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407 lines
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Plaintext
407 lines
14 KiB
Plaintext
[section:nmp Non-Member Properties]
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Properties that are common to all distributions are accessed via non-member
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getter functions: non-membership allows more of these functions to be added over time,
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as the need arises. Unfortunately the literature uses many different and
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confusing names to refer to a rather small number of actual concepts; refer
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to the [link math_toolkit.dist_ref.nmp.concept_index concept index] to find the property you
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want by the name you are most familiar with.
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Or use the [link math_toolkit.dist_ref.nmp.function_index function index]
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to go straight to the function you want if you already know its name.
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[h4:function_index Function Index]
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* __cdf.
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* __ccdf.
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* __chf.
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* __hazard.
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* __kurtosis.
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* __kurtosis_excess
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* __mean.
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* __median.
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* __mode.
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* __pdf.
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* __range.
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* __quantile.
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* __quantile_c.
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* __skewness.
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* __sd.
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* __support.
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* __variance.
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[h4:concept_index Conceptual Index]
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* __ccdf.
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* __cdf.
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* __chf.
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* [link math_toolkit.dist_ref.nmp.cdf_inv Inverse Cumulative Distribution Function].
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* [link math_toolkit.dist_ref.nmp.survival_inv Inverse Survival Function].
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* __hazard
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* [link math_toolkit.dist_ref.nmp.lower_critical Lower Critical Value].
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* __kurtosis.
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* __kurtosis_excess
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* __mean.
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* __median.
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* __mode.
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* [link math_toolkit.dist_ref.nmp.cdfPQ P].
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* [link math_toolkit.dist_ref.nmp.percent Percent Point Function].
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* __pdf.
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* [link math_toolkit.dist_ref.nmp.pmf Probability Mass Function].
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* __range.
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* [link math_toolkit.dist_ref.nmp.cdfPQ Q].
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* __quantile.
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* [link math_toolkit.dist_ref.nmp.quantile_c Quantile from the complement of the probability].
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* __skewness.
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* __sd
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* [link math_toolkit.dist_ref.nmp.survival Survival Function].
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* [link math_toolkit.dist_ref.nmp.support support].
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* [link math_toolkit.dist_ref.nmp.upper_critical Upper Critical Value].
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* __variance.
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[h4:cdf Cumulative Distribution Function]
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template <class RealType, class ``__Policy``>
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RealType cdf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);
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The __cdf is the probability that
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the variable takes a value less than or equal to x. It is equivalent
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to the integral from -infinity to x of the __pdf.
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This function may return a __domain_error if the random variable is outside
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the defined range for the distribution.
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For example, the following graph shows the cdf for the
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normal distribution:
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[$../graphs/cdf.png]
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[h4:ccdf Complement of the Cumulative Distribution Function]
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template <class Distribution, class RealType>
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RealType cdf(const ``['Unspecified-Complement-Type]``<Distribution, RealType>& comp);
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The complement of the __cdf
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is the probability that
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the variable takes a value greater than x. It is equivalent
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to the integral from x to infinity of the __pdf, or 1 minus the __cdf of x.
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This is also known as the survival function.
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This function may return a __domain_error if the random variable is outside
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the defined range for the distribution.
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In this library, it is obtained by wrapping the arguments to the `cdf`
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function in a call to `complement`, for example:
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// standard normal distribution object:
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boost::math::normal norm;
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// print survival function for x=2.0:
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std::cout << cdf(complement(norm, 2.0)) << std::endl;
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For example, the following graph shows the __complement of the cdf for the
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normal distribution:
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[$../graphs/survival.png]
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See __why_complements for why the complement is useful and when it should be used.
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[h4:hazard Hazard Function]
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template <class RealType, class ``__Policy``>
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RealType hazard(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);
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Returns the __hazard of /x/ and distibution /dist/.
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This function may return a __domain_error if the random variable is outside
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the defined range for the distribution.
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[equation hazard]
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[caution
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Some authors refer to this as the conditional failure
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density function rather than the hazard function.]
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[h4:chf Cumulative Hazard Function]
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template <class RealType, class ``__Policy``>
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RealType chf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);
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Returns the __chf of /x/ and distibution /dist/.
