mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-11-17 17:42:02 -05:00
369 lines
14 KiB
Plaintext
369 lines
14 KiB
Plaintext
[section:ibeta_inv_function The Incomplete Beta Function Inverses]
|
|
|
|
``
|
|
#include <boost/math/special_functions/beta.hpp>
|
|
``
|
|
|
|
namespace boost{ namespace math{
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3, class T4>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py);
|
|
|
|
template <class T1, class T2, class T3, class T4, class ``__Policy``>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3, class T4>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py);
|
|
|
|
template <class T1, class T2, class T3, class T4, class ``__Policy``>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibetac_inva(T1 b, T2 x, T3 q);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibetac_inva(T1 b, T2 x, T3 q, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibeta_invb(T1 a, T2 x, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibeta_invb(T1 a, T2 x, T3 p, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibetac_invb(T1 a, T2 x, T3 q);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibetac_invb(T1 a, T2 x, T3 q, const ``__Policy``&);
|
|
|
|
}} // namespaces
|
|
|
|
[h4 Description]
|
|
|
|
|
|
There are six [@http://functions.wolfram.com/GammaBetaErf/ incomplete beta function inverses]
|
|
which allow you solve for
|
|
any of the three parameters to the incomplete beta, starting from either
|
|
the result of the incomplete beta (p) or its complement (q).
|
|
|
|
[optional_policy]
|
|
|
|
[tip When people normally talk about the inverse of the incomplete
|
|
beta function, they are talking about inverting on parameter /x/.
|
|
These are implemented here as ibeta_inv and ibetac_inv, and are by
|
|
far the most efficient of the inverses presented here.
|
|
|
|
The inverses on the /a/ and /b/ parameters find use in some statistical
|
|
applications, but have to be computed by rather brute force numerical
|
|
techniques and are consequently several times slower.
|
|
These are implemented here as ibeta_inva and ibeta_invb,
|
|
and complement versions ibetac_inva and ibetac_invb.]
|
|
|
|
The return type of these functions is computed using the __arg_promotion_rules
|
|
when called with arguments T1...TN of different types.
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3, class T4>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py);
|
|
|
|
template <class T1, class T2, class T3, class T4, class ``__Policy``>
|
|
``__sf_result`` ibeta_inv(T1 a, T2 b, T3 p, T4* py, const ``__Policy``&);
|
|
|
|
Returns a value /x/ such that: `p = ibeta(a, b, x);`
|
|
and sets `*py = 1 - x` when the `py` parameter is provided and is non-null.
|
|
Note that internally this function computes whichever is the smaller of
|
|
`x` and `1-x`, and therefore the value assigned to `*py` is free from
|
|
cancellation errors. That means that even if the function returns `1`, the
|
|
value stored in `*py` may be non-zero, albeit very small.
|
|
|
|
Requires: /a,b > 0/ and /0 <= p <= 1/.
|
|
|
|
[optional_policy]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, const ``__Policy``&);
|
|
|
|
template <class T1, class T2, class T3, class T4>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py);
|
|
|
|
template <class T1, class T2, class T3, class T4, class ``__Policy``>
|
|
``__sf_result`` ibetac_inv(T1 a, T2 b, T3 q, T4* py, const ``__Policy``&);
|
|
|
|
Returns a value /x/ such that: `q = ibetac(a, b, x);`
|
|
and sets `*py = 1 - x` when the `py` parameter is provided and is non-null.
|
|
Note that internally this function computes whichever is the smaller of
|
|
`x` and `1-x`, and therefore the value assigned to `*py` is free from
|
|
cancellation errors. That means that even if the function returns `1`, the
|
|
value stored in `*py` may be non-zero, albeit very small.
|
|
|
|
Requires: /a,b > 0/ and /0 <= q <= 1/.
|
|
|
|
[optional_policy]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibeta_inva(T1 b, T2 x, T3 p, const ``__Policy``&);
|
|
|
|
Returns a value /a/ such that: `p = ibeta(a, b, x);`
|
|
|
|
Requires: /b > 0/, /0 < x < 1/ and /0 <= p <= 1/.
|
|
|
|
[optional_policy]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibetac_inva(T1 b, T2 x, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibetac_inva(T1 b, T2 x, T3 p, const ``__Policy``&);
|
|
|
|
Returns a value /a/ such that: `q = ibetac(a, b, x);`
|
|
|
|
Requires: /b > 0/, /0 < x < 1/ and /0 <= q <= 1/.
|
|
|
|
[optional_policy]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibeta_invb(T1 b, T2 x, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibeta_invb(T1 b, T2 x, T3 p, const ``__Policy``&);
|
|
|
|
Returns a value /b/ such that: `p = ibeta(a, b, x);`
|
|
|
|
Requires: /a > 0/, /0 < x < 1/ and /0 <= p <= 1/.
