LCVT_DATASET
Latin Hypercubes using CVT Startup

LCVT_DATASET is a C++ program, using double precision arithmetic, which computes a Latin Hypercube in M dimensions, with N points, using a CVT dataset as the initial estimate. The resulting dataset can be written to a file.

A Latin Square dataset is typically a two dimensional dataset of N points in the unit square, with the property that, if both the x and y axes are divided up into N equal subintervals, exactly one dataset point has an x or y coordinate in each subinterval. Latin squares can easily be extended to the case of M dimensions, and may be pedantically called Latin Hypersquares or Latin Hypercubes in such a case. Statisticians like Latin Squares, as do experiment designers, and and people who need to approximate scalar functions of many variables.

The fact that the projection of a Latin Square dataset onto any coordinate axis is either exactly evenly spaced, or approximately so (depending on the algorithm), turns out to be an attractive feature for many uses.

However, a CVT dataset in a regular domain, such as the unit hypercube, has the tendency for the projections of the points to cluster together in any coordinate axis. This program is mainly an attempt to explore whether a dataset can be computed using techniques similar to those of a CVT, but with the constraint (whether imposed or expected) that the point projections do not clump up.

The approach used here is quite simple. First we compute a CVT in M dimensions, comprising N points. We assume that the bounding region is the unit hypercube. We are now going to adjust the coordinates of the points to achieve the Latin Hypercube property. For each coordinate direction, we simply sort the points by that coordinate, and then overwrite the original values by the values we'd expect to get for a centered Latin Hypercube, namely, 1/(2*N), 3/(2*N), ..., (2*N-1)/(2*N).

Now this process guarantees that we get a Latin Hypercube. Our hope is that the process of adjusting the point coordinates does not too severely damage the nice dispersion properties inherent in the CVT point placement.

An earlier version of this program was "very" interactive, allowing the user to enter input in any order. This turned out to be a little too confusing. The new version of the program asks the user for input in a strict order. If you find this procedure too restrictive, you can try out the old program.

Briefly the user needs to specify the following:

1. The spatial dimension M of the points;
2. The number of points N to be generated.
3. The random number seed;
4. How the initial points are chosen. If you have no preference, choose UNIFORM.
   • GRID, use a grid of points;
   • HALTON, use Halton points;
   • RANDOM, use RAND (C++ intrinsic);
   • UNIFORM, use a simple uniform random number generator;
   • USER, call the "user" routine;
   • (file_name), read the initial points from a file.
5. The number of CVT iterations. If you have no preference, try 5, 10 or 20;
6. How the sampling is done. If you have no preference, use UNIFORM.
   • GRID, use a grid of points;
   • HALTON, use Halton points;
   • RANDOM, use RAND (C++ intrinsic);
   • UNIFORM, use a simple uniform random number generator;
   • USER, call the "user" routine;
7. The number of sampling points to use. Think of this as a sampling of the unit hypercube. So to compare it to N, the number of points, you need to take its M-th root. In 2D, if you're using 10 generators, and 100 sample points, to get area and sampling computations twice as good requires 4 times the sampling. It never hurts to use more sampling points.
8. The "batch size". This parameter controls how many sampling points are to be generated at one time. You can set this value equal to the number of sampling points, but if you are having memory problems, it can be set lower. In such a case, a smaller value might be 1000, for instance.
9. The number of CVT iterations to carry out. It's not really necessary to compute the CVT super accurately, since we're just going to perturb it anyway. This value could be anywhere from 10 to 500. Convergence of the CVT is typically slow, especially if the starting positions are poor.
10. The number of Latin Hypercube iterations to carry out. Actually, the iterations don't seem to improve the data much, so a value of 1 or 2 can be reasonable.
11. The name of a file into which the final pointset should be written.

Related Data and Programs:

CVT is a C++ library of routines that can compute a CVT (Centroidal Voronoi Tessellation).

CVT_DATASET is an interactive executable C++ program for specifying and computing a CVT (Centroidal Voronoi Tessellation).

LCVT is a C++ library of routines used by LCVT_DATASET; a compiled copy of that library must be available to build the program.

LCVT is a dataset directory which contains a collection of sample LCVT datasets created by LCVT_DATASET.

LCVT_DATASET is also available in FORTRAN90 version and MATLAB version.

PLOT_POINTS is an executable FORTRAN90 program which can read a TABLE file of points (in 2D) and plot them.

TABLE is a file format for storing M by N data, which is used by LCVT_DATASET.

TABLE_DISCREPANCY is an executable C++ program which can read a TABLE file of points (presumed to lie in the unit hypercube) and compute bounds on the star discrepancy, a measure of dispersion.

TABLE_LATINIZE is an executable C++ program which can read a TABLE file of points and "latinize" the points, that is, "gently" rearranging them so that they are regularly spaced in every coordinate direction.

TABLE_QUALITY is an executable C++ program which can read a TABLE file of points and compute various measures of the quality of dispersion.

