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If your program is a subroutine library, you +may consider it more useful to permit linking proprietary applications with +the library. If this is what you want to do, use the GNU Lesser General +Public License instead of this License. But first, please read +. + diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/StatisticalMethods.cabal b/StatisticalMethods.cabal new file mode 100644 --- /dev/null +++ b/StatisticalMethods.cabal @@ -0,0 +1,50 @@ +name: StatisticalMethods +version: 0.0.0.1 +author: Christian Hoener zu Siederdissen +maintainer: choener@tbi.univie.ac.at +homepage: http://www.tbi.univie.ac.at/~choener/Haskell/ +copyright: Christian Hoener zu Siederdissen, 2011 +category: Statistics +synopsis: Collection of useful statistical methods. +license: GPL-3 +license-file: LICENSE +build-type: Simple +stability: experimental +cabal-version: >= 1.4.0 +description: + High-level statistical methods. + . + * Confusion matrix + . + * Confusion matrix dependent statistics (sensitivity, specificity, F-measure, mcc) + . + * EM algorithm for two-component Gaussian mixture. + . + * GMM (Gaussian Mixture Models) with >=1 Gaussians fitted to the data. + . + . + . + Note that some methods are for testing only (two-component + Gaussian mixture EM). + + + +extra-source-files: + +library + build-depends: + base >=4 && <5, + vector, + statistics, + tuple + + exposed-modules: + Statistics.ConfusionMatrix + Statistics.ConfusionMatrix.Instances + Statistics.EM.GMM + Statistics.EM.TwoGaussian + Statistics.PerformanceMetrics + TestData.Elements + + ghc-options: + -O2 diff --git a/Statistics/ConfusionMatrix.hs b/Statistics/ConfusionMatrix.hs new file mode 100644 --- /dev/null +++ b/Statistics/ConfusionMatrix.hs @@ -0,0 +1,30 @@ +{-# LANGUAGE GeneralizedNewtypeDeriving #-} +{-# LANGUAGE FlexibleInstances #-} + +-- | The confusion matrix contains four data points: the true and false +-- positives and the true and false negatives. From these four data points, +-- other statistics can be extracted. +-- +-- Fawcett, ROC Graphs: Notes and Practical Considerations for Researchers, +-- 2004, Kluwer Academic Publishers + +module Statistics.ConfusionMatrix where + + + +-- | The confusion matrix. + +data ConfusionMatrix = ConfusionMatrix + { fn :: WrappedDouble + , fp :: WrappedDouble + , tn :: WrappedDouble + , tp :: WrappedDouble + } deriving (Read,Show,Eq) + +type WrappedDouble = Either String Double + +-- | Given a certain data-set, create a confusion matrix. + +class MkConfusionMatrix a where + mkConfusionMatrix :: a -> ConfusionMatrix + diff --git a/Statistics/ConfusionMatrix/Instances.hs b/Statistics/ConfusionMatrix/Instances.hs new file mode 100644 --- /dev/null +++ b/Statistics/ConfusionMatrix/Instances.hs @@ -0,0 +1,24 @@ + +-- | In general, it is not easy to define the whole confusion matrix +-- generically without knowing anything about the source data. For certain +-- elements however, it is possible. These instances are all defined on +-- newtypes in order to not create instances on generic data types like lists. + +module Statistics.ConfusionMatrix.Instances where + +import Statistics.ConfusionMatrix + + + +-- | The ctor expects the total number of possibilities first, then a list of +-- true positive elements, followed by a list of predicted elements. + +newtype (Eq a, Ord a) => ListSimilar a = ListSimilar (Int,[a],[a]) + +instance (Eq a, Ord a) => MkConfusionMatrix (ListSimilar a) where + mkConfusionMatrix (ListSimilar (count,tp,pred)) = ConfusionMatrix + { fn = error "fn undefined" + , fp = error "fp undefined" + , tn = error "tn undefined" + , tp = error "tp undefined" + } diff --git a/Statistics/EM/GMM.hs b/Statistics/EM/GMM.hs new file mode 100644 --- /dev/null +++ b/Statistics/EM/GMM.hs @@ -0,0 +1,100 @@ + +-- | EM for a mixture of k one-dimensional Gaussians. This procedure tends to +-- produce "NaN"s whenever more Gaussians are being selected than are called +-- for. This is rather convenient. ;-) +-- +-- TODO cite paper + +module Statistics.EM.GMM + ( emFix + , emStarts + ) where + +import Control.Monad.Fix (fix) +import