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|
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module : Statistics.Distribution.FDistribution
-- Copyright : (c) 2011 Aleksey Khudyakov
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- Fisher F distribution
module Statistics.Distribution.FDistribution (
FDistribution
-- * Constructors
, fDistribution
, fDistributionE
, fDistributionReal
, fDistributionRealE
-- * Accessors
, fDistributionNDF1
, fDistributionNDF2
) where
import Control.Applicative
import Data.Aeson (FromJSON(..), ToJSON, Value(..), (.:))
import Data.Binary (Binary(..))
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import Numeric.SpecFunctions (
logBeta, incompleteBeta, invIncompleteBeta, digamma)
import Numeric.MathFunctions.Constants (m_neg_inf)
import qualified Statistics.Distribution as D
import Statistics.Function (square)
import Statistics.Internal
-- | F distribution
data FDistribution = F { fDistributionNDF1 :: {-# UNPACK #-} !Double
, fDistributionNDF2 :: {-# UNPACK #-} !Double
, _pdfFactor :: {-# UNPACK #-} !Double
}
deriving (Eq, Typeable, Data, Generic)
instance Show FDistribution where
showsPrec i (F n m _) = defaultShow2 "fDistributionReal" n m i
instance Read FDistribution where
readPrec = defaultReadPrecM2 "fDistributionReal" fDistributionRealE
instance ToJSON FDistribution
instance FromJSON FDistribution where
parseJSON (Object v) = do
n <- v .: "fDistributionNDF1"
m <- v .: "fDistributionNDF2"
maybe (fail $ errMsgR n m) return $ fDistributionRealE n m
parseJSON _ = empty
instance Binary FDistribution where
put (F n m _) = put n >> put m
get = do
n <- get
m <- get
maybe (fail $ errMsgR n m) return $ fDistributionRealE n m
fDistribution :: Int -> Int -> FDistribution
fDistribution n m = maybe (error $ errMsg n m) id $ fDistributionE n m
fDistributionReal :: Double -> Double -> FDistribution
fDistributionReal n m = maybe (error $ errMsgR n m) id $ fDistributionRealE n m
fDistributionE :: Int -> Int -> Maybe FDistribution
fDistributionE 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 Just $ F n' m' f'
| otherwise = Nothing
fDistributionRealE :: Double -> Double -> Maybe FDistribution
fDistributionRealE n m
| n > 0 && m > 0 =
let f' = 0.5 * (log m * m + log n * n) - logBeta (0.5*n) (0.5*m)
in Just $ F n m f'
| otherwise = Nothing
errMsg :: Int -> Int -> String
errMsg _ _ = "Statistics.Distribution.FDistribution.fDistribution: non-positive number of degrees of freedom"
errMsgR :: Double -> Double -> String
errMsgR _ _ = "Statistics.Distribution.FDistribution.fDistribution: non-positive number of degrees of freedom"
instance D.Distribution FDistribution where
cumulative = cumulative
complCumulative = complCumulative
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
-- Only matches +∞
| isInfinite x = 1
-- NOTE: Here we rely on implementation detail of incompleteBeta. It
-- computes using series expansion for sufficiently small x
-- and uses following identity otherwise:
--
-- I(x; a, b) = 1 - I(1-x; b, a)
--
-- Point is we can compute 1-x as m/(m+y) without loss of
-- precision for large x. Sadly this switchover point is
-- implementation detail.
| n >= (n+m)*bx = incompleteBeta (0.5 * n) (0.5 * m) bx
| otherwise = 1 - incompleteBeta (0.5 * m) (0.5 * n) bx1
where
y = n * x
bx = y / (m + y)
bx1 = m / (m + y)
complCumulative :: FDistribution -> Double -> Double
complCumulative (F n m _) x
| x <= 0 = 1
-- Only matches +∞
| isInfinite x = 0
-- See NOTE at cumulative
| m >= (n+m)*bx = incompleteBeta (0.5 * m) (0.5 * n) bx
| otherwise = 1 - incompleteBeta (0.5 * n) (0.5 * m) bx1
where
y = n*x
bx = m / (m + y)
bx1 = 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 * square m * (m + n - 2) / (n * square (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.genContinuous
|