TABLE_TOP is an executable FORTRAN90 program which can read a TABLE file of points in M dimensions (where M is likely to be more than 2!) and make plots of all 2D projections onto pairs of coordinate axes.

Reference:

1. Franz Aurenhammer,
   Voronoi diagrams - a study of a fundamental geometric data structure,
   ACM Computing Surveys,
   Volume 23, Number 3, September 1991, pages 345-405.
2. Franz Aurenhammer, Rolf Klein,
   Voronoi Diagrams,
   in Handbook of Computational Geometry,
   edited by J Sack, J Urrutia,
   Elsevier, 1999,
   LC: QA448.D38H36.
3. John Burkardt, Max Gunzburger, Janet Peterson, Rebecca Brannon,
   User Manual and Supporting Information for Library of Codes for Centroidal Voronoi Placement and Associated Zeroth, First, and Second Moment Determination,
   Sandia National Laboratories Technical Report SAND2002-0099,
   February 2002,
   ../../publications/bgpb_2002.pdf
4. Qiang Du, Vance Faber, Max Gunzburger,
   Centroidal Voronoi Tessellations: Applications and Algorithms,
   SIAM Review,
   Volume 41, Number 4, December 1999, pages 637-676.
5. Vicente Romero, John Burkardt, Max Gunzburger, Janet Peterson,
   Initial Evaluation of Pure and "Latinized" Centroidal Voronoi Tessellation for Non-Uniform Statistical Sampling,
   Sensitivity Analysis of Model Output (SAMO 2004) Conference, Santa Fe, March 8-11, 2004,
   rbgp_2004.pdf.
6. Yuki Saka, Max Gunzburger, John Burkardt,
   Latinized, improved LHS, and CVT point sets in hypercubes,
   submitted to IEEE Transactions on Information Theory,
   sgb_submitted.pdf.

Source Code:

• lcvt_dataset.C, the source code.
• lcvt_dataset.csh, commands to compile, link and load the source code.

Examples and Tests:

Example 1 is a dataset of N=85 points with spatial dimension M=2, using UNIFORM initialization and sampling, and 10,000 sample points:

• lcvt01.inp, input commands.
• lcvt01.out, printed output.
• lcvt01.txt, the LCVT dataset.
• lcvt01.png, a PNG image of the LCVT dataset, created by PLOT_POINTS.

Example 2 is a dataset of N=85 points with spatial dimension M=2, using RANDOM initialization and sampling, and 250,000 sample points, 10 CVT iterations and 2 Latinization iterations:

• lcvt02.inp, input commands.
• lcvt02.out, printed output.
• lcvt02.txt, the LCVT dataset.
• lcvt02.png, a PNG image of the LCVT dataset, created by PLOT_POINTS.

Example 3 is a dataset of N=200 points with spatial dimension M=7, using UNIFORM initialization and sampling, and 20,000 sample points, 5 CVT iterations and 2 Latinization iterations:

• lcvt03.inp, input commands.
• lcvt03.out, printed output.
• lcvt03.txt, the LCVT dataset.
• lcvt03_page1.png, "page 1" of a PNG image of pairs of coordinates of the LCVT dataset, created by TABLE_TOP.
• lcvt03_page2.png, "page 2" of a PNG image of pairs of coordinates of the LCVT dataset, created by TABLE_TOP.
• lcvt03_page3.png, "page 3" of a PNG image of pairs of coordinates of the LCVT dataset, created by TABLE_TOP.
• lcvt03_page4.png, "page 4" of a PNG image of pairs of coordinates of the LCVT dataset, created by TABLE_TOP.

List of Routines:

• MAIN is the main program for LCVT_DATASET.
• CH_CAP capitalizes a single character.
• CH_EQI is true if two characters are equal, disregarding case.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• CLUSTER_ENERGY returns the energy of a dataset.
• CVT_ITERATION takes one step of the CVT iteration.
• FIND_CLOSEST finds the Voronoi cell generator closest to a point X.
• I4_MAX returns the maximum of two I4's.
• I4_MIN returns the smaller of two I4's.
• I4_TO_HALTON computes an element of a Halton sequence.
• LCVT_WRITE writes a Latinized CVT dataset to a file.
• PRIME returns any of the first PRIME_MAX prime numbers.
• R8_EPSILON returns the roundoff unit for R8 arithmetic.
• R8_UNIFORM_01 returns a unit pseudorandom R8.
• R8MAT_LATINIZE "Latinizes" an R8MAT.
• R8TABLE_DATA_READ reads the data from an R8TABLE file.
• R8VEC_SORT_HEAP_INDEX_A does an indexed heap ascending sort of an R8VEC.
• REGION_SAMPLER returns a sample point in the physical region.
• S_EQI reports whether two strings are equal, ignoring case.
• S_LEN_TRIM returns the length of a string to the last nonblank.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.
• TIMESTRING returns the current YMDHMS date as a string.
• TUPLE_NEXT_FAST computes the next element of a tuple space, "fast".

You can go up one level to the C++ source codes.