module Statistics.Distribution.Normal
(
NormalDistribution
, normalDistr
, normalFromSample
, standard
) where
import Data.Aeson (FromJSON, ToJSON)
import Control.Applicative ((<$>), (<*>))
import Data.Binary (Binary)
import Data.Binary (put, get)
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import Numeric.MathFunctions.Constants (m_sqrt_2, m_sqrt_2_pi)
import Numeric.SpecFunctions (erfc, invErfc)
import qualified Statistics.Distribution as D
import qualified Statistics.Sample as S
import qualified System.Random.MWC.Distributions as MWC
data NormalDistribution = ND {
mean :: !Double
, stdDev :: !Double
, ndPdfDenom :: !Double
, ndCdfDenom :: !Double
} deriving (Eq, Read, Show, Typeable, Data, Generic)
instance FromJSON NormalDistribution
instance ToJSON NormalDistribution
instance Binary NormalDistribution where
put (ND w x y z) = put w >> put x >> put y >> put z
get = ND <$> get <*> get <*> get <*> get
instance D.Distribution NormalDistribution where
cumulative = cumulative
complCumulative = complCumulative
instance D.ContDistr NormalDistribution where
logDensity = logDensity
quantile = quantile
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 :: NormalDistribution
standard = ND { mean = 0.0
, stdDev = 1.0
, ndPdfDenom = log m_sqrt_2_pi
, ndCdfDenom = m_sqrt_2
}
normalDistr :: Double
-> Double
-> NormalDistribution
normalDistr m sd
| sd > 0 = ND { mean = m
, stdDev = sd
, ndPdfDenom = log $ m_sqrt_2_pi * sd
, ndCdfDenom = m_sqrt_2 * sd
}
| otherwise =
error $ "Statistics.Distribution.Normal.normalDistr: standard deviation must be positive. Got " ++ show sd
normalFromSample :: S.Sample -> NormalDistribution
normalFromSample xs
= 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