|
|
|
|
[optional_policy]
|
|
|
|
template <class T1, class T2, class T3>
|
|
``__sf_result`` ibetac_invb(T1 b, T2 x, T3 p);
|
|
|
|
template <class T1, class T2, class T3, class ``__Policy``>
|
|
``__sf_result`` ibetac_invb(T1 b, T2 x, T3 p, const ``__Policy``&);
|
|
|
|
Returns a value /b/ such that: `q = ibetac(a, b, x);`
|
|
|
|
Requires: /a > 0/, /0 < x < 1/ and /0 <= q <= 1/.
|
|
|
|
[optional_policy]
|
|
|
|
[h4 Accuracy]
|
|
|
|
The accuracy of these functions should closely follow that
|
|
of the regular forward incomplete beta functions. However,
|
|
note that in some parts of their domain, these functions can
|
|
be extremely sensitive to changes in input, particularly when
|
|
the argument /p/ (or it's complement /q/) is very close to `0` or `1`.
|
|
|
|
Comparisons to other libraries are shown below, note that our test data
|
|
exercises some rather extreme cases in the incomplete beta function
|
|
which many other libraries fail to handle:
|
|
|
|
[table_ibeta_inv]
|
|
|
|
[table_ibetac_inv]
|
|
|
|
[table_ibeta_inva]
|
|
|
|
[table_ibetac_inva]
|
|
|
|
[table_ibeta_invb]
|
|
|
|
[table_ibetac_invb]
|
|
|
|
[h4 Testing]
|
|
|
|
There are two sets of tests:
|
|
|
|
* Basic sanity checks attempt to "round-trip" from
|
|
/a, b/ and /x/ to /p/ or /q/ and back again. These tests have quite
|
|
generous tolerances: in general both the incomplete beta and its
|
|
inverses change so rapidly, that round tripping to more than a couple
|
|
of significant digits isn't possible. This is especially true when
|
|
/p/ or /q/ is very near one: in this case there isn't enough
|
|
"information content" in the input to the inverse function to get
|
|
back where you started.
|
|
* Accuracy checks using high precision test values. These measure
|
|
the accuracy of the result, given exact input values.
|
|
|
|
[h4 Implementation of ibeta_inv and ibetac_inv]
|
|
|
|
These two functions share a common implementation.
|
|
|
|
First an initial approximation to x is computed then the
|
|
last few bits are cleaned up using
|
|
[@http://en.wikipedia.org/wiki/Simple_rational_approximation Halley iteration].
|
|
The iteration limit is set to 1/2 of the number of bits in T, which by experiment is
|
|
sufficient to ensure that the inverses are at least as accurate as the normal
|
|
incomplete beta functions. Up to 5 iterations may be
|
|
required in extreme cases, although normally only one or two are required.
|
|
Further, the number of iterations required decreases with increasing /a/ and
|
|
/b/ (which generally form the more important use cases).
|
|
|
|
The initial guesses used for iteration are obtained as follows:
|
|
|
|
Firstly recall that:
|
|
|
|
[equation ibeta_inv5]
|
|
|
|
We may wish to start from either p or q, and to calculate either x or y.
|
|
In addition at
|
|
any stage we can exchange a for b, p for q, and x for y if it results in a
|
|
more manageable problem.
|
|
|
|
For `a+b >= 5` the initial guess is computed using the methods described in:
|
|
|
|
Asymptotic Inversion of the Incomplete Beta Function,
|
|
by N. M. [@http://homepages.cwi.nl/~nicot/ Temme].
|
|
Journal of Computational and Applied Mathematics 41 (1992) 145-157.
|
|
|
|
The nearly symmetrical case (section 2 of the paper) is used for
|
|
|
|
[equation ibeta_inv2]
|
|
|
|
and involves solving the inverse error function first. The method is accurate
|
|
to at least 2 decimal digits when [^a = 5] rising to at least 8 digits when
|
|
[^a = 10[super 5]].
|
|
|
|
The general error function case (section 3 of the paper) is used for
|
|
|
|
[equation ibeta_inv3]
|
|
|
|
and again expresses the inverse incomplete beta in terms of the
|
|
inverse of the error function. The method is accurate to at least
|
|
2 decimal digits when [^a+b = 5] rising to 11 digits when [^a+b = 10[super 5]].
|
|
However, when the result is expected to be very small, and when a+b is
|
|
also small, then its accuracy tails off, in this case when p[super 1/a] < 0.0025
|
|
then it is better to use the following as an initial estimate:
|
|
|
|
[equation ibeta_inv4]
|
|
|
|
Finally the for all other cases where `a+b > 5` the method of section
|
|
4 of the paper is used. This expresses the inverse incomplete beta in terms
|
|
of the inverse of the incomplete gamma function, and is therefore significantly
|
|
more expensive to compute than the other cases. However the method is accurate
|
|
to at least 3 decimal digits when [^a = 5] rising to at least 10 digits when
|
|
[^a = 10[super 5]]. This method is limited to a > b, and therefore we need to perform
|
|
an exchange a for b, p for q and x for y when this is not the case. In addition
|
|
when p is close to 1 the method is inaccurate should we actually want y rather
|
|
than x as output. Therefore when q is small ([^q[super 1/p] < 10[super -3]]) we use:
|
|
|
|
[equation ibeta_inv6]
|
|
|
|
which is both cheaper to compute than the full method, and a more accurate
|
|
estimate on q.
