ConClusion (empty) → 0.0.1

raw patch · 8 files changed

+3008/−0 lines, 8 filesdep +ConClusiondep +PSQueuedep +aeson

Dependencies added: ConClusion, PSQueue, aeson, attoparsec, base, cmdargs, containers, formatting, hmatrix, massiv, optics, rio, text

Files

• Changelog.md +8/−0
• ConClusion.cabal +119/−0
• LICENSE.md +662/−0
• README.md +123/−0
• app/ConClusion.hs +742/−0
• src/ConClusion/Chemistry/Topology.hs +210/−0
• src/ConClusion/Numeric/Data.hs +197/−0
• src/ConClusion/Numeric/Statistics.hs +947/−0

@@ -0,0 +1,8 @@+# Changelog++## 0.0.1+  - raising upper bounds of optics and attoparsec++## 0.0.0+Initial version, providing PCA, DBScan and hierarchical clustering.+Command line interface to CREST XYZ trajectories.

@@ -0,0 +1,119 @@+cabal-version: 1.12++-- This file has been generated from package.yaml by hpack version 0.34.4.+--+-- see: https://github.com/sol/hpack++name:           ConClusion+version:        0.0.1+synopsis:       Cluster algorithms, PCA, and chemical conformere analysis+description:    Please see the README on GitLab at <https://gitlab.com/theoretical-chemistry-jena/quantum-chemistry/ConfoCluster>+category:       Statistics, Chemistry+bug-reports:    https://gitlab.com/theoretical-chemistry-jena/quantum-chemistry/ConfoCluster/-/issues+author:         Phillip Seeber+maintainer:     phillip.seeber@googlemail.com+copyright:      2021 Phillip Seeber+license:        AGPL-3+license-file:   LICENSE.md+build-type:     Simple+extra-source-files:+    README.md+    LICENSE.md+    Changelog.md++source-repository head+  type: git+  location: https://gitlab.com/theoretical-chemistry-jena/quantum-chemistry/ConfoCluster++library+  exposed-modules:+      ConClusion.Chemistry.Topology+      ConClusion.Numeric.Data+      ConClusion.Numeric.Statistics+  other-modules:+      Paths_ConClusion+  hs-source-dirs:+      src+  default-extensions:+      BangPatterns+      OverloadedStrings+      NoImplicitPrelude+      FlexibleContexts+      ScopedTypeVariables+      OverloadedLabels+      DataKinds+      FlexibleInstances+      MultiParamTypeClasses+      UndecidableInstances+      TypeFamilies+      DuplicateRecordFields+      ScopedTypeVariables+      DataKinds+      DeriveAnyClass+      DeriveDataTypeable+      DeriveGeneric+      DeriveTraversable+      FlexibleInstances+      GeneralizedNewtypeDeriving+      TypeApplications+      RecordWildCards+      NamedFieldPuns+  ghc-options: -Wall -Wno-unused-top-binds -Wcompat -Widentities -Wincomplete-record-updates -Wincomplete-uni-patterns -Wpartial-fields -Wredundant-constraints+  build-depends:+      PSQueue >=1.1.0.1 && <1.2+    , aeson ==1.5.*+    , attoparsec >=0.13.0.0 && <0.15+    , base >=4.7 && <4.15+    , containers >=0.6.0.0 && <0.7+    , formatting >=7.1.0 && <7.2+    , hmatrix >=0.20.0 && <0.21+    , massiv >=0.6.0.0 && <0.7+    , rio >=0.1.13.0 && <0.2+  default-language: Haskell2010++executable conclusion+  main-is: ConClusion.hs+  other-modules:+      Paths_ConClusion+  hs-source-dirs:+      app+  default-extensions:+      BangPatterns+      OverloadedStrings+      NoImplicitPrelude+      FlexibleContexts+      ScopedTypeVariables+      OverloadedLabels+      DataKinds+      FlexibleInstances+      MultiParamTypeClasses+      UndecidableInstances+      TypeFamilies+      DuplicateRecordFields+      ScopedTypeVariables+      DataKinds+      DeriveAnyClass+      DeriveDataTypeable+      DeriveGeneric+      DeriveTraversable+      FlexibleInstances+      GeneralizedNewtypeDeriving+      TypeApplications+      RecordWildCards+      NamedFieldPuns+  ghc-options: -Wall -Wno-unused-top-binds -Wcompat -Widentities -Wincomplete-record-updates -Wincomplete-uni-patterns -Wpartial-fields -Wredundant-constraints -threaded -rtsopts -with-rtsopts=-N+  build-depends:+      ConClusion+    , PSQueue >=1.1.0.1 && <1.2+    , aeson ==1.5.*+    , attoparsec >=0.13.0.0 && <0.15+    , base >=4.7 && <4.15+    , cmdargs >=0.10.0 && <0.11+    , containers >=0.6.0.0 && <0.7+    , formatting >=7.1.0 && <7.2+    , hmatrix >=0.20.0 && <0.21+    , massiv >=0.6.0.0 && <0.7+    , optics >=0.3 && <0.5+    , rio >=0.1.13.0 && <0.2+    , text >=1.2.0.0 && <1.3+  default-language: Haskell2010

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@@ -0,0 +1,123 @@+# ConClusion+ConClusion provides principal component analysis, hierarchical clustering and DBScan in Haskell.+There is also a command line interface for processing of [CREST](https://github.com/grimme-lab/crest) conformere trajectories.+Hence the name: CONformere CLUStering.+The procedure to analyse conformere data has three steps:++  1. Read the trajectory and calculate a set of features for each conformere.+     The features can include the energy, a set of bond lengths, a set of bond angles, and a set of dihedral angles in arbitrary combination.+     Those descriptors form a feature matrix.+  2. A [principal component analysis](https://en.wikipedia.org/wiki/Principal_component_analysis) of the feature matrix might be perfomed to reduce the number of dimensions and remove redundancies.+  3. The (potentially PCA-processed) feature matrix is being clustered.+     Different distance measures are available.+     Either [DBScan](https://en.wikipedia.org/wiki/DBSCAN) or [hierarchical clustering](https://en.wikipedia.org/wiki/Hierarchical_clustering) can be used to group different conformeres.++While the command line interface only fits the work flow described above, the underlying clustering algorithms and PCA are implemented in a general way and can be utilised independently as library.++## Installation+### Bundled Archive+A self-contained executable archive is build for the main branch and for releases.+This can be executed directly on any Linux and has just to be downloaded.+Go to the [page of releaes](https://gitlab.com/theoretical-chemistry-jena/quantum-chemistry/ConfoCluster/-/releases) and download an archive.+Make it executable (e.g. `chmod +x conclusion`) and you are done!++### From Source+If you have the Haskell toolchain intalled and therefore working Cabal and GHC, you may build ConClusion from source.+This also requires working BLAS and LAPACK libraries on your system.++```bash+git clone https://gitlab.com/theoretical-chemistry-jena/quantum-chemistry/ConfoCluster.git ConClusion+cd ConClusion+cabal install --installdir=$(PREFIX)+```+Choose a `PREFIX` where to install the executable.+`$HOME/.local/bin/` is often a good choice.++If you would like to use ConClusion on systems where Nix is not available (Windows, BSD, ...) this is the way to go.+++### With Nix+When you have Nix available on your system, everything can be build by Nix:++```bash+git clone https://gitlab.com/theoretical-chemistry-jena/quantum-chemistry/ConfoCluster.git ConClusion+cd ConClusion/nix+nix-build -A ConClusion.components.exes.conclusion+```+++## Usage+The command line interface to the `conclusion` executable offers full control about all three steps described above.++  - `-x --xyz` takes the path to the XYZ trajectory that is to be processed.+    It must contain the energy as comment line (which is the case for CREST trajectories).