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{-# LANGUAGE DeriveDataTypeable #-}
-- |
-- Module : Statistics.Distribution.Gamma
-- Copyright : (c) 2009, 2011 Bryan O'Sullivan
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- The gamma distribution. This is a continuous probability
-- distribution with two parameters, /k/ and ϑ. If /k/ is
-- integral, the distribution represents the sum of /k/ independent
-- exponentially distributed random variables, each of which has a
-- mean of ϑ.
module Statistics.Distribution.Gamma
(
GammaDistribution
-- * Constructors
, gammaDistr
, improperGammaDistr
-- * Accessors
, gdShape
, gdScale
) where
import Data.Typeable (Typeable)
import Numeric.MathFunctions.Constants (m_pos_inf, m_NaN)
import Numeric.SpecFunctions (incompleteGamma, invIncompleteGamma)
import Statistics.Distribution.Poisson.Internal as Poisson
import qualified Statistics.Distribution as D
import qualified System.Random.MWC.Distributions as MWC
-- | The gamma distribution.
data GammaDistribution = GD {
gdShape :: {-# UNPACK #-} !Double -- ^ Shape parameter, /k/.
, gdScale :: {-# UNPACK #-} !Double -- ^ Scale parameter, ϑ.
} deriving (Eq, Read, Show, Typeable)
-- | Create gamma distribution. Both shape and scale parameters must
-- be positive.
gammaDistr :: Double -- ^ Shape parameter. /k/
-> Double -- ^ Scale parameter, ϑ.
-> GammaDistribution
gammaDistr k theta
| k <= 0 = error $ msg ++ "shape must be positive. Got " ++ show k
| theta <= 0 = error $ msg ++ "scale must be positive. Got " ++ show theta
| otherwise = improperGammaDistr k theta
where msg = "Statistics.Distribution.Gamma.gammaDistr: "
{-# INLINE gammaDistr #-}
-- | Create gamma distribution. This constructor do not check whether
-- parameters are valid
improperGammaDistr :: Double -- ^ Shape parameter. /k/
-> Double -- ^ Scale parameter, ϑ.
-> GammaDistribution
improperGammaDistr = GD
{-# INLINE improperGammaDistr #-}
instance D.Distribution GammaDistribution where
cumulative = cumulative
instance D.ContDistr GammaDistribution where
density = density
quantile = quantile
instance D.Variance GammaDistribution where
variance (GD a l) = a * l * l
{-# INLINE variance #-}
instance D.Mean GammaDistribution where
mean (GD a l) = a * l
{-# INLINE mean #-}
instance D.MaybeMean GammaDistribution where
maybeMean = Just . D.mean
instance D.MaybeVariance GammaDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.ContGen GammaDistribution where
genContVar (GD a l) = MWC.gamma a l
density :: GammaDistribution -> Double -> Double
density (GD a l) x
| a < 0 || l <= 0 = m_NaN
| x <= 0 = 0
| a == 0 = if x == 0 then m_pos_inf else 0
| x == 0 = if a < 1 then m_pos_inf else if a > 1 then 0 else 1/l
| a < 1 = Poisson.probability (x/l) a * a / x
| otherwise = Poisson.probability (x/l) (a-1) / l
{-# INLINE density #-}
cumulative :: GammaDistribution -> Double -> Double
cumulative (GD k l) x
| x <= 0 = 0
| otherwise = incompleteGamma k (x/l)
{-# INLINE cumulative #-}
quantile :: GammaDistribution -> Double -> Double
quantile (GD k l) p
| p == 0 = 0
| p == 1 = 1/0
| p > 0 && p < 1 = l * invIncompleteGamma k p
| otherwise =
error $ "Statistics.Distribution.Gamma.quantile: p must be in [0,1] range. Got: "++show p
{-# INLINE quantile #-}
|
