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|
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE OverloadedStrings #-}
{-# LANGUAGE DeriveDataTypeable, DeriveGeneric #-}
-- |
-- Module : Statistics.Distribution.Laplace
-- Copyright : (c) 2015 Mihai Maruseac
-- License : BSD3
--
-- Maintainer : mihai.maruseac@maruseac.com
-- Stability : experimental
-- Portability : portable
--
-- The Laplace distribution. This is the continuous probability
-- defined as the difference of two iid exponential random variables
-- or a Brownian motion evaluated as exponentially distributed times.
-- It is used in differential privacy (Laplace Method), speech
-- recognition and least absolute deviations method (Laplace's first
-- law of errors, giving a robust regression method)
--
module Statistics.Distribution.Laplace
(
LaplaceDistribution
-- * Constructors
, laplace
, laplaceE
-- * Accessors
, ldLocation
, ldScale
) where
import Control.Applicative
import Data.Aeson (FromJSON(..), ToJSON, Value(..), (.:))
import Data.Binary (Binary(..))
import Data.Data (Data, Typeable)
import GHC.Generics (Generic)
import qualified Data.Vector.Generic as G
import qualified Statistics.Distribution as D
import qualified Statistics.Quantile as Q
import qualified Statistics.Sample as S
import Statistics.Internal
data LaplaceDistribution = LD {
ldLocation :: {-# UNPACK #-} !Double
-- ^ Location.
, ldScale :: {-# UNPACK #-} !Double
-- ^ Scale.
} deriving (Eq, Typeable, Data, Generic)
instance Show LaplaceDistribution where
showsPrec i (LD l s) = defaultShow2 "laplace" l s i
instance Read LaplaceDistribution where
readPrec = defaultReadPrecM2 "laplace" laplaceE
instance ToJSON LaplaceDistribution
instance FromJSON LaplaceDistribution where
parseJSON (Object v) = do
l <- v .: "ldLocation"
s <- v .: "ldScale"
maybe (fail $ errMsg l s) return $ laplaceE l s
parseJSON _ = empty
instance Binary LaplaceDistribution where
put (LD l s) = put l >> put s
get = do
l <- get
s <- get
maybe (fail $ errMsg l s) return $ laplaceE l s
instance D.Distribution LaplaceDistribution where
cumulative = cumulative
complCumulative = complCumulative
instance D.ContDistr LaplaceDistribution where
density (LD l s) x = exp (- abs (x - l) / s) / (2 * s)
logDensity (LD l s) x = - abs (x - l) / s - log 2 - log s
quantile = quantile
complQuantile = complQuantile
instance D.Mean LaplaceDistribution where
mean (LD l _) = l
instance D.Variance LaplaceDistribution where
variance (LD _ s) = 2 * s * s
instance D.MaybeMean LaplaceDistribution where
maybeMean = Just . D.mean
instance D.MaybeVariance LaplaceDistribution where
maybeStdDev = Just . D.stdDev
maybeVariance = Just . D.variance
instance D.Entropy LaplaceDistribution where
entropy (LD _ s) = 1 + log (2 * s)
instance D.MaybeEntropy LaplaceDistribution where
maybeEntropy = Just . D.entropy
instance D.ContGen LaplaceDistribution where
genContVar = D.genContinuous
cumulative :: LaplaceDistribution -> Double -> Double
cumulative (LD l s) x
| x <= l = 0.5 * exp ( (x - l) / s)
| otherwise = 1 - 0.5 * exp ( - (x - l) / s )
complCumulative :: LaplaceDistribution -> Double -> Double
complCumulative (LD l s) x
| x <= l = 1 - 0.5 * exp ( (x - l) / s)
| otherwise = 0.5 * exp ( - (x - l) / s )
quantile :: LaplaceDistribution -> Double -> Double
quantile (LD l s) p
| p == 0 = -inf
| p == 1 = inf
| p == 0.5 = l
| p > 0 && p < 0.5 = l + s * log (2 * p)
| p > 0.5 && p < 1 = l - s * log (2 - 2 * p)
| otherwise =
error $ "Statistics.Distribution.Laplace.quantile: p must be in [0,1] range. Got: "++show p
where
inf = 1 / 0
complQuantile :: LaplaceDistribution -> Double -> Double
complQuantile (LD l s) p
| p == 0 = inf
| p == 1 = -inf
| p == 0.5 = l
| p > 0 && p < 0.5 = l - s * log (2 * p)
| p > 0.5 && p < 1 = l + s * log (2 - 2 * p)
| otherwise =
error $ "Statistics.Distribution.Laplace.quantile: p must be in [0,1] range. Got: "++show p
where
inf = 1 / 0
-- | Create an Laplace distribution.
laplace :: Double -- ^ Location
-> Double -- ^ Scale
-> LaplaceDistribution
laplace l s = maybe (error $ errMsg l s) id $ laplaceE l s
-- | Create an Laplace distribution.
laplaceE :: Double -- ^ Location
-> Double -- ^ Scale
-> Maybe LaplaceDistribution
laplaceE l s
| s >= 0 = Just (LD l s)
| otherwise = Nothing
errMsg :: Double -> Double -> String
errMsg _ s = "Statistics.Distribution.Laplace.laplace: scale parameter must be positive. Got " ++ show s
-- | Create Laplace distribution from sample. The location is estimated
-- as the median of the sample, and the scale as the mean absolute
-- deviation of the median.
instance D.FromSample LaplaceDistribution Double where
fromSample xs
| G.null xs = Nothing
| otherwise = Just $! LD s l
where
s = Q.median Q.medianUnbiased xs
l = S.mean $ G.map (\x -> abs $ x - s) xs
|