++  - `--dim` specifies the characterising features; therefore a set of internal coordinates and maybe the energy.+    A comma separated list of an arbitrary amount of features can be specified with the following syntax:++    - `e` for the energy+    - `b m n` for a bond length, where `m` and `n` are the 0-based atom indices+    - `a m n o` for an angle, where `m`, `n` and `o` are 0-based atom indices.+      Calculates the angle around `n`+    - `d m n o p` for a dihedral angle, where `m`, `n`, `o` and `p` are 0-based atom indices.+      Calculates the rotation of the bond between `n` and `o`.+      Dihedrals use a metric, that respects periodicity and direction of the rotation.+      For each dihedral there will be two rows in the feature matrix, therefore.+      See [this paper](https://doi.org/10.1002/prot.20310).+    - (indices are 0-based)++  - `-p --pca` activates dimensionalty reduction by principal component analysis.+    Give an integer to specify how many principal components are kept.+    During the execution of ConClusion the error introduced by PCA will be printed.++  - `-c --cluster` activates clustering of the results.+    The clustering algorithm can be selected by giving either `dbscan` or `hca`++  - `--measure` specifies the distance measure between the conformeres.+    By default an euclidean distance is used, but Manhattan and Mahalanobis distances are also available, as well as a general L_p norm.+    If Mahalanobis distances are used, it might be worth a try to disable PCA.++  - `--joinstrat` controls how inter-cluster distances are calculated in hierarchical clustering.+    `single` might be the best choice to get dense groups of conformeres.++  - `--distance` gives the search radius in DBScan or the dendrogram cut distance in hierarchical clustering.++  - `--minsize` is the minimum size of a cluster in DBScan.+    If `--forcemin` is given, the clusters obtained by HCA are also filtered for a minimum size.++  - `--forcemin` forces filtering of HCA clusters for their minimum size as given by `--minsize`.+    Disabled by default.++Each processing step will produce a Gnuplot compatible file (space separated columns). The pure feature matrix will be `features.dat`, the results of the PCA will be in `pca.dat` and the clustering results will be in `cluster.dat`. The first column in `cluster.dat` will be an integer giving the cluster number this point belongs to, that can be used for colour-coding in Gnuplot.++## Example+A perylene dye with four phenoxy groups has different conformeres, that have different spectral properties.+For solubility the dye has also some alkyl groups.+Crest finds about 1400 conformeres, most of them being different only in the alkyl side-chains, that do not influence spectral properties.+Therefore, a much smaller group of different conformeres with respect to different positions of phenoxy groups exist.+From each of those groups the lowest energy conformere shall be obtained.+We therefore select eight dihedral angles and the energy as features; 2 dihedrals for each phenoxy group.+One dihedral per phenoxy group describing the rotation of the perylene-O bond, the second one describing the rotation around the O-Ph bond.+As the dihedral angles are not independent from each other, as some orientations of phenoxy groups are not possible, we use a PCA to reduce dimensionalty and remove redundancies.+After the PCA, DBScan is used to obtain clusters of similar conformeres.+The lowest index in each cluster is also the lowest energy conformere in each group, as CREST sorts conformeres by energy.++```bash+conclusion \+  --xyz=crest_conformers.xyz \+  --pca=3 \+  --dim="e, d 19 18 2 56, d 18 2 56 65, d 16 15 1 45, d 15 1 45 46, d 11 13 0 34, d 13 0 34 43, d 29 31 3 67, d31 3 67 76" \+  --measure=manhattan \+  --cluster=dbscan \+  --distance=0.3 \+  --minsize=5+```++## Library/Haskell Package+ConClusion provides principal components analysis and the clustering algorithms DBScan and hierarchical clustering.+The algorithms are implemented in efficient parallel arrays and perform quite well.+For the API see the haddock documentation, which can be generated by:++```bash+cabal haddock+```

@@ -0,0 +1,742 @@+-- |+-- Module      : Main+-- Description : Command Line Interface to analyse CREST results.+-- Copyright   : Phillip Seeber, 2021+-- License     : AGPL-3+-- Maintainer  : phillip.seeber@uni-jena.de+-- Stability   : experimental+-- Portability : POSIX, Windows+module Main (main) where++import ConClusion.Chemistry.Topology hiding (xyz)+import qualified ConClusion.Numeric.Statistics as Statistics+import Data.Aeson+import Data.Attoparsec.Text hiding (D)+import Data.Foldable+import qualified Data.IntSet as IntSet+import Data.Massiv.Array as Massiv hiding (B, D)+import qualified Data.Massiv.Array as Massiv+import Data.Text.Lazy (toStrict)+import Data.Text.Lazy.Builder as TB+import Data.Version (showVersion)+import Formatting hiding (char, (%))+import qualified Formatting as F+import Optics hiding (view)+import Paths_ConClusion (version)+import RIO hiding (Lens', Vector, lens, view, (^.))+import qualified RIO.Text as Text+import System.Console.CmdArgs+import System.IO.Unsafe (unsafePerformIO)++main :: IO ()+main = do+  app <- cmdArgs conClusionCmd >>= cmdToEnv+  runRIO app $ do+    logInfo $ "ConClusion, version " <> (displayShow . showVersion $ version) <> "\n"++    -- Processing.+    (featureMat, _trj) <- processTrajectory+    pcaMat <- dimReduction featureMat+    _clusters <- clusterFeatures pcaMat++    logInfo "Done. Good Luck with the results!"++----------------------------------------------------------------------------------------------------+-- Processing Logic++-- | Read the trajectory and calculate the specified features.+{-# SCC processTrajectory #-}+processTrajectory :: (HasXYZ env, HasDim env, HasLogFunc env) => RIO env (Matrix DL Double, Trajectory)+processTrajectory = do+  -- Obtain environment+  file <- view xyzL+  dims <- view dimL+  let outputFile = "features.dat"++  -- Logging+  {- ORMOLU_DISABLE -}+  logInfo $+    "+-----------------------+\n\+    \| Trajectory Processing |\n\+    \+-----------------------+\n\n\+    \  Trajectory File     : "  <> displayShow file <> "\n\+    \  Feature Matrix File : "  <> displayShow outputFile <> "\n\+    \  Features            :\n" <> displayFeatures dims+  {- ORMOLU_ENABLE -}++  -- Processing.+  raw <- readFileUtf8 file+  trj <- handleFailure $ parse' trajectory raw+  features <- handleFailure $ getFeatures dims trj++  -- Output.+  toGnuPlot Nothing (Just . Massiv.fromList Seq $ dims) (compute features)+    >>= writeFileUtf8 outputFile++  logInfo "\n\n\n"+  return (features, trj)++-- | Possibly perform a PCA.+{-# SCC dimReduction #-}+dimReduction :: (HasPrincipalComponentAnalysis env, HasLogFunc env) => Matrix DL Double -> RIO env (Matrix U Double)+dimReduction mat = do+  doPCA <- view pcaL+  case doPCA of+    Nothing -> return . compute $ featureMat+    Just PrincipalComponentAnalysis {keep} -> do+      let outputFile = "pca.dat"++      -- Logging.