Data.List (sort,maximumBy,tails,inits,genericLength) +import Data.Ord +import qualified Data.Vector as V +import qualified Data.Vector.Unboxed as VU +import Statistics.Distribution +import Statistics.Distribution.Normal +import Data.Tuple.Select (sel2) + + + +type Data = VU.Vector Double +type Theta = VU.Vector (Double,Double,Double) -- weight, mean, variance +type ThetaL = [(Double,Double,Double)] -- weight, mean, variance + +-- | Perform one EM step given the data. In General, emSteps should be iterated +-- until some convergence criterion is met. + +emStep :: Data -> Theta -> Theta +emStep xs ts = ts' where + -- E-step + -- TODO this could be made easier by using hmatrix with enabled "vector" + resps = V.map (\i -> calcResV (ts `VU.unsafeIndex` i)) $ V.enumFromN 0 tlen + calcResV t = VU.map (calcRes t) xs + calcRes (w,mu,s) x = (w * density (normalDistr mu s) x) / (VU.sum . VU.map (\(w',mu',s') -> w'* density (normalDistr mu' s') x) $ ts) + ns = VU.map (\i -> VU.sum $ resps `V.unsafeIndex` i) $ VU.enumFromN 0 tlen + -- M-step + ws = VU.map (\w -> w / fromIntegral (VU.length xs)) ns + mus = VU.map (\i -> + (1/ ns `VU.unsafeIndex` i) * + (VU.sum $ VU.zipWith (*) (resps `V.unsafeIndex` i) xs)) + $ VU.enumFromN 0 tlen + ss = VU.map (\i -> + (1/ ns `VU.unsafeIndex` i) * + (VU.sum $ VU.zipWith (*) (resps `V.unsafeIndex` i) (VU.map (\x -> (x - (mus `VU.unsafeIndex` i))^2) xs))) + $ VU.enumFromN 0 tlen + ts' = VU.zip3 ws mus ss + tlen = VU.length ts + +-- | Produces an infinite list of 'Theta's that will (should) convergence +-- toward a local optimum. + +emIter :: Data -> Theta -> [Theta] +emIter xs ts = iterate (emStep xs) ts + +-- | Find an optimal set of parameters 'Theta'. The additional "takeWhile (not +-- . isnan . fst)" makes sure that in cases of overfitting, 'emFix' does +-- terminate. Due to the way we check and take, in case of NaNs, the returned +-- values will be NaNs (checking fst, returning snd). + +emFix :: Data -> Theta -> Theta +emFix xs ts = res where + res = last . map snd . takeWhile (not . isnan . fst) . takeWhile (not . converged) $ zip ys zs + ys = emIter xs ts + zs = tail ys + converged (y,z) = abs (logLikelihood y xs - logLikelihood z xs) < epsilon + epsilon = 10 ^^ (-10) + isnan ns = let (ws,_,_) = VU.unzip3 ns in VU.any isNaN ws + +-- | Calculate the log-likelihood for a given set of parameters 'Theta' and +-- some data 'Data'. Used by 'emFix' to estimate if convergence is reached. +-- +-- TODO could be useful in a more general setting within StatisticalMethods. + +logLikelihood :: Theta -> Data -> Double +logLikelihood ts xs = (VU.sum . VU.map lls $ xs) / (fromIntegral $ VU.length xs) where + lls x = VU.sum . VU.map (\t -> ll t x) $ ts + ll (w,mu,s) x = w * density (normalDistr mu s) x + +-- | Given a set of 'Data' and a number 'k' of Gaussian peaks, try to find the +-- optimal GMM. This is done by trying each data point as mu for each Gaussian. +-- Note that this will be rather slow for larger 'k' (larger than, say 2 or 3). +-- In that case, a random-drawing method should be chosen. +-- +-- TODO xs' -> xs sorting makes me cry! + +emStarts :: Int -> Data -> Theta +emStarts k xs' = maximumBy (comparing (\t -> logLikelihood t xs)) . map (emFix xs) $ ts where + ts = map VU.fromList . f k . VU.toList $ xs + mkT mu = (w,mu,sampleVar) + f l zs + | l< 1 = error "emStarts called with k<1" + | l==1 = map (\z -> [mkT z]) zs + | l> 1 = [mkT y : ys | y <- zs, ys <- f (l-1) (dropWhile ( (x-sampleMu)^2) $ xs) / (fromIntegral $ VU.length xs) + w = 1 / fromIntegral k + xs = VU.fromList . sort . VU.toList $ xs' diff --git a/Statistics/EM/TwoGaussian.hs b/Statistics/EM/TwoGaussian.hs new file mode 100644 --- /dev/null +++ b/Statistics/EM/TwoGaussian.hs @@ -0,0 +1,70 @@ + +-- | This is a simplified version of the Expectation-Maximization algorithm for +-- a two-component Gaussian mixture model. Cf. Hastie et al, The Elements of +-- Statistical Learning, Springer. Chapter 8.5.1. + +module Statistics.EM.TwoGaussian + ( emFix + , emStarts + ) where + +import qualified Data.Vector.Unboxed as VU +import Statistics.Distribution +import Statistics.Distribution.Normal +import