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https://issues.apache.org/jira/browse/STATISTICS-100?page=com.atlassian.jira.plugin.system.issuetabpanels:all-tabpanel
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Alex Herbert resolved STATISTICS-100.
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Fix Version/s: 1.4
Resolution: Implemented
Added in commit:
615995edc843b7cd1cb8ca18206dea4e21902fc6
> Zipf distribution can use the Hurwitz zeta function to avoid summation over
> all PDF values
> ------------------------------------------------------------------------------------------
>
> Key: STATISTICS-100
> URL: https://issues.apache.org/jira/browse/STATISTICS-100
> Project: Commons Statistics
> Issue Type: Improvement
> Components: distribution
> Affects Versions: 1.3
> Reporter: Alex Herbert
> Assignee: Alex Herbert
> Priority: Minor
> Fix For: 1.4
>
>
> The [Zipf distribution
> (Wikipedia)|https://en.wikipedia.org/wiki/Zipf%27s_law#Formal_definition]
> uses a probability mass function (PMF) of:
> {noformat}
> 1 1
> pmf(k; N, s) = -- . ----
> s H
> k N,s
> {noformat}
> The normalising constant H_N,s is the N-th generalised harmonic number of
> order N of s.
> The cumulative probability for range [a, b] is computed by summation of the
> power term k^-s for k in [a, b]. This can be millions of power terms for one
> cumulative probability when the support [1, N] is large.
> When s is above 1 the cumulative probability can use the [Hurwitz zeta
> function (Wikipedia)|https://en.wikipedia.org/wiki/Hurwitz_zeta_function]:
> {noformat}
> oo 1
> zeta(s, a) = sum ------
> k=0 s
> (k+a)
> cdf(k; N, s) = (zeta(s, 1) - zeta(s, k+1)) / H_N,s
> sf(k; N, s) = (zeta(s, k+1) - zeta(s, N+1)) / H_N,s
> H_N,s = zeta(s, 1) - zeta(s, N+1)
> {noformat}
> The Hurwitz zeta function for s > 1 is absolutely convergent. This function
> can be used to significantly improve the performance of the Zipf distribution.
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