+      {- ORMOLU_DISABLE -}+      logInfo+        "+------------------------------+\n\+        \| Principal Component Analysis |\n\+        \+------------------------------+\n"+      {- ORMOLU_ENABLE -}++      -- Processing+      pcaData <- Statistics.pca keep featureMat+      let behaviourPercent = Statistics.remaining pcaData++      -- Logging.+      {- ORMOLU_DISABLE -}+      logInfo $+        "  Dimensions         : " <> display keep <> "\n\+        \  Eigenvalues        : " <> (displayEigenvalues . Statistics.allEigenValues $ pcaData) <> "\n\+        \  Mean Squared Error : " <> (displayMSE . Statistics.mse $ pcaData) <> "\n\+        \  Behaviour Captured : " <> displayPercent behaviourPercent <> "\n\+        \  PCA Matrix File    : " <> displayShow outputFile+      {- ORMOLU_ENABLE -}++      -- Output+      toGnuPlot Nothing Nothing (Statistics.y pcaData) >>= writeFileUtf8 outputFile++      logInfo "\n\n\n"+      return . Statistics.y $ pcaData+  where+    featureMat = compute @U mat++-- | Cluster analysis of the (dimensionalty reduced) feature matrix.+{-# SCC clusterFeatures #-}+clusterFeatures ::+  (HasClustering env, HasLogFunc env) => Matrix U Double -> RIO env Statistics.Clusters+clusterFeatures mat = do+  doClustering <- view clusteringL+  case doClustering of+    Nothing -> let Sz (_ :. n) = size mat in return $ makeArray @Massiv.B @Ix1 @IntSet Par (Sz n) $ \i -> IntSet.singleton i+    Just cl -> do+      let outputFile = "cluster.dat"+      -- Logging+      {- ORMOLU_DISABLE -}+      logInfo+        "+------------+\n\+        \| Clustering |\n\+        \+------------+\n"+      {- ORMOLU_ENABLE -}++      clusters <- case cl of+        DBScan {distance, measure, minSize} -> clusterDB distance measure minSize+        HCA {distance, measure, joinstrat, forcemin} -> clusterHCA distance measure joinstrat forcemin++      {- ORMOLU_DISABLE -}+      logInfo $+        "  Number of Clusters    : " <> (display . Massiv.elemsCount $ clusters) <> "\n\+        \  Cluster Data File     : " <> displayShow outputFile <> "\n\+        \  Cluster               : " <> displayClusters clusters+      {- ORMOLU_ENABLE -}++      -- Output+      toGnuPlot (Just clusters) Nothing mat >>= writeFileUtf8 outputFile++      return clusters+  where+    -- Make the distance function for clustering.+    distFn m = case m of+      Lr i -> Statistics.lpNorm i+      Manhattan -> Statistics.manhattan+      Euclidean -> Statistics.euclidean+      Mahalanobis -> Statistics.mahalanobis++    -- Cluster data with DBScan algorithm.+    clusterDB d m s = do+      clusters <- Statistics.dbscan (distFn m) s d mat+      {- ORMOLU_DISABLE -}+      logInfo $+        "  Algorithm            : DBScan\n\+        \  Distance Measure     : " <> displayShow m <> "\n\+        \  Search Distance ε    : " <> display d <> "\n\+        \  Minimal Cluster Size : " <> display s+      {- ORMOLU_ENABLE -}+      return clusters++    -- Cluster data with Hierarchical Cluster Analysis.+    clusterHCA d m j fm = do+      dendrogram <- Statistics.hca (distFn m) j mat+      let allClusters = Statistics.cutDendroAt dendrogram d+          dendroFile = "dendrogram.json"+          clusters = case fm of+            Nothing -> allClusters+            Just minSize -> compute . sfilter (\c -> IntSet.size c >= minSize) $ allClusters++      -- Write dendrogram file to disk.+      liftIO $ encodeFile dendroFile dendrogram++      {- ORMOLU_DISABLE -}+      logInfo $+        "  Algorithm             : Hierarchical Cluster Analysis\n\+        \  Distance Measure      : " <> displayShow m <> "\n\+        \  Search Distance       : " <> display d <> "\n\+        \  Cluster Join Strategy : " <> displayShow j <> "\n" <>+        (case fm of+          Nothing -> mempty+          Just minSize -> "  Minimal Cluster Size  : " <> display minSize <> "\n") <>+        "  Dendrogram File       : " <> displayShow dendroFile+      {- ORMOLU_ENABLE -}+      return clusters++----------------------------------------------------------------------------------------------------+-- Command line interfaces++-- | Command line arguments for ConClusion.+data ConClusionCmd = ConClusionCmd+  { -- | Filepath to the CREST xyz trajectory file.+    xyz :: FilePath,+    -- | Dimensions to process.+    dim :: String,+    -- | If PCA should be used to reduce dimensionalty of the problem. If so, the number of+    -- dimensions to keep.+    pca :: Maybe Int,+    -- | Cluster algorithm to be used for clustering, if any.+    cluster :: Maybe String,+    -- | Distance measure of points.+    measure :: String,+    -- | Cluster joining strategy for hierarchical clustering.+    joinstrat :: String,+    -- | Search distance in case of DBScan or distance for the Dendogram cut in case of hierarchical+    -- clustering.+    distance :: Double,+    -- | Mininmal size of a cluster in case of DBScan.+    minsize :: Int,+    -- | If to force a minimum size of clusters in HCA.+    forcemin :: Bool,+    -- | Do verbose logging.+    verbose :: Bool+  }+  deriving (Show, Data, Typeable)++conClusionCmd :: ConClusionCmd+conClusionCmd =+  ConClusionCmd+    { xyz =+        "crest_conformers.xyz"+          &= typFile+          &= help "Path to CREST trajectory.",+      dim =+        "e"+          &= help+            "Expression of dimensions to analyse:\n\+            \  - \"e\"         Energy\n\+            \  - \"b m n\"     Bond length between the atoms with\n\+            \                indices m and n\n\+            \  - \"a m n o\"   Angle between the atoms with indices\n\+            \                m, n and o\n\+            \  - \"d m n o p\" Sinus of the dihedral between the atoms\n\+            \                m, n, o and p\n\+            \Multiple features may be combined by giving a comma separated list of them.\n\+            \Default: e",+      pca =+        Nothing+          &= typ "INT"+          &= help+            "Perform a principal component analysis on the features.\+            \Optionally give the number of dimensions you want to keep.\n\+            \Default: 2"+          &= opt (2 :: Int),+      cluster =+        Nothing+          &= help+            "Cluster algorithm to cluster conformeres.\n\+            \  - \"dbscan\" DBScan algorithm. Depends on \n\+            \             \"measure\", \"distance\" and \"minSize\"\n\+            \  - \"hca\"    Hierarchical Cluster Analysis. Depends on\n\+            \             \"measure\", \"joinstrat\" and \"distance\"\n\+            \Default: dbscan"+          &= opt ("dbscan" :: String),+      measure =+        "mahalanobis"+          &= help+            "Distance measure for distance between conformeres.\n\+            \  - \"lrX\"         General L_r norm, where X is a value\n\+            \                  between 3 and 9\n\+            \  - \"manhattan\"   Manhattan distances\n\+            \  - \"euclidean\"   Euclidean distances\n\+            \  - \"mahalanobis\" Mahalanobis distances\n\+            \Default: mahalanobis",+      joinstrat =+        "ward"+          &= help+            "Strategy for inter-cluster distance in hierarchical clustering.