mirror of
https://github.com/saitohirga/WSJT-X.git
synced 2024-09-06 21:36:44 -04:00
155 lines
5.1 KiB
Plaintext
155 lines
5.1 KiB
Plaintext
[section:cauchy_dist Cauchy-Lorentz Distribution]
|
|
|
|
``#include <boost/math/distributions/cauchy.hpp>``
|
|
|
|
template <class RealType = double,
|
|
class ``__Policy`` = ``__policy_class`` >
|
|
class cauchy_distribution;
|
|
|
|
typedef cauchy_distribution<> cauchy;
|
|
|
|
template <class RealType, class ``__Policy``>
|
|
class cauchy_distribution
|
|
{
|
|
public:
|
|
typedef RealType value_type;
|
|
typedef Policy policy_type;
|
|
|
|
cauchy_distribution(RealType location = 0, RealType scale = 1);
|
|
|
|
RealType location()const;
|
|
RealType scale()const;
|
|
};
|
|
|
|
The [@http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy-Lorentz distribution]
|
|
is named after Augustin Cauchy and Hendrik Lorentz.
|
|
It is a [@http://en.wikipedia.org/wiki/Probability_distribution continuous probability distribution]
|
|
with [@http://en.wikipedia.org/wiki/Probability_distribution probability distribution function PDF]
|
|
given by:
|
|
|
|
[equation cauchy_ref1]
|
|
|
|
The location parameter x[sub 0][space] is the location of the
|
|
peak of the distribution (the mode of the distribution),
|
|
while the scale parameter [gamma][space] specifies half the width
|
|
of the PDF at half the maximum height. If the location is
|
|
zero, and the scale 1, then the result is a standard Cauchy
|
|
distribution.
|
|
|
|
The distribution is important in physics as it is the solution
|
|
to the differential equation describing forced resonance,
|
|
while in spectroscopy it is the description of the line shape
|
|
of spectral lines.
|
|
|
|
The following graph shows how the distributions moves as the
|
|
location parameter changes:
|
|
|
|
[graph cauchy_pdf1]
|
|
|
|
While the following graph shows how the shape (scale) parameter alters
|
|
the distribution:
|
|
|
|
[graph cauchy_pdf2]
|
|
|
|
[h4 Member Functions]
|
|
|
|
cauchy_distribution(RealType location = 0, RealType scale = 1);
|
|
|
|
Constructs a Cauchy distribution, with location parameter /location/
|
|
and scale parameter /scale/. When these parameters take their default
|
|
values (location = 0, scale = 1)
|
|
then the result is a Standard Cauchy Distribution.
|
|
|
|
Requires scale > 0, otherwise calls __domain_error.
|
|
|
|
RealType location()const;
|
|
|
|
Returns the location parameter of the distribution.
|
|
|
|
RealType scale()const;
|
|
|
|
Returns the scale parameter of the distribution.
|
|
|
|
[h4 Non-member Accessors]
|
|
|
|
All the [link math_toolkit.dist_ref.nmp usual non-member accessor functions]
|
|
that are generic to all distributions are supported: __usual_accessors.
|
|
|
|
Note however that the Cauchy distribution does not have a mean,
|
|
standard deviation, etc. See __math_undefined
|
|
[/link math_toolkit.pol_ref.assert_undefined mathematically undefined function]
|
|
to control whether these should fail to compile with a BOOST_STATIC_ASSERTION_FAILURE,
|
|
which is the default.
|
|
|
|
Alternately, the functions __mean, __sd,
|
|
__variance, __skewness, __kurtosis and __kurtosis_excess will all
|
|
return a __domain_error if called.
|
|
|
|
The domain of the random variable is \[-[max_value], +[min_value]\].
|
|
|
|
[h4 Accuracy]
|
|
|
|
The Cauchy distribution is implemented in terms of the
|
|
standard library `tan` and `atan` functions,
|
|
and as such should have very low error rates.
|
|
|
|
[h4 Implementation]
|
|
|
|
[def __x0 x[sub 0 ]]
|
|
|
|
In the following table __x0 is the location parameter of the distribution,
|
|
[gamma][space] is its scale parameter,
|
|
/x/ is the random variate, /p/ is the probability and /q = 1-p/.
|
|
|
|
[table
|
|
[[Function][Implementation Notes]]
|
|
[[pdf][Using the relation: pdf = 1 / ([pi] * [gamma] * (1 + ((x - __x0) / [gamma])[super 2]) ]]
|
|
[[cdf and its complement][
|
|
The cdf is normally given by:
|
|
|
|
p = 0.5 + atan(x)/[pi]
|
|
|
|
But that suffers from cancellation error as x -> -[infin].
|
|
So recall that for `x < 0`:
|
|
|
|
atan(x) = -[pi]/2 - atan(1/x)
|
|
|
|
Substituting into the above we get:
|
|
|
|
p = -atan(1/x) ; x < 0
|
|
|
|
So the procedure is to calculate the cdf for -fabs(x)
|
|
using the above formula. Note that to factor in the location and scale
|
|
parameters you must substitute (x - __x0) / [gamma][space] for x in the above.
|
|
|
|
This procedure yields the smaller of /p/ and /q/, so the result
|
|
may need subtracting from 1 depending on whether we want the complement
|
|
or not, and whether /x/ is less than __x0 or not.
|
|
]]
|
|
[[quantile][The same procedure is used irrespective of whether we're starting
|
|
from the probability or its complement. First the argument /p/ is
|
|
reduced to the range \[-0.5, 0.5\], then the relation
|
|
|
|
x = __x0 [plusminus] [gamma][space] / tan([pi] * p)
|
|
|
|
is used to obtain the result. Whether we're adding
|
|
or subtracting from __x0 is determined by whether we're
|
|
starting from the complement or not.]]
|
|
[[mode][The location parameter.]]
|
|
]
|
|
|
|
[h4 References]
|
|
|
|
* [@http://en.wikipedia.org/wiki/Cauchy_distribution Cauchy-Lorentz distribution]
|
|
* [@http://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm NIST Exploratory Data Analysis]
|
|
* [@http://mathworld.wolfram.com/CauchyDistribution.html Weisstein, Eric W. "Cauchy Distribution." From MathWorld--A Wolfram Web Resource.]
|
|
|
|
[endsect][/section:cauchy_dist Cauchi]
|
|
|
|
[/ cauchy.qbk
|
|
Copyright 2006, 2007 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).
|
|
]
|