edesign (version 1.0-13)

maxentropy: Maximum entropy sampling

Description

Starting point is a network $A[F]$ with $nf$ points. Now one has to select $ns$ points of a set of candidate sites to augment the existing network. The aim of maximum entropy sampling is to select a feasible D-optimal design that maximizes the logarithm of the determinant of all principal submatrices of $A$ arising by this expansion.

This algorithm is based on the interlacing property of eigenvalues. It starts with an initial solution given directly or provided by the greedy or dual-greedy approach. It uses a branch-and-bound strategy to calculate an optimal solution.

It is also possible to construct a completely new network, that means $nf=0$.

Usage

maxentropy(A,nf,ns,method="d",S.start=NULL,rtol=1e-6,mattest=TRUE,etol=0,verbose=FALSE)

Arguments

A

Spatial covariance matrix $A$.

nf

Number of stations are forced into every feasible solution.

ns

Number of stations have to be added to the existing network.

method

Method to determine the initial solution: "d"=dual-greedy algorithm, "g"=greedy algorithm, "dc"=dual-greedy + interchange algorithm, "gc"=greedy + interchange algorithm, "c"=interchange algorithm + directly given initial solution. Otherwise this algorithm has to be started with an directly given initial solution.

S.start

Vector that gives the $ns$ indices contained in the initial solution of dimension $dim(A)[1]-nf$ that should to be improved.

rtol

The algorithm terminates if the optimal solution is obtained with a tolerance of $rtol$.

mattest

Logical, if TRUE a tes for for symmetry and positive definiteness of the matrix $A$ isperformed (default is TRUE).

etol

Tolerance for checking positve definiteness (default 0).

verbose

Logical, if TRUE some information is printed per iteration (default is FALSE).

Value

A object of class monet containing the following elements:

S

Vector containing the indices of the added sites in the initial solution or 0 for the other sites.

det.start

Determinant of the principal submatrix indexed by the initial solution.

det

Determinant of the principal submatrix indexed by the optimal solution.

maxcount

Maximum of active subproblems.

iter

Number of iterations.

Details

$A[F]$ denotes the principal submatrix of $A$ having rows and columns indexed by $1..nf$.

References

Ko, Lee, Queyranne, An exact algorithm for maximum entropy sampling, Operations Research 43 (1995), 684-691.

Gebhardt, C.: Bayessche Methoden in der geostatistischen Versuchsplanung. PhD Thesis, Univ. Klagenfurt, Austria, 2003

O.P. Baume, A. Gebhardt, C. Gebhardt, G.B.M. Heuvelink and J. Pilz: Network optimization algorithms and scenarios in the context of automatic mapping. Computers & Geosciences 37 (2011) 3, 289-294

See Also

greedy, dualgreedy, interchange

Examples

x <- c(0.97900601,0.82658702,0.53105628,0.91420190,0.35304969,
       0.14768239,0.58000004,0.60690101,0.36289026,0.82022147,
       0.95290664,0.07928365,0.04833764,0.55631735,0.06427738,
       0.31216689,0.43851418,0.34433556,0.77699357,0.84097327)
y <- c(0.36545512,0.72144122,0.95688671,0.25422154,0.48199229,
       0.43874199,0.90166634,0.60898628,0.82634713,0.29670695,
       0.86879093,0.45277452,0.09386800,0.04788365,0.20557817,
       0.61149264,0.94643855,0.78219937,0.53946353,0.70946842)
A <- outer(x, x, "-")^2 + outer(y, y, "-")^2
A <- (2 - A)/10
diag(A) <- 0
diag(A) <- 1/20 + apply(A, 2, sum)

S.entrp<-c(0,7,0,9,0,11,0,13,14,0,0,0,0,0,0)
maxentropy(A,5,5,S.start=S.entrp)
maxentropy(A,5,5,method="g")
maxentropy(A,5,5)