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|
{-# LANGUAGE OverloadedStrings, PatternGuards,
DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module : Statistics.Distribution.NegativeBinomial
-- Copyright : (c) 2022 Lorenz Minder
-- License : BSD3
--
-- Maintainer : lminder@gmx.net
-- Stability : experimental
-- Portability : portable
--
-- The negative binomial distribution. This is the discrete probability
-- distribution of the number of failures in a sequence of independent
-- yes\/no experiments before a specified number of successes /r/. Each
-- Bernoulli trial has success probability /p/ in the range (0, 1]. The
-- parameter /r/ must be positive, but does not have to be integer.
module Statistics.Distribution.NegativeBinomial (
NegativeBinomialDistribution
-- * Constructors
, negativeBinomial
, negativeBinomialE
-- * Accessors
, nbdSuccesses
, nbdProbability
) where
import Control.Applicative
import Data.Aeson (FromJSON(..), ToJSON, Value(..), (.:))
import Data.Binary (Binary(..))
import Data.Data (Data, Typeable)
import Data.Foldable (foldl')
import GHC.Generics (Generic)
import Numeric.SpecFunctions (incompleteBeta, log1p)
import Numeric.SpecFunctions.Extra (logChooseFast)
import Numeric.MathFunctions.Constants (m_epsilon, m_tiny)
import qualified Statistics.Distribution as D
import Statistics.Internal
-- Math helper functions
-- | Generalized binomial coefficients.
--
-- These computes binomial coefficients with the small generalization
-- that the /n/ need not be integer, but can be real.
gChoose :: Double -> Int -> Double
gChoose n k
| k < 0 = 0
| k' >= 50 = exp $ logChooseFast n k'
| otherwise = foldl' (*) 1 factors
where factors = [ (n - k' + j) / j | j <- [1..k'] ]
k' = fromIntegral k
-- Implementation of Negative Binomial
-- | The negative binomial distribution.
data NegativeBinomialDistribution = NBD {
nbdSuccesses :: {-# UNPACK #-} !Double
-- ^ Number of successes until stop
, nbdProbability :: {-# UNPACK #-} !Double
-- ^ Success probability.
} deriving (Eq, Typeable, Data, Generic)
instance Show NegativeBinomialDistribution where
showsPrec i (NBD r p) = defaultShow2 "negativeBinomial" r p i
instance Read NegativeBinomialDistribution where
readPrec = defaultReadPrecM2 "negativeBinomial" negativeBinomialE
instance ToJSON NegativeBinomialDistribution
instance FromJSON NegativeBinomialDistribution where
parseJSON (Object v) = do
r <- v .: "nbdSuccesses"
p <- v .: "nbdProbability"
maybe (fail $ errMsg r p) return $ negativeBinomialE r p
parseJSON _ = empty
instance Binary NegativeBinomialDistribution where
put (NBD r p) = put r >> put p
get = do
r <- get
p <- get
maybe (fail $ errMsg r p) return $ negativeBinomialE r p
instance D.Distribution NegativeBinomialDistribution where
cumulative = cumulative
complCumulative = complCumulative
instance D.DiscreteDistr NegativeBinomialDistribution where
probability = probability
logProbability = logProbability
instance D.Mean NegativeBinomialDistribution where
mean = mean
instance D.Variance NegativeBinomialDistribution where
variance = variance
instance D.MaybeMean NegativeBinomialDistribution where
maybeMean = Just . D.mean
instance D.MaybeVariance NegativeBinomialDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.Entropy NegativeBinomialDistribution where
entropy = directEntropy
instance D.MaybeEntropy NegativeBinomialDistribution where
maybeEntropy = Just . D.entropy
-- This could be slow for big n
probability :: NegativeBinomialDistribution -> Int -> Double
probability d@(NBD r p) k
| k < 0 = 0
-- Switch to log domain for large k + r to avoid overflows.
--
-- We also want to avoid underflow when computing (1-p)^k &
-- p^r.
| k' + r < 1000
, pK >= m_tiny
, pR >= m_tiny = gChoose (k' + r - 1) k * pK * pR
| otherwise = exp $ logProbability d k
where
pK = exp $ log1p (-p) * k'
pR = p**r
k' = fromIntegral k
logProbability :: NegativeBinomialDistribution -> Int -> Double
logProbability (NBD r p) k
| k < 0 = (-1)/0
| otherwise = logChooseFast (k' + r - 1) k'
+ log1p (-p) * k'
+ log p * r
where k' = fromIntegral k
cumulative :: NegativeBinomialDistribution -> Double -> Double
cumulative (NBD r p) x
| isNaN x = error "Statistics.Distribution.NegativeBinomial.cumulative: NaN input"
| isInfinite x = if x > 0 then 1 else 0
| k < 0 = 0
| otherwise = incompleteBeta r (fromIntegral (k+1)) p
where
k = floor x :: Integer
complCumulative :: NegativeBinomialDistribution -> Double -> Double
complCumulative (NBD r p) x
| isNaN x = error "Statistics.Distribution.NegativeBinomial.complCumulative: NaN input"
| isInfinite x = if x > 0 then 0 else 1
| k < 0 = 1
| otherwise = incompleteBeta (fromIntegral (k+1)) r (1 - p)
where
k = (floor x)::Integer
mean :: NegativeBinomialDistribution -> Double
mean (NBD r p) = r * (1 - p)/p
variance :: NegativeBinomialDistribution -> Double
variance (NBD r p) = r * (1 - p)/(p * p)
directEntropy :: NegativeBinomialDistribution -> Double
directEntropy d =
negate . sum $
takeWhile (< -m_epsilon) $
dropWhile (>= -m_epsilon) $
[ let x = probability d k in x * log x | k <- [0..]]
-- | Construct negative binomial distribution. Number of failures /r/
-- must be positive and probability must be in (0,1] range
negativeBinomial :: Double -- ^ Number of successes.
-> Double -- ^ Success probability.
-> NegativeBinomialDistribution
negativeBinomial r p = maybe (error $ errMsg r p) id $ negativeBinomialE r p
-- | Construct negative binomial distribution. Number of failures /r/
-- must be positive and probability must be in (0,1] range
negativeBinomialE :: Double -- ^ Number of successes.
-> Double -- ^ Success probability.
-> Maybe NegativeBinomialDistribution
negativeBinomialE r p
| r > 0 && 0 < p && p <= 1 = Just (NBD r p)
| otherwise = Nothing
errMsg :: Double -> Double -> String
errMsg r p
= "Statistics.Distribution.NegativeBinomial.negativeBinomial: r=" ++ show r
++ " p=" ++ show p ++ ", but need r>0 and p in (0,1]"
|
