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|
{-# LANGUAGE DeriveDataTypeable #-}
-- |
-- 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
, fromParams
, fromSample
, standard
) where
import Control.Exception (assert)
import Data.Number.Erf (erfc)
import Data.Typeable (Typeable)
import Statistics.Constants (m_sqrt_2, m_sqrt_2_pi)
import qualified Statistics.Distribution as D
import qualified Statistics.Sample as S
-- | The normal distribution.
data NormalDistribution = ND {
mean :: {-# UNPACK #-} !Double
, variance :: {-# UNPACK #-} !Double
, ndPdfDenom :: {-# UNPACK #-} !Double
, ndCdfDenom :: {-# UNPACK #-} !Double
} deriving (Eq, Read, Show, Typeable)
instance D.Distribution NormalDistribution where
density = density
cumulative = cumulative
quantile = quantile
instance D.Variance NormalDistribution where
variance = variance
instance D.Mean NormalDistribution where
mean = mean
-- | Standard normal distribution with mean equal to 0 and variance equal to 1
standard :: NormalDistribution
standard = ND {
mean = 0.0
, variance = 1.0
, ndPdfDenom = m_sqrt_2_pi
, ndCdfDenom = m_sqrt_2
}
-- | Create normal distribution from parameters
fromParams :: Double -- ^ Mean of distribution
-> Double -- ^ Variance of distribution
-> NormalDistribution
fromParams m v = assert (v > 0)
ND {
mean = m
, variance = v
, ndPdfDenom = m_sqrt_2_pi * sv
, ndCdfDenom = m_sqrt_2 * sv
}
where sv = sqrt v
-- | Create distribution using parameters estimated from
-- sample. Variance is estimated using maximum likelihood method
-- (biased estimation).
fromSample :: S.Sample -> NormalDistribution
fromSample a = fromParams (S.mean a) (S.variance a)
density :: NormalDistribution -> Double -> Double
density d x = exp (-xm * xm / (2 * variance d)) / ndPdfDenom d
where xm = x - mean d
cumulative :: NormalDistribution -> Double -> Double
cumulative d x = erfc (-(x-mean d) / ndCdfDenom d) / 2
quantile :: NormalDistribution -> Double -> Double
quantile d p
| p < 0 || p > 1 = inf/inf
| p == 0 = -inf
| p == 1 = inf
| p == 0.5 = mean d
| otherwise = x * sqrt (variance d) + mean d
where x = D.findRoot standard p 0 (-100) 100
inf = 1/0
|