Control.Monad.Fix (fix) +import Data.List (sort,maximumBy,tails) +import Data.Ord + +type Weight = Double +type Normal = (Double,Double) + +-- | Perform one EM step + +emStep :: VU.Vector Double -> (Weight,Normal,Normal) -> (Weight,Normal,Normal) +emStep xs (w,(mu1,s1),(mu2,s2)) = (w',(mu1',s1'),(mu2',s2')) where + -- this is the expectation step + ys = VU.map responsibility xs + n1 = normalDistr mu1 s1 + n2 = normalDistr mu2 s2 + responsibility y = let + n1y = density n1 y + n2y = density n2 y + in w * n2y / ((1-w) * n1y + w * n2y) + -- this is the maximization step + w' = VU.sum ys / fromIntegral (VU.length ys) + -- new Gaussian 1 + div1 = VU.sum . VU.map (\y -> 1-y) $ ys + mu1' = (VU.sum $ VU.zipWith (\x y -> (1-y) * x) xs ys) / div1 + s1' = (VU.sum $ VU.zipWith (\x y -> (1-y) * (x-mu1')^2) xs ys) / div1 + -- new Gaussian 2 + div2 = VU.sum $ ys + mu2' = (VU.sum $ VU.zipWith (\x y -> y * x) xs ys) / div2 + s2' = (VU.sum $ VU.zipWith (\x y -> y * (x-mu2')^2) xs ys) / div2 + +-- | Performs an infinite number of EM steps, iterating towards converging +-- parameters. + +emIter :: VU.Vector Double -> (Weight,Normal,Normal) -> [(Weight,Normal,Normal)] +emIter xs theta = iterate (emStep xs) theta + +-- | Finds the fix-points of the EM step iterations. + +emFix :: VU.Vector Double -> (Weight,Normal,Normal) -> (Weight,Normal,Normal) +emFix xs theta = last . map fst . takeWhile f $ zip zs (tail zs) where + zs = emIter xs theta + f ( (w1,(mu11,s11),(mu12,s12)) , (w2,(mu21,s21),(mu22,s22)) ) = w1 =/= w2 && mu11 =/= mu21 && mu12 =/= mu22 && s11 =/= s21 && s12 =/= s22 + a =/= b = abs (a-b) > epsilon + epsilon = 10.0 ^^ (-10) + +-- | Finds the best fix-point with all elements 'xs' as starting points for the +-- means. It holds that mu_1 < mu_2. + +emStarts :: VU.Vector Double -> (Weight,Normal,Normal) +emStarts xs = maximumBy (comparing loglikelihood) . map (emFix xs) $ [f xs mu1 mu2 | mu1 <- VU.toList xs, mu2 <- VU.toList xs, mu1 (x-sampleMu)^2) $ xs) / (fromIntegral $ VU.length xs) diff --git a/Statistics/PerformanceMetrics.hs b/Statistics/PerformanceMetrics.hs new file mode 100644 --- /dev/null +++ b/Statistics/PerformanceMetrics.hs @@ -0,0 +1,69 @@ +{-# LANGUAGE DoAndIfThenElse #-} +{-# LANGUAGE RecordWildCards #-} + +-- | Common performance metrics which can be calculated using the confusion +-- matrix. +-- +-- Fawcett, ROC Graphs: Notes and Practical Considerations for Researchers, +-- 2004, Kluwer Academic Publishers + +module Statistics.PerformanceMetrics where + +import Control.Monad.Instances + +import Statistics.ConfusionMatrix + + + +-- | sensitivity + +sensitivity :: ConfusionMatrix -> WrappedDouble +sensitivity ConfusionMatrix{..} = do + tp <- tp + fn <- fn + if tp>0 + then Right $ tp / (tp+fn) + else Right 0 + +-- | specificity + +specificity :: ConfusionMatrix -> WrappedDouble +specificity ConfusionMatrix{..} = do + tp <- tp + fp <- fp + if tp>0 + then Right $ tp / (tp+fp) + else Right 0 + +-- | positive predictive value + +ppv :: ConfusionMatrix -> WrappedDouble +ppv ConfusionMatrix{..} = do + tp <- tp + fp <- fp + if tp>0 + then Right $ tp / (tp+fp) + else Right 0 + +-- | mathews correlation coefficient + +mcc :: ConfusionMatrix -> WrappedDouble +mcc ConfusionMatrix{..} = do + tp <- tp + tn <- tn + fp <- fp + fn <- fn + let d' = (tp+fp) * (tp+fn) * (tn+fp) * (tn+fn) + let d = if d'==0 then 1 else sqrt d' + return $ (tp*tn - fp*fn) / d + +-- | F-measure + +fmeasure :: ConfusionMatrix -> WrappedDouble +fmeasure ConfusionMatrix{..} = do + tp <- tp + fp <- fp + fn <- fn + if tp>0 + then Right $ 2*tp / (2*tp + fp + fn) + else Right 0 diff --git a/TestData/Elements.hs b/TestData/Elements.hs new file mode 100644 --- /dev/null +++ b/TestData/Elements.hs @@ -0,0 +1,14 @@ + +-- | This module contains test data taken from "Elements of Statistical +-- Learning". +-- +-- TODO correct citation + +module TestData.Elements where + +import qualified Data.Vector.Unboxed as VU +import Statistics.EM.GMM + +table_8_1 :: VU.Vector Double +table_8_1 = VU.fromList [ -0.39, 0.12, 0.94, 1.67, 1.76, 2.44, 3.72, 4.28, 4.92, 5.53 + , 0.06, 0.48, 1.01, 1.68, 1.80, 3.25, 4.12, 4.60, 5.28, 6.22 ]