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

DIAPHONY is a MATLAB 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

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:

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

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

Reference:

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.
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.
Peter Heelekalek, Hannes Leeb,
Dyadic Diaphony,
Acta Arithmetica,
Volume 80, Number 2, 1997, pages 187-196.

Source Code:

diaphony.m, the source code.
timestamp.m, prints the YMDHMS date as a timestamp.