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This function may return a __domain_error if the random variable is outside
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the defined range for the distribution.
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[equation chf]
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[caution
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Some authors refer to this as simply the "Hazard Function".]
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[h4:mean mean]
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template<class RealType, class ``__Policy``>
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RealType mean(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the mean of the distribution /dist/.
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This function may return a __domain_error if the distribution does not have
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a defined mean (for example the Cauchy distribution).
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[h4:median median]
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template<class RealType, class ``__Policy``>
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RealType median(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the median of the distribution /dist/.
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[h4:mode mode]
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template<class RealType, ``__Policy``>
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RealType mode(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the mode of the distribution /dist/.
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This function may return a __domain_error if the distribution does not have
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a defined mode.
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[h4:pdf Probability Density Function]
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template <class RealType, class ``__Policy``>
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RealType pdf(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& x);
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For a continuous function, the probability density function (pdf) returns
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the probability that the variate has the value x.
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Since for continuous distributions the probability at a single point is actually zero,
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the probability is better expressed as the integral of the pdf between two points:
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see the __cdf.
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For a discrete distribution, the pdf is the probability that the
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variate takes the value x.
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This function may return a __domain_error if the random variable is outside
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the defined range for the distribution.
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For example, for a standard normal distribution the pdf looks like this:
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[$../graphs/pdf.png]
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[h4:range Range]
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template<class RealType, class ``__Policy``>
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std::pair<RealType, RealType> range(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the valid range of the random variable over distribution /dist/.
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[h4:quantile Quantile]
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template <class RealType, class ``__Policy``>
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RealType quantile(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist, const RealType& p);
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The quantile is best viewed as the inverse of the __cdf, it returns
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a value /x/ such that `cdf(dist, x) == p`.
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This is also known as the /percent point function/, or /percentile/, or /fractile/,
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it is also the same as calculating the ['lower critical value] of a distribution.
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This function returns a __domain_error if the probability lies outside [0,1].
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The function may return an __overflow_error if there is no finite value
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that has the specified probability.
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The following graph shows the quantile function for a standard normal
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distribution:
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[$../graphs/quantile.png]
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[h4:quantile_c Quantile from the complement of the probability.]
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See also [link math_toolkit.stat_tut.overview.complements complements].
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template <class Distribution, class RealType>
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RealType quantile(const ``['Unspecified-Complement-Type]``<Distribution, RealType>& comp);
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This is the inverse of the __ccdf. It is calculated by wrapping
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the arguments in a call to the quantile function in a call to
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/complement/. For example:
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// define a standard normal distribution:
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boost::math::normal norm;
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// print the value of x for which the complement
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// of the probability is 0.05:
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std::cout << quantile(complement(norm, 0.05)) << std::endl;
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The function computes a value /x/ such that
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`cdf(complement(dist, x)) == q` where /q/ is complement of the
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probability.
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[link why_complements Why complements?]
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This function is also called the inverse survival function, and is the
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same as calculating the ['upper critical value] of a distribution.
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This function returns a __domain_error if the probablity lies outside [0,1].
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The function may return an __overflow_error if there is no finite value
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that has the specified probability.
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The following graph show the inverse survival function for the normal
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distribution:
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[$../graphs/survival_inv.png]
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[h4:sd Standard Deviation]
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template <class RealType, class ``__Policy``>
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RealType standard_deviation(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the standard deviation of distribution /dist/.
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This function may return a __domain_error if the distribution does not have
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a defined standard deviation.
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[h4:support support]
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template<class RealType, class ``__Policy``>
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std::pair<RealType, RealType> support(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the supported range of random variable over the distribution /dist/.
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The distribution is said to be 'supported' over a range that is
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[@http://en.wikipedia.org/wiki/Probability_distribution
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"the smallest closed set whose complement has probability zero"].
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Non-mathematicians might say it means the 'interesting' smallest range
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of random variate x that has the cdf going from zero to unity.
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Outside are uninteresting zones where the pdf is zero, and the cdf zero or unity.
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[h4:variance Variance]
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template <class RealType, class ``__Policy``>
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RealType variance(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the variance of the distribution /dist/.
