covariance (empty) → 0.1.0.0

raw patch · 9 files changed

+1136/−0 lines, 9 filesdep +basedep +covariancedep +hmatrix

Dependencies added: base, covariance, hmatrix, tasty, tasty-hunit, vector

Files

• CHANGELOG.md +11/−0
• LICENSE +674/−0
• covariance.cabal +50/−0
• src/Statistics/Covariance.hs +40/−0
• src/Statistics/Covariance/Internal/Tools.hs +56/−0
• src/Statistics/Covariance/LedoitWolf.hs +120/−0
• src/Statistics/Covariance/OracleApproximatingShrinkage.hs +63/−0
• src/Statistics/Covariance/RaoBlackwellLedoitWolf.hs +64/−0
• test/Test.hs +58/−0

@@ -0,0 +1,11 @@++# Revision history for covariance+++## Unreleased changes+++## 0.1.0.0++-   Shrinkage based estimators.+

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@@ -0,0 +1,50 @@+cabal-version:      2.4+name:               covariance+version:            0.1.0.0+synopsis:+    Well-conditioned estimation of large-dimensional covariance matrices++-- A longer description of the package.+description:        Please see the README on GitHub at <https://github.com/dschrempf/covariance#readme>+homepage:           https://github.com/dschrempf/covariance++-- A URL where users can report bugs.+-- bug-reports:+license:            GPL-3.0-or-later+license-file:       LICENSE+author:             Dominik Schrempf+maintainer:         dominik.schrempf@gmail.com++-- A copyright notice.+-- copyright:+category:+    Math+  , Statistics+extra-source-files: CHANGELOG.md++library+    exposed-modules:+      Statistics.Covariance+    other-modules:+      Statistics.Covariance.LedoitWolf+      Statistics.Covariance.RaoBlackwellLedoitWolf+      Statistics.Covariance.OracleApproximatingShrinkage+      Statistics.Covariance.Internal.Tools+    ghc-options: -Wall -Wunused-packages+    build-depends:    base ^>=4.14.3.0+                    , hmatrix+                    , vector+    hs-source-dirs:   src+    default-language: Haskell2010++test-suite covariance-test+    default-language: Haskell2010+    type:             exitcode-stdio-1.0+    hs-source-dirs:   test+    main-is:          Test.hs+    ghc-options: -Wall -Wunused-packages+    build-depends:    base ^>=4.14.3.0+                    , covariance+                    , hmatrix+                    , tasty+                    , tasty-hunit

@@ -0,0 +1,40 @@+-- |+-- Module      :  Statistics.Covariance+-- Description :  Estimate covariance matrices from sample data+-- Copyright   :  (c) 2021 Dominik Schrempf+-- License     :  GPL-3.0-or-later+--+-- Maintainer  :  dominik.schrempf@gmail.com+-- Stability   :  experimental+-- Portability :  portable+--+-- Creation date: Tue Sep 14 13:02:15 2021.+module Statistics.Covariance+  ( empiricalCovariance,++    -- * Shrinkage based estimators+    --+    -- | See the overview on shrinkage estimators provided by+    -- [scikit-learn](https://scikit-learn.org/dev/modules/covariance.html#shrunk-covariance).+    module Statistics.Covariance.LedoitWolf,+    module Statistics.Covariance.RaoBlackwellLedoitWolf,+    module Statistics.Covariance.OracleApproximatingShrinkage,+  )+where++import qualified Numeric.LinearAlgebra as L+import Statistics.Covariance.LedoitWolf+import Statistics.Covariance.OracleApproximatingShrinkage+import Statistics.Covariance.RaoBlackwellLedoitWolf++-- | Empirical or sample covariance.+--+-- Classical maximum-likelihood estimator; asymptotically unbiased but sensitive+-- to outliers.+--+-- Re-export of the empirical covariance 'L.meanCov' provided by+-- [hmatrix](https://hackage.haskell.org/package/hmatrix).+--+-- NOTE: This function may call 'error'.+empiricalCovariance :: L.Matrix Double -> L.Herm Double+empiricalCovariance = snd . L.meanCov

