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https://issues.apache.org/jira/browse/MATH-699?page=com.atlassian.jira.plugin.system.issuetabpanels:comment-tabpanel&focusedCommentId=13145211#comment-13145211
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Sébastien Brisard commented on MATH-699:
----------------------------------------
Hi,
I thought I would keep you up to date with the problems I'm facing at the
moment.
First of all, I see no smart way to check that for sure,
{{inverseCumulativeProbability(0)}} (resp. {{inverseCumulativeProbability(1)}})
should return {{Double.NEGATIVE_INFINITY}} (resp.
{{Double.POSITIVE_INFINITY}}). What could be done would be to test for a
neighbouring value, and check that the cumulative probability is slightly above
zero (resp. below one). But this makes no sense, because the only neighbouring
value which would make sense would be {{+/-Double.MAX_VALUE}}, and this surely
return 0.0 or 1.0. So this is my first problem.
The way I see things is as follows: users might have no clue about the inverse
cumulative probability, but they *must* know the values of this inverse for p =
0 and p = 1. So I would suggest to change the contract of
{{getInitialDomain(p)}}. I would make clear in the javadoc that
{{getInitialDomain(p)}} should return {{Double.NEGATIVE_INFINITY}} *if, and
only if* {{inverseCumulativeProbability(0) == Double.NEGATIVE_INFINITY}}.
Similarly, {{getInitialDomain(p)}} should return {{Double.POSITIVE_INFINITY}}
*if, and only if* {{inverseCumulativeProbability(1) ==
Double.POSITIVE_INFINITY}}.
My second problem is to check for the presence of plateaux, consistently with
finite precision. Here is my initial idea. I first define the two absolute
accuracies
* {{dx = getSolverAbsoluteAccuracy()}},
* {{dp = getSolverFunctionValueAccuracy()}}.
Then, if {{x}} is the root found by the solver: we do have
{{cumulativeProbability( x ) == p}} (within a specified accuracy). The problem
is to check whether there is a *smaller* value which also satisfies this
requirement (in which case, the smallest such value must be returned).
My initial test was {{cumulativeProbability(x - dx) == p}}. Then, I tried
{{FastMath.abs(cumulativeProbability(x - dx) - p) <= dp}}. Although more
consistent with finite precision, this is not fully satisfactory, because in
simple cases (where there is no plateau), it might lead to the solver moving to
a somewhat less good point. At the very least, it would lead to additional
iterations... to finally get back to the initial point {{x}}.
Then, I thought of checking for *exact* nullity of the pdf at x. The problem is
that the pdf might have discontinuities, in which case, this simple test might
fail (if {{x}} turns out to be the higher-end of the plateau, and there is a
slope discontinuity here).
So, I'm now back to this test: {{cumulativeProbability(x - dx) ==
cumulativeProbability( x )}}. Please note
* *exact* equality test,
* I'm no longer testing for equality with the target value {{p}}, but with its
estimate {{cumulativeProbability( x )}}.
I would be grateful for some feedback on these issues. Also, I think it is
clear that
* my code will require careful reviewers!!!
* it won't be completely fool-proof.
> inverseCumulativeDistribution fails with cumulative distribution having a
> plateau
> ---------------------------------------------------------------------------------
>
> Key: MATH-699
> URL: https://issues.apache.org/jira/browse/MATH-699
> Project: Commons Math
> Issue Type: Bug
> Affects Versions: 3.0
> Reporter: Sébastien Brisard
> Assignee: Sébastien Brisard
> Priority: Minor
> Attachments: AbstractContinuousDistributionTest.java
>
>
> This bug report follows MATH-692. The attached unit test fails. As required
> by the definition in MATH-692, the lower-bound of the interval on which the
> cdf is constant should be returned. This is not so at the moment.
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