From 0b36f3e8c799cfc823983b882c0eee25f910e1ea Mon Sep 17 00:00:00 2001 From: raibread Date: Mon, 28 Nov 2016 19:05:41 -0500 Subject: [PATCH 1/6] Added linear regression function that returns further statistics when there are normal errors --- .gitignore | 1 + Statistics/Distribution/StudentT.hs | 2 +- Statistics/Matrix.hs | 11 +++ Statistics/Regression.hs | 112 +++++++++++++++++++++++----- Statistics/Types.hs | 40 ++++++++++ statistics.cabal | 68 ++++++++++------- 6 files changed, 184 insertions(+), 50 deletions(-) diff --git a/.gitignore b/.gitignore index cb592045..062903d8 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1,3 @@ dist +**/.DS_Store cabal.sandbox.config diff --git a/Statistics/Distribution/StudentT.hs b/Statistics/Distribution/StudentT.hs index c982152f..3770884d 100644 --- a/Statistics/Distribution/StudentT.hs +++ b/Statistics/Distribution/StudentT.hs @@ -21,6 +21,7 @@ module Statistics.Distribution.StudentT ( ) where import Control.Applicative +import Control.DeepSeq (NFData(..)) import Data.Aeson (FromJSON(..), ToJSON, Value(..), (.:)) import Data.Binary (Binary(..)) import Data.Data (Data, Typeable) @@ -32,7 +33,6 @@ import qualified Statistics.Distribution as D import Statistics.Distribution.Transform (LinearTransform (..)) import Statistics.Internal - -- | Student-T distribution newtype StudentT = StudentT { studentTndf :: Double } deriving (Eq, Typeable, Data, Generic) diff --git a/Statistics/Matrix.hs b/Statistics/Matrix.hs index 2b016074..9ae9d02b 100644 --- a/Statistics/Matrix.hs +++ b/Statistics/Matrix.hs @@ -1,4 +1,5 @@ {-# LANGUAGE PatternGuards #-} + -- | -- Module : Statistics.Matrix -- Copyright : 2011 Aleksey Khudyakov, 2014 Bryan O'Sullivan @@ -29,6 +30,7 @@ module Statistics.Matrix , generateSym , ident , diag + , toDiag , dimension , center , multiply @@ -169,6 +171,7 @@ generateSym n f = runST $ do ident :: Int -> Matrix ident n = diag $ U.replicate n 1.0 + -- | Create a square matrix with given diagonal, other entries default to 0 diag :: Vector -> Matrix diag v @@ -180,6 +183,14 @@ diag v where n = U.length v +-- | Return diagonal of a square matrix +toDiag :: Matrix -> Vector +toDiag m + | rs /= cs = error $ "matrix is not square, dimension = " ++ show d + | otherwise = U.map (\i -> unsafeIndex m i i) $ U.fromList [0..(rs-1)] + where + d@(rs,cs) = dimension m + -- | Return the dimensions of this matrix, as a (row,column) pair. dimension :: Matrix -> (Int, Int) dimension (Matrix r c _ _) = (r, c) diff --git a/Statistics/Regression.hs b/Statistics/Regression.hs index 7815467f..7ba59545 100644 --- a/Statistics/Regression.hs +++ b/Statistics/Regression.hs @@ -9,6 +9,7 @@ module Statistics.Regression ( olsRegress , ols + , normalRegress , rSquare , bootstrapRegress ) where @@ -24,9 +25,11 @@ import Statistics.Function as F import Statistics.Matrix hiding (map) import Statistics.Matrix.Algorithms (qr) import Statistics.Resampling (splitGen) -import Statistics.Types (Estimate(..),ConfInt,CL,estimateFromInterval,getPValue) +import Statistics.Types (Estimate(..),ConfInt,CL,estimateFromInterval, + getPValue, TestStatistic(..), TErr(..)) import Statistics.Sample (mean) import Statistics.Sample.Internal (sum) +import Statistics.Distribution.FDistribution import System.Random.MWC (GenIO, uniformR) import qualified Data.Vector as V import qualified Data.Vector.Generic as G @@ -59,24 +62,78 @@ olsRegress preds@(_:_) resps show (G.length resps, n) | otherwise = (coeffs, rSquare mxpreds resps coeffs) where - coeffs = ols mxpreds resps - mxpreds = transpose . - fromVector (length lss + 1) n . - G.concat $ preds ++ [G.replicate n 1] + coeffs = ols mxpreds resps + mxpreds = transpose . + fromVector (length lss + 1) n . + G.concat $ preds ++ [G.replicate n 1] lss@(n:ls) = map G.length preds olsRegress _ _ = error "no predictors given" + +-- | +-- +normalRegress :: [Vector] + -> Vector + -> (V.Vector (Estimate TErr Double) + ,(TestStatistic Double FDistribution) + ,Double) +normalRegress preds@(_:_) resps + | any (/=n) ls = error $ "predictor vector length mismatch " ++ + show lss + | G.length resps /= n = error $ "responder/predictor length mismatch " ++ + show (G.length resps, n) + | otherwise = (U.convert est,fStat,rSq) + where + est = U.zipWith Estimate coeffs $ + U.map (\se -> TErr se (fromIntegral df1)) stdErr + rSq = 1 - ssr / sst + fStat = TestStatistic ((sst - ssr) / (fromIntegral df1) / sigma2) + (fDistribution df1 df2) + stdErr = U.map (\x -> sqrt $ x * sigma2) $ varCoeff r + sigma2 = ssr / (fromIntegral df2) + (ssr,sst) = sumSquares mxpreds resps coeffs + (coeffs,r) = olsR mxpreds resps + df1 = cols mxpreds - 1 + df2 = n - cols mxpreds + mxpreds = transpose . + fromVector (length lss + 1) n . + G.concat $ preds ++ [G.replicate n 1] + lss@(n:ls) = map G.length preds +normalRegress _ _ = error "no predictors given" + -- | Compute the ordinary least-squares solution to /A x = b/. -ols :: Matrix -- ^ /A/ has at least as many rows as columns. - -> Vector -- ^ /b/ has the same length as columns in /A/. - -> Vector -ols a b +-- We also return the /R/ matrix of the /QR/ +-- decompotion of /A/ so that we can compute standard errors. +olsR :: Matrix -- ^ /A/ has at least as many rows as columns. + -> Vector -- ^ /b/ has the same length as columns in /A/. + -> (Vector,Matrix) +olsR a b | rs < cs = error $ "fewer rows than columns " ++ show d - | otherwise = solve r (transpose