@@ -0,0 +1,56 @@+-- |+-- Module      :  Statistics.Covariance.Internal.Tools+-- Description :  Common functions+-- Copyright   :  (c) 2021 Dominik Schrempf+-- License     :  GPL-3.0-or-later+--+-- Maintainer  :  dominik.schrempf@gmail.com+-- Stability   :  experimental+-- Portability :  portable+--+-- Creation date: Fri Sep 10 09:47:00 2021.+module Statistics.Covariance.Internal.Tools+  ( centerWith,+    shrinkWith,+    trace,+  )+where++import qualified Data.Vector.Storable as VS+import qualified Numeric.LinearAlgebra as L+import qualified Numeric.LinearAlgebra.Devel as L++centerWith ::+  -- Mean vector of dimension P.+  L.Vector Double ->+  -- Data matrix of dimension N x P.+  L.Matrix Double ->+  -- Data matrix with means 0.+  L.Matrix Double+centerWith ms = L.mapMatrixWithIndex (\(_, j) x -> x - ms VS.! j)++-- Shrinkage a covariance matrix.+shrinkWith ::+  -- Shrinkage factor.+  Double ->+  -- Sample covariance matrix.+  L.Herm Double ->+  -- Scale of identity matrix (trace of sample covariance matrix divided by+  -- dimension). See Chen2010b, Equation 3.+  Double ->+  -- Identity matrix.+  L.Herm Double ->+  L.Herm Double+shrinkWith rho sigma mu im+  | rho < 0.0 = error "shrinkWith: Bug! Shrinkage factor is negative."+  | rho > 1.0 = error "shrinkWith: Bug! Shrinkage factor is larger than 1.0."+  | mu < 0.0 = error "shrinkWith: Bug! Scaling factor of identity matrix is negative."+  | rho == 1.0 = L.trustSym $ L.scale mu (L.unSym im)+  | otherwise =+    L.trustSym $+      L.scale (1.0 - rho) (L.unSym sigma)+        + L.scale (rho * mu) (L.unSym im)++-- Trace of a matrix.+trace :: L.Matrix Double -> Double+trace = L.sumElements . L.takeDiag

@@ -0,0 +1,120 @@+-- |+-- Module      :  Statistics.Covariance.LedoitWolf+-- Description :  Shrinkage based covariance estimator by Ledoit and Wolf+-- Copyright   :  (c) 2021 Dominik Schrempf+-- License     :  GPL-3.0-or-later+--+-- Maintainer  :  dominik.schrempf@gmail.com+-- Stability   :  experimental+-- Portability :  portable+--+-- Creation date: Thu Sep  9 14:08:26 2021.+module Statistics.Covariance.LedoitWolf+  ( ledoitWolf,+  )+where++import Data.Foldable+import qualified Numeric.LinearAlgebra as L+import Statistics.Covariance.Internal.Tools++-- | Shrinkage based covariance estimator by Ledoit and Wolf.+--+-- See Ledoit, O., & Wolf, M., A well-conditioned estimator for+-- large-dimensional covariance matrices, Journal of Multivariate Analysis,+-- 88(2), 365–411 (2004). http://dx.doi.org/10.1016/s0047-259x(03)00096-4.+--+-- Return 'Left' if+--+-- - only one sample is available.+--+-- - no parameters are available.+--+-- NOTE: This function may call 'error' due to partial library functions.+ledoitWolf ::+  -- | Sample data matrix of dimension \(n \times p\), where \(n\) is the number+  -- of samples (rows), and \(p\) is the number of parameters (columns).+  L.Matrix Double ->+  Either String (L.Herm Double)+ledoitWolf xs+  | n < 2 = Left "ledoitWolf: Need more than one sample."+  | p < 1 = Left "ledoitWolf: Need at least one parameter."+  -- The Ledoit and Wolf shrinkage estimator of the covariance matrix+  -- (Equation 14). However, a different, more general formula avoiding a2 is+  -- used. See Equation (4) in Chen2010b.+  | otherwise = Right $ shrinkWith rho sigma mu im+  where+    n = L.rows xs+    p = L.cols xs+    (means, sigma) = L.meanCov xs+    xsCentered = centerWith means xs+    im = L.trustSym $ L.ident p+    mu = muE sigma+    d2 = d2E im sigma mu+    b2 = b2E xsCentered sigma d2+    rho = b2 / d2++-- Inner product for symmetric matrices based on an adjusted Frobenius norm (p+-- 376).+--+-- NOTE: This function is commutative (and therefor qualified as an inner+-- product) for symmetric matrices only, and not in the general case.+frobenius :: L.Matrix Double -> L.Matrix Double -> Double+frobenius xs ys+  | xsRows /= xsCols = error "frobenius: Bug! Left matrix is not square."+  | ysRows /= ysCols = error "frobenius: Bug! Right matrix is not square."+  | xsRows /= ysRows = error "frobenius: Bug! Matrices have different size."+  | otherwise = recip (fromIntegral xsRows) * trace (xs L.<> L.tr' ys)+  where+    xsRows = L.rows xs+    xsCols = L.cols xs+    ysRows = L.rows ys+    ysCols = L.cols ys++-- Estimator of mu (Lemma 3.2).+muE ::+  -- Sample covariance matrix.+  L.Herm Double ->+  Double+-- Avoid matrix multiplication in Frobenius norm.+muE sigma' = recip xsRows * trace sigma+  where+    sigma = L.unSym sigma'+    xsRows = fromIntegral $ L.rows sigma++-- Estimator of d2 (Lemma 3.3).+d2E ::+  -- Identity matrix.+  L.Herm Double ->+  -- Sample covariance matrix.+  L.Herm Double ->+  -- Estimate of mu.+  Double ->+  Double+d2E im sigma mu = frobenius m m+  where+    m = L.unSym sigma - L.scale mu (L.unSym im)++-- Estimator of b2 (Lemma 3.4).+b2E ::+  -- Data matrix.+  L.Matrix Double ->+  -- Sample covariance matrix.+  L.Herm Double ->+  -- Estimate of d2.+  Double ->+  Double+b2E xs sigma = min b2m+  where+    n = fromIntegral $ L.rows xs+    -- NOTE: The authors use a transposed data matrix. They refer to one out of+    -- n columns, each with p rows. Here, we have one out of n rows, each with p+    -- columns.+    --+    -- Long story short. Each y is an observation, a vector of length p.+    ys =+      [ frobenius d d+        | y <- L.toRows xs,+          let d = (L.asColumn y L.<> L.asRow y) - L.unSym sigma+      ]+    b2m = recip (n * n) * foldl' (+) 0 ys

