TABLE_DIAPHONY
the "Diaphony" (dispersion)
of an M-dimensional Pointset

TABLE_DIAPHONY is a FORTRAN90 program which computes the "diaphony" of an M-dimensional pointset.

The "diaphony" of an M-dimensional pointset is a numeric measure of the uniformity of the dispersion of the points throughout the unit hypercube.

The point sets we are interested in have been generated by a variety of algorithms, and are stored as datasets, of spatial dimension 2, 7 and 16, and sample sizes 10, 100, 1,000 and 10,000. The point set types include:

• CVT (Centroidal Voronoi Tessellation) values,
• Faure quasirandom values,
• points on a uniform grid, with five centering options,
• Halton quasirandom values,
• Hammersley quasirandom values,
• IHS (Improved Distributed Hypercube Sampling) values,
• Latin Center Square values,
• Latin Edge Square values,
• Latin Random Square values,
• LHS (Latin Hypercube Sampling) due to Vicente Romero,
• Niederreiter quasirandom values (base 2),
• Sobol quasirandom values,
• uniform random values.

Regarded as a random variable itself, the diaphony of a set of N points has an expected value of 1/sqrt(N).

For the Halton datasets in 2D, 7D and 16D, here is the table of the number of points versus the diaphony:

Diaphony(M,N)	M=2D	M=7D	M=16D	1/Sqrt(N)
N=10	0.246	0.316	0.316	0.316
N=100	0.043	0.099	0.099	0.100
N=1000	0.006	0.031	0.031	0.031
N=10000	0.001	0.009	0.009	0.001

Related Data and Programs:

The TABLE format is used for the input to this program.

TABLE_BARPLOT_PPMA is an executable program which reads a table file and creates a PPMA bargraph of the data.

TABLE_BORDER is an executable program which can be used to add a border (of zero values) to a table file.

TABLE_COLUMNS is an executable program which can extract specific columns of data from a table file.

TABLE_COLUMNS_PERMUTE is an executable FORTRAN90 program which permutes the columns of a table file.

TABLE_DELAUNAY computes the Delaunay triangulation of a set of points.

TABLE_DIAPHONY computes the diaphony of a set of points stored in a table file.

TABLE_HISTOGRAM can make a histogram of a set of points stored in a table file.

The TABLE_IO library supplies the routines used to read the TABLE file.

TABLE_LATINIZE is a FORTRAN90 program that reads a file of points and creates a "latinized" version by adjusting the data.

TABLE_MERGE is a FORTRAN90 program that reads a file of points, and removes duplicates, and points that are close to each other.

TABLE_ORTHONORMALIZE is a FORTRAN90 program that reads a file of points and orthonormalizes the columns.

TABLE_QUALITY is a FORTRAN90 program that reads a file of points and computes the quality of dispersion.

TABLE_READ is a MATLAB script that can read in a TABLE file.

TABLE_RECORD_MATCH is an executable program that can be used to find close records in a table file.

TABLE_SCALE is an executable program that can be used to multiply the entries of a table file by a scale vector.

TABLE_SHIFT is an executable program that can be used to shift the entries of a table file by a shift vector.

TABLE_STATS is an executable program that can read a table file and compute certain statistics.

TABLE_TET_MESH is an executable program that can read a table file of 3D data, and compute a tetrahedral mesh.

TABLE_TOP is an executable FORTRAN90 program that can read a table file of M-dimensional data and make a table of plots of all pairs of coordinates.

TABLE_UNBORDER is an executable program that can be used to remove the border from a table file.

TABLE_UNIFORM_NOISE is an executable program that can be used to add a uniform noise term to the data in a table file.

TABLE_VORONOI can be used to compute information about the Voronoi diagram of the points.

Reference:

1. Peter Heelekalek,
   On Correlation Analysis of Pseudorandom Numbers, in Monte Carlo and Quasi-Monte Carlo Methods 1996, edited by Harald Niederreiter, Peter Hellekalek, Gerhard Larcher, and Peter Zinterhof,
   Volume 127 of Lecture Notes in Statistics,
   Springer Verlag, 1997, pages 251-265.
2. Peter Heelekalek, Harald Niederreiter,
   The Weighted Spectral Test: Diaphony,
   ACM Transactions on Modeling and Computer Simulation,
   Volume 8, Number 1, January 1998, pages 43-60.
3. Peter Heelekalek, Hannes Leeb,
   Dyadic Diaphony,
   Acta Arithmetica,
   Volume 80, Number 2, 1997, pages 187-196.

