# IHS Improved Hypercube Sampling

IHS, a MATLAB library which carries out the Improved Hypercube Sampling (IHS) algorithm.

N Points in an M dimensional Latin hypercube are to be selected. Each of the M coordinate dimensions is discretized to the values 1 through N. The points are to be chosen in such a way that no two points have any coordinate value in common. This is a standard Latin hypercube requirement, and there are many solutions.

This algorithm differs in that it tries to pick a solution which has the property that the points are "spread out" as evenly as possible. It does this by determining an optimal even spacing, and using the duplication factor D to allow it to choose the best of the various options available to it (the more duplication, the better chance of optimization).

One drawback to this algorithm is that it requires an internal real distance array of dimension D * N * N. For a relatively moderate problem with N = 1000, this can exceed the easily accessible memory. Moreover, the program is inherently quadratic in N in execution time as well as memory; the computation of the I-th point in the set of N requires a consideration of the value of the coordinates used up by the previous points, and the distances from each of those points to the candidates for the next point.

### Languages:

IHS is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

CVT, a MATLAB library which computes elements of a Centroidal Voronoi Tessellation.

FAURE, a MATLAB library which computes elements of a Faure quasirandom sequence.

GRID, a MATLAB library which computes elements of a grid sequence.

HALTON, a MATLAB library which computes elements of a Halton quasirandom sequence.

HAMMERSLEY, a MATLAB library which computes elements of a Hammersley Quasi Monte Carlo (QMC) sequence, using a simple interface.

ihs, a MATLAB library which computes an Improved Hypercube Sampling (ihs) Quasi Monte Carlo (QMC) sequence;

LATIN_CENTER, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing center points.

LATIN_EDGE, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing edge points.

LATIN_RANDOM, a MATLAB library which computes elements of a Latin Hypercube dataset, choosing points at random.

LATTICE_RULE, a MATLAB library which approximates multidimensional integrals using lattice rules.

LCVT, a MATLAB library which computes a latinized Centroidal Voronoi Tessellation.

NIEDERREITER2, a MATLAB library which computes elements of a Niederreiter quasirandom sequence using base 2.

NORMAL, a MATLAB library which computes elements of a sequence of pseudorandom normally distributed values.

SOBOL, a MATLAB library which computes elements of a Sobol quasirandom sequence.

TABLE_QUALITY, a MATLAB program which measures the dispersion quality of points in a dataset read from a file.

UNIFORM, a MATLAB library which computes elements of a uniform pseudorandom sequence.

VAN_DER_CORPUT, a MATLAB library which computes elements of a 1D van der Corput sequence.

### Author:

John Burkardt; MATLAB performance enhancements courtesy of Jeremy Dewar, Tulane University.

### Reference:

1. Brian Beachkofski, Ramana Grandhi,
Improved Distributed Hypercube Sampling,
American Institute of Aeronautics and Astronautics Paper 2002-1274.

### Source Code:

• covariance.m, does a covariance calculation for IHS solutions.
• ihs.m, implements the improved hypercube sampling algorithm.
• ihs_write.m, writes an IHS dataset to a file.
• r8vec_std.m, returns the standard deviation of an R8VEC.
• timestamp.m, prints the current YMDHMS date as a timestamp.

### Examples and Tests:

• ihs_test01.m, demonstrates IHS for several dimensions.
• ihs_test02.m, demonstrates IHS for several duplication values.
• ihs_test03.m, demonstrates IHS for several numbers of points.
• ihs_test04.m, demonstrates IHS for several random number seeds.

Last revised on 02 February 2019.