@@ -0,0 +1,63 @@+-- |+-- Module      :  Statistics.Covariance.OracleApproximatingShrinkage+-- Description :  Iterative shrinkage based covariance estimator+-- Copyright   :  (c) 2021 Dominik Schrempf+-- License     :  GPL-3.0-or-later+--+-- Maintainer  :  dominik.schrempf@gmail.com+-- Stability   :  experimental+-- Portability :  portable+--+-- Creation date: Fri Sep 10 13:58:15 2021.+module Statistics.Covariance.OracleApproximatingShrinkage+  ( oracleApproximatingShrinkage,+  )+where++import Debug.Trace hiding (trace)+import qualified Numeric.LinearAlgebra as L+import Statistics.Covariance.Internal.Tools++-- | Iterative shrinkage based covariance estimator.+--+-- See Chen, Y., Wiesel, A., Eldar, Y. C., & Hero, A. O., Shrinkage algorithms+-- for mmse covariance estimation, IEEE Transactions on Signal Processing,+-- 58(10), 5016–5029 (2010). http://dx.doi.org/10.1109/tsp.2010.2053029.+--+-- Return 'Left' if+--+-- - only one sample is available.+--+-- - no parameters are available.+--+-- NOTE: This function may call 'error' due to partial library functions.+oracleApproximatingShrinkage ::+  -- | Sample data matrix of dimension \(n \times p\), where \(n\) is the number+  -- of samples (rows), and \(p\) is the number of parameters (columns).+  L.Matrix Double ->+  Either String (L.Herm Double)+oracleApproximatingShrinkage xs+  | n < 2 = Left "oracleApproximatingShrinkage: Need more than one sample."+  | p < 1 = Left "oracleApproximatingShrinkage: Need at least one parameter."+  | otherwise = Right $ shrinkWith rho sigma mu im+  where+    n = L.rows xs+    p = L.cols xs+    (_, sigma) = L.meanCov xs+    im = L.trustSym $ L.ident p+    -- Trace and squared trace of sigma.+    trS = trace $ L.unSym sigma+    tr2S = trS * trS+    -- Trace of (sigma squared).+    s2 = let s = L.unSym sigma in s L.<> s+    trS2 = trace s2+    -- NOTE: Equation 25 specifying phi is erroneous.+    -- Shrinkage factor (Equation 23).+    n' = fromIntegral n+    p' = fromIntegral p+    rhoNominator = tr2S - recip p' * trS2+    rhoDenominator = (n' - 1 / p') * ( trS2 - recip p' * tr2S)+    rho' = traceShowId $ rhoNominator / rhoDenominator+    rho = min rho' 1.0+    -- Scaling factor of the identity matrix (Equation 3).+    mu = trS / p'

