diaphony

diaphony, a C++ code 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 information on this web page is distributed under the MIT 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:

diaphony_test

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 C++ program which reads a file of points and creates a "latinized" version by adjusting the data.

TABLE_QUALITY, a C++ 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.cpp, the source code.
• diaphony.sh, compiles the source code.