Source Code:

• diaphony_test.f90, the source code.
• diaphony_test.csh, commands to compile the source code.

Examples and Tests:

• diaphony_cvt.csh, commands to compute the diaphony of the CVT datasets.
• diaphony_faure.csh, commands to compute the diaphony of the FAURE datasets.
• diaphony_grid_1.csh, commands to compute the diaphony of the GRID 1 datasets.
• diaphony_grid_2.csh, commands to compute the diaphony of the GRID 2 datasets.
• diaphony_grid_3.csh, commands to compute the diaphony of the GRID 3 datasets.
• diaphony_grid_4.csh, commands to compute the diaphony of the GRID 4 datasets.
• diaphony_grid_5.csh, commands to compute the diaphony of the GRID 5 datasets.
• diaphony_halton.csh, commands to compute the diaphony of the HALTON datasets.
• diaphony_hammersley.csh, commands to compute the diaphony of the HAMMERSLEY datasets.
• diaphony_ihs.csh, commands to compute the diaphony of the IHS datasets.
• diaphony_latin_center.csh, commands to compute the diaphony of the LATIN CENTER datasets.
• diaphony_latin_edge.csh, commands to compute the diaphony of the LATIN EDGE datasets.
• diaphony_latin_random.csh, commands to compute the diaphony of the LATIN RANDOM datasets.
• diaphony_lhs.csh, commands to compute the diaphony of the LHS datasets.
• diaphony_niederreiter2.csh, commands to compute the diaphony of the NIEDERREITER2 datasets.
• diaphony_sobol.csh, commands to compute the diaphony of the SOBOL datasets.
• diaphony_uniform.csh, commands to compute the diaphony of the UNIFORM datasets.

• diaphony_cvt.txt diaphony results for the points in the CVT dataset.
• diaphony_faure.txt diaphony results for the points in the FAURE dataset.
• diaphony_grid_1.txt diaphony results for the points in the GRID dataset with CENTER = 1.
• diaphony_grid_2.txt diaphony results for the points in the GRID dataset with CENTER = 2
• diaphony_grid_3.txt diaphony results for the points in the GRID dataset with CENTER = 3.
• diaphony_grid_4.txt diaphony results for the points in the GRID dataset with CENTER = 4.
• diaphony_grid_5.txt diaphony results for the points in the GRID dataset with CENTER = 5.
• diaphony_halton.txt diaphony results for the points in the HALTON dataset.
• diaphony_hammersley.txt diaphony results for the points in the HAMMERSLEY dataset.
• diaphony_ihs.txt diaphony results for the points in the IHS dataset.
• diaphony_latin_center.txt diaphony results for the points in the LATIN CENTER dataset.
• diaphony_latin_edge.txt diaphony results for the points in the LATIN EDGE dataset.
• diaphony_latin_random.txt diaphony results for the points in the LATIN_RANDOM dataset.
• diaphony_lhs.txt diaphony results for the points in the LHS dataset.
• diaphony_niederreiter2.txt diaphony results for the points in the NIEDERREITER2 dataset.
• diaphony_sobol.txt diaphony results for the points in the SOBOL dataset.
• diaphony_uniform.txt diaphony results for the points in the UNIFORM dataset.

List of Routines:

• MAIN is the main program for TABLE_DIAPHONY.
• DIAPHONY evaluates the diaphony of a N-dimensional point set.
• F evaluates the function used to calculate the diaphony.
• FILE_COLUMN_COUNT counts the number of columns in the first line of a file.
• FILE_LINE_COUNT counts the number of lines in a file.
• GET_FILE_NAME gets the input file name.
• GET_UNIT returns a free FORTRAN unit number.
• R8_MODP returns the nonnegative remainder of real division.
• READ_INPUT_FILE reads the data from the input file.
• S_WORD_COUNT counts the number of "words" in a string.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.