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|
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- 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
, gammaDistrE
, improperGammaDistr
, improperGammaDistrE
-- * Accessors
, gdShape
, gdScale
) where
import Control.Applicative
import Data.Aeson (FromJSON(..), ToJSON, Value(..), (.:))
import Data.Binary (Binary(..))
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import Numeric.MathFunctions.Constants (m_pos_inf, m_NaN, m_neg_inf)
import Numeric.SpecFunctions (incompleteGamma, invIncompleteGamma, logGamma, digamma)
import qualified System.Random.MWC.Distributions as MWC
import qualified Numeric.Sum as Sum
import Statistics.Distribution.Poisson.Internal as Poisson
import qualified Statistics.Distribution as D
import Statistics.Internal
-- | The gamma distribution.
data GammaDistribution = GD {
gdShape :: {-# UNPACK #-} !Double -- ^ Shape parameter, /k/.
, gdScale :: {-# UNPACK #-} !Double -- ^ Scale parameter, ϑ.
} deriving (Eq, Typeable, Data, Generic)
instance Show GammaDistribution where
showsPrec i (GD k theta) = defaultShow2 "improperGammaDistr" k theta i
instance Read GammaDistribution where
readPrec = defaultReadPrecM2 "improperGammaDistr" improperGammaDistrE
instance ToJSON GammaDistribution
instance FromJSON GammaDistribution where
parseJSON (Object v) = do
k <- v .: "gdShape"
theta <- v .: "gdScale"
maybe (fail $ errMsgI k theta) return $ improperGammaDistrE k theta
parseJSON _ = empty
instance Binary GammaDistribution where
put (GD x y) = put x >> put y
get = do
k <- get
theta <- get
maybe (fail $ errMsgI k theta) return $ improperGammaDistrE k theta
-- | Create gamma distribution. Both shape and scale parameters must
-- be positive.
gammaDistr :: Double -- ^ Shape parameter. /k/
-> Double -- ^ Scale parameter, ϑ.
-> GammaDistribution
gammaDistr k theta
= maybe (error $ errMsg k theta) id $ gammaDistrE k theta
errMsg :: Double -> Double -> String
errMsg k theta
= "Statistics.Distribution.Gamma.gammaDistr: "
++ "k=" ++ show k
++ "theta=" ++ show theta
++ " but must be positive"
-- | Create gamma distribution. Both shape and scale parameters must
-- be positive.
gammaDistrE :: Double -- ^ Shape parameter. /k/
-> Double -- ^ Scale parameter, ϑ.
-> Maybe GammaDistribution
gammaDistrE k theta
| k > 0 && theta > 0 = Just (GD k theta)
| otherwise = Nothing
-- | Create gamma distribution. Both shape and scale parameters must
-- be non-negative.
improperGammaDistr :: Double -- ^ Shape parameter. /k/
-> Double -- ^ Scale parameter, ϑ.
-> GammaDistribution
improperGammaDistr k theta
= maybe (error $ errMsgI k theta) id $ improperGammaDistrE k theta
errMsgI :: Double -> Double -> String
errMsgI k theta
= "Statistics.Distribution.Gamma.gammaDistr: "
++ "k=" ++ show k
++ "theta=" ++ show theta
++ " but must be non-negative"
-- | Create gamma distribution. Both shape and scale parameters must
-- be non-negative.
improperGammaDistrE :: Double -- ^ Shape parameter. /k/
-> Double -- ^ Scale parameter, ϑ.
-> Maybe GammaDistribution
improperGammaDistrE k theta
| k >= 0 && theta >= 0 = Just (GD k theta)
| otherwise = Nothing
instance D.Distribution GammaDistribution where
cumulative = cumulative
instance D.ContDistr GammaDistribution where
density = density
logDensity (GD k theta) x
| x <= 0 = m_neg_inf
| otherwise = Sum.sum Sum.kbn [ log x * (k - 1)
, - (x / theta)
, - logGamma k
, - log theta * k
]
quantile = quantile
instance D.Variance GammaDistribution where
variance (GD a l) = a * l * l
instance D.Mean GammaDistribution where
mean (GD a l) = a * l
instance D.MaybeMean GammaDistribution where
maybeMean = Just . D.mean
instance D.MaybeVariance GammaDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.MaybeEntropy GammaDistribution where
maybeEntropy (GD a l)
| a > 0 && l > 0 =
Just $
a
+ log l
+ logGamma a
+ (1-a) * digamma a
| otherwise = Nothing
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
cumulative :: GammaDistribution -> Double -> Double
cumulative (GD k l) x
| x <= 0 = 0
| otherwise = incompleteGamma k (x/l)
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
|