q `multiplyV` b) + | otherwise = (solve r (transpose q `multiplyV` b),r) where d@(rs,cs) = dimension a (q,r) = qr a +-- | Compute the ordinary least-squares solution to /A x = b/. +ols :: Matrix -- ^ /A/ has at least as many rows as columns. + -> Vector -- ^ /b/ has the same length as columns in /A/. + -> Vector +ols a b = fst $ olsR a b + +-- | Compute standard errors for regression coefficients +-- in on σ scale +varCoeff :: Matrix + -> Vector +varCoeff r = toDiag $ rinv `multiply` (transpose rinv) + where rinv = invU r + +-- | Efficient inversion of upper triangular matrix +-- using back substitution algorithm +invU :: Matrix + -> Matrix +invU r = fromColumns $ map (\e -> solve r e) $ toColumns $ ident n + where n = rows r + -- | Solve the equation /R x = b/. solve :: Matrix -- ^ /R/ is an upper-triangular square matrix. -> Vector -- ^ /b/ is of the same length as rows\/columns in /R/. @@ -85,28 +142,43 @@ solve r b | n /= l = error $ "row/vector mismatch " ++ show (n,l) | otherwise = U.create $ do s <- U.thaw b - rfor n 0 $ \i -> do + rfor n0 0 $ \i -> do si <- (/ unsafeIndex r i i) <$> M.unsafeRead s i M.unsafeWrite s i si for 0 i $ \j -> F.unsafeModify s j $ subtract (unsafeIndex r j i * si) return s - where n = rows r - l = U.length b + where n = rows r + n0 = (fst . U.head $ U.dropWhile (\x -> snd x == 0) $ + U.reverse $ U.indexed b) + 1 + -- ^ Starting at the last non-zero element of b provides a constant + -- factor speed up when using solve to perform backward substitution + -- to invert upper-triangualr matrices. + l = U.length b + + +-- | +-- +sumSquares :: Matrix -- ^ Predictors (regressors). + -> Vector -- ^ Responders. + -> Vector -- ^ Regression coefficients. + -> (Double,Double) +sumSquares pred resp coeff = (r,t) + where + r = sum $ flip U.imap resp $ \i x -> square (x - p i) + t = sum $ flip U.map resp $ \x -> square (x - mean resp) + p i = sum . flip U.imap coeff $ \j -> (* unsafeIndex pred i j) -- | Compute /R²/, the coefficient of determination that -- indicates goodness-of-fit of a regression. -- -- This value will be 1 if the predictors fit perfectly, dropping to 0 -- if they have no explanatory power. -rSquare :: Matrix -- ^ Predictors (regressors). - -> Vector -- ^ Responders. - -> Vector -- ^ Regression coefficients. +rSquare :: Matrix + -> Vector + -> Vector -> Double rSquare pred resp coeff = 1 - r / t - where - r = sum $ flip U.imap resp $ \i x -> square (x - p i) - t = sum $ flip U.map resp $ \x -> square (x - mean resp) - p i = sum . flip U.imap coeff $ \j -> (* unsafeIndex pred i j) + where (r,t) = sumSquares pred resp coeff -- | Bootstrap a regression function. Returns both the results of the -- regression and the requested confidence interval values. diff --git a/Statistics/Types.hs b/Statistics/Types.hs index ab72dcb3..60562052 100644 --- a/Statistics/Types.hs +++ b/Statistics/Types.hs @@ -46,6 +46,7 @@ module Statistics.Types -- * Estimates and upper/lower limits , Estimate(..) , NormalErr(..) + , TErr(..) , ConfInt(..) , UpperLimit(..) , LowerLimit(..) @@ -60,6 +61,7 @@ module Statistics.Types , Scale(..) -- * Other , Sample + , TestStatistic(..) , WeightedSample , Weights ) where @@ -78,6 +80,7 @@ import Statistics.Internal import Statistics.Types.Internal import Statistics.Distribution import Statistics.Distribution.Normal +import Statistics.Distribution.StudentT ---------------------------------------------------------------- @@ -350,6 +353,21 @@ instance ToJSON a => ToJSON (NormalErr a) instance NFData a => NFData (NormalErr a) where rnf (NormalErr x) = rnf x +-- | +-- t distributed errors. Soread as 1σ errors with degrees of freedom +-- which corresponds to 100 x Pr(|t_{df}| <= σ) CL. +data TErr a = TErr + { tError :: a + , tDf :: Double + } + deriving (Eq, Read, Show, Typeable, Data, Generic) + +instance Binary a => Binary (TErr a) +instance FromJSON a => FromJSON (TErr a) +instance ToJSON a => ToJSON (TErr a) +instance NFData a => NFData (TErr a) where + rnf (TErr x tdist) = rnf x `seq` rnf tdist + -- | Confidence interval. It assumes that confidence interval forms -- single interval and isn't set of disjoint intervals. @@ -388,6 +406,14 @@ estimateNormErr x dx = Estimate x (NormalErr dx) -> Estimate NormalErr a (±) = estimateNormErr + +-- | Create estimate with t errors +estimateTErr :: a -- ^ Point estimate + -> a -- ^ 1σ error + -> Double -- ^ degrees of freedom + -> Estimate TErr a +estimateTErr x dx df = Estimate x (TErr dx df) + -- | Create estimate with asymmetric error. estimateFromErr :: a -- ^ Central estimate @@ -477,6 +503,15 @@ instance NFData a => NFData (LowerLimit a) where rnf (LowerLimit x cl) = rnf x `seq` rnf cl +---------------------------------------------------------------- +-- Test Statistic +---------------------------------------------------------------- + +data TestStatistic a b = TestStatistic + { testStat :: !a + , refDist :: !b + } deriving (Eq, Read, Show, Typeable, Data) + ---------------------------------------------------------------- -- Deriving unbox instances ---------------------------------------------------------------- @@ -501,6 +536,11 @@ derivingUnbox "NormalErr" [| \(NormalErr a) -> a |] [| NormalErr |] +derivingUnbox "TErr" + [t| forall a. Unbox a => TErr a -> (a, Double) |] + [| \(TErr a df) -> (a,df) |] + [| \(a,df) -> TErr a df |] + derivingUnbox "ConfInt" [t| forall a. Unbox a => ConfInt a -> (a, a, CL