edesign (version 1.0-7)

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"=intercha
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:
  • SVector containing the indices of the added sites in the initial solution or 0 for the other sites.
  • det.startDeterminant of the principal submatrix indexed by the initial solution.
  • detDeterminant of the principal submatrix indexed by the optimal solution.
  • maxcountMaximum of active subproblems.
  • iterNumber 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

Run this code
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)

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