@@ -0,0 +1,64 @@+-- |+-- Module      :  Statistics.Covariance.RaoBlackwellLedoitWolf+-- Description :  Improved shrinkage based covariance estimator+-- Copyright   :  (c) 2021 Dominik Schrempf+-- License     :  GPL-3.0-or-later+--+-- Maintainer  :  dominik.schrempf@gmail.com+-- Stability   :  experimental+-- Portability :  portable+--+-- Creation date: Fri Sep 10 09:26:58 2021.+module Statistics.Covariance.RaoBlackwellLedoitWolf+  ( raoBlackwellLedoitWolf,+  )+where++import qualified Numeric.LinearAlgebra as L+import Statistics.Covariance.Internal.Tools++-- | Improved shrinkage based covariance estimator by Ledoit and Wolf using the+-- Rao-Blackwell theorem.+--+-- See Chen, Y., Wiesel, A., Eldar, Y. C., & Hero, A. O., Shrinkage algorithms+-- for mmse covariance estimation, IEEE Transactions on Signal Processing,+-- 58(10), 5016–5029 (2010). http://dx.doi.org/10.1109/tsp.2010.2053029.+--+-- Return 'Left' if+--+-- - only one sample is available.+--+-- - no parameters are available.+--+-- NOTE: This function may call 'error' due to partial library functions.+raoBlackwellLedoitWolf ::+  -- | Sample data matrix of dimension \(n \times p\), where \(n\) is the number+  -- of samples (rows), and \(p\) is the number of parameters (columns).+  L.Matrix Double ->+  Either String (L.Herm Double)+raoBlackwellLedoitWolf xs+  | n < 2 = Left "raoBlackwellLedoitWolf: Need more than one sample."+  | p < 1 = Left "raoBlackwellLedoitWolf: Need at least one parameter."+  -- Rao-Blackwell Ledoit and Wolf shrinkage estimator of the covariance matrix+  -- (Equation 16).+  | otherwise = Right $ shrinkWith rho sigma mu im+  where+    n = L.rows xs+    p = L.cols xs+    (_, sigma) = L.meanCov xs+    im = L.trustSym $ L.ident p+    -- Trace and squared trace of sigma.+    trS = trace $ L.unSym sigma+    tr2S = trS * trS+    -- Trace of (sigma squared).+    s2 = let s = L.unSym sigma in s L.<> s+    trS2 = trace s2+    -- Shrinkage factor (Equation 17, and 19).+    n' = fromIntegral n+    p' = fromIntegral p+    rhoNominator = ((n' - 2) / n') * trS2 + tr2S+    rhoDenominator = (n' + 2) * (trS2 - recip p' * tr2S)+    rho' = rhoNominator / rhoDenominator+    rho = min rho' 1.0+    -- Scaling factor of the identity matrix (Equation 3).+    mu = trS / p'

@@ -0,0 +1,58 @@+-- |+-- Module      :  Spec+-- Description :  Covariance test suite+-- Copyright   :  (c) 2021 Dominik Schrempf+-- License     :  GPL-3.0-or-later+--+-- Maintainer  :  dominik.schrempf@gmail.com+-- Stability   :  experimental+-- Portability :  portable+--+-- Creation date: Thu Sep  9 21:59:03 2021.+module Main+  ( main,+  )+where++import Data.Either+import qualified Numeric.LinearAlgebra as L+import Statistics.Covariance.LedoitWolf+import Statistics.Covariance.OracleApproximatingShrinkage+import Statistics.Covariance.RaoBlackwellLedoitWolf+import Test.Tasty+import Test.Tasty.HUnit++main :: IO ()+main = defaultMain unitTests++emptyM :: L.Matrix Double+emptyM = L.fromLists [[]]++oneSampleM :: L.Matrix Double+oneSampleM = L.fromLists [[1, 2, 3, 4, 5]]++type Estimator = (L.Matrix Double -> Either String (L.Herm Double), String)++unitTestsForEstimator :: Estimator -> [TestTree]+unitTestsForEstimator (e, m) =+  [ testCase (m <> " fails on empty data matrices.") $+      assertEqual "" True (isLeft $ e emptyM),+    testCase (m <> " fails on one sample data matrices.") $+      assertEqual "" True (isLeft $ raoBlackwellLedoitWolf oneSampleM)+  ]++unitTestsForEstimators :: [Estimator] -> [TestTree]+unitTestsForEstimators = concatMap unitTestsForEstimator++estimators :: [Estimator]+estimators =+  [ (ledoitWolf, "ledoitWolf"),+    (raoBlackwellLedoitWolf, "raoBlackwellLedoitWolf"),+    (oracleApproximatingShrinkage, "oracleApproximatingShrinkage")+  ]++unitTests :: TestTree+unitTests = testGroup "Unit tests" $ unitTestsForEstimators estimators++-- import qualified Numeric.LinearAlgebra as L+-- let m = L.gaussianSample 666 30 (L.fromList [0..40]) (L.trustSym $ L.diagl [1..41])