bump version to 12.2.2-pve1

[ceph.git] / ceph / src / boost / libs / math / doc / policies / policy_tutorial.qbk

[section:pol_tutorial Policy Tutorial]

[section:what_is_a_policy So Just What is a Policy Anyway?]

A policy is a compile-time mechanism for customising the behaviour of a
special function, or a statistical distribution.  With Policies you can
control:

* What action to take when an error occurs.
* What happens when you call a function that is mathematically undefined
(for example, if you ask for the mean of a Cauchy distribution).
* What happens when you ask for a quantile of a discrete distribution.
* Whether the library is allowed to internally promote `float` to `double`
and `double` to `long double` in order to improve precision.
* What precision to use when calculating the result.

Some of these policies could arguably be runtime variables, but then we couldn't
use compile-time dispatch internally to select the best evaluation method
for the given policies.

For this reason a Policy is a /type/: in fact it's an instance of the
class template `boost::math::policies::policy<>`.  This class is just a
compile-time-container of user-selected policies (sometimes called a type-list):

   using namespace boost::math::policies;
   //
   // Define a policy that sets ::errno on overflow, and does
   // not promote double to long double internally:
   //
   typedef policy<domain_error<errno_on_error>, promote_double<false> > mypolicy;

[endsect] [/section:what_is_a_policy So Just What is a Policy Anyway?]

[section:policy_tut_defaults Policies Have Sensible Defaults]

Most of the time you can just ignore the policy framework.

['*The defaults for the various policies are as follows,
if these work OK for you then you can stop reading now!]

[variablelist
[[Domain Error][Throws a `std::domain_error` exception.]]
[[Pole Error][Occurs when a function is evaluated at a pole: throws a `std::domain_error` exception.]]
[[Overflow Error][Throws a `std::overflow_error` exception.]]
[[Underflow][Ignores the underflow, and returns zero.]]
[[Denormalised Result][Ignores the fact that the result is denormalised, and returns it.]]
[[Rounding Error][Throws a `boost::math::rounding_error` exception.]]
[[Internal Evaluation Error][Throws a `boost::math::evaluation_error` exception.]]
[[Indeterminate Result Error][Returns a result that depends on the function where the error occurred.]]
[[Promotion of float to double][Does occur by default - gives full float precision results.]]
[[Promotion of double to long double][Does occur by default if long double offers
   more precision than double.]]
[[Precision of Approximation Used][By default uses an approximation that
   will result in the lowest level of error for the type of the result.]]
[[Behaviour of Discrete Quantiles]
   [
   The quantile function will by default return an integer result that has been
   /rounded outwards/.  That is to say lower quantiles (where the probability is
   less than 0.5) are rounded downward, and upper quantiles (where the probability
   is greater than 0.5) are rounded upwards.  This behaviour
   ensures that if an X% quantile is requested, then /at least/ the requested
   coverage will be present in the central region, and /no more than/
   the requested coverage will be present in the tails.

This behaviour can be changed so that the quantile functions are rounded
   differently, or even return a real-valued result using
   [link math_toolkit.pol_overview Policies].  It is strongly
   recommended that you read the tutorial
   [link math_toolkit.pol_tutorial.understand_dis_quant
   Understanding Quantiles of Discrete Distributions] before
   using the quantile function on a discrete distribution.  The
   [link math_toolkit.pol_ref.discrete_quant_ref reference docs]
   describe how to change the rounding policy
   for these distributions.
]]
]

What's more, if you define your own policy type, then it automatically
inherits the defaults for any policies not explicitly set, so given:

   using namespace boost::math::policies;
   //
   // Define a policy that sets ::errno on overflow, and does
   // not promote double to long double internally:
   //
   typedef policy<domain_error<errno_on_error>, promote_double<false> > mypolicy;

then `mypolicy` defines a policy where only the overflow error handling and
`double`-promotion policies differ from the defaults.

[endsect][/section:policy_tut_defaults Policies Have Sensible Defaults]

[section:policy_usage So How are Policies Used Anyway?]

The details follow later, but basically policies can be set by either:

* Defining some macros that change the default behaviour: [*this is the
   recommended method for setting installation-wide policies].
* By instantiating a distribution object with an explicit policy:
   this is mainly reserved for ad hoc policy changes.
* By passing a policy to a special function as an optional final argument:
   this is mainly reserved for ad hoc policy changes.
* By using some helper macros to define a set of functions or distributions
in the current namespace that use a specific policy: [*this is the
recommended method for setting policies on a project- or translation-unit-wide
basis].

