```{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module    : Statistics.Distribution.FDistribution
-- Copyright : (c) 2011 Aleksey Khudyakov
--
-- Maintainer  : bos@serpentine.com
-- Stability   : experimental
-- Portability : portable
--
-- Fisher F distribution
module Statistics.Distribution.FDistribution (
FDistribution
, fDistribution
, fDistributionNDF1
, fDistributionNDF2
) where

import Data.Binary (Binary)
import Data.Data (Data, Typeable)
import Numeric.MathFunctions.Constants (m_neg_inf)
import GHC.Generics (Generic)
import qualified Statistics.Distribution as D
import Numeric.SpecFunctions (
logBeta, incompleteBeta, invIncompleteBeta, digamma)
import Data.Binary (put, get)
import Control.Applicative ((<\$>), (<*>))

-- | F distribution
data FDistribution = F { fDistributionNDF1 :: {-# UNPACK #-} !Double
, fDistributionNDF2 :: {-# UNPACK #-} !Double
, _pdfFactor        :: {-# UNPACK #-} !Double
}
deriving (Eq, Show, Read, Typeable, Data, Generic)

instance Binary FDistribution where
get = F <\$> get <*> get <*> get
put (F x y z) = put x >> put y >> put z

fDistribution :: Int -> Int -> FDistribution
fDistribution n m
| n > 0 && m > 0 =
let n' = fromIntegral n
m' = fromIntegral m
f' = 0.5 * (log m' * m' + log n' * n') - logBeta (0.5*n') (0.5*m')
in F n' m' f'
| otherwise =
error "Statistics.Distribution.FDistribution.fDistribution: non-positive number of degrees of freedom"

instance D.Distribution FDistribution where
cumulative = cumulative

instance D.ContDistr FDistribution where
density d x
| x <= 0    = 0
| otherwise = exp \$ logDensity d x
logDensity d x
| x <= 0    = m_neg_inf
| otherwise = logDensity d x
quantile = quantile

cumulative :: FDistribution -> Double -> Double
cumulative (F n m _) x
| x <= 0       = 0
| isInfinite x = 1            -- Only matches +∞
| otherwise    = let y = n*x in incompleteBeta (0.5 * n) (0.5 * m) (y / (m + y))

logDensity :: FDistribution -> Double -> Double
logDensity (F n m fac) x
= fac + log x * (0.5 * n - 1) - log(m + n*x) * 0.5 * (n + m)

quantile :: FDistribution -> Double -> Double
quantile (F n m _) p
| p >= 0 && p <= 1 =
let x = invIncompleteBeta (0.5 * n) (0.5 * m) p
in m * x / (n * (1 - x))
| otherwise =
error \$ "Statistics.Distribution.Uniform.quantile: p must be in [0,1] range. Got: "++show p

instance D.MaybeMean FDistribution where
maybeMean (F _ m _) | m > 2     = Just \$ m / (m - 2)
| otherwise = Nothing

instance D.MaybeVariance FDistribution where
maybeStdDev (F n m _)
| m > 4     = Just \$ 2 * sqr m * (m + n - 2) / (n * sqr (m - 2) * (m - 4))
| otherwise = Nothing

instance D.Entropy FDistribution where
entropy (F n m _) =
let nHalf = 0.5 * n
mHalf = 0.5 * m in
log (n/m)
+ logBeta nHalf mHalf
+ (1 - nHalf) * digamma nHalf
- (1 + mHalf) * digamma mHalf
+ (nHalf + mHalf) * digamma (nHalf + mHalf)

instance D.MaybeEntropy FDistribution where
maybeEntropy = Just . D.entropy

instance D.ContGen FDistribution where
genContVar = D.genContinous

sqr :: Double -> Double
sqr x = x * x
{-# INLINE sqr #-}
```