-- | -- 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 assesing -- whether given sample could be described by distribution or whether -- two samples have the same distribution. It's only applicable to -- continous distributions. module Statistics.Test.KolmogorovSmirnov ( -- * Kolmogorov-Smirnov test kolmogorovSmirnovTest , kolmogorovSmirnovTestCdf , kolmogorovSmirnovTest2 -- * Evaluate statistics , kolmogorovSmirnovCdfD , kolmogorovSmirnovD , kolmogorovSmirnov2D -- * Probablities , kolmogorovSmirnovProbability -- * Data types , TestType(..) , TestResult(..) -- * References -- $references ) where import Control.Monad (when) import Prelude hiding (exponent, sum) import Statistics.Distribution (Distribution(..)) import Statistics.Function (sort, unsafeModify) import Statistics.Matrix (center, exponent, for, fromVector, power) import Statistics.Test.Types (TestResult(..), TestType(..), significant) import Statistics.Types (Sample) import qualified Data.Vector.Unboxed as U import qualified Data.Vector.Unboxed.Mutable as M ---------------------------------------------------------------- -- Test ---------------------------------------------------------------- -- | Check that sample could be described by -- distribution. 'Significant' means distribution is not compatible -- with data for given p-value. -- -- This test uses Marsaglia-Tsang-Wang exact alogorithm for -- calculation of p-value. kolmogorovSmirnovTest :: Distribution d => d -- ^ Distribution -> Double -- ^ p-value -> Sample -- ^ Data sample -> TestResult kolmogorovSmirnovTest d = kolmogorovSmirnovTestCdf (cumulative d) -- | Variant of 'kolmogorovSmirnovTest' which uses CFD in form of -- function. kolmogorovSmirnovTestCdf :: (Double -> Double) -- ^ CDF of distribution -> Double -- ^ p-value -> Sample -- ^ Data sample -> TestResult kolmogorovSmirnovTestCdf cdf p sample | p > 0 && p < 1 = significant $ 1 - prob < p | otherwise = error "Statistics.Test.KolmogorovSmirnov.kolmogorovSmirnovTestCdf:bad p-value" where d = kolmogorovSmirnovCdfD cdf sample prob = kolmogorovSmirnovProbability (U.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. -- -- This test uses approxmate formula for computing p-value. kolmogorovSmirnovTest2 :: Double -- ^ p-value -> Sample -- ^ Sample 1 -> Sample -- ^ Sample 2 -> TestResult kolmogorovSmirnovTest2 p xs1 xs2 | p > 0 && p < 1 = significant $ 1 - prob( d*(en + 0.12 + 0.11/en) ) < p | otherwise = error "Statistics.Test.KolmogorovSmirnov.kolmogorovSmirnovTest2:bad p-value" where d = kolmogorovSmirnov2D xs1 xs2 -- Effective number of data points n1 = fromIntegral (U.length xs1) n2 = fromIntegral (U.length xs2) en = sqrt $ n1 * n2 / (n1 + n2) -- prob z | z < 0 = error "kolmogorovSmirnov2D: internal error" | z == 0 = 1 | 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) -- 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 :: (Double -> Double) -- ^ CDF function -> Sample -- ^ Sample -> Double kolmogorovSmirnovCdfD cdf sample | U.null sample = 0 | otherwise = U.maximum $ U.zipWith3 (\p a b -> abs (p-a) `max` abs (p-b)) ps steps (U.tail steps) where xs = sort sample n = U.length xs -- ps = U.map cdf xs steps = U.map ((/ fromIntegral n) . fromIntegral) $ U.generate (n+1) id -- | Calculate Kolmogorov's statistic /D/ for given cumulative -- distribution function (CDF) and data sample. If sample is empty -- returns 0. kolmogorovSmirnovD :: (Distribution d) => d -- ^ Distribution -> Sample -- ^ Sample -> Double kolmogorovSmirnovD d = kolmogorovSmirnovCdfD (cumulative d) -- | Calculate Kolmogorov's statistic /D/ for two data samples. If -- either of samples is empty returns 0. kolmogorovSmirnov2D :: Sample -- ^ First sample -> Sample -- ^ Second sample -> Double kolmogorovSmirnov2D sample1 sample2 | U.null sample1 || U.null sample2 = 0 | otherwise = worker 0 0 0 where xs1 = sort sample1 xs2 = sort sample2 n1 = U.length xs1 n2 = U.length xs2 en1 = fromIntegral n1 en2 = fromIntegral n2 -- Find new index skip x i xs = go (i+1) where go n | n >= U.length xs = n | xs U.! 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 U.! i1 d2 = xs2 U.! 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) -- | 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 potencially 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 $ 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 m = loop 1 (center m) (exponent m) 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 ---------------------------------------------------------------- -- $references -- -- * G. Marsaglia, W. W. Tsang, J. Wang (2003) Evaluating Kolmogorov's -- distribution, Journal of Statistical Software, American -- Statistical Association, vol. 8(i18).