The following sections introduce these methods in more detail.

[endsect] [/section:policy_usage So How are Policies Used Anyway?]

[section:changing_policy_defaults Changing the Policy Defaults]

The default policies used by the library are changed by the usual
configuration macro method.

For example, passing `-DBOOST_MATH_DOMAIN_ERROR_POLICY=errno_on_error` to
your compiler will cause domain errors to set `::errno` and return a __NaN
rather than the usual default behaviour of throwing a `std::domain_error`
exception.

[tip For Microsoft Visual Studio,you can add to the Project Property Page,
C/C++, Preprocessor, Preprocessor definitions like:

``BOOST_MATH_ASSERT_UNDEFINED_POLICY=0
BOOST_MATH_OVERFLOW_ERROR_POLICY=errno_on_error``

This may be helpful to avoid complications with pre-compiled headers
that may mean that the equivalent definitions in source code:

``#define BOOST_MATH_ASSERT_UNDEFINED_POLICY false
#define BOOST_MATH_OVERFLOW_ERROR_POLICY errno_on_error``

*may be ignored*.

The compiler command line shows:

``/D "BOOST_MATH_ASSERT_UNDEFINED_POLICY=0"
/D "BOOST_MATH_OVERFLOW_ERROR_POLICY=errno_on_error"``
] [/MSVC tip]

There is however a very important caveat to this:

[important
[*['Default policies changed by setting configuration macros must be changed
uniformly in every translation unit in the program.]]

Failure to follow this rule may result in violations of the "One
Definition Rule (ODR)" and result in unpredictable program behaviour.]

That means there are only two safe ways to use these macros:

* Edit them in [@../../../../boost/math/tools/user.hpp boost/math/tools/user.hpp],
so that the defaults are set on an installation-wide basis.
Unfortunately this may not be convenient if
you are using a pre-installed Boost distribution (on Linux for example).
* Set the defines in your project's Makefile or build environment, so that they
are set uniformly across all translation units.

What you should *not* do is:

* Set the defines in the source file using `#define` as doing so
almost certainly will break your program, unless you're absolutely
certain that the program is restricted to a single translation unit.

And, yes, you will find examples in our test programs where we break this
rule: but only because we know there will always be a single
translation unit only: ['don't say that you weren't warned!]

[import ../../example/error_handling_example.cpp]

[error_handling_example]

[endsect] [/section:changing_policy_defaults Changing the Policy Defaults]

[section:ad_hoc_dist_policies Setting Policies for Distributions on an Ad Hoc Basis]

All of the statistical distributions in this library are class templates
that accept two template parameters:
real type (float, double ...) and policy (how to handle exceptional events),
both with sensible defaults, for example:

   namespace boost{ namespace math{

   template <class RealType = double, class Policy = policies::policy<> >
   class fisher_f_distribution;

   typedef fisher_f_distribution<> fisher_f;

   }}

This policy gets used by all the accessor functions that accept
a distribution as an argument, and forwarded to all the functions called
by these.  So if you use the shorthand-typedef for the distribution, then you get
`double` precision arithmetic and all the default policies.

However, say for example we wanted to evaluate the quantile
of the binomial distribution at float precision, without internal
promotion to double, and with the result rounded to the /nearest/
integer, then here's how it can be done:

[import ../../example/policy_eg_3.cpp]

[policy_eg_3]

Which outputs:

[pre quantile is: 40]

[endsect][/section:ad_hoc_dist_policies Setting Policies for Distributions on an Ad Hoc Basis]

[section:ad_hoc_sf_policies Changing the Policy on an Ad Hoc Basis for the Special Functions]

All of the special functions in this library come in two overloaded forms,
one with a final "policy" parameter, and one without.  For example:

   namespace boost{ namespace math{

   template <class RealType, class Policy>
   RealType tgamma(RealType, const Policy&);

   template <class RealType>
   RealType tgamma(RealType);

   }} // namespaces

Normally, the second version is just a forwarding wrapper to the first
like this:

   template <class RealType>
   inline RealType tgamma(RealType x)
   {
      return tgamma(x, policies::policy<>());
   }

