# statistics / Statistics / Distribution / Normal.hs

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122``` ```{-# LANGUAGE BangPatterns, 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 , normalFromSample , standard ) where 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 Statistics.Distribution as D import qualified Statistics.Sample as S import qualified System.Random.MWC.Distributions as MWC -- | The normal distribution. data NormalDistribution = ND { mean :: {-# UNPACK #-} !Double , stdDev :: {-# UNPACK #-} !Double , ndPdfDenom :: {-# UNPACK #-} !Double , ndCdfDenom :: {-# UNPACK #-} !Double } deriving (Eq, Read, Show, Typeable, Data, Generic) instance Binary NormalDistribution instance D.Distribution NormalDistribution where cumulative = cumulative complCumulative = complCumulative instance D.ContDistr NormalDistribution where density = density 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.ContGen NormalDistribution where genContVar d = MWC.normal (mean d) (stdDev d) {-# INLINE genContVar #-} -- | Standard normal distribution with mean equal to 0 and variance equal to 1 standard :: NormalDistribution standard = ND { mean = 0.0 , stdDev = 1.0 , ndPdfDenom = 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 | sd > 0 = ND { mean = m , stdDev = sd , ndPdfDenom = m_sqrt_2_pi * sd , ndCdfDenom = m_sqrt_2 * sd } | otherwise = error \$ "Statistics.Distribution.Normal.normalDistr: standard deviation must be positive. Got " ++ show sd -- | Create distribution using parameters estimated from -- sample. Variance is estimated using maximum likelihood method -- (biased estimation). normalFromSample :: S.Sample -> NormalDistribution normalFromSample xs = normalDistr m (sqrt v) where (m,v) = S.meanVariance xs density :: NormalDistribution -> Double -> Double density d x = exp (-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 * (1 - p) inf = 1/0 ```