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{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module : Statistics.Distribution.Exponential
-- Copyright : (c) 2009 Bryan O'Sullivan
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- The exponential distribution. This is the continuous probability
-- distribution of the times between events in a Poisson process, in
-- which events occur continuously and independently at a constant
-- average rate.
module Statistics.Distribution.Exponential
(
ExponentialDistribution
-- * Constructors
, exponential
, exponentialE
-- * Accessors
, edLambda
) where
import Control.Applicative
import Data.Aeson (FromJSON(..),ToJSON,Value(..),(.:))
import Data.Binary (Binary, put, get)
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import Numeric.SpecFunctions (log1p,expm1)
import Numeric.MathFunctions.Constants (m_neg_inf)
import qualified System.Random.MWC.Distributions as MWC
import qualified Statistics.Distribution as D
import qualified Statistics.Sample as S
import Statistics.Internal
newtype ExponentialDistribution = ED {
edLambda :: Double
} deriving (Eq, Typeable, Data, Generic)
instance Show ExponentialDistribution where
showsPrec n (ED l) = defaultShow1 "exponential" l n
instance Read ExponentialDistribution where
readPrec = defaultReadPrecM1 "exponential" exponentialE
instance ToJSON ExponentialDistribution
instance FromJSON ExponentialDistribution where
parseJSON (Object v) = do
l <- v .: "edLambda"
maybe (fail $ errMsg l) return $ exponentialE l
parseJSON _ = empty
instance Binary ExponentialDistribution where
put = put . edLambda
get = do
l <- get
maybe (fail $ errMsg l) return $ exponentialE l
instance D.Distribution ExponentialDistribution where
cumulative = cumulative
complCumulative = complCumulative
instance D.ContDistr ExponentialDistribution where
density (ED l) x
| x < 0 = 0
| otherwise = l * exp (-l * x)
logDensity (ED l) x
| x < 0 = m_neg_inf
| otherwise = log l + (-l * x)
quantile = quantile
complQuantile = complQuantile
instance D.Mean ExponentialDistribution where
mean (ED l) = 1 / l
instance D.Variance ExponentialDistribution where
variance (ED l) = 1 / (l * l)
instance D.MaybeMean ExponentialDistribution where
maybeMean = Just . D.mean
instance D.MaybeVariance ExponentialDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.Entropy ExponentialDistribution where
entropy (ED l) = 1 - log l
instance D.MaybeEntropy ExponentialDistribution where
maybeEntropy = Just . D.entropy
instance D.ContGen ExponentialDistribution where
genContVar = MWC.exponential . edLambda
cumulative :: ExponentialDistribution -> Double -> Double
cumulative (ED l) x | x <= 0 = 0
| otherwise = - expm1 (-l * x)
complCumulative :: ExponentialDistribution -> Double -> Double
complCumulative (ED l) x | x <= 0 = 1
| otherwise = exp (-l * x)
quantile :: ExponentialDistribution -> Double -> Double
quantile (ED l) p
| p >= 0 && p <= 1 = - log1p(-p) / l
| otherwise =
error $ "Statistics.Distribution.Exponential.quantile: p must be in [0,1] range. Got: "++show p
complQuantile :: ExponentialDistribution -> Double -> Double
complQuantile (ED l) p
| p == 0 = 0
| p >= 0 && p < 1 = -log p / l
| otherwise =
error $ "Statistics.Distribution.Exponential.quantile: p must be in [0,1] range. Got: "++show p
-- | Create an exponential distribution.
exponential :: Double -- ^ Rate parameter.
-> ExponentialDistribution
exponential l = maybe (error $ errMsg l) id $ exponentialE l
-- | Create an exponential distribution.
exponentialE :: Double -- ^ Rate parameter.
-> Maybe ExponentialDistribution
exponentialE l
| l > 0 = Just (ED l)
| otherwise = Nothing
errMsg :: Double -> String
errMsg l = "Statistics.Distribution.Exponential.exponential: scale parameter must be positive. Got " ++ show l
-- | Create exponential distribution from sample. Estimates the rate
-- with the maximum likelihood estimator, which is biased. Returns
-- @Nothing@ if the sample mean does not exist or is not positive.
instance D.FromSample ExponentialDistribution Double where
fromSample xs = let m = S.mean xs
in if m > 0 then Just (ED (1/m)) else Nothing
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