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|
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module : Statistics.Distribution.Lognormal
-- Copyright : (c) 2020 Ximin Luo
-- License : BSD3
--
-- Maintainer : infinity0@pwned.gg
-- Stability : experimental
-- Portability : portable
--
-- The log normal distribution. This is a continuous probability
-- distribution that describes data whose log is clustered around a
-- mean. For example, the multiplicative product of many independent
-- positive random variables.
module Statistics.Distribution.Lognormal
(
LognormalDistribution
-- * Constructors
, lognormalDistr
, lognormalDistrErr
, lognormalDistrMeanStddevErr
, lognormalStandard
) where
import Data.Aeson (FromJSON, ToJSON)
import Data.Binary (Binary (..))
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import Numeric.MathFunctions.Constants (m_huge, m_sqrt_2_pi)
import Numeric.SpecFunctions (expm1, log1p)
import qualified Data.Vector.Generic as G
import qualified Statistics.Distribution as D
import qualified Statistics.Distribution.Normal as N
import Statistics.Internal
-- | The lognormal distribution.
newtype LognormalDistribution = LND N.NormalDistribution
deriving (Eq, Typeable, Data, Generic)
instance Show LognormalDistribution where
showsPrec i (LND d) = defaultShow2 "lognormalDistr" m s i
where
m = D.mean d
s = D.stdDev d
instance Read LognormalDistribution where
readPrec = defaultReadPrecM2 "lognormalDistr" $
(either (const Nothing) Just .) . lognormalDistrErr
instance ToJSON LognormalDistribution
instance FromJSON LognormalDistribution
instance Binary LognormalDistribution where
put (LND d) = put m >> put s
where
m = D.mean d
s = D.stdDev d
get = do
m <- get
sd <- get
either fail return $ lognormalDistrErr m sd
instance D.Distribution LognormalDistribution where
cumulative = cumulative
complCumulative = complCumulative
instance D.ContDistr LognormalDistribution where
logDensity = logDensity
quantile = quantile
complQuantile = complQuantile
instance D.MaybeMean LognormalDistribution where
maybeMean = Just . D.mean
instance D.Mean LognormalDistribution where
mean (LND d) = exp (m + v / 2)
where
m = D.mean d
v = D.variance d
instance D.MaybeVariance LognormalDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.Variance LognormalDistribution where
variance (LND d) = expm1 v * exp (2 * m + v)
where
m = D.mean d
v = D.variance d
instance D.Entropy LognormalDistribution where
entropy (LND d) = logBase 2 (s * exp (m + 0.5) * m_sqrt_2_pi)
where
m = D.mean d
s = D.stdDev d
instance D.MaybeEntropy LognormalDistribution where
maybeEntropy = Just . D.entropy
instance D.ContGen LognormalDistribution where
genContVar d = D.genContinuous d
-- | Standard log normal distribution with mu 0 and sigma 1.
--
-- Mean is @sqrt e@ and variance is @(e - 1) * e@.
lognormalStandard :: LognormalDistribution
lognormalStandard = LND N.standard
-- | Create log normal distribution from parameters.
lognormalDistr
:: Double -- ^ Mu
-> Double -- ^ Sigma
-> LognormalDistribution
lognormalDistr mu sig = either error id $ lognormalDistrErr mu sig
-- | Create log normal distribution from parameters.
lognormalDistrErr
:: Double -- ^ Mu
-> Double -- ^ Sigma
-> Either String LognormalDistribution
lognormalDistrErr mu sig
| sig >= sqrt (log m_huge - 2 * mu) = Left $ errMsg mu sig
| otherwise = LND <$> N.normalDistrErr mu sig
errMsg :: Double -> Double -> String
errMsg mu sig =
"Statistics.Distribution.Lognormal.lognormalDistr: sigma must be > 0 && < "
++ show lim ++ ". Got " ++ show sig
where lim = sqrt (log m_huge - 2 * mu)
-- | Create log normal distribution from mean and standard deviation.
lognormalDistrMeanStddevErr
:: Double -- ^ Mu
-> Double -- ^ Sigma
-> Either String LognormalDistribution
lognormalDistrMeanStddevErr m sd = LND <$> N.normalDistrErr mu sig
where r = sd / m
sig2 = log1p (r * r)
sig = sqrt sig2
mu = log m - sig2 / 2
-- | Variance is estimated using maximum likelihood method
-- (biased estimation) over the log of the data.
--
-- Returns @Nothing@ if sample contains less than one element or
-- variance is zero (all elements are equal)
instance D.FromSample LognormalDistribution Double where
fromSample = fmap LND . D.fromSample . G.map log
logDensity :: LognormalDistribution -> Double -> Double
logDensity (LND d) x
| x > 0 = let lx = log x in D.logDensity d lx - lx
| otherwise = 0
cumulative :: LognormalDistribution -> Double -> Double
cumulative (LND d) x
| x > 0 = D.cumulative d $ log x
| otherwise = 0
complCumulative :: LognormalDistribution -> Double -> Double
complCumulative (LND d) x
| x > 0 = D.complCumulative d $ log x
| otherwise = 1
quantile :: LognormalDistribution -> Double -> Double
quantile (LND d) = exp . D.quantile d
complQuantile :: LognormalDistribution -> Double -> Double
complQuantile (LND d) = exp . D.complQuantile d
|
