interchange: Interchange algorithm for 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.

The interchange algorithm improves a feasible initial solution directly given or obtained by the greedy or dual greedy algorithm for maximum entropy sampling.

It is also possible to improve the initial solution for the construction of a completely new network, that means $nf=0$, but in this case the interchange algorithm fails for $ns=1$.

Usage

interchange(A, nf, ns, S.start, etol=0, mattest = TRUE)

Arguments

Spatial covariance matrix $A$.

Number of stations are forced into every feasible solution.

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

S.start

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

etol

Tolerance for checking positve definiteness (default 0)

mattest

Toggles testing matrix A for symmetry and positive definiteness (default T)

Value

S.start: Vector containing the indices of the added sites in the initial solution or 0 for the other sites.
S: Vector containing the indices of the added sites in the solution or 0 for the other sites.
det: Determinant of the principal submatrix indexed by the initial solution.

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

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.c<-c(0,7,0,9,0,11,0,13,14,0,0,0,0,0,0)
interchange(A,5,5,S.c)
interchange(A,5,5,greedy(A,5,5)$S)
interchange(A,5,5,dualgreedy(A,5,5)$S)

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