```{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module    : Statistics.Distribution.CauchyLorentz
-- Copyright : (c) 2011 Aleksey Khudyakov
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- The Cauchy-Lorentz distribution. It's also known as Lorentz
-- distribution or Breit–Wigner distribution.
--
-- It doesn't have mean and variance.
module Statistics.Distribution.CauchyLorentz (
CauchyDistribution
, cauchyDistribMedian
, cauchyDistribScale
-- * Constructors
, cauchyDistribution
, standardCauchy
) where

import Data.Aeson (FromJSON, ToJSON)
import Data.Binary (Binary)
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import qualified Statistics.Distribution as D
import Data.Binary (put, get)
import Control.Applicative ((<\$>), (<*>))

-- | Cauchy-Lorentz distribution.
data CauchyDistribution = CD {
-- | Central value of Cauchy-Lorentz distribution which is its
--   mode and median. Distribution doesn't have mean so function
--   is named after median.
cauchyDistribMedian :: {-# UNPACK #-} !Double
-- | Scale parameter of Cauchy-Lorentz distribution. It's
--   different from variance and specify half width at half
--   maximum (HWHM).
, cauchyDistribScale  :: {-# UNPACK #-} !Double
}
deriving (Eq, Show, Read, Typeable, Data, Generic)

instance FromJSON CauchyDistribution
instance ToJSON CauchyDistribution

instance Binary CauchyDistribution where
put (CD x y) = put x >> put y
get = CD <\$> get <*> get

-- | Cauchy distribution
cauchyDistribution :: Double    -- ^ Central point
-> Double    -- ^ Scale parameter (FWHM)
-> CauchyDistribution
cauchyDistribution m s
| s > 0     = CD m s
| otherwise =
error \$ "Statistics.Distribution.CauchyLorentz.cauchyDistribution: FWHM must be positive. Got " ++ show s

standardCauchy :: CauchyDistribution
standardCauchy = CD 0 1

instance D.Distribution CauchyDistribution where
cumulative (CD m s) x = 0.5 + atan( (x - m) / s ) / pi

instance D.ContDistr CauchyDistribution where
density (CD m s) x = (1 / pi) / (s * (1 + y*y))
where y = (x - m) / s
quantile (CD m s) p
| p > 0 && p < 1 = m + s * tan( pi * (p - 0.5) )
| p == 0         = -1 / 0
| p == 1         =  1 / 0
| otherwise      =
error \$ "Statistics.Distribution.CauchyLorentz..quantile: p must be in [0,1] range. Got: "++show p

instance D.ContGen CauchyDistribution where
genContVar = D.genContinous

instance D.Entropy CauchyDistribution where
entropy (CD _ s) = log s + log (4*pi)

instance D.MaybeEntropy CauchyDistribution where
maybeEntropy = Just . D.entropy
```