multistageoptimum.nlm: Function for optimizing n-stage selection with the NLM algorithm for a given correlation matrix

Description

This function is used to calculate the maximum of $\Delta G$ with given correlation matrix by non-linear minimization algorithm.

Usage

multistageoptimum.nlm(corr, ini.value, 
Budget, CostC, CostTv, Nf, iterlim, alg,N.upper, N.lower)

Arguments

corr

is the correlation matrix of y and X, which is introduced in function multistagecorr. The correlation matrix must be symmetric and positive-definite. Before starting the calculations, the user is recommended to check the correlation matrix.

ini.value

is a vector, which stores the number of candidates in each stage for the algorithm to begin with. As default, it will use $N={N_1,N_2,...,N_n}={a+1,...,a+n}$, where a is defined as $Budget/(CostC+sum(CostTv)+1)$

Budget

contains the value of total budget.

CostC

contains the costs of producing or identifying a candidate.

CostTv

contains a vector with length n reflecting the cost of evaluating a candidate in the tests performed at stage i, i=1,...,n. The cost might vary in different stages.

is the number of final selected candidates.

iterlim

is the maximum number of iterations to be executed before the Newton algorithm is terminated. By default it is equal to 20. If the $\texttt{Budget}$ increases 10 times for making the selection, the value of $\texttt{iterlim}$ has to be increased $lg(10)$

alg

is used to switch between two algorithms. If alg = GenzBretz(), which is by default, the quasi-Monte Carlo algorithm from Genz (2009) will be used. If alg = Miwa(), the program will use the Miwa algorithm (Mi et. al., 2009), whic

N.upper

is the vector of up limits of number of candidates X.

N.lower

is the vector of low limits of number of candidates X.

Value

The output of this function is a vector similar as in multistageoptimal.grid() if detail = FALSE. However, the optimal number of candidates in each stage determined by the NLM algorithm is clearly not an integer, because the function uses a numerical algorithm, which depends on derivatives.

References

A. Genz and F. Bretz. Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195, Springer-Verlag, Heidelberg, 2009. A. Genz, F. Bretz, T. Miwa, X. Mi, F. Leisch, F. Scheipl and T. Hothorn. mvtnorm: Multivariate normal and t distributions. R package version 0.9-9, 2010. G.M. Tallis. Moment generating function of truncated multi-normal distribution. Journal of the Royal Statistical Society, Series B, 23(1):223-229, 1961. H.F. Utz. Mehrstufenselecktion in der Pflanzenzuechtung. Doctor thesis, University Hohenheim, 1969. 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) W.G. Cochran. Improvent by means of selection. In: Proceedings Second Berkeley Symposium on Math Stat Prof, pp449-470, 1951.

Examples

Run this code

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  
)

multistageoptimum.nlm(N.upper=rep(100,3), corr=corr, Budget=200,
 CostC=0.5, Nf=5)