maxentropy

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Maximum entropy sampling

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 $nf=0$.

Keywords
spatial , design
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).
Details

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

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.

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

Aliases
  • maxentropy
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)

Documentation reproduced from package edesign, version 1.0-13, License: GPL (>= 2.0)

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