\n\+            \  - \"single\"     Single-Linkage (minimal distance of\n\+            \                 points between clusters)\n\+            \  - \"complete\"   Complete-Linkage (maximal distance of\n\+            \                 points between clusters)\n\+            \  - \"median\"      Median distance linkage between\n\+            \                 all points between clusters)\n\+            \  - \"upgma\"      Average-Group-Linkage (average distance\n\+            \                 of all points of joined clusters)\n\+            \  - \"wpgma\"      Average-Linkage (average distance\n\+            \                 of all points between clusters)\n\+            \  - \"centroid\" Distance of cluster centroids\n\+            \  - \"ward\"     Ward distance of minimal residues\n\+            \Default: ward",+      distance =+        0.1+          &= typ "FLOAT"+          &= help+            "Search distance in DBScan or Dendogram cut distance in hierarchical clustering.\n\+            \Default: 0.1",+      minsize =+        5+          &= typ "INT"+          &= help+            "Minimal size of a cluster in DBScan.\n\+            \Default: 5",+      forcemin =+        False+          &= help+            "Force a minimal cluster size in hierarchical clustering.\n\+            \Default: False",+      verbose =+        False+          &= help "Verbose logging."+    }+    &= summary+      "Analysis of conformere ensembles.\n\+      \Processes an annotated XYZ trajectory (as generated by CREST, for example), where the energy\+      \is provided in the comment line. The analysis is a three-step procedure:\n\+      \  1. Calculate selected features from the ensemble, that are meant to\n\+      \     characterise the conformeres.\n\+      \     Relevant settings: \"--dim\"\n\+      \  2. (Optional) Perform a PCA analysis on the feature matrix to reduce the\n\+      \     number of dimensions.\n\+      \     Relevant settings: \"--pca\"\n\+      \  3. (Optional) Cluster the feature matrix into groups of conformeres. If a PCA\n\+      \     has been performed, use the feature matrix with reduced dimensionalty.\n\+      \     Relevant settins: \"--cluster\", \"--measure\", \"joinstrat\", \"--distance\",\n\+      \     \"--minsize\""++----------------------------------------------------------------------------------------------------+-- Runtime Environment.++-- Reader classes+class HasXYZ a where+  xyzL :: Lens' a FilePath++class HasDim a where+  dimL :: Lens' a [Feature]++class HasPrincipalComponentAnalysis a where+  pcaL :: Lens' a (Maybe PrincipalComponentAnalysis)++class HasClustering a where+  clusteringL :: Lens' a (Maybe Clustering)++-- | Full runtime environment of ConClusion with all settings combined.+data App = App+  { -- | Filepath to the CREST xyz trajectory file.+    xyz :: FilePath,+    -- | Features that are to be analysed.+    dim :: [Feature],+    -- | Settings for principal component analysis.+    pca :: Maybe PrincipalComponentAnalysis,+    -- | Settings for Clustering.+    clustering :: Maybe Clustering,+    -- | RIO's 'logFunc'.+    logFunc :: LogFunc+  }++-- Lenses+instance (k ~ A_Lens, a ~ FilePath, b ~ a) => LabelOptic "xyz" k App App a b where+  labelOptic = lens (xyz :: App -> FilePath) $ \s b -> (s {xyz = b} :: App)++instance (k ~ A_Lens, a ~ [Feature], b ~ a) => LabelOptic "dim" k App App a b where+  labelOptic = lens (dim :: App -> [Feature]) $ \s b -> (s {dim = b} :: App)++instance (k ~ A_Lens, a ~ Maybe PrincipalComponentAnalysis, b ~ a) => LabelOptic "pca" k App App a b where+  labelOptic = lens (pca :: App -> Maybe PrincipalComponentAnalysis) $ \s b -> (s {pca = b} :: App)++instance (k ~ A_Lens, a ~ Maybe Clustering, b ~ a) => LabelOptic "clustering" k App App a b where+  labelOptic = lens clustering $ \s b -> s {clustering = b}++instance (k ~ A_Lens, a ~ LogFunc, b ~ a) => LabelOptic "logFunc" k App App a b where+  labelOptic = lens logFunc $ \s b -> s {logFunc = b}++-- Reader Classes+instance HasXYZ App where+  xyzL = castOptic #xyz++instance HasDim App where+  dimL = castOptic #dim++instance HasPrincipalComponentAnalysis App where+  pcaL = castOptic #pca++instance HasClustering App where+  clusteringL = castOptic #clustering++instance HasLogFunc App where+  logFuncL = toLensVL #logFunc++-- | Settings for principal component analysis.+newtype PrincipalComponentAnalysis = PrincipalComponentAnalysis+  { -- | Number of dimensions to keep from PCA.+    keep :: Int+  }++-- Lenses+instance (k ~ A_Lens, a ~ Int, b ~ a) => LabelOptic "keep" k PrincipalComponentAnalysis PrincipalComponentAnalysis a b where+  labelOptic = lens keep $ \s b -> s {keep = b}++-- | Settings for clustering.+data Clustering+  = -- | DBScan.+    DBScan+      { -- | Search distance around a point.+        distance :: Double,+        -- | Distance measure in clustering.+        measure :: Measure,+        -- | Minimal size of a cluster.+        minSize :: Int+      }+  | -- | Hierarchical clustering.+    HCA+      { -- | Cut distance in the dendogram to select the number of clusters.+        distance :: Double,+        -- | Distance measure in clustering.+        measure :: Measure,+        -- | Join strategy for clusters.+        joinstrat :: Statistics.JoinStrat Double,+        -- | A minimum cluster size might be forced in post processing for HCA.+        forcemin :: Maybe Int+      }++-- | Distance measure between clusters.+data Measure+  = Lr Int+  | Manhattan+  | Euclidean+  | Mahalanobis+  deriving (Show)++----------------------------------------------------------------------------------------------------+-- Utilities++-- | Conversion of command line arguments into runtime environment.+cmdToEnv :: ConClusionCmd -> IO App+cmdToEnv ConClusionCmd {xyz, dim, pca, cluster, measure, joinstrat, distance, minsize, forcemin, verbose} = do+  -- Dimensionalty/Feature construction.+  dimApp <- case parseOnly dimParser (Text.pack dim) of+    Left err -> throwM $ ArgumentException "dim" err "a comma separated list of dimension specifications"+    Right res -> return res++  -- PCA construction.+  pcaSettings <- case pca of+    Nothing -> return Nothing+    Just x ->+      if x >= 1+        then return . Just $ PrincipalComponentAnalysis {keep = x}+        else throwM $ ArgumentException "pca" (show x) "a positive integer smaller than the number of clusters."++  -- Clustering construction.+  clDistance <-+    if distance <= 0+      then throwM $ ArgumentException "distance" (show distance) "a positive number"+      else return distance+  clMeasure <- case measure of+    "lr3" -> return $ Lr 3+    "lr4" -> return $ Lr 4+    "lr5" -> return $ Lr 5+    "lr6" -> return $ Lr 6+    "lr7" -> return $ Lr 7+    "lr8" -> return $ Lr 8+    "lr9" -> return $ Lr 9+    "manhattan" -> return Manhattan+    "euclidean" -> return Euclidean+    "mahalanobis" -> return Mahalanobis+    _ -> throwM $ ArgumentException "measure" measure "one of \"manhattan | euclidean | mahalanobis | lrX\" with X = [ 3 .. 