So calling a special function with a specific policy
is just a matter of defining the policy type to use
and passing it as the final parameter.  For example,
suppose we want `tgamma` to behave in a C-compatible
fashion and set `::errno` when an error occurs, and never
throw an exception:

[import ../../example/policy_eg_1.cpp]

[policy_eg_1]

which outputs:

[pre
Result of tgamma(30000) is: 1.#INF
errno = 34
Result of tgamma(-10) is: 1.#QNAN
errno = 33
]

Alternatively, for ad hoc use, we can use the `make_policy`
helper function to create a policy for us: this usage is more
verbose, so is probably only preferred when a policy is going
to be used once only:

[import ../../example/policy_eg_2.cpp]

[policy_eg_2]

[endsect] [/section:ad_hoc_sf_policies Changing the Policy on an Ad Hoc Basis for the Special Functions]

[section:namespace_policies Setting Policies at Namespace or Translation Unit Scope]

Sometimes what you want to do is just change a set of policies within
the current scope: *the one thing you should not do in this situation
is use the configuration macros*, as this can lead to "One Definition
Rule" violations.  Instead this library provides a pair of macros
especially for this purpose.

Let's consider the special functions first: we can declare a set of
forwarding functions that all use a specific policy using the
macro BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS(['Policy]).  This
macro should be used either inside a unique namespace set aside for the
purpose (for example, a C namespace for a C-style policy),
or an unnamed namespace if you just want the functions
visible in global scope for the current file only.

[import ../../example/policy_eg_4.cpp]

[policy_eg_4]

The same mechanism works well at file scope as well, by using an unnamed
namespace, we can ensure that these declarations don't conflict with any
alternate policies present in other translation units:

[import ../../example/policy_eg_5.cpp]

[policy_eg_5]

Handling policies for the statistical distributions is very similar except that now
the macro BOOST_MATH_DECLARE_DISTRIBUTIONS accepts two parameters: the
floating point type to use, and the policy type to apply.  For example:

   BOOST_MATH_DECLARE_DISTRIBUTIONS(double, mypolicy)

Results a set of typedefs being defined like this:

   typedef boost::math::normal_distribution<double, mypolicy> normal;

The name of each typedef is the same as the name of the distribution
class template, but without the "_distribution" suffix.

[import ../../example/policy_eg_6.cpp]

[policy_eg_6]

[note
There is an important limitation to note: you can *not use the macros
BOOST_MATH_DECLARE_DISTRIBUTIONS and BOOST_MATH_DECLARE_SPECIAL_FUNCTIONS
['in the same namespace]*,  as doing so creates ambiguities between functions
and distributions of the same name.
]

As before, the same mechanism works well at file scope as well: by using an unnamed
namespace, we can ensure that these declarations don't conflict with any
alternate policies present in other translation units:

[import ../../example/policy_eg_7.cpp]

[policy_eg_7]

[endsect][/section:namespace_policies Setting Policies at Namespace or Translation Unit Scope]

[section:user_def_err_pol Calling User Defined Error Handlers]

[import ../../example/policy_eg_8.cpp]

[policy_eg_8]

[import ../../example/policy_eg_9.cpp]

[policy_eg_9]

[endsect] [/section:user_def_err_pol Calling User Defined Error Handlers]

[section:understand_dis_quant Understanding Quantiles of Discrete Distributions]

Discrete distributions present us with a problem when calculating the
quantile: we are starting from a continuous real-valued variable - the
probability - but the result (the value of the random variable)
should really be discrete.

Consider for example a Binomial distribution, with a sample size of
50, and a success fraction of 0.5.  There are a variety of ways
we can plot a discrete distribution, but if we plot the PDF
as a step-function then it looks something like this:

[$../graphs/binomial_pdf.png]

Now lets suppose that the user asks for a the quantile that corresponds
to a probability of 0.05, if we zoom in on the CDF for that region here's
what we see:

[$../graphs/binomial_quantile_1.png]

As can be seen there is no random variable that corresponds to
a probability of exactly 0.05, so we're left with two choices as
shown in the figure:

* We could round the result down to 18.
* We could round the result up to 19.

In fact there's actually a third choice as well: we could "pretend" that the
distribution was continuous and return a real valued result: in this case we
would calculate a result of approximately 18.701 (this accurately
reflects the fact that the result is nearer to 19 than 18).

By using policies we can offer any of the above as options, but that
still leaves the question: ['What is actually the right thing to do?]

