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Discrete Discrepancy


Given a point set P={x_n}_(n=0)^(N-1) in the s-dimensional unit cube [0,1)^s, the star discrepancy is defined as

 D_N^*(P)=sup_(J in Upsilon^*)D(J,P),
(1)

where the local discrepancy is defined as

 D(J,P)=|(number of x_n in J)/N-Vol(J)|,
(2)

Vol(J) is the content of J, and Upsilon^* is the class of all discrete subintervals J of I of the form

 J=product_(i=1)^s[0,a_i/M)
(3)

with 0<=a_i<M.


See also

Discrepancy, Local Discrepancy, Star Discrepancy

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References

Niederreiter, H. Random Number Generation and Quasi-Monte Carlo Methods. Philadelphia, PA: SIAM, 1992.

Referenced on Wolfram|Alpha

Discrete Discrepancy

Cite this as:

Weisstein, Eric W. "Discrete Discrepancy." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DiscreteDiscrepancy.html

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