Double) |] [| \(ConfInt a b c) -> (a,b,c) |] diff --git a/statistics.cabal b/statistics.cabal index b0bfb192..f1bf6bfc 100644 --- a/statistics.cabal +++ b/statistics.cabal @@ -40,6 +40,8 @@ extra-source-files: examples/kde/data/faithful.csv examples/kde/kde.html examples/kde/kde.tpl + examples/regression/regression.hs + examples/regression/data/wine.csv tests/Tests/Math/Tables.hs tests/Tests/Math/gen.py tests/utils/Makefile @@ -100,19 +102,23 @@ library Statistics.Test.Internal Statistics.Types.Internal build-depends: - aeson >= 0.6.0.0, - base >= 4.4 && < 5, - binary >= 0.5.1.0, - deepseq >= 1.1.0.2, - erf, - math-functions >= 0.1.7, - monad-par >= 0.3.4, - mwc-random >= 0.13.0.0, - primitive >= 0.3, - vector >= 0.10, - vector-algorithms >= 0.4, - vector-th-unbox, - vector-binary-instances >= 0.2.1 + aeson >= 0.6.0.0, + base >= 4.4 && < 5, + binary >= 0.5.1.0, + bytestring >= 0.10.6.0, + cassava >= 0.4.5.0, + containers >= 0.5.6.2, + deepseq >= 1.1.0.2, + erf, + math-functions >= 0.1.7, + monad-par >= 0.3.4, + mwc-random >= 0.13.0.0, + primitive >= 0.3, + table-layout >= 0.7.0.0, + vector >= 0.10, + vector-algorithms >= 0.4, + vector-binary-instances >= 0.2.1, + vector-th-unbox if impl(ghc < 7.6) build-depends: ghc-prim @@ -146,22 +152,26 @@ test-suite tests -Wall -threaded -rtsopts -fsimpl-tick-factor=500 build-depends: - HUnit, - QuickCheck >= 2.7.5, - base, - binary, - erf, - aeson, - ieee754 >= 0.7.3, - math-functions, - mwc-random, - primitive, - statistics, - test-framework, - test-framework-hunit, - test-framework-quickcheck2, - vector, - vector-algorithms + HUnit, + QuickCheck >= 2.7.5, + aeson, + base, + binary, + bytestring >= 0.10.6.0, + cassava >= 0.4.5.0, + containers >= 0.5.6.2, + erf, + ieee754 >= 0.7.3, + math-functions, + mwc-random, + primitive, + statistics, + table-layout >= 0.7.0.0, + test-framework, + test-framework-hunit, + test-framework-quickcheck2, + vector, + vector-algorithms source-repository head type: git From a7d52d82830c4bf0ff884456424c92c55cb101b4 Mon Sep 17 00:00:00 2001 From: raibread Date: Mon, 28 Nov 2016 19:12:06 -0500 Subject: [PATCH 2/6] example regression --- .gitignore | 1 + examples/regression/data/wine.csv | 4899 +++++++++++++++++++++++++++++ examples/regression/regression.hs | 75 + 3 files changed, 4975 insertions(+) create mode 100644 examples/regression/data/wine.csv create mode 100644 examples/regression/regression.hs diff --git a/.gitignore b/.gitignore index 062903d8..83d36b4f 100644 --- a/.gitignore +++ b/.gitignore @@ -1,3 +1,4 @@ dist **/.DS_Store +TAGS cabal.sandbox.config diff --git a/examples/regression/data/wine.csv b/examples/regression/data/wine.csv new file mode 100644 index 00000000..b199abcf --- /dev/null +++ b/examples/regression/data/wine.csv @@ -0,0 +1,4899 @@ +fixedAcidity,volatileAcidity,citricAcid,residualSugar,chlorides,freeSulfurDioxide,totalSulfurDioxide,density,pH,sulphates,alcohol,quality 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+5.7,0.21,0.32,0.9,0.038,38,121,0.99074,3.24,0.46,10.6,6 +6.5,0.23,0.38,1.3,0.032,29,112,0.99298,3.29,0.54,9.7,5 +6.2,0.21,0.29,1.6,0.039,24,92,0.99114,3.27,0.5,11.2,6 +6.6,0.32,0.36,8,0.047,57,168,0.9949,3.15,0.46,9.6,5 +6.5,0.24,0.19,1.2,0.041,30,111,0.99254,2.99,0.46,9.4,6 +5.5,0.29,0.3,1.1,0.022,20,110,0.98869,3.34,0.38,12.8,7 +6,0.21,0.38,0.8,0.02,22,98,0.98941,3.26,0.32,11.8,6 \ No newline at end of file diff --git a/examples/regression/regression.hs b/examples/regression/regression.hs new file mode 100644 index 00000000..b4d2711b --- /dev/null +++ b/examples/regression/regression.hs @@ -0,0 +1,75 @@ +{-# LANGUAGE OverloadedStrings, DeriveGeneric #-} + +import Control.Applicative +import qualified Data.ByteString.Lazy as BL +import Data.Csv +import qualified Data.Vector as V +import qualified Data.Vector.Unboxed as U +import Statistics.Regression +import Statistics.Matrix hiding (map) +import Data.Foldable +import System.IO +import System.Exit +import Data.List +import Data.Ord + +data Wine = Wine + { fixedAcidity :: !Double + , volatileAcidity :: !Double + , citricAcid :: !Double + , residualSugar :: !Double + , chlorides :: !Double + , freeSulfurDioxide :: !Double + , totalSulferDioxide :: !Double + , density :: !Double + , pH :: !Double + , sulphates :: !Double + , alcohol :: !Double + , quality :: !Double + } deriving Show + + +instance FromNamedRecord Wine where + parseNamedRecord r = Wine <$> r .: "fixedAcidity" + <*> r .: "volatileAcidity" + <*> r .: "citricAcid" + <*> r .: "residualSugar" + <*> r .: "chlorides" + <*> r .: "freeSulfurDioxide" + <*> r .: "totalSulfurDioxide" + <*> r .: "density" + <*> r .: "pH" + <*> r .: "sulphates" + <*> r .: "alcohol" + <*> r .: "quality" + +toXY :: V.Vector Wine -> ([Vector],Vector) +toXY wine = (toX wine, toY wine) + where toX :: V.Vector Wine -> [Vector] + toX w = V.convert <$> (mapX <*> [w]) + mapX = V.map <$> [ fixedAcidity + , volatileAcidity + , citricAcid + , residualSugar + , chlorides + , freeSulfurDioxide + , totalSulferDioxide + , density + , pH + , sulphates + , alcohol + ] + toY :: V.Vector Wine -> Vector + toY w = V.convert $ V.map quality w + +main :: IO() +main = do + wineCSV <- BL.readFile "data/wine.csv" + wine <- case decodeByName wineCSV of + Left error -> do + hPutStrLn stderr error + exitFailure + Right (_header, wine) -> return wine + + let (preds,y) = toXY wine + putStrLn $ show (normalRegress preds y) From 