|
|
|
|
When a and b are both small there is a distinct lack of information in the
|
|
literature on how to proceed. I am extremely grateful to Prof Nico Temme
|
|
who provided the following information with a great deal of patience and
|
|
explanation on his part. Any errors that follow are entirely my own, and not
|
|
Prof Temme's.
|
|
|
|
When a and b are both less than 1, then there is a point of inflection in
|
|
the incomplete beta at point `xs = (1 - a) / (2 - a - b)`. Therefore if
|
|
[^p > I[sub x](a,b)] we swap a for b, p for q and x for y, so that now we always
|
|
look for a point x below the point of inflection `xs`, and on a convex curve.
|
|
An initial estimate for x is made with:
|
|
|
|
[equation ibeta_inv7]
|
|
|
|
which is provably below the true value for x:
|
|
[@http://en.wikipedia.org/wiki/Newton%27s_method Newton iteration] will
|
|
therefore smoothly converge on x without problems caused by overshooting etc.
|
|
|
|
When a and b are both greater than 1, but a+b is too small to use the other
|
|
methods mentioned above, we proceed as follows. Observe that there is a point
|
|
of inflection in the incomplete beta at `xs = (1 - a) / (2 - a - b)`. Therefore
|
|
if [^p > I[sub x](a,b)] we swap a for b, p for q and x for y, so that now we always
|
|
look for a point x below the point of inflection `xs`, and on a concave curve.
|
|
An initial estimate for x is made with:
|
|
|
|
[equation ibeta_inv4]
|
|
|
|
which can be improved somewhat to:
|
|
|
|
[equation ibeta_inv1]
|
|
|
|
when b and x are both small (I've used b < a and x < 0.2). This
|
|
actually under-estimates x, which drops us on the wrong side of x for Newton
|
|
iteration to converge monotonically. However, use of higher derivatives
|
|
and Halley iteration keeps everything under control.
|
|
|
|
The final case to be considered if when one of a and b is less than or equal
|
|
to 1, and the other greater that 1. Here, if b < a we swap a for b, p for q
|
|
and x for y. Now the curve of the incomplete beta is convex with no points
|
|
of inflection in [0,1]. For small p, x can be estimated using
|
|
|
|
[equation ibeta_inv4]
|
|
|
|
which under-estimates x, and drops us on the right side of the true value
|
|
for Newton iteration to converge monotonically. However, when p is large
|
|
this can quite badly underestimate x. This is especially an issue when we
|
|
really want to find y, in which case this method can be an arbitrary number
|
|
of order of magnitudes out, leading to very poor convergence during iteration.
|
|
|
|
Things can be improved by considering the incomplete beta as a distorted
|
|
quarter circle, and estimating y from:
|
|
|
|
[equation ibeta_inv8]
|
|
|
|
This doesn't guarantee that we will drop in on the right side of x for
|
|
monotonic convergence, but it does get us close enough that Halley iteration
|
|
rapidly converges on the true value.
|
|
|
|
[h4 Implementation of inverses on the a and b parameters]
|
|
|
|
These four functions share a common implementation.
|
|
|
|
First an initial approximation is computed for /a/ or /b/:
|
|
where possible this uses a Cornish-Fisher expansion for the
|
|
negative binomial distribution to get within around 1 of the
|
|
result. However, when /a/ or /b/ are very small the Cornish Fisher
|
|
expansion is not usable, in this case the initial approximation
|
|
is chosen so that
|
|
I[sub x](a, b) is near the middle of the range [0,1].
|
|
|
|
This initial guess is then
|
|
used as a starting value for a generic root finding algorithm. The
|
|
algorithm converges rapidly on the root once it has been
|
|
bracketed, but bracketing the root may take several iterations.
|
|
A better initial approximation for /a/ or /b/ would improve these
|
|
functions quite substantially: currently 10-20 incomplete beta
|
|
function invocations are required to find the root.
|
|
|
|
[endsect][/section:ibeta_inv_function The Incomplete Beta Function Inverses]
|
|
|
|
[/
|
|
Copyright 2006 John Maddock and Paul A. Bristow.
|
|
Distributed under the Boost Software License, Version 1.0.
|
|
(See accompanying file LICENSE_1_0.txt or copy at
|
|
http://www.boost.org/LICENSE_1_0.txt).
|
|
]
|