```{-# LANGUAGE BangPatterns, ScopedTypeVariables #-}
-- |
-- Module    : Statistics.Distribution
-- Copyright : (c) 2009 Bryan O'Sullivan
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Types classes for probability distrubutions

module Statistics.Distribution
(
-- * Type classes
Distribution(..)
, DiscreteDistr(..)
, ContDistr(..)
-- ** Distribution statistics
, MaybeMean(..)
, Mean(..)
, MaybeVariance(..)
, Variance(..)
, MaybeEntropy(..)
, Entropy(..)
-- ** Random number generation
, ContGen(..)
, DiscreteGen(..)
, genContinous
-- * Helper functions
, findRoot
, sumProbabilities
) where

import Control.Applicative     ((<\$>), Applicative(..))

import qualified Data.Vector.Unboxed as U
import System.Random.MWC

-- | Type class common to all distributions. Only c.d.f. could be
-- defined for both discrete and continous distributions.
class Distribution d where
-- | Cumulative distribution function.  The probability that a
-- random variable /X/ is less or equal than /x/,
-- i.e. P(/X/&#8804;/x/). Cumulative should be defined for
-- infinities as well:
--
-- > cumulative d +∞ = 1
-- > cumulative d -∞ = 0
cumulative :: d -> Double -> Double

-- | One's complement of cumulative distibution:
--
-- > complCumulative d x = 1 - cumulative d x
--
-- It's useful when one is interested in P(/X/</x/) and
-- expression on the right side begin to lose precision. This
-- function have default implementation but implementors are
-- encouraged to provide more precise implementation.
complCumulative :: d -> Double -> Double
complCumulative d x = 1 - cumulative d x

-- | Discrete probability distribution.
class Distribution  d => DiscreteDistr d where
-- | Probability of n-th outcome.
probability :: d -> Int -> Double
probability d = exp . logProbability d
{-# INLINE probability #-}

-- | Logarithm of probability of n-th outcome
logProbability :: d -> Int -> Double
logProbability d = log . probability d
{-# INLINE logProbability #-}

-- | Continuous probability distributuion.
--
--   Minimal complete definition is 'quantile' and either 'density' or
--   'logDensity'.
class Distribution d => ContDistr d where
-- | Probability density function. Probability that random
-- variable /X/ lies in the infinitesimal interval
-- [/x/,/x+/&#948;/x/) equal to /density(x)/&#8901;&#948;/x/
density :: d -> Double -> Double
density d = exp . logDensity d
{-# INLINE density #-}

-- | Inverse of the cumulative distribution function. The value
-- /x/ for which P(/X/&#8804;/x/) = /p/. If probability is outside
-- of [0,1] range function should call 'error'
quantile :: d -> Double -> Double

-- | Natural logarithm of density.
logDensity :: d -> Double -> Double
logDensity d = log . density d
{-# INLINE logDensity #-}

-- | Type class for distributions with mean. 'maybeMean' should return
--   'Nothing' if it's undefined for current value of data
class Distribution d => MaybeMean d where
maybeMean :: d -> Maybe Double

-- | Type class for distributions with mean. If distribution have
--   finite mean for all valid values of parameters it should be
--   instance of this type class.
class MaybeMean d => Mean d where
mean :: d -> Double

-- | Type class for distributions with variance. If variance is
--   undefined for some parameter values both 'maybeVariance' and
--   'maybeStdDev' should return Nothing.
--
--   Minimal complete definition is 'maybeVariance' or 'maybeStdDev'
class MaybeMean d => MaybeVariance d where
maybeVariance :: d -> Maybe Double
maybeVariance d = (*) <\$> x <*> x where x = maybeStdDev d
maybeStdDev   :: d -> Maybe Double
maybeStdDev = fmap sqrt . maybeVariance

-- | Type class for distributions with variance. If distibution have
--   finite variance for all valid parameter values it should be
--   instance of this type class.
--
--   Minimal complete definition is 'variance' or 'stdDev'
class (Mean d, MaybeVariance d) => Variance d where
variance :: d -> Double
variance d = x * x where x = stdDev d
stdDev   :: d -> Double
stdDev = sqrt . variance

-- | Type class for distributions with entropy, meaning Shannon entropy
--   in the case of a discrete distribution, or differential entropy in the
--   case of a continuous one.  'maybeEntropy' should return 'Nothing' if
--   entropy is undefined for the chosen parameter values.
class (Distribution d) => MaybeEntropy d where
-- | Returns the entropy of a distribution, in nats, if such is defined.
maybeEntropy :: d -> Maybe Double

-- | Type class for distributions with entropy, meaning Shannon
--   entropy in the case of a discrete distribution, or differential
--   entropy in the case of a continuous one.  If the distribution has
--   well-defined entropy for all valid parameter values then it
--   should be an instance of this type class.
class (MaybeEntropy d) => Entropy d where
-- | Returns the entropy of a distribution, in nats.
entropy :: d -> Double

-- | Generate discrete random variates which have given
--   distribution.
class Distribution d => ContGen d where
genContVar :: PrimMonad m => d -> Gen (PrimState m) -> m Double

-- | Generate discrete random variates which have given
--   distribution. 'ContGen' is superclass because it's always possible
--   to generate real-valued variates from integer values
class (DiscreteDistr d, ContGen d) => DiscreteGen d where
genDiscreteVar :: PrimMonad m => d -> Gen (PrimState m) -> m Int

-- | Generate variates from continous distribution using inverse
--   transform rule.
genContinous :: (ContDistr d, PrimMonad m) => d -> Gen (PrimState m) -> m Double
genContinous d gen = do
x <- uniform gen
return \$! quantile d x
{-# INLINE genContinous #-}

data P = P {-# UNPACK #-} !Double {-# UNPACK #-} !Double

-- | Approximate the value of /X/ for which P(/x/>/X/)=/p/.
--
-- This method uses a combination of Newton-Raphson iteration and
-- bisection with the given guess as a starting point.  The upper and
-- lower bounds specify the interval in which the probability
-- distribution reaches the value /p/.
findRoot :: ContDistr d =>
d                   -- ^ Distribution
-> Double              -- ^ Probability /p/
-> Double              -- ^ Initial guess
-> Double              -- ^ Lower bound on interval
-> Double              -- ^ Upper bound on interval
-> Double
findRoot d prob = loop 0 1
where
loop !(i::Int) !dx !x !lo !hi
| abs dx <= accuracy || i >= maxIters = x
| otherwise                           = loop (i+1) dx'' x'' lo' hi'
where
err                   = cumulative d x - prob
P lo' hi' | err < 0   = P x hi
| otherwise = P lo x
pdf                   = density d x
P dx' x' | pdf /= 0   = P (err / pdf) (x - dx)
| otherwise  = P dx x
P dx'' x''
| x' < lo' || x' > hi' || pdf == 0 = let y = (lo' + hi') / 2
in  P (y-x) y
| otherwise                        = P dx' x'
accuracy = 1e-15
maxIters = 150

-- | Sum probabilities in inclusive interval.
sumProbabilities :: DiscreteDistr d => d -> Int -> Int -> Double
sumProbabilities d low hi =
-- Return value is forced to be less than 1 to guard againist roundoff errors.
-- ATTENTION! this check should be removed for testing or it could mask bugs.
min 1 . U.sum . U.map (probability d) \$ U.enumFromTo low hi
{-# INLINE sumProbabilities #-}
```