9]"+  clMinsize <-+    if minsize <= 0+      then throwM $ ArgumentException "minsize" (show minsize) ">= 1"+      else return minsize+  clJoinstrat <- case joinstrat of+    "single" -> return Statistics.SingleLinkage+    "complete" -> return Statistics.CompleteLinkage+    "median" -> return Statistics.Median+    "upgma" -> return Statistics.UPGMA+    "wpgma" -> return Statistics.WPGMA+    "centroid" -> return Statistics.Centroid+    "ward" -> return Statistics.Ward+    _ -> throwM $ ArgumentException "joinstrat" joinstrat "one of \"single | complete | upgma | wpgma | centroid | ward\""+  clustering <- case cluster of+    Just "dbscan" ->+      return . Just $+        DBScan+          { distance = clDistance,+            measure = clMeasure,+            minSize = clMinsize+          }+    Just "hca" ->+      return . Just $+        HCA+          { distance = clDistance,+            measure = clMeasure,+            joinstrat = clJoinstrat,+            forcemin = if forcemin then Just clMinsize else Nothing+          }+    Nothing -> return Nothing+    Just c -> throwM $ ArgumentException "cluster" c "one of \"dbscan | hca\""++  -- Log function setup.+  logOptions <-+    id+      . setLogUseTime False+      . setLogUseColor True+      . setLogVerboseFormat False+      . setLogUseLoc False+      <$> logOptionsHandle stdout verbose+  (lf, _ :: IO ()) <- newLogFunc logOptions+  return+    App+      { xyz = xyz,+        dim = dimApp,+        pca = pcaSettings,+        clustering = clustering,+        logFunc = lf+      }++-- | Parser for dimensionalty expressions.+dimParser :: Parser [Feature]+dimParser = many1 $ do+  skipMany (char ' ')+  fType <- eParser <|> bParser <|> aParser <|> dParser+  skipMany (char ' ')+  _ <- option ',' (char ',')+  return fType+  where+    eParser = char 'e' *> return Energy+    bParser = do+      _ <- char 'b'+      skipMany (char ' ')+      a <- decimal+      skipMany (char ' ')+      b <- decimal+      return . Bond $ B a b+    aParser = do+      _ <- char 'a'+      skipMany (char ' ')+      a <- decimal+      skipMany (char ' ')+      b <- decimal+      skipMany (char ' ')+      c <- decimal+      return . Angle $ A a b c+    dParser = do+      _ <- char 'd'+      skipMany (char ' ')+      a <- decimal+      skipMany (char ' ')+      b <- decimal+      skipMany (char ' ')+      c <- decimal+      skipMany (char ' ')+      d <- decimal+      return . Dihedral $ D a b c d++-- | Substitution for RIO's @view@ for optics.+view :: (Is k A_Getter, MonadReader s f) => Optic' k is s b -> f b+view l = (^. l) <$> ask++-- | Parsers lifted to 'MonadThrow'.+parse' :: MonadThrow m => Parser x -> Text -> m x+parse' p t = case parseOnly p t of+  Left err -> throwM $ ParseException err+  Right res -> return res++-- | Logging a problem as an error and then rethrow the underlying exception.+handleFailure :: (Exception e, HasLogFunc env) => Either e r -> RIO env r+handleFailure (Right r) = return r+handleFailure (Left e) = do+  logError . displayShow $ e+  throwM e++-- | Display features for a trajectory in a nice way.+displayFeatures :: Foldable f => f Feature -> Utf8Builder+displayFeatures features =+  foldl+    ( \acc f ->+        acc+          <> ( case f of+                 Energy -> "    - E"+                 Bond (B a b) -> "    - B " <> display a <> " " <> display b+                 Angle (A a b c) -> "    - A " <> display a <> " " <> display b <> " " <> display c+                 Dihedral (D a b c d) -> "    - D " <> display a <> " " <> display b <> " " <> display c <> " " <> display d+             )+          <> "\n"+    )+    mempty+    features++-- | Display a percent value.+displayPercent :: Double -> Utf8Builder+displayPercent v = display $ sformat (fixed 1 F.% " %") v++-- | Display a mean squared error.+displayMSE :: Double -> Utf8Builder+displayMSE v = display $ sformat (fixed 2) v++-- | Display eigenvalues+displayEigenvalues :: Vector U Double -> Utf8Builder+displayEigenvalues vec = display . Massiv.fold . Massiv.map fmtD $ vec+  where+    fmtD d = format (right 6 ' ' F.%. fixed 2) d++-- | Display best cluster members.+displayBestMember :: Vector U Int -> Utf8Builder+displayBestMember vec =+  display . toLazyText . Massiv.fold . Massiv.map (\i -> bformat ("  - " F.% int F.% "\n") i) $ vec++-- | Converts a matrix to a Gnuplot matrix. Prints a header line as a comment with the features, if+-- available. Adds the assigned cluster number in the first column if available.+{-# SCC toGnuPlot #-}+toGnuPlot ::+  ( MonadThrow m+  ) =>+  Maybe Statistics.Clusters ->+  Maybe (Vector Massiv.B Feature) ->+  Matrix U Double ->+  m Text+toGnuPlot clusters features mat = do+  -- Build the lines representing the points. If cluster information is available, the first column+  -- is the cluster, that this point is assigned to.+  valueLines <- case clusters of+    Nothing -> return . Massiv.fold . Massiv.map (writeLine . Left) . innerSlices $ mat+    Just cl -> do+      let points = innerSlices mat :: Vector Massiv.D (Vector M Double)+          labeledCl = compute . Massiv.imap (,) $ cl :: Vector Massiv.B (Int, IntSet)++      -- Anotate all points with their assignment to a cluster number. If a point is from DBScan, it+      -- is not necessarily assigned to a cluster.+      annoPoints <- Massiv.iforM @Massiv.B points $ \pointIx vec -> do+        let filteredCl = Massiv.dropWhile (\(_, clPoints) -> not $ pointIx `IntSet.member` clPoints) labeledCl+        let assignedCl = filteredCl !? 0 :: Maybe (Int, IntSet)+            clNumber = fst <$> assignedCl :: Maybe Int+        return (clNumber, vec)++      -- return annoClusters+      return . Massiv.fold . Massiv.map (writeLine . Right) $ annoPoints++  featureLine <- case features of+    Nothing -> return mempty+    Just f -> do+      let Sz nFeatureLabels = actualFeatureSize f+          featureLine = writeFeatures f+      if nFeatures == nFeatureLabels+        then return featureLine+        else throwM $ SizeMismatchException (Sz nFeatureLabels) (Sz nFeatures)+  return . toStrict . TB.toLazyText $ featureLine <> valueLines+  where+    Sz (nFeatures :. _) = size mat++    -- The column width in the Gnuplot file.+    cw = 14++    -- Formatter for doubles.+    dForm a = bformat (left cw ' ' F.%. fixed 4) a++    -- Formatter for Integers.+    iForm = fitLeft 3 F.%. left 3 ' ' F.%. int++    -- Formatter for Features+    fForm f = case f of+      Energy -> bformat (left cw ' ' F.%. builder) "E"+      Bond (B a b) -> bformat (left cw ' ' F.%. (builder F.% iForm F.% "," F.% iForm)) "B" a b+      Angle (A a b c) -> bformat (left cw ' ' F.%. (builder F.% iForm F.% "," F.% iForm F.% "," F.% iForm)) "A" a b c+      Dihedral (D a b c d) ->+        bformat (left cw ' ' F.%. (builder F.% iForm F.% "," F.% iForm F.% "," F.% iForm F.% "," F.% iForm)) "A sin" a b c d+          <> bformat (left cw ' ' F.%. (builder F.% iForm F.% "," F.% iForm F.% "," F.% iForm F.% "," F.% iForm)) "A cos" a b c d++    -- Print a vector of numeric values.+    doubleVecB :: Source r Ix1 Double => Vector r Double -> TB.Builder+    doubleVecB v = Massiv.fold . Massiv.map dForm $ v++    -- Writer for a value line.