7dd60f5a98016dc81f238f47d7092eceefb60acf Mon Sep 17 00:00:00 2001 From: raibread Date: Mon, 5 Dec 2016 23:25:17 -0500 Subject: [PATCH 3/6] Regression functions added to address point 1. in issue #67 Added two type classes, 'WeightedNormalRegress' and 'NormalRegress', which have primary methods 'weightedNormalRegress' and 'normalRegress', respectively, which run weigihted and ordinary least squares, again respectively. These methods extend 'olsRegress' and compute regression coefficient standard errors and an overall model fit test statistic in addition to estimating the coefficients and returning the R-squared goodness of fit metric. Both type classes have instances for when the noise variance is unknown and when it is known. In both instances, it is assumed that the noise is Gaussian. Some auxiliary changes were needed to make these extensions to the Regression module: **(list below is of important addtions but is not exhaustive)** - A new error data type, 'TErr', which stores t-distributed errors was added to Statistics/Types.hs for coefficient standard errors when the noise variance is unknown. - A function 'diagOf' in the Statistics/Matrix.hs returns the diagonal of a square matrix. This is used to retrieve the standard errors of each regression coefficient in 'varCoeff'. - 'inv' in Statistics/Regression.hs can take the inverse of upper or lower triangular matrices (also needed in 'varCoeff'. Some future-looking changes were made as well: - In anticipation of implementing generalized least square which allows for specifcation of arbitrary linear correlation structure across the noise component, this commit has implemented the Cholesky factorization 'chol' in Statistics/Matrix/Algorithms.hs. - 'solve' can now handle lower triangular matrices as well. --- Statistics/Distribution/StudentT.hs | 2 +- Statistics/Matrix.hs | 10 +- Statistics/Matrix/Algorithms.hs | 25 + Statistics/Matrix/Types.hs | 11 + Statistics/Regression.hs | 212 +- Statistics/Types.hs | 13 +- examples/regression/data/weights.csv | 4899 ++++++++++++++++++++++++++ examples/regression/regression.hs | 50 +- statistics.cabal | 68 +- 9 files changed, 5187 insertions(+), 103 deletions(-) create mode 100644 examples/regression/data/weights.csv diff --git a/Statistics/Distribution/StudentT.hs b/Statistics/Distribution/StudentT.hs index 3770884d..c982152f 100644 --- a/Statistics/Distribution/StudentT.hs +++ b/Statistics/Distribution/StudentT.hs @@ -21,7 +21,6 @@ module Statistics.Distribution.StudentT ( ) where import Control.Applicative -import Control.DeepSeq (NFData(..)) import Data.Aeson (FromJSON(..), ToJSON, Value(..), (.:)) import Data.Binary (Binary(..)) import Data.Data (Data, Typeable) @@ -33,6 +32,7 @@ import qualified Statistics.Distribution as D import Statistics.Distribution.Transform (LinearTransform (..)) import Statistics.Internal + -- | Student-T distribution newtype StudentT = StudentT { studentTndf :: Double } deriving (Eq, Typeable, Data, Generic) diff --git a/Statistics/Matrix.hs b/Statistics/Matrix.hs index 9ae9d02b..13c2b02a 100644 --- a/Statistics/Matrix.hs +++ b/Statistics/Matrix.hs @@ -13,6 +13,8 @@ module Statistics.Matrix ( -- * Data types Matrix(..) + , TMatrix(..) + , UpperLower(..) , Vector -- * Conversion from/to lists/vectors , fromVector @@ -30,7 +32,7 @@ module Statistics.Matrix , generateSym , ident , diag - , toDiag + , diagOf , dimension , center , multiply @@ -184,10 +186,10 @@ diag v n = U.length v -- | Return diagonal of a square matrix -toDiag :: Matrix -> Vector -toDiag m +diagOf :: Matrix -> Vector +diagOf m | rs /= cs = error $ "matrix is not square, dimension = " ++ show d - | otherwise = U.map (\i -> unsafeIndex m i i) $ U.fromList [0..(rs-1)] + | otherwise = U.generate rs (\i -> unsafeIndex m i i) where d@(rs,cs) = dimension m diff --git a/Statistics/Matrix/Algorithms.hs b/Statistics/Matrix/Algorithms.hs index 5feef227..3c1dc8c7 100644 --- a/Statistics/Matrix/Algorithms.hs +++ b/Statistics/Matrix/Algorithms.hs @@ -12,6 +12,7 @@ module Statistics.Matrix.Algorithms import Control.Applicative ((<$>), (<*>)) import Control.Monad.ST (ST, runST) +import Control.Monad (when) import Prelude hiding (sum, replicate) import Statistics.Matrix (Matrix, column, dimension, for, norm) import qualified Statistics.Matrix.Mutable as M @@ -37,6 +38,30 @@ qr mat = runST $ do M.unsafeModify a i jj $ subtract (p * aij) (,) <$> M.unsafeFreeze a <*> M.unsafeFreeze r +-- | /O(n^3)/ Compute the Cholesky factorization and return +-- the lower triangular Cholesky factor (/L/). Note: does +-- not check whether matrix is positive definite or not. +-- Will fail if diagonal elements are non-positive. +chol :: Matrix -> Matrix +chol mat + | m /= n = error "Matrix must be square." + | otherwise = runST $ do + l <- M.thaw mat + for 0 n $ \j -> do + M.unsafeModify l j j sqrt + for (j+1) n $ \i -> do + ljj <- M.unsafeRead l j j + M.unsafeModify l i j (/ ljj) + M.unsafeWrite l j i 0 + for (j+1) (i+1) $ \jj -> do + ljjj <- M.unsafeRead l jj j + lij <- M.unsafeRead l i j + M.unsafeModify l i jj (subtract $ lij*ljjj) + when (i /= jj) $ + M.unsafeModify l jj i (subtract $ lij*ljjj) + M.unsafeFreeze l + where (m,n) = dimension mat + innerProduct :: M.MMatrix s -> Int -> Int -> ST s Double innerProduct mmat j k = M.immutably mmat $ \mat -> sum $ U.zipWith (*) (column mat j) (column mat k) diff --git a/Statistics/Matrix/Types.hs b/Statistics/Matrix/Types.hs index 90fc6250..54b7ebc5 100644 --- a/Statistics/Matrix/Types.hs +++ b/Statistics/Matrix/Types.hs @@ -14,6 +14,8 @@ module Statistics.Matrix.Types , MVector , Matrix(..) , MMatrix(..) + , TMatrix(..) + , UpperLower(..) , debug ) where @@ -42,6 +44,15 @@ data MMatrix s = MMatrix {-# UNPACK #-} !Int !(MVector s) +-- | Data type records whether a triangular matrix is upper or lower triangular. +data UpperLower = Upper | Lower + deriving (Eq, Show) + +-- | Triangular matrix, stored as a Matrix with indication of whether it is +-- upper or lower triangular. +data TMatrix = TMatrix Matrix UpperLower + deriving (Eq, Show) + -- The Show instance is useful only for debugging. instance Show Matrix where show = debug diff --git a/Statistics/Regression.hs b/Statistics/Regression.hs index 7ba59545..df8bda0b 100644 --- a/Statistics/Regression.hs +++ b/Statistics/Regression.hs @@ -1,3 +1,4 @@ +{-# LANGUAGE FunctionalDependencies #-} -- | -- Module : Statistics.Regression -- Copyright : 2014 Bryan O'Sullivan @@ -10,6 +11,7 @@ module Statistics.Regression olsRegress , ols , normalRegress + , weightedNormalRegress , rSquare , bootstrapRegress ) where @@ -26,11 +28,14 @@ import Statistics.Matrix hiding (map) import Statistics.Matrix.Algorithms (qr) import Statistics.Resampling (splitGen) import Statistics.Types (Estimate(..),ConfInt,CL,estimateFromInterval, - getPValue, TestStatistic(..), TErr(..)) + getPValue, TestStatistic(..), TErr(..), NormalErr(..), + estimateTErr, estimateNormErr, normalError) import Statistics.Sample (mean) import Statistics.Sample.Internal (sum) import Statistics.Distribution.FDistribution +import Statistics.Distribution.ChiSquared import System.Random.MWC (GenIO, uniformR) +import Data.Maybe (fromMaybe, isJust) import qualified Data.Vector as V import qualified Data.Vector.Generic as G import qualified Data.Vector.Unboxed as U @@ -70,37 +75,121 @@ olsRegress preds@(_:_) resps olsRegress _ _ = error "no predictors given" --- | --- -normalRegress :: [Vector] - -> Vector - -> (V.Vector (Estimate TErr Double) - ,(TestStatistic Double FDistribution) - ,Double) -normalRegress preds@(_:_) resps - | any (/=n) ls = error $ "predictor vector length mismatch " ++ +-- | Special case of WeightedNormalRegress where regression noise/error is +-- assumed to be homoskedastic (i.e. noise in all observations have the same variance). +-- That is, noise e has /var(e) = σ²I/. +class (WeightedNormalRegress a b) => NormalRegress a b | a -> b, b -> a where + -- | Perform OLS regression with homoskedastic errors. + -- See 'weightedNormalRegress' for more details. + normalRegress :: [Vector] -- ^ list of predictor variables + -> Vector -- ^ responder vector + -> a Double -- ^ coefficient estimate error type + -> (V.Vector (Estimate a Double) + ,(TestStatistic Double b) + ,Double) + normalRegress preds resp err = weightedNormalRegress preds resp Nothing err + + +instance NormalRegress TErr FDistribution +instance NormalRegress NormalErr ChiSquared + +-- | Class with main method 'weightedNormalRegress' that performs a weighted +-- least squares regression with Gaussian noise, e, with /var(e) = σ²diag(w)/ +-- where /w/ is known. +class WeightedNormalRegress a b | a -> b, b -> a where + -- | Takes degrees of freedom and return a reference + -- distribution for the overall fit test statistic. + testDist :: Int -- ^ degrees of freedom 1 + -> Int -- ^ degrees of freedom 2 (optional -- not needed for 'ChiSquared') + -> b + + -- | Returns variance parameter for regression noise/error + -- component. + errVar :: Double -- ^ sum of square residuals + -> Int -- ^ residual degrees of freedom + -> a Double -- ^ either 'Unknown' ('TErr' datatype) if we are to estimate + -- the variance or 'NormalErr Double' if it is known a priori. + -> Double + + -- | Return a vector of type 'Estimate a Double' given coefficient estimates, + -- error estimates, and (optional) degrees of freedom. + estimateErrs :: Vector -- ^ coefficient estimates + -> Vector -- ^ coefficient standard errors + -> Int -- ^ degrees of freedom (optional -- not needed for 'NormalErr') + -> V.Vector (Estimate a Double) + + -- | Perform a weighted least squares regression and calculate coefficient + -- standard errors, goodness-of-fit, and test statistic for overall fit of + -- the model. Can handle gaussian noise setting when variance is known and + -- when it is unknown and needs to be estimated + -- + -- The returned tuple consists of: + -- + -- * vector of type 'Estimate a Double' which contains point estimates for + -- regression coefficients, standard errors, and (optional) degrees of freedom. + -- + -- * overall model test statistic of type 'TestStatistic Double b' which contains + -- value of test statistic for overall model fit and reference distribution. + -- + -- * /R²/, the coefficient of determination (see 'rSquare' for details). + weightedNormalRegress :: [Vector] -- ^ list of predictor vectors + -> Vector -- ^ responder vector + -> Maybe Vector -- ^ w in /var(e) = σ²diag(w)/ + -> a Double -- ^ coefficient estimate error type + -> (V.Vector (Estimate a Double) + ,(TestStatistic Double b) + ,Double) + weightedNormalRegress preds@(_:_) resps w err + | any (/=n) ls = error $ "predictor vector length mismatch " ++ show lss - | G.length resps /= n = error $ "responder/predictor length mismatch " ++ + | G.length resps /= n = error $ "responder/predictor length mismatch " ++ show (G.length resps, n) - | otherwise = (U.convert est,fStat,rSq) - where - est = U.zipWith Estimate coeffs $ - U.map (\se -> TErr se (fromIntegral df1)) stdErr - rSq = 1 - ssr / sst - fStat = TestStatistic ((sst - ssr) / (fromIntegral df1) / sigma2) - (fDistribution df1 df2) - stdErr = U.map (\x -> sqrt $ x * sigma2) $ varCoeff r - sigma2 = ssr / (fromIntegral df2) - (ssr,sst) = sumSquares mxpreds resps coeffs - (coeffs,r) = olsR mxpreds resps - df1 = cols mxpreds - 1 - df2 = n - cols mxpreds - mxpreds = transpose . - fromVector (length lss + 1) n . - G.concat $ preds ++ [G.replicate n 1] - lss@(n:ls) = map G.length preds -normalRegress _ _ = error "no predictors given" + | (G.length <$> w) /= Just n + && isJust w = error $ "weight vector length mismatch" + | otherwise = (est,fitStat,rSq) + where + est = estimateErrs coeffs stdErr df1 + rSq = 1 - ssr / sst + fitStat = TestStatistic ((sst - ssr) / (fromIntegral df1) / sigma2) + (testDist df1 df2) + stdErr = U.map (\x -> sqrt $ x * sigma2) $ varCoeff r + sigma2 = errVar ssr df2 err + (ssr,sst0) = sumSquares mxpreds wResps coeffs + sst = fromMaybe sst0 $ + sum <$> U.zipWith (\x y-> square $ y * (x - mean resps)) resps <$> weights + (coeffs,r) = olsR mxpreds wResps + df1 = cols mxpreds - 1 + df2 = n - cols mxpreds + mxpreds = transpose . + fromVector (length lss + 1) n . + G.concat $ wPreds ++ [fromMaybe (G.replicate n 1) weights] + weights = U.map (\s -> 1 / sqrt s) <$> w + wResps = fromMaybe resps $ U.zipWith (*) resps <$> weights + wPreds = fromMaybe preds $ (\x -> map (U.zipWith (*) x)) <$> weights <*> Just preds + lss@(n:ls) = map G.length preds + weightedNormalRegress _ _ _ _ = error "no predictors given" + +-- | Instance of 'WeightedNormalRegress' when noise variance is to be estimated +-- and thus regression coefficients follow a 'StudentT' distribution and overall +-- model fit test statistic has reference distribution 'FDistribution'. +instance WeightedNormalRegress TErr FDistribution where + testDist df1 df2 = fDistribution df1 df2 + errVar ssr df err + | err /= Unknown = error $ "for t-distributed errors, error type " ++ + "must be specified as Unknown" + | otherwise = ssr / (fromIntegral df) + estimateErrs ps se df = U.convert $ + U.zipWith (\p s -> estimateTErr p s (fromIntegral df)) ps se +-- | Instnace of 'WeightedNormalRegerss' when noise variance is known and +-- thus regression coefficients follow a 'NormalDistribution' distribution +-- and overall model fit test statistic has reference distribution 'ChiSquared'. +instance WeightedNormalRegress NormalErr ChiSquared where + testDist df1 _ = chiSquared df1 + errVar _ _ err = (normalError err)**2 + estimateErrs ps se _ = U.convert $ + U.zipWith (\p s -> estimateNormErr p s) ps se + -- | Compute the ordinary least-squares solution to /A x = b/. -- We also return the /R/ matrix of the /QR/ -- decompotion of /A/ so that we can compute standard errors. @@ -109,12 +198,13 @@ olsR :: Matrix -- ^ /A/ has at least as many rows as columns. -> (Vector,Matrix) olsR a b | rs < cs = error $ "fewer rows than columns " ++ show d - | otherwise = (solve r (transpose q `multiplyV` b),r) + | otherwise = (solve (TMatrix r Upper) (transpose q `multiplyV` b),r) where d@(rs,cs) = dimension a (q,r) = qr a -- | Compute the ordinary least-squares solution to /A x = b/. +-- This is a wrapper for ols :: Matrix -- ^ /A/ has at least as many rows as columns. -> Vector -- ^ /b/ has the same length as columns in /A/. -> Vector @@ -122,42 +212,58 @@ ols a b = fst $ olsR a b -- | Compute standard errors for regression coefficients -- in on σ scale -varCoeff :: Matrix +varCoeff :: Matrix -- ^ /R/ part of QR decomposition of design matrix /A/ -> Vector -varCoeff r = toDiag $ rinv `multiply` (transpose rinv) - where rinv = invU r - --- | Efficient inversion of upper triangular matrix --- using back substitution algorithm -invU :: Matrix - -> Matrix -invU r = fromColumns $ map (\e -> solve r e) $ toColumns $ ident n - where n = rows r +varCoeff r = diagOf $ rinv `multiply` (transpose rinv) + where rinv = inv $ TMatrix r Upper +-- | Efficient inversion of triangular matrix +-- using forward or backward substitution algorithm +inv :: TMatrix -- ^ upper or lower triangular matrix + -> Matrix +inv tr = fromColumns $ map (\e -> solve tr e) $ toColumns $ ident n + where TMatrix r _ = tr + n = rows r + -- | Solve the equation /R x = b/. -solve :: Matrix -- ^ /R/ is an upper-triangular square matrix. +solve :: TMatrix -- ^ /R/ is a (upper or lower) triangular square matrix. -> Vector -- ^ /b/ is of the same length as rows\/columns in /R/. -> Vector -solve r b +solve tr b | n /= l = error $ "row/vector mismatch " ++ show (n,l) | otherwise = U.create $ do s <- U.thaw b - rfor n0 0 $ \i -> do + rfor n0 n1 $ \i -> do si <- (/ unsafeIndex r i i) <$> M.unsafeRead s i M.unsafeWrite s i si for 0 i $ \j -> F.unsafeModify s j $ subtract (unsafeIndex r j i * si) return s - where n = rows r - n0 = (fst . U.head $ U.dropWhile (\x -> snd x == 0) $ + where TMatrix r ul = tr + n = rows r + n0 = if ul == Upper + then (fst . U.head $ U.dropWhile (\x -> snd x == 0) $ U.reverse $ U.indexed b) + 1 - -- ^ Starting at the last non-zero element of b provides a constant - -- factor speed up when using solve to perform backward substitution - -- to invert upper-triangualr matrices. + -- ^ Starting at the last non-zero element of b provides a constant + -- factor speed up when using solve to perform backward substitution + -- to invert upper-triangualr matrices. + else (fst . U.head $ U.dropWhile (\x -> snd x == 0) $ + U.indexed b) + n1 = if ul == Upper + then 0 + else n l = U.length b - --- | +-- | Compute sum of squares decomposition for a given +-- fitted regression model. +-- +-- The return pair consists of: +-- +-- * residual sum of squares (/SSR/). +-- +-- * total sum of squares (/SST/). -- +-- model sum of squares (/SSM/) can be derived +-- from the identity /SST = SSR + SSM/. sumSquares :: Matrix -- ^ Predictors (regressors). -> Vector -- ^ Responders. -> Vector -- ^ Regression coefficients. @@ -173,9 +279,9 @@ sumSquares pred resp coeff = (r,t) -- -- This value will be 1 if the predictors fit perfectly, dropping to 0 -- if they have no explanatory power. -rSquare :: Matrix - -> Vector - -> Vector +rSquare :: Matrix -- ^ Predictors (regressors). + -> Vector -- ^ Responders. + -> Vector -- ^ Regression coefficients -> Double rSquare pred resp coeff = 1 - r / t where (r,t) = sumSquares pred resp coeff diff --git a/Statistics/Types.hs b/Statistics/Types.hs index 60562052..eef73aa7 100644 --- a/Statistics/Types.hs +++ b/Statistics/Types.hs @@ -53,6 +53,7 @@ module Statistics.Types -- ** Constructors , estimateNormErr , (±) + , estimateTErr , estimateFromInterval , estimateFromErr -- ** Accessors @@ -337,7 +338,6 @@ instance (NFData (e a), NFData a) => NFData (Estimate e a) where rnf (Estimate x dx) = rnf x `seq` rnf dx - -- | -- Normal errors. They are stored as 1σ errors which corresponds to -- 68.8% CL. Since we can recalculate them to any confidence level if @@ -354,12 +354,16 @@ instance NFData a => NFData (NormalErr a) where rnf (NormalErr x) = rnf x -- | --- t distributed errors. Soread as 1σ errors with degrees of freedom +-- t distributed errors. Stored as 1σ errors with degrees of freedom -- which corresponds to 100 x Pr(|t_{df}| <= σ) CL. +-- +-- Can also be set as 'Unknown'. For example, when running linear +-- regression with 'normalRegress' we can input regression coefficient +-- error type as 'Unknown' if we want to estimate it. data TErr a = TErr { tError :: a , tDf :: Double - } + } | Unknown deriving (Eq, Read, Show, Typeable, Data, Generic) instance Binary a => Binary (TErr a) @@ -367,6 +371,7 @@ instance FromJSON a => FromJSON (TErr a) instance ToJSON a => ToJSON (TErr a) instance NFData a => NFData (TErr a) where rnf (TErr x tdist) = rnf x `seq` rnf tdist + rnf Unknown = () -- | Confidence interval. 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Wine where <*> r .: "alcohol" <*> r .: "quality" +instance FromNamedRecord Weights where + parseNamedRecord r = Weights <$> r .: "weights" + toXY :: V.Vector Wine -> ([Vector],Vector) toXY wine = (toX wine, toY wine) where toX :: V.Vector Wine -> [Vector] @@ -66,10 +72,48 @@ main :: IO() main = do wineCSV <- BL.readFile "data/wine.csv" wine <- case decodeByName wineCSV of - Left error -> do - hPutStrLn stderr error + Left err -> do + hPutStrLn stderr err exitFailure Right (_header, wine) -> return wine + + -- To run example with homoskedastic errors, + -- comment out from here... + + wCSV <- BL.readFile "data/weights.csv" + w <- case decodeByName wCSV of + Left err -> do + hPutStrLn stderr err + exitFailure + Right (_header, w) -> return w + let w1 = V.convert $ V.map weights w let (preds,y) = toXY wine - putStrLn $ show (normalRegress preds y) + putStrLn $ show (weightedNormalRegress preds y (Just w1) Unknown) + + -- putStrLn $ show (weightedNormalRegress preds y (Just w1) $ Normal 0.5382708) + -- ^ This will give (numerically) equivalent results with different + -- reference distributions. + + -- ...to here, and uncomment lines below. + + -- let (preds,y) = toXY wine + -- putStrLn $ show (normalRegress preds y Unknown) + + -- putStrLn $ show (normalRegerss preds y $ NormalErr 0.7513569) + -- ^ This will give (numerically) equivalent results with different + -- reference distributions. + + +{- +Equivalent R code to check for correctness: + +wine.df <- read.csv("data/wine.csv") +w.df <- read.csv("data/weights.csv") + +fit.weighted <- lm(quality ~ ., data = wine.df, weights = 1/w.df$weights) +summary(fit.weighted) + +fit.ordinary <- lm(quality ~ ., data = wine.df) +summary(fit.ordinary) +-} diff --git a/statistics.cabal b/statistics.cabal index f1bf6bfc..b0bfb192 100644 --- a/statistics.cabal +++ b/statistics.cabal @@ -40,8 +40,6 @@ extra-source-files: examples/kde/data/faithful.csv examples/kde/kde.html examples/kde/kde.tpl - examples/regression/regression.hs - examples/regression/data/wine.csv tests/Tests/Math/Tables.hs tests/Tests/Math/gen.py tests/utils/Makefile @@ -102,23 +100,19 @@ library Statistics.Test.Internal Statistics.Types.Internal build-depends: - aeson >= 0.6.0.0, - base >= 4.4 && < 5, - binary >= 0.5.1.0, - bytestring >= 0.10.6.0, - cassava >= 0.4.5.0, - containers >= 0.5.6.2, - deepseq >= 1.1.0.2, - erf, - math-functions >= 0.1.7, - monad-par >= 0.3.4, - mwc-random >= 0.13.0.0, - primitive >= 0.3, - table-layout >= 0.7.0.0, - vector >= 0.10, - vector-algorithms >= 0.4, - vector-binary-instances >= 0.2.1, - vector-th-unbox + aeson >= 0.6.0.0, + base >= 4.4 && < 5, + binary >= 0.5.1.0, + deepseq >= 1.1.0.2, + erf, + math-functions >= 0.1.7, + monad-par >= 0.3.4, + mwc-random >= 0.13.0.0, + primitive >= 0.3, + vector >= 