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 $nf=0$.
- Spatial covariance matrix $A$.
- Number of stations are forced into every feasible solution.
- Number of stations have to be added to the existing network.
- Method to determine the initial solution:
"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.
- Vector that gives the $ns$ indices contained in the initial solution of dimension $dim(A)-nf$ that should to be improved.
- The algorithm terminates if the optimal solution is obtained with a tolerance of $rtol$.
- Logical, if
TRUEa tes for for symmetry and positive definiteness of the matrix $A$ isperformed (default is
- Tolerance for checking positve definiteness (default 0).
- Logical, if
TRUEsome information is printed per iteration (default is
$A[F]$ denotes the principal submatrix of $A$ having rows and columns indexed by $1..nf$.
A object of class
- Vector containing the indices of the added sites in the initial solution or 0 for the other sites.
- Determinant of the principal submatrix indexed by the initial solution.
- Determinant of the principal submatrix indexed by the optimal solution.
- Maximum of active subproblems.
- Number of iterations.
monetcontaining the following elements:
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
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)