1     ,0.5630,
               0.4979,  0.5630,0.5630,1), 
              nrow=4  
)


multistagevariance(Q=Q,corr=corr,alg=Miwa)

var.time.miwa=system.time (var.miwa<-multistagevariance(Q=Q,corr=corr,alg=Miwa))

var.time.bretz=system.time (var.bretz<-multistagevariance(Q=Q,corr=corr))


var.time.miwa
var.miwa[1]
var.time.bretz
var.bretz[1]

# further examples 1 


Q= c(0.9674216, 1.6185430)
corr=matrix( c(1,      0.7071068, 0.9354143,
               0.7071068, 1,      0.7559289,
               0.9354143, 0.7559289, 1    
             ), 
              nrow=3  
)


multistagevariance(Q=Q,corr=corr,alg=Miwa)

var.time.miwa=system.time (var.miwa<-multistagevariance(Q=Q,corr=corr,alg=Miwa))

var.time.bretz=system.time (var.bretz<-multistagevariance(Q=Q,corr=corr))


var.time.miwa
var.miwa[1]
var.time.bretz
var.bretz[1]


# further examples 

 alpha1<- 1/(24)^0.5
 alpha2<- 1/(24)^0.5
 Q=multistagetp(alpha=c(alpha1,alpha2),corx=corr[2:3,2:3])


corr=matrix( c(1,      0.7071068, 0.9354143,
               0.7071068, 1,      0.7559289,
               0.9354143, 0.7559289, 1    
             ), 
              nrow=3  
)

multistagevariance(Q=Q,corr=corr,alg=Miwa)