0.10, + vector-algorithms >= 0.4, + vector-th-unbox, + vector-binary-instances >= 0.2.1 if impl(ghc < 7.6) build-depends: ghc-prim @@ -152,26 +146,22 @@ test-suite tests -Wall -threaded -rtsopts -fsimpl-tick-factor=500 build-depends: - HUnit, - QuickCheck >= 2.7.5, - aeson, - base, - binary, - bytestring >= 0.10.6.0, - cassava >= 0.4.5.0, - containers >= 0.5.6.2, - erf, - ieee754 >= 0.7.3, - math-functions, - mwc-random, - primitive, - statistics, - table-layout >= 0.7.0.0, - test-framework, - test-framework-hunit, - test-framework-quickcheck2, - vector, - vector-algorithms + HUnit, + QuickCheck >= 2.7.5, + base, + binary, + erf, + aeson, + ieee754 >= 0.7.3, + math-functions, + mwc-random, + primitive, + statistics, + test-framework, + test-framework-hunit, + test-framework-quickcheck2, + vector, + vector-algorithms source-repository head type: git From b2ffd1a8a03ae19c95a10ba1dde30f4be6188272 Mon Sep 17 00:00:00 2001 From: raibread Date: Mon, 5 Dec 2016 23:38:21 -0500 Subject: [PATCH 4/6] Fixed typo in Regression.hs; Added regression examples --- statistics.cabal | 3 +++ 1 file changed, 3 insertions(+) diff --git a/statistics.cabal b/statistics.cabal index b0bfb192..fa766e41 100644 --- a/statistics.cabal +++ b/statistics.cabal @@ -40,6 +40,9 @@ extra-source-files: examples/kde/data/faithful.csv examples/kde/kde.html examples/kde/kde.tpl + examples/regression/regression.hs + examples/regression/data/wine.csv + examples/regression/data/weights.csv tests/Tests/Math/Tables.hs tests/Tests/Math/gen.py tests/utils/Makefile From 68af0ec243952c74ce7dbb3e3d6cd0a48cefec25 Mon Sep 17 00:00:00 2001 From: raibread Date: Tue, 6 Dec 2016 20:01:12 -0500 Subject: [PATCH 5/6] Fix for loop error in solve method for lower triangular matrices. Previously extension of `solve` in Regression module to deal with lower triangular matrices incorrectly used `rfor` for a forward for loop. To resolve this issue and to still allow `solve` to work for both with maximum code reusability I wrote a wrapper for `rfor` and `for` in the Functions modules called `for_` that checks the the start and end indices to determine which direction to run the loop. --- Statistics/Function.hs | 6 ++++++ Statistics/Regression.hs | 8 ++++---- 2 files changed, 10 insertions(+), 4 deletions(-) diff --git a/Statistics/Function.hs b/Statistics/Function.hs index b0300f19..40308e29 100644 --- a/Statistics/Function.hs +++ b/Statistics/Function.hs @@ -37,6 +37,7 @@ module Statistics.Function -- * Combinators , for , rfor + , for_ ) where #include "MachDeps.h" @@ -141,6 +142,11 @@ rfor n0 !n f = loop n0 | otherwise = let i' = i-1 in f i' >> loop i' {-# INLINE rfor #-} +for_ :: Monad m => Int -> Int -> (Int -> m ()) -> m () +for_ n0 !n f | n0 > n = rfor n0 n f + | otherwise = for n0 n f +{-# INLINE for_ #-} + unsafeModify :: M.MVector s Double -> Int -> (Double -> Double) -> ST s () unsafeModify v i f = do k <- M.unsafeRead v i diff --git a/Statistics/Regression.hs b/Statistics/Regression.hs index df8bda0b..0bc10637 100644 --- a/Statistics/Regression.hs +++ b/Statistics/Regression.hs @@ -181,7 +181,7 @@ instance WeightedNormalRegress TErr FDistribution where estimateErrs ps se df = U.convert $ U.zipWith (\p s -> estimateTErr p s (fromIntegral df)) ps se --- | Instnace of 'WeightedNormalRegerss' when noise variance is known and +-- | Instance of 'WeightedNormalRegerss' when noise variance is known and -- thus regression coefficients follow a 'NormalDistribution' distribution -- and overall model fit test statistic has reference distribution 'ChiSquared'. instance WeightedNormalRegress NormalErr ChiSquared where @@ -233,7 +233,7 @@ solve tr b | n /= l = error $ "row/vector mismatch " ++ show (n,l) | otherwise = U.create $ do s <- U.thaw b - rfor n0 n1 $ \i -> do + for_ n0 n1 $ \i -> do si <- (/ unsafeIndex r i i) <$> M.unsafeRead s i M.unsafeWrite s i si for 0 i $ \j -> F.unsafeModify s j $ subtract (unsafeIndex r j i * si) @@ -246,8 +246,8 @@ solve tr b -- ^ Starting at the last non-zero element of b provides a constant -- factor speed up when using solve to perform backward substitution -- to invert upper-triangualr matrices. - else (fst . U.head $ U.dropWhile (\x -> snd x == 0) $ - U.indexed b) + else fst . U.head $ U.dropWhile (\x -> snd x == 0) $ + U.indexed b n1 = if ul == Upper then 0 else n From c454953d61f6841aae357780b185341a227fa104 Mon Sep 17 00:00:00 2001 From: raibread Date: Tue, 6 Dec 2016 20:48:48 -0500 Subject: [PATCH 6/6] fix typo in regression.hs example --- examples/regression/regression.hs | 5 ++--- 1 file changed, 2 insertions(+), 3 deletions(-) diff --git a/examples/regression/regression.hs b/examples/regression/regression.hs index c88bf274..fffa3fe2 100644 --- a/examples/regression/regression.hs +++ b/examples/regression/regression.hs @@ -90,8 +90,7 @@ main = do let w1 = V.convert $ V.map weights w let (preds,y) = toXY wine putStrLn $ show (weightedNormalRegress preds y (Just w1) Unknown) - - -- putStrLn $ show (weightedNormalRegress preds y (Just w1) $ Normal 0.5382708) + --putStrLn $ show (weightedNormalRegress preds y (Just w1) $ NormalErr 0.5382708) -- ^ This will give (numerically) equivalent results with different -- reference distributions. @@ -106,7 +105,7 @@ main = do {- -Equivalent R code to check for correctness: +qEquivalent R code to check for correctness: wine.df <- read.csv("data/wine.csv") w.df <- read.csv("data/weights.csv")