And in particular: ['What policy should we use by default?]

In coming to an answer we should realise that:

* Calculating an integer result is often much faster than
calculating a real-valued result: in fact in our tests it
was up to 20 times faster.
* Normally people calculate quantiles so that they can perform
a test of some kind: ['"If the random variable is less than N
then we can reject our null-hypothesis with 90% confidence."]

So there is a genuine benefit to calculating an integer result
as well as it being "the right thing to do" from a philosophical
point of view.  What's more if someone asks for a quantile at 0.05,
then we can normally assume that they are asking for
['[*at least] 95% of the probability to the right of the value chosen,
and [*no more than] 5% of the probability to the left of the value chosen.]

In the above binomial example we would therefore round the result down to 18.

The converse applies to upper-quantiles: If the probability is greater than
0.5 we would want to round the quantile up, ['so that [*at least] the requested
probability is to the left of the value returned, and [*no more than] 1 - the
requested probability is to the right of the value returned.]

Likewise for two-sided intervals, we would round lower quantiles down,
and upper quantiles up.  This ensures that we have ['at least the requested
probability in the central region] and ['no more than 1 minus the requested
probability in the tail areas.]

For example, taking our 50 sample binomial distribution with a success fraction
of 0.5, if we wanted a two sided 90% confidence interval, then we would ask
for the 0.05 and 0.95 quantiles with the results ['rounded outwards] so that
['at least 90% of the probability] is in the central area:

[$../graphs/binomial_pdf_3.png]

So far so good, but there is in fact a trap waiting for the unwary here:

   quantile(binomial(50, 0.5), 0.05);

returns 18 as the result, which is what we would expect from the graph above,
and indeed there is no x greater than 18 for which:

   cdf(binomial(50, 0.5), x) <= 0.05;

However:

   quantile(binomial(50, 0.5), 0.95);

returns 31, and indeed while there is no x less than 31 for which:

   cdf(binomial(50, 0.5), x) >= 0.95;

We might naively expect that for this symmetrical distribution the result
would be 32 (since 32 = 50 - 18), but we need to remember that the cdf of
the binomial is /inclusive/ of the random variable.  So while the left tail
area /includes/ the quantile returned, the right tail area always excludes
an upper quantile value: since that "belongs" to the central area.

Look at the graph above to see what's going on here: the lower quantile
of 18 belongs to the left tail, so any value <= 18 is in the left tail.
The upper quantile of 31 on the other hand belongs to the central area,
so the tail area actually starts at 32, so any value > 31 is in the
right tail.

Therefore if U and L are the upper and lower quantiles respectively, then
a random variable X is in the tail area - where we would reject the null
hypothesis if:

   X <= L || X > U

And the a variable X is inside the central region if:

   L < X <= U

The moral here is to ['always be very careful with your comparisons
when dealing with a discrete distribution], and if in doubt,
['base your comparisons on CDF's instead].

[heading Other Rounding Policies are Available]

As you would expect from a section on policies, you won't be surprised
to know that other rounding options are available:

[variablelist

[[integer_round_outwards]
   [This is the default policy as described above: lower quantiles
   are rounded down (probability < 0.5), and upper quantiles
   (probability > 0.5) are rounded up.

   This gives /no more than/ the requested probability
   in the tails, and /at least/ the requested probability
   in the central area.]]
[[integer_round_inwards]
   [This is the exact opposite of the default policy:
   lower quantiles
   are rounded up (probability < 0.5),
   and upper quantiles (probability > 0.5) are rounded down.

   This gives /at least/ the requested probability
   in the tails, and /no more than/ the requested probability
   in the central area.]]
[[integer_round_down][This policy will always round the result down
   no matter whether it is an upper or lower quantile]]
[[integer_round_up][This policy will always round the result up
   no matter whether it is an upper or lower quantile]]
[[integer_round_nearest][This policy will always round the result
   to the nearest integer
   no matter whether it is an upper or lower quantile]]
[[real][This policy will return a real valued result
   for the quantile of a discrete distribution: this is
   generally much slower than finding an integer result
   but does allow for more sophisticated rounding policies.]]

]

[import ../../example/policy_eg_10.cpp]

[policy_eg_10]

[endsect]

[endsect] [/section:pol_Tutorial Policy Tutorial]


[/ math.qbk
  Copyright 2007, 2013 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).
]