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This function may return a __domain_error if the distribution does not have
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a defined variance.
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[h4:skewness Skewness]
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template <class RealType, class ``__Policy``>
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RealType skewness(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the skewness of the distribution /dist/.
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This function may return a __domain_error if the distribution does not have
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a defined skewness.
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[h4:kurtosis Kurtosis]
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template <class RealType, class ``__Policy``>
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RealType kurtosis(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the 'proper' kurtosis (normalized fourth moment) of the distribution /dist/.
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kertosis = [beta][sub 2][space]= [mu][sub 4][space] / [mu][sub 2][super 2]
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Where [mu][sub i][space] is the i'th central moment of the distribution, and
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in particular [mu][sub 2][space] is the variance of the distribution.
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The kurtosis is a measure of the "peakedness" of a distribution.
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Note that the literature definition of kurtosis is confusing.
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The definition used here is that used by for example
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[@http://mathworld.wolfram.com/Kurtosis.html Wolfram MathWorld]
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(that includes a table of formulae for kurtosis excess for various distributions)
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but NOT the definition of
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[@http://en.wikipedia.org/wiki/Kurtosis kurtosis used by Wikipedia]
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which treats "kurtosis" and "kurtosis excess" as the same quantity.
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kurtosis_excess = 'proper' kurtosis - 3
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This subtraction of 3 is convenient so that the ['kurtosis excess]
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of a normal distribution is zero.
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This function may return a __domain_error if the distribution does not have
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a defined kurtosis.
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'Proper' kurtosis can have a value from zero to + infinity.
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[h4:kurtosis_excess Kurtosis excess]
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template <class RealType, ``__Policy``>
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RealType kurtosis_excess(const ``['Distribution-Type]``<RealType, ``__Policy``>& dist);
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Returns the kurtosis excess of the distribution /dist/.
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kurtosis excess = [gamma][sub 2][space]= [mu][sub 4][space] / [mu][sub 2][super 2][space]- 3 = kurtosis - 3
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Where [mu][sub i][space] is the i'th central moment of the distribution, and
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in particular [mu][sub 2][space] is the variance of the distribution.
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The kurtosis excess is a measure of the "peakedness" of a distribution, and
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is more widely used than the "kurtosis proper". It is defined so that
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the kurtosis excess of a normal distribution is zero.
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This function may return a __domain_error if the distribution does not have
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a defined kurtosis excess.
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Kurtosis excess can have a value from -2 to + infinity.
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kurtosis = kurtosis_excess +3;
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The kurtosis excess of a normal distribution is zero.
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[h4:cdfPQ P and Q]
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The terms P and Q are sometimes used to refer to the __cdf
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and its [link math_toolkit.dist_ref.nmp.ccdf complement] respectively.
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Lowercase p and q are sometimes used to refer to the values returned
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by these functions.
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[h4:percent Percent Point Function or Percentile]
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The percent point function, also known as the percentile, is the same as
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the __quantile.
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[h4:cdf_inv Inverse CDF Function.]
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The inverse of the cumulative distribution function, is the same as the
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__quantile.
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[h4:survival_inv Inverse Survival Function.]
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The inverse of the survival function, is the same as computing the
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[link math_toolkit.dist_ref.nmp.quantile_c quantile
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from the complement of the probability].
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[h4:pmf Probability Mass Function]
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The Probability Mass Function is the same as the __pdf.
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The term Mass Function is usually applied to discrete distributions,
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while the term __pdf applies to continuous distributions.
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[h4:lower_critical Lower Critical Value.]
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The lower critical value calculates the value of the random variable
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given the area under the left tail of the distribution.
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It is equivalent to calculating the __quantile.
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[h4: upper_critical Upper Critical Value.]
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The upper critical value calculates the value of the random variable
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given the area under the right tail of the distribution. It is equivalent to
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calculating the [link math_toolkit.dist_ref.nmp.quantile_c quantile from the complement of the
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probability].
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[h4:survival Survival Function]
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Refer to the __ccdf.
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[endsect][/section:nmp Non-Member Properties]
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[/ non_members.qbk
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Copyright 2006 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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