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

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.

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	0.010

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

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

Related Data and Programs:

STAR_DISCREPANCY, a C++ program which reads a TABLE file of points (presumed to lie in the unit hypercube) and computes bounds on the star discrepancy, a measure of dispersion, by Eric Thiemard.

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

TABLE_QUALITY, a FORTRAN90 program which reads a file of points and computes the quality of dispersion.

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.f90, the source code.

List of Routines:

• MAIN is the main program for DIAPHONY.
• CH_CAP capitalizes a single character.
• CH_EQI is a case insensitive comparison of two characters for equality.
• CH_TO_DIGIT returns the integer value of a base 10 digit.
• 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_ROW_COUNT counts the number of row records in a file.
• GET_UNIT returns a free FORTRAN unit number.
• R8_MODP returns the nonnegative remainder of real division.
• R8MAT_DATA_READ reads data from an R8MAT file.
• R8MAT_HEADER_READ reads the header from an R8MAT file.
• S_TO_R8 reads an R8 from a string.
• S_TO_R8VEC reads an R8VEC from a string.
• 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.