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|
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-----------------------------------------------------------------------------
-- |
-- Module : Statistics.Distribution.Beta
-- Copyright : (C) 2012 Edward Kmett,
-- License : BSD-style (see the file LICENSE)
--
-- Maintainer : Edward Kmett <ekmett@gmail.com>
-- Stability : provisional
-- Portability : DeriveDataTypeable
--
----------------------------------------------------------------------------
module Statistics.Distribution.Beta
( BetaDistribution
-- * Constructor
, betaDistr
, betaDistrE
, improperBetaDistr
, improperBetaDistrE
-- * Accessors
, bdAlpha
, bdBeta
) 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.SpecFunctions (
incompleteBeta, invIncompleteBeta, logBeta, digamma, log1p)
import Numeric.MathFunctions.Constants (m_NaN,m_neg_inf)
import qualified Statistics.Distribution as D
import Statistics.Internal
-- | The beta distribution
data BetaDistribution = BD
{ bdAlpha :: {-# UNPACK #-} !Double
-- ^ Alpha shape parameter
, bdBeta :: {-# UNPACK #-} !Double
-- ^ Beta shape parameter
} deriving (Eq, Typeable, Data, Generic)
instance Show BetaDistribution where
showsPrec n (BD a b) = defaultShow2 "improperBetaDistr" a b n
instance Read BetaDistribution where
readPrec = defaultReadPrecM2 "improperBetaDistr" improperBetaDistrE
instance ToJSON BetaDistribution
instance FromJSON BetaDistribution where
parseJSON (Object v) = do
a <- v .: "bdAlpha"
b <- v .: "bdBeta"
maybe (fail $ errMsgI a b) return $ improperBetaDistrE a b
parseJSON _ = empty
instance Binary BetaDistribution where
put (BD a b) = put a >> put b
get = do
a <- get
b <- get
maybe (fail $ errMsgI a b) return $ improperBetaDistrE a b
-- | Create beta distribution. Both shape parameters must be positive.
betaDistr :: Double -- ^ Shape parameter alpha
-> Double -- ^ Shape parameter beta
-> BetaDistribution
betaDistr a b = maybe (error $ errMsg a b) id $ betaDistrE a b
-- | Create beta distribution. Both shape parameters must be positive.
betaDistrE :: Double -- ^ Shape parameter alpha
-> Double -- ^ Shape parameter beta
-> Maybe BetaDistribution
betaDistrE a b
| a > 0 && b > 0 = Just (BD a b)
| otherwise = Nothing
errMsg :: Double -> Double -> String
errMsg a b = "Statistics.Distribution.Beta.betaDistr: "
++ "shape parameters must be positive. Got a = "
++ show a
++ " b = "
++ show b
-- | Create beta distribution. Both shape parameters must be
-- non-negative. So it allows to construct improper beta distribution
-- which could be used as improper prior.
improperBetaDistr :: Double -- ^ Shape parameter alpha
-> Double -- ^ Shape parameter beta
-> BetaDistribution
improperBetaDistr a b
= maybe (error $ errMsgI a b) id $ improperBetaDistrE a b
-- | Create beta distribution. Both shape parameters must be
-- non-negative. So it allows to construct improper beta distribution
-- which could be used as improper prior.
improperBetaDistrE :: Double -- ^ Shape parameter alpha
-> Double -- ^ Shape parameter beta
-> Maybe BetaDistribution
improperBetaDistrE a b
| a >= 0 && b >= 0 = Just (BD a b)
| otherwise = Nothing
errMsgI :: Double -> Double -> String
errMsgI a b
= "Statistics.Distribution.Beta.betaDistr: "
++ "shape parameters must be non-negative. Got a = " ++ show a
++ " b = " ++ show b
instance D.Distribution BetaDistribution where
cumulative (BD a b) x
| x <= 0 = 0
| x >= 1 = 1
| otherwise = incompleteBeta a b x
complCumulative (BD a b) x
| x <= 0 = 1
| x >= 1 = 0
-- For small x we use direct computation to avoid precision loss
-- when computing (1-x)
| x < 0.5 = 1 - incompleteBeta a b x
-- Otherwise we use property of incomplete beta:
-- > I(x,a,b) = 1 - I(1-x,b,a)
| otherwise = incompleteBeta b a (1-x)
instance D.Mean BetaDistribution where
mean (BD a b) = a / (a + b)
instance D.MaybeMean BetaDistribution where
maybeMean = Just . D.mean
instance D.Variance BetaDistribution where
variance (BD a b) = a*b / (apb*apb*(apb+1))
where apb = a + b
instance D.MaybeVariance BetaDistribution where
maybeVariance = Just . D.variance
instance D.Entropy BetaDistribution where
entropy (BD a b) =
logBeta a b
- (a-1) * digamma a
- (b-1) * digamma b
+ (a+b-2) * digamma (a+b)
instance D.MaybeEntropy BetaDistribution where
maybeEntropy = Just . D.entropy
instance D.ContDistr BetaDistribution where
density (BD a b) x
| a <= 0 || b <= 0 = m_NaN
| x <= 0 = 0
| x >= 1 = 0
| otherwise = exp $ (a-1)*log x + (b-1) * log1p (-x) - logBeta a b
logDensity (BD a b) x
| a <= 0 || b <= 0 = m_NaN
| x <= 0 = m_neg_inf
| x >= 1 = m_neg_inf
| otherwise = (a-1)*log x + (b-1)*log1p (-x) - logBeta a b
quantile (BD a b) p
| p == 0 = 0
| p == 1 = 1
| p > 0 && p < 1 = invIncompleteBeta a b p
| otherwise =
error $ "Statistics.Distribution.Gamma.quantile: p must be in [0,1] range. Got: "++show p
instance D.ContGen BetaDistribution where
genContVar = D.genContinuous
|