+    writeLine ::+      (Source r Ix1 Double) => Either (Vector r Double) (Maybe Int, Vector r Double) -> TB.Builder+    writeLine (Left vec) = doubleVecB vec <> "\n"+    writeLine (Right (cl, vec)) = bformat iForm (fromMaybe (-1) cl) <> doubleVecB vec <> "\n"++    -- Writer for the Feature line.+    writeFeatures :: Vector Massiv.B Feature -> TB.Builder+    writeFeatures vec = "#" <> (Massiv.fold . Massiv.map fForm $ vec) <> "\n"++    -- Actual feature size.+    actualFeatureSize vec =+      unsafePerformIO $+        Massiv.foldlP+          ( \acc f -> case f of+              Dihedral _ -> 2 + acc+              _ -> 1 + acc+          )+          0+          (+)+          0+          vec++-- | Pretty printer for clusters.+displayClusters :: Statistics.Clusters -> Utf8Builder+displayClusters clusters = display . toLazyText . Massiv.fold . Massiv.map printCl $ clusters+  where+    -- Print clusters as list of indices.+    iForm = fitLeft 5 F.%. left 5 ' ' F.%. int++    -- Printer of a single cluster.+    printCl :: IntSet -> TB.Builder+    printCl cl =+      fst $+        IntSet.foldl+          ( \(bAcc, counter) i ->+              let this = bformat iForm i <> if counter `mod` (16 :: Int) == 0 then "\n      " else mempty+               in (bAcc <> this, counter + 1)+          )+          ("\n\n    - ", 1)+          cl++----------------------------------------------------------------------------------------------------+-- Exceptions++-- | Exception thrown on unexcpected command line arguments.+data ArgumentException = ArgumentException+  { argName :: String,+    is :: String,+    should :: String+  }++instance Show ArgumentException where+  show ArgumentException {argName, is, should} =+    "ArgumentException: Field \""+      <> argName+      <> "\" got \""+      <> is+      <> "\" but must be "+      <> should+      <> "."++instance Exception ArgumentException++-- | Exceptions thrown on failed parsing.+newtype ParseException = ParseException String++instance Show ParseException where+  show (ParseException err) = "ParseException: \"" <> err <> "\""++instance Exception ParseException

@@ -0,0 +1,210 @@+-- |+-- Module      : ConClusion.Chemistry.Topology+-- Description : Principal Component Analysis+-- Copyright   : Phillip Seeber, 2021+-- License     : AGPL-3+-- Maintainer  : phillip.seeber@googlemail.com+-- Stability   : experimental+-- Portability : POSIX, Windows+--+-- This module implements routines to work with simple molden style XYZ trajectories. This includes+-- parsers as well as functions to obtain structural features in internal coordinates.+--+-- For an introduction into PCA see <https://www.cs.cmu.edu/~elaw/papers/pca.pdf>.+--+-- Diherdrals require a special metric, see <https://onlinelibrary.wiley.com/doi/full/10.1002/prot.20310)>.+module ConClusion.Chemistry.Topology+  ( Molecule,+    Trajectory,+    xyz,+    trajectory,+    B (..),+    A (..),+    D (..),+    Feature (..),+    getFeatures,+  )+where++import ConClusion.Numeric.Data hiding (angle)+import qualified ConClusion.Numeric.Data as ND+import Data.Attoparsec.Text hiding (D)+import Data.Foldable+import Data.Massiv.Array as Massiv hiding (B, D, forM)+import qualified Numeric.LinearAlgebra as LA+import RIO+import qualified RIO.Seq as Seq+import qualified RIO.Text as Text+import qualified RIO.Vector.Boxed as VectorB++-- | A Molecule in cartesian coordinates.+data Molecule = Molecule+  { -- | The energy of the molecule.+    energy :: !Double,+    -- | Chemical symbols or names of the atoms. \(N\) vector.+    atoms :: !(VectorB.Vector Text),+    -- | Cartesian coordinates. Atoms as rows, xyz as columns. \(N \times 3\) matrix.+    coordinates :: !(Massiv.Matrix S Double)+  }++type Trajectory = Seq Molecule++-- | Parser for molecules in Molden XYZ format.+xyz :: Parser Molecule+xyz = do+  -- Starting line with number of atoms.+  nAtoms <- do+    _ <- skipMany . char $ ' '+    n <- decimal+    _ <- skipMany . char $ ' '+    endOfLine+    return n++  -- Comment line. Must contain the energy.+  energy <- do+    _ <- skipMany . char $ ' '+    e <- double+    _ <- skipMany . char $ ' '+    endOfLine+    return e++  -- Multiple lines of atoms.+  atoms <- count nAtoms $ do+    _ <- skipMany . char $ ' '+    name <- many1 $ letter <|> digit+    _ <- skipMany1 . char $ ' '+    x <- double+    _ <- skipMany1 . char $ ' '+    y <- double+    _ <- skipMany1 . char $ ' '+    z <- double+    _ <- skipMany . char $ ' '+    endOfLine+    let row = Massiv.expandOuter (Sz 1) const . Massiv.fromList @S Seq $ [x, y, z]+    return (Text.pack name, row)++  return+    Molecule+      { energy = energy,+        atoms = VectorB.fromListN nAtoms . fmap fst $ atoms,+        coordinates = Massiv.compute . Massiv.concat' (Dim 2) . fmap snd $ atoms+      }++-- | Parser for trajectories in XYZ format as produced by CREST.+{-# SCC trajectory #-}+trajectory :: Parser Trajectory+trajectory = do+  mols <- Seq.fromList <$> many1 xyz+  skipSpace+  endOfInput+  return mols++-- | Selection of a bond between two atoms.+data B = B Int Int++-- | Selection of an angle between three atoms.+data A = A Int Int Int++-- | Selection of a dihedral angle between four atoms. Rotation around the central two.+data D = D Int Int Int Int++-- | Selections+data Feature+  = Energy+  | Bond B+  | Angle A+  | Dihedral D++-- | Calculate a bond distance.+bond :: MonadThrow m => B -> Molecule -> m Double+bond (B a b) Molecule {coordinates}+  | a == b = throwM . IndexException $ "selected atoms are identical"+  | otherwise = do+    coordA <- compute @U <$> (coordinates !?> a)+    coordB <- compute @U <$> (coordinates !?> b)+    vecAB <- coordA .-. coordB+    return . sqrt . Massiv.sum . Massiv.map (^ (2 :: Int)) $ vecAB++-- | Calculates the sinus of an angle defined by three atoms.+angle :: MonadThrow m => A -> Molecule -> m Double+angle (A a b c) Molecule {coordinates}+  | a == b || b == c || a == c = throwM . IndexException $ "selected atoms are identical"+  | otherwise = do+    coordA <- compute @U <$> (coordinates !?> a)+    coordB <- compute @U <$> (coordinates !?> b)+    coordC <- compute @U <$> (coordinates !?> c)+    vecAB <- coordA .-. coordB+    vecCB <- coordC .-. coordB+    return $ ND.angle vecAB vecCB++-- | Calculates the dihedral angle defined by four atoms. Respects rotation direction. Obtains the+-- result in radian.+dihedral' :: MonadThrow m => D -> Molecule -> m Double+dihedral' (D a b c d) Molecule {coordinates}+  | a == b || a == c || a == d || b == c || b == d || c == d = throwM . IndexException $ "selected atoms are indentical"+  | otherwise = do+    coordA <- compute @U <$> (coordinates !?> a)+    coordB <- compute @U <$> (coordinates !?> b)+    coordC <- compute @U <$> (coordinates !?> c)+    coordD <- compute @U <$> (coordinates !?> d)+    vecAB <- coordA .-. coordB+    vecBC <- coordB .-. coordC+    vecCD <- coordC .