{-# LANGUAGE FlexibleContexts #-} -- | -- Module : Statistics.Test.KolmogorovSmirnov -- Copyright : (c) 2011 Aleksey Khudyakov -- License : BSD3 -- -- Maintainer : bos@serpentine.com -- Stability : experimental -- Portability : portable -- -- Kolmogov-Smirnov tests are non-parametric tests for assessing -- whether given sample could be described by distribution or whether -- two samples have the same distribution. It's only applicable to -- continuous distributions. module Statistics.Test.KolmogorovSmirnov ( -- * Kolmogorov-Smirnov test kolmogorovSmirnovTest , kolmogorovSmirnovTestCdf , kolmogorovSmirnovTest2 -- * Evaluate statistics , kolmogorovSmirnovCdfD , kolmogorovSmirnovD , kolmogorovSmirnov2D -- * Probablities , kolmogorovSmirnovProbability -- * References -- \$references , module Statistics.Test.Types ) where import Control.Monad (when) import Prelude hiding (exponent, sum) import Statistics.Distribution (Distribution(..)) import Statistics.Function (gsort, unsafeModify) import Statistics.Matrix (center, for, fromVector) import qualified Statistics.Matrix as Mat import Statistics.Test.Types import Statistics.Types (mkPValue) import qualified Data.Vector as V import qualified Data.Vector.Storable as S import qualified Data.Vector.Unboxed as U import qualified Data.Vector.Generic as G import Data.Vector.Generic ((!)) import qualified Data.Vector.Unboxed.Mutable as M ---------------------------------------------------------------- -- Test ---------------------------------------------------------------- -- | Check that sample could be described by distribution. Returns -- @Nothing@ is sample is empty -- -- This test uses Marsaglia-Tsang-Wang exact algorithm for -- calculation of p-value. kolmogorovSmirnovTest :: (Distribution d, G.Vector v Double) => d -- ^ Distribution -> v Double -- ^ Data sample -> Maybe (Test ()) {-# INLINE kolmogorovSmirnovTest #-} kolmogorovSmirnovTest d = kolmogorovSmirnovTestCdf (cumulative d) -- | Variant of 'kolmogorovSmirnovTest' which uses CDF in form of -- function. kolmogorovSmirnovTestCdf :: (G.Vector v Double) => (Double -> Double) -- ^ CDF of distribution -> v Double -- ^ Data sample -> Maybe (Test ()) {-# INLINE kolmogorovSmirnovTestCdf #-} kolmogorovSmirnovTestCdf cdf sample | G.null sample = Nothing | otherwise = Just Test { testSignificance = mkPValue \$ 1 - prob , testStatistics = d , testDistribution = () } where d = kolmogorovSmirnovCdfD cdf sample prob = kolmogorovSmirnovProbability (G.length sample) d -- | Two sample Kolmogorov-Smirnov test. It tests whether two data -- samples could be described by the same distribution without -- making any assumptions about it. If either of samples is empty -- returns Nothing. -- -- This test uses approximate formula for computing p-value. kolmogorovSmirnovTest2 :: (G.Vector v Double) => v Double -- ^ Sample 1 -> v Double -- ^ Sample 2 -> Maybe (Test ()) kolmogorovSmirnovTest2 xs1 xs2 | G.null xs1 || G.null xs2 = Nothing | otherwise = Just Test { testSignificance = mkPValue \$ 1 - prob d , testStatistics = d , testDistribution = () } where d = kolmogorovSmirnov2D xs1 xs2 * (en + 0.12 + 0.11/en) -- Effective number of data points n1 = fromIntegral (G.length xs1) n2 = fromIntegral (G.length xs2) en = sqrt \$ n1 * n2 / (n1 + n2) -- prob z | z < 0 = error "kolmogorovSmirnov2D: internal error" | z == 0 = 0 | z < 1.18 = let y = exp( -1.23370055013616983 / (z*z) ) in 2.25675833419102515 * sqrt( -log y ) * (y + y**9 + y**25 + y**49) | otherwise = let x = exp(-2 * z * z) in 1 - 2*(x - x**4 + x**9) {-# INLINABLE kolmogorovSmirnovTest2 #-} {-# SPECIALIZE kolmogorovSmirnovTest2 :: U.Vector Double -> U.Vector Double -> Maybe (Test ()) #-} {-# SPECIALIZE kolmogorovSmirnovTest2 :: V.Vector Double -> V.Vector Double -> Maybe (Test ()) #-} {-# SPECIALIZE kolmogorovSmirnovTest2 :: S.Vector Double -> S.Vector Double -> Maybe (Test ()) #-} -- FIXME: Find source for approximation for D ---------------------------------------------------------------- -- Kolmogorov's statistic ---------------------------------------------------------------- -- | Calculate Kolmogorov's statistic /D/ for given cumulative -- distribution function (CDF) and data sample. If sample is empty -- returns 0. kolmogorovSmirnovCdfD :: G.Vector v Double => (Double -> Double) -- ^ CDF function -> v Double -- ^ Sample -> Double kolmogorovSmirnovCdfD cdf sample | G.null sample = 0 | otherwise = G.maximum \$ G.zipWith3 (\p a b -> abs (p-a) `max` abs (p-b)) ps steps (G.tail steps) where xs = gsort sample n = G.length xs -- ps = G.map cdf xs steps = G.map (/ fromIntegral n) \$ G.generate (n+1) fromIntegral {-# INLINABLE kolmogorovSmirnovCdfD #-} {-# SPECIALIZE kolmogorovSmirnovCdfD :: (Double -> Double) -> U.Vector Double -> Double #-} {-# SPECIALIZE kolmogorovSmirnovCdfD :: (Double -> Double) -> V.Vector Double -> Double #-} {-# SPECIALIZE kolmogorovSmirnovCdfD :: (Double -> Double) -> S.Vector Double -> Double #-} -- | Calculate Kolmogorov's statistic /D/ for given cumulative -- distribution function (CDF) and data sample. If sample is empty -- returns 0. kolmogorovSmirnovD :: (Distribution d, G.Vector v Double) => d -- ^ Distribution -> v Double -- ^ Sample -> Double kolmogorovSmirnovD d = kolmogorovSmirnovCdfD (cumulative d) {-# INLINE kolmogorovSmirnovD #-} -- | Calculate Kolmogorov's statistic /D/ for two data samples. If -- either of samples is empty returns 0. kolmogorovSmirnov2D :: (G.Vector v Double) => v Double -- ^ First sample -> v Double -- ^ Second sample -> Double kolmogorovSmirnov2D sample1 sample2 | G.null sample1 || G.null sample2 = 0 | otherwise = worker 0 0 0 where xs1 = gsort sample1 xs2 = gsort sample2 n1 = G.length xs1 n2 = G.length xs2 en1 = fromIntegral n1 en2 = fromIntegral n2 -- Find new index skip x i xs = go (i+1) where go n | n >= G.length xs = n | xs ! n == x = go (n+1) | otherwise = n -- Main loop worker d i1 i2 | i1 >= n1 || i2 >= n2 = d | otherwise = worker d' i1' i2' where d1 = xs1 ! i1 d2 = xs2 ! i2 i1' | d1 <= d2 = skip d1 i1 xs1 | otherwise = i1 i2' | d2 <= d1 = skip d2 i2 xs2 | otherwise = i2 d' = max d (abs \$ fromIntegral i1' / en1 - fromIntegral i2' / en2) {-# INLINABLE kolmogorovSmirnov2D #-} {-# SPECIALIZE kolmogorovSmirnov2D :: U.Vector Double -> U.Vector Double -> Double #-} {-# SPECIALIZE kolmogorovSmirnov2D :: V.Vector Double -> V.Vector Double -> Double #-} {-# SPECIALIZE kolmogorovSmirnov2D :: S.Vector Double -> S.Vector Double -> Double #-} -- | Calculate cumulative probability function for Kolmogorov's -- distribution with /n/ parameters or probability of getting value -- smaller than /d/ with n-elements sample. -- -- It uses algorithm by Marsgalia et. al. and provide at least -- 7-digit accuracy. kolmogorovSmirnovProbability :: Int -- ^ Size of the sample -> Double -- ^ D value -> Double kolmogorovSmirnovProbability n d -- Avoid potentially lengthy calculations for large N and D > 0.999 | s > 7.24 || (s > 3.76 && n > 99) = 1 - 2 * exp( -(2.000071 + 0.331 / sqrt n' + 1.409 / n') * s) -- Exact computation | otherwise = fini \$ KSMatrix 0 matrix `power` n where s = n' * d * d n' = fromIntegral n size = 2*k - 1 k = floor (n' * d) + 1 h = fromIntegral k - n' * d -- Calculate initial matrix matrix = let m = U.create \$ do mat <- M.new (size*size) -- Fill matrix with 0 and 1s for 0 size \$ \row -> for 0 size \$ \col -> do let val | row + 1 >= col = 1 | otherwise = 0 :: Double M.write mat (row * size + col) val -- Correct left column/bottom row for 0 size \$ \i -> do let delta = h ^^ (i + 1) unsafeModify mat (i * size) (subtract delta) unsafeModify mat (size * size - 1 - i) (subtract delta) -- Correct corner element if needed when (2*h > 1) \$ do unsafeModify mat ((size - 1) * size) (+ ((2*h - 1) ^ size)) -- Divide diagonals by factorial let divide g num | num == size = return () | otherwise = do for num size \$ \i -> unsafeModify mat (i * (size + 1) - num) (/ g) divide (g * fromIntegral (num+2)) (num+1) divide 2 1 return mat in fromVector size size m -- Last calculation fini (KSMatrix e m) = loop 1 (center m) e where loop i ss eQ | i > n = ss * 10 ^^ eQ | ss' < 1e-140 = loop (i+1) (ss' * 1e140) (eQ - 140) | otherwise = loop (i+1) ss' eQ where ss' = ss * fromIntegral i / fromIntegral n data KSMatrix = KSMatrix Int Mat.Matrix multiply :: KSMatrix -> KSMatrix -> KSMatrix multiply (KSMatrix e1 m1) (KSMatrix e2 m2) = KSMatrix (e1+e2) (Mat.multiply m1 m2) power :: KSMatrix -> Int -> KSMatrix power mat 1 = mat power mat n = avoidOverflow res where mat2 = power mat (n `quot` 2) pow = multiply mat2 mat2 res | odd n = multiply pow mat | otherwise = pow avoidOverflow :: KSMatrix -> KSMatrix avoidOverflow ksm@(KSMatrix e m) | center m > 1e140 = KSMatrix (e + 140) (Mat.map (* 1e-140) m) | otherwise = ksm ---------------------------------------------------------------- -- \$references -- -- * G. Marsaglia, W. W. Tsang, J. Wang (2003) Evaluating Kolmogorov's -- distribution, Journal of Statistical Software, American -- Statistical Association, vol. 8(i18).