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|
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module : Statistics.Distribution.Normal
-- Copyright : (c) 2009 Bryan O'Sullivan
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- The normal distribution. This is a continuous probability
-- distribution that describes data that cluster around a mean.
module Statistics.Distribution.Normal
(
NormalDistribution
-- * Constructors
, normalDistr
, normalDistrE
, normalDistrErr
, standard
) 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_sqrt_2, m_sqrt_2_pi)
import Numeric.SpecFunctions (erfc, invErfc)
import qualified System.Random.MWC.Distributions as MWC
import qualified Data.Vector.Generic as G
import qualified Statistics.Distribution as D
import qualified Statistics.Sample as S
import Statistics.Internal
-- | The normal distribution.
data NormalDistribution = ND {
mean :: {-# UNPACK #-} !Double
, stdDev :: {-# UNPACK #-} !Double
, ndPdfDenom :: {-# UNPACK #-} !Double
, ndCdfDenom :: {-# UNPACK #-} !Double
} deriving (Eq, Typeable, Data, Generic)
instance Show NormalDistribution where
showsPrec i (ND m s _ _) = defaultShow2 "normalDistr" m s i
instance Read NormalDistribution where
readPrec = defaultReadPrecM2 "normalDistr" normalDistrE
instance ToJSON NormalDistribution
instance FromJSON NormalDistribution where
parseJSON (Object v) = do
m <- v .: "mean"
sd <- v .: "stdDev"
either fail return $ normalDistrErr m sd
parseJSON _ = empty
instance Binary NormalDistribution where
put (ND m sd _ _) = put m >> put sd
get = do
m <- get
sd <- get
either fail return $ normalDistrErr m sd
instance D.Distribution NormalDistribution where
cumulative = cumulative
complCumulative = complCumulative
instance D.ContDistr NormalDistribution where
logDensity = logDensity
quantile = quantile
complQuantile = complQuantile
instance D.MaybeMean NormalDistribution where
maybeMean = Just . D.mean
instance D.Mean NormalDistribution where
mean = mean
instance D.MaybeVariance NormalDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.Variance NormalDistribution where
stdDev = stdDev
instance D.Entropy NormalDistribution where
entropy d = 0.5 * log (2 * pi * exp 1 * D.variance d)
instance D.MaybeEntropy NormalDistribution where
maybeEntropy = Just . D.entropy
instance D.ContGen NormalDistribution where
genContVar d = MWC.normal (mean d) (stdDev d)
-- | Standard normal distribution with mean equal to 0 and variance equal to 1
standard :: NormalDistribution
standard = ND { mean = 0.0
, stdDev = 1.0
, ndPdfDenom = log m_sqrt_2_pi
, ndCdfDenom = m_sqrt_2
}
-- | Create normal distribution from parameters.
--
-- IMPORTANT: prior to 0.10 release second parameter was variance not
-- standard deviation.
normalDistr :: Double -- ^ Mean of distribution
-> Double -- ^ Standard deviation of distribution
-> NormalDistribution
normalDistr m sd = either error id $ normalDistrErr m sd
-- | Create normal distribution from parameters.
--
-- IMPORTANT: prior to 0.10 release second parameter was variance not
-- standard deviation.
normalDistrE :: Double -- ^ Mean of distribution
-> Double -- ^ Standard deviation of distribution
-> Maybe NormalDistribution
normalDistrE m sd = either (const Nothing) Just $ normalDistrErr m sd
-- | Create normal distribution from parameters.
--
normalDistrErr :: Double -- ^ Mean of distribution
-> Double -- ^ Standard deviation of distribution
-> Either String NormalDistribution
normalDistrErr m sd
| sd > 0 = Right $ ND { mean = m
, stdDev = sd
, ndPdfDenom = log $ m_sqrt_2_pi * sd
, ndCdfDenom = m_sqrt_2 * sd
}
| otherwise = Left $ errMsg m sd
errMsg :: Double -> Double -> String
errMsg _ sd = "Statistics.Distribution.Normal.normalDistr: standard deviation must be positive. Got " ++ show sd
-- | Variance is estimated using maximum likelihood method
-- (biased estimation).
--
-- Returns @Nothing@ if sample contains less than one element or
-- variance is zero (all elements are equal)
instance D.FromSample NormalDistribution Double where
fromSample xs
| G.length xs <= 1 = Nothing
| v == 0 = Nothing
| otherwise = Just $! normalDistr m (sqrt v)
where
(m,v) = S.meanVariance xs
logDensity :: NormalDistribution -> Double -> Double
logDensity d x = (-xm * xm / (2 * sd * sd)) - ndPdfDenom d
where xm = x - mean d
sd = stdDev d
cumulative :: NormalDistribution -> Double -> Double
cumulative d x = erfc ((mean d - x) / ndCdfDenom d) / 2
complCumulative :: NormalDistribution -> Double -> Double
complCumulative d x = erfc ((x - mean d) / ndCdfDenom d) / 2
quantile :: NormalDistribution -> Double -> Double
quantile d p
| p == 0 = -inf
| p == 1 = inf
| p == 0.5 = mean d
| p > 0 && p < 1 = x * ndCdfDenom d + mean d
| otherwise =
error $ "Statistics.Distribution.Normal.quantile: p must be in [0,1] range. Got: "++show p
where x = - invErfc (2 * p)
inf = 1/0
complQuantile :: NormalDistribution -> Double -> Double
complQuantile d p
| p == 0 = inf
| p == 1 = -inf
| p == 0.5 = mean d
| p > 0 && p < 1 = x * ndCdfDenom d + mean d
| otherwise =
error $ "Statistics.Distribution.Normal.complQuantile: p must be in [0,1] range. Got: "++show p
where x = invErfc (2 * p)
inf = 1/0
|