-. coordD+    let planeABC = vecH2M $ LA.cross (vecM2H vecAB) (vecM2H vecBC) :: Massiv.Vector U Double+        planeBCD = vecH2M $ LA.cross (vecM2H vecBC) (vecM2H vecCD) :: Massiv.Vector U Double+        normVecRot = LA.cross (vecM2H vecCD) (vecM2H vecBC) :: LA.Vector Double+        rotDir =+          if vecH2M normVecRot !.! vecAB < 0+            then -1+            else 1+    return $ rotDir * ND.angle planeABC planeBCD++-- | Calculates the a metric value of the dihedral angle defined by four atoms. This must create 2+-- values in the feature matrix, instead of one.+-- See <https://onlinelibrary.wiley.com/doi/full/10.1002/prot.20310)>+dihedral :: MonadThrow m => D -> Molecule -> m (Double, Double)+dihedral d mol = do+  dihedRad <- dihedral' d mol+  return (sin dihedRad, cos dihedRad)++-- | Get all selected features from a molecule.+getFeature ::+  ( Traversable t,+    MonadThrow m+  ) =>+  -- | Selection of multiple features.+  t Feature ->+  -- | A Molecule where to apply to.+  Molecule ->+  m (Massiv.Vector Massiv.DL Double)+getFeature sel mol = do+  features <- for sel $ \s -> case s of+    Energy -> return . Left . energy $ mol+    Bond b -> Left <$> bond b mol+    Angle a -> Left <$> angle a mol+    Dihedral d -> Right <$> dihedral d mol++  let featureVec =+        foldl+          ( \acc val -> case val of+              Left s -> acc `Massiv.snoc` s+              Right (a, b) -> acc `Massiv.snoc` a `Massiv.snoc` b+          )+          mempty+          features++  return featureVec++-- | Obtains the feature matrix \(\mathbf{X}\) for a principal component analysis. Given \(m\)+-- features to analyse in \(n\) measurements, \(\mathbf{X}\) will be a \(m \times n\) matrix.+{-# SCC getFeatures #-}+getFeatures ::+  ( MonadThrow m,+    Traversable f+  ) =>+  f Feature ->+  Trajectory ->+  m (Massiv.Matrix DL Double)+getFeatures sel trj = traverse toCols trj >>= concatM (Dim 1)+  where+    toCols v = expandInner @Ix2 (Sz 1) const . compute @U <$> getFeature sel v

@@ -0,0 +1,197 @@+-- |+-- Module      : ConClusion.Numeric.Data+-- Description : Type castings, conversions and utilities+-- Copyright   : Phillip Seeber, 2021+-- License     : AGPL-3+-- Maintainer  : phillip.seeber@googlemail.com+-- Stability   : experimental+-- Portability : POSIX, Windows+module ConClusion.Numeric.Data+  ( -- * Conversion of array types.++    -- Conversion between HMatrix and Massiv's array types+    IndexException (..),+    vecH2M,+    vecM2H,+    matH2M,+    matM2H,++    -- * Array Processing+    magnitude,+    normalise,+    angle,+    minDistAt,+    minDistAtVec,+    iMinimumM,++    -- * Utilities+    printMat,++    -- * Binary Trees+    BinTree (..),+    root,+    takeBranchesWhile,+    takeLeafyBranchesWhile,+  )+where++import Data.Aeson hiding (Array)+import Data.Massiv.Array as Massiv hiding (IndexException)+import Data.Massiv.Array.Manifest.Vector as Massiv+import Formatting+import Numeric.LinearAlgebra as LA hiding (magnitude, (<>))+import RIO+import qualified RIO.Vector.Storable as VectorS+import System.IO.Unsafe (unsafePerformIO)++-- | Exception regarding indexing in some kind of aaray.+newtype IndexException = IndexException String deriving (Show)++instance Exception IndexException++-- | Converts a vector from the HMatrix package to the Massiv representation.+{-# SCC vecH2M #-}+vecH2M :: (Element e, Mutable r Ix1 e) => VectorS.Vector e -> Massiv.Vector r e+vecH2M hVec = fromVector' Seq (Sz $ VectorS.length hVec) hVec++-- | Converts a vector from the Massiv representation to the HMatrix representation.+{-# SCC vecM2H #-}+vecM2H :: (Manifest r Ix1 e, Element e) => Massiv.Vector r e -> LA.Vector e+vecM2H = Massiv.toVector++-- | Converts a matrix from the HMatrix representation to the Massiv representation.+{-# SCC matH2M #-}+matH2M :: (Mutable r Ix1 e, Element e) => LA.Matrix e -> Massiv.Matrix r e+matH2M hMat = Massiv.resize' (Sz $ nRows :. nCols) . vecH2M . LA.flatten $ hMat+  where+    nRows = LA.rows hMat+    nCols = LA.cols hMat++-- | Converts a matrix from Massiv to HMatrix representation.+{-# SCC matM2H #-}+matM2H :: (Manifest r Ix1 e, Element e, Resize r Ix2, Load r Ix2 e) => Massiv.Matrix r e -> LA.Matrix e+matM2H mMat = LA.reshape nCols . vecM2H . Massiv.flatten $ mMat+  where+    Sz (_nRows :. nCols) = Massiv.size mMat++-- | Magnitude of a vector (length).+magnitude :: (Massiv.Numeric r e, Source r Ix1 e, Floating e) => Massiv.Vector r e -> e+magnitude v = sqrt $ v !.! v++-- | Normalise a vector.+normalise :: (Massiv.Numeric r e, Source r Ix1 e, Floating e) => Massiv.Vector r e -> Massiv.Vector r e+normalise v = v .* (1 / magnitude v)++-- | Angle between two vectors.+angle :: (Massiv.Numeric r e, Source r Ix1 e, Floating e) => Massiv.Vector r e -> Massiv.Vector r e -> e+angle a b = acos $ a !.! b / (magnitude a * magnitude b)++-- | Find the minimal distance in a distance matrix, which is not the main diagonal.+{-# SCC minDistAt #-}+minDistAt ::+  ( Manifest r Ix2 e,+    MonadThrow m,+    Ord e+  ) =>+  Massiv.Matrix r e ->+  m (e, Ix2)+minDistAt arr+  | isEmpty arr = throwM $ SizeEmptyException (Massiv.size arr)+  | otherwise = return . unsafePerformIO $ ifoldlP minFold start chFold start arr+  where+    ix0 = pureIndex 0+    e0 = arr Massiv.! ix0+    start = (e0, ix0)+    minFold acc@(eA, _) ix@(m :. n) e = if e < eA && m > n then (e, ix) else acc+    chFold acc@(eA, _) ch@(e, _) = if e < eA then ch else acc++-- | Find the minimal element of a vector, which is at a larger than the supplied index.+minDistAtVec ::+  ( Manifest r Ix1 e,+    MonadThrow m,+    Ord e+  ) =>+  Ix1 ->+  Massiv.Vector r e ->+  m (e, Ix1)+minDistAtVec ixStart vec+  | isEmpty vec = throwM $ SizeEmptyException (Massiv.size vec)+  | ixStart >= nElems = throwM $ IndexOutOfBoundsException (Sz nElems) ixStart+  | otherwise = do+    let (minE, minIx) = unsafePerformIO $ ifoldlP minFold startAcc chFold startAcc searchVec+    return (minE, minIx + ixStart + 1)+  where+    Sz nElems = Massiv.size vec+    searchVec = Massiv.drop (Sz $ ixStart + 1) vec+    ix0 = 0+    e0 = searchVec Massiv.! ix0+    startAcc = (e0, ix0)+    minFold acc@(eA, _) ix e = if e < eA then (e, ix) else acc+    chFold acc ch = min acc ch++-- | Like 'Massiv.minimumM' but also returns the index of the minimal element.+iMinimumM ::+  ( Manifest r ix a,+    MonadThrow m,+    Ord a+  ) =>+  Array r ix a ->+  m (a, ix)+iMinimumM arr+  | isEmpty arr = throwM $ SizeEmptyException (Massiv.size arr)+  | otherwise = return . unsafePerformIO $ ifoldlP minFold start chFold start arr+  where+    ix0 = pureIndex 0+    e0 = arr Massiv.! ix0+    start = (e0, ix0)++    minFold acc@(eA, _) ix e = if e < eA then (e, ix) else acc+    chFold acc@(eA, _) ch@(e, _) = if e < eA then ch else acc++-- | Quickly print a matrix with numerical values+printMat :: (Source r Ix2 e, Real e) => Massiv.Matrix r e -> Massiv.Matrix D Text+printMat mat = Massiv.map (sformat (left 4 ' ' %. fixed 2)) mat++----------------------------------------------------------------------------------------------------+-- Binary Trees.++-- | A binary tree.+data BinTree e = Leaf e | Node e (BinTree e) (BinTree e)+  deriving (Eq, Show, Generic)++instance (FromJSON e) => FromJSON (BinTree e)++instance (ToJSON e) => ToJSON (BinTree e)++instance Functor BinTree where+  fmap f (Leaf a) = Leaf (f a)+  fmap f (Node a l r) = Node (f a) (fmap f l) (fmap f r)++-- | Look at the root of a binary tree.+root :: BinTree e -> e+root (Leaf e) = e+root (Node e _ _) = e++-- | Steps down each branch of a tree until some criterion is satisfied or the end of the branch is+-- reached. Each end of the branch is added to a result.+takeBranchesWhile :: (a -> Bool) -> BinTree a -> Massiv.Vector DL a+takeBranchesWhile chk tree = go tree (Massiv.empty @DL)+  where+    go (Leaf v) acc = if chk v then acc `snoc` v else acc+    go (Node v l r) acc =+      let vAcc = if chk v then acc `snoc` v else acc+          lAcc = go l vAcc+          rAcc = go r lAcc+       in if chk v then rAcc else vAcc++-- | Takes the first value in each branch, that does not fullfill the criterion anymore and adds it+-- to the result. Terminal leafes of the branches are always taken.+takeLeafyBranchesWhile :: (a -> Bool) -> BinTree a -> Massiv.Vector DL a+takeLeafyBranchesWhile chk tree = go tree (Massiv.empty @DL)+  where+    go (Leaf v) acc = acc `snoc` v+    go (Node v l r) acc =+      let vAcc = if chk v then acc else acc `snoc` v+          lAcc = go l vAcc+          rAcc = go r lAcc+       in if chk v then rAcc else vAcc

@@ -0,0 +1,947 @@+-- |+-- Module      : ConClusion.Numeric.Statistics+-- Description : Statistical Functions+-- Copyright   : Phillip Seeber, 2021+-- License     : AGPL-3+-- Maintainer  : phillip.seeber@googlemail.com+-- Stability   : experimental+-- Portability : POSIX, Windows+module ConClusion.Numeric.Statistics+  ( -- * PCA+    PCA (..),+    pca,++    -- * Variance+    normalise,+    meanDeviation,+    covariance,++    -- * Distance Metrics+    DistFn,+    lpNorm,+    manhattan,+    euclidean,+    mahalanobis,++    -- * Cluster Algorithms+    Clusters,++    -- ** DBScan+    DistanceInvalidException (..),+    dbscan,++    -- ** Hierarchical Cluster Analysis+    Dendrogram,+    JoinStrat (..),+    hca,+    cutDendroAt,+  )+where++import ConClusion.Numeric.Data hiding (normalise)+import Data.Aeson hiding (Array)+import Data.Complex+import qualified Data.IntSet as IntSet+import Data.Massiv.Array as Massiv+import Data.Massiv.Array.Unsafe as Massiv+import qualified Data.PSQueue as PQ+import qualified Numeric.LinearAlgebra as LA+import RIO hiding (Vector)+import System.IO.Unsafe (unsafePerformIO)++----------------------------------------------------------------------------------------------------+-- Others/Helpers++-- | Solves eigenvalue problem of a square matrix and obtains its eigenvalues and eigenvectors.+{-# SCC eig #-}+eig ::+  ( Mutable r1 Ix1 (Complex Double),+    Mutable r2 Ix1 (Complex Double),+    LA.Field e,+    Manifest r3 Ix1 e,+    Resize r3 Ix2,+    Load r3 Ix2 e,+    MonadThrow m+  ) =>+  Matrix r3 e ->+  m (Vector r1 (Complex Double), Matrix r2 (Complex Double))+eig covM+  | m /= n = throwM $ IndexException "eigenvalue problems can only be solved for square matrix"+  | otherwise = return . bimap vecH2M matH2M . LA.eig $ cov+  where+    Sz (m :. n) = size covM+    cov = matM2H covM++-- | Sort eigenvalues and eigenvectors by magnitude of the eigenvalues in descending order (largest+-- eigenvalues first). Eigenvectors are the columns of the input matrix.+{-# SCC eigSort #-}+eigSort ::+  ( Load r2 Ix2 e,+    MonadThrow m,+    Source r1 Ix1 e,+    Source r2 Ix2 e,+    Mutable r1 Ix1 e,+    Mutable r2 Ix2 e,+    Unbox e,+    Ord e+  ) =>+  (Vector r1 e, Matrix r2 e) ->+  m (Vector r1 e, Matrix r2 e)+eigSort (vec, mat)+  | m /= n = throwM $ IndexException "matrix of the eigenvectors is not a square matrix"+  | n /= n' = throwM $ IndexException "different number of eigenvalues and eigenvectors"+  | otherwise = do+    let ixedEigenvalues = Massiv.zip vec ixVec+        (eigenValSortAsc, ixSort) = (\a -> (get fst a, get snd a)) . quicksort . compute @U $ ixedEigenvalues+        eigenVecSortAsc = backpermute' (Sz $ m :. n) (\(r :. c) -> r :. (ixSort ! c)) mat+        eigenValSort = reverse' (Dim 1) eigenValSortAsc+        eigenVecSort = reverse' (Dim 1) eigenVecSortAsc+    return (compute eigenValSort, compute eigenVecSort)+  where+    Sz (m :. n) = size mat+    Sz n' = size vec+    ixVec = makeArrayLinear @D Seq (Sz n') id+    get acc = compute @U . Massiv.map acc++----------------------------------------------------------------------------------------------------+-- Principal Component Analysis++data PCA = PCA+  { -- | Original feature matrix.+    x :: Matrix U Double,+    -- | Feature matrix in mean deviation form.+    x' :: Matrix U Double,+    -- | Transformed data.+    y :: Matrix U Double,+    -- | Transformation matrix to transform feature matrix into PCA result matrix.+    a :: Matrix U Double,+    -- | Mean squared error introduced by PCA.+    mse :: Double,+    -- | Percentage of the behaviour captured in the remaining dimensions.+    remaining :: Double,+    -- | All eigenvalues from the diagonalisation of the covariance matrix.+    allEigenValues :: Vector U Double,+    -- | Eigenvalues that were kept for PCA.+    pcaEigenValues :: Vector U Double,+    -- | All eigenvectors from the diagonalisation of the covariance matrix.+    allEigenVecs :: Matrix U Double,+    -- | Eigenvectors that were kept for PCA.+    pcaEigenVecs :: Matrix U Double+  }++-- | Transform the input values with a transformation matrix \(\mathbf{A}\), where \(\mathbf{A}\) is+-- constructed from the eigenvectors associated to the largest eigenvalues.+{-# SCC transformToPCABasis #-}+transformToPCABasis ::+  ( Source (R r) Ix2 e,+    Extract r Ix2 e,+    Mutable r Ix2 e,+    Numeric r e,+    MonadThrow m+  ) =>+  -- | Number of dimensions to keep from PCA.+  Int ->+  -- | Matrix of the eigenvectors, sorted descendingly by eigenvalues, where the eigenvectors are+  -- the columns of the matrix.+  Matrix r e ->+  -- | Feature matrix in mean deviation form.+  Matrix r e ->+  -- | Input data transformed by PCA to lower dimensions, and the transformation matrix+  -- \(\mathbf{A}\).+  m (Matrix r e, Matrix r e)+transformToPCABasis nDim eigenVecMat featureMat+  | mE /= nE = throwM $ IndexException "the matrix of the eigenvectors must be a quadratic matrix"+  | nDim <= 0 = throwM $ IndexException "the number of dimensions of the PCA is smaller than or zero"+  | nDim >= nE = throwM $ IndexException "more than the possible amount of dimensions has been selected"+  | mE /= mF = throwM $ IndexException "eigenvector matrix and feature matrix have mismatching dimensions"+  | otherwise = do+    matA <- compute . transpose <$> extractM (0 :. 0) (Sz $ mE :. nDim) eigenVecMat+    pcaData <- matA .><. featureMat+    return (pcaData, matA)+  where+    Sz (mE :. nE) = size eigenVecMat+    Sz (mF :. _nF) = size featureMat++-- | Performs a PCA on the feature matrix \(\mathbf{X}\) by solving the eigenproblem of the+-- covariance matrix. The function takes the feature matrix directly and perfoms the conversion+-- to mean deviation form, the calculation of the covariance matrix and the eigenvalue problem+-- automatically.+{-# SCC pca #-}+pca ::+  ( Numeric r Double,+    Mutable r Ix2 Double,+    Manifest r Ix1 Double,+    Source (R r) Ix2 Double,+    Extract r Ix2 Double,+    MