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|
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE PatternGuards #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module : Statistics.Distribution.Binomial
-- Copyright : (c) 2009 Bryan O'Sullivan
-- License : BSD3
--
-- Maintainer : bos@serpentine.com
-- Stability : experimental
-- Portability : portable
--
-- The binomial distribution. This is the discrete probability
-- distribution of the number of successes in a sequence of /n/
-- independent yes\/no experiments, each of which yields success with
-- probability /p/.
module Statistics.Distribution.Binomial
(
BinomialDistribution
-- * Constructors
, binomial
, binomialE
-- * Accessors
, bdTrials
, bdProbability
) 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 (choose,logChoose,incompleteBeta,log1p)
import Numeric.MathFunctions.Constants (m_epsilon,m_tiny)
import qualified Statistics.Distribution as D
import qualified Statistics.Distribution.Poisson.Internal as I
import Statistics.Internal
-- | The binomial distribution.
data BinomialDistribution = BD {
bdTrials :: {-# UNPACK #-} !Int
-- ^ Number of trials.
, bdProbability :: {-# UNPACK #-} !Double
-- ^ Probability.
} deriving (Eq, Typeable, Data, Generic)
instance Show BinomialDistribution where
showsPrec i (BD n p) = defaultShow2 "binomial" n p i
instance Read BinomialDistribution where
readPrec = defaultReadPrecM2 "binomial" binomialE
instance ToJSON BinomialDistribution
instance FromJSON BinomialDistribution where
parseJSON (Object v) = do
n <- v .: "bdTrials"
p <- v .: "bdProbability"
maybe (fail $ errMsg n p) return $ binomialE n p
parseJSON _ = empty
instance Binary BinomialDistribution where
put (BD x y) = put x >> put y
get = do
n <- get
p <- get
maybe (fail $ errMsg n p) return $ binomialE n p
instance D.Distribution BinomialDistribution where
cumulative = cumulative
complCumulative = complCumulative
instance D.DiscreteDistr BinomialDistribution where
probability = probability
logProbability = logProbability
instance D.Mean BinomialDistribution where
mean = mean
instance D.Variance BinomialDistribution where
variance = variance
instance D.MaybeMean BinomialDistribution where
maybeMean = Just . D.mean
instance D.MaybeVariance BinomialDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.Entropy BinomialDistribution where
entropy (BD n p)
| n == 0 = 0
| n <= 100 = directEntropy (BD n p)
| otherwise = I.poissonEntropy (fromIntegral n * p)
instance D.MaybeEntropy BinomialDistribution where
maybeEntropy = Just . D.entropy
-- This could be slow for big n
probability :: BinomialDistribution -> Int -> Double
probability (BD n p) k
| k < 0 || k > n = 0
| n == 0 = 1
-- choose could overflow Double for n >= 1030 so we switch to
-- log-domain to calculate probability
--
-- We also want to avoid underflow when computing p^k &
-- (1-p)^(n-k).
| n < 1000
, pK >= m_tiny
, pNK >= m_tiny = choose n k * pK * pNK
| otherwise = exp $ logChoose n k + log p * k' + log1p (-p) * nk'
where
pK = p^k
pNK = (1-p)^(n-k)
k' = fromIntegral k
nk' = fromIntegral $ n - k
logProbability :: BinomialDistribution -> Int -> Double
logProbability (BD n p) k
| k < 0 || k > n = (-1)/0
| n == 0 = 0
| otherwise = logChoose n k + log p * k' + log1p (-p) * nk'
where
k' = fromIntegral k
nk' = fromIntegral $ n - k
cumulative :: BinomialDistribution -> Double -> Double
cumulative (BD n p) x
| isNaN x = error "Statistics.Distribution.Binomial.cumulative: NaN input"
| isInfinite x = if x > 0 then 1 else 0
| k < 0 = 0
| k >= n = 1
| otherwise = incompleteBeta (fromIntegral (n-k)) (fromIntegral (k+1)) (1 - p)
where
k = floor x
complCumulative :: BinomialDistribution -> Double -> Double
complCumulative (BD n p) x
| isNaN x = error "Statistics.Distribution.Binomial.complCumulative: NaN input"
| isInfinite x = if x > 0 then 0 else 1
| k < 0 = 1
| k >= n = 0
| otherwise = incompleteBeta (fromIntegral (k+1)) (fromIntegral (n-k)) p
where
k = floor x
mean :: BinomialDistribution -> Double
mean (BD n p) = fromIntegral n * p
variance :: BinomialDistribution -> Double
variance (BD n p) = fromIntegral n * p * (1 - p)
directEntropy :: BinomialDistribution -> Double
directEntropy d@(BD n _) =
negate . sum $
takeWhile (< negate m_epsilon) $
dropWhile (not . (< negate m_epsilon)) $
[ let x = probability d k in x * log x | k <- [0..n]]
-- | Construct binomial distribution. Number of trials must be
-- non-negative and probability must be in [0,1] range
binomial :: Int -- ^ Number of trials.
-> Double -- ^ Probability.
-> BinomialDistribution
binomial n p = maybe (error $ errMsg n p) id $ binomialE n p
-- | Construct binomial distribution. Number of trials must be
-- non-negative and probability must be in [0,1] range
binomialE :: Int -- ^ Number of trials.
-> Double -- ^ Probability.
-> Maybe BinomialDistribution
binomialE n p
| n < 0 = Nothing
| p >= 0 && p <= 1 = Just (BD n p)
| otherwise = Nothing
errMsg :: Int -> Double -> String
errMsg n p
= "Statistics.Distribution.Binomial.binomial: n=" ++ show n
++ " p=" ++ show p ++ "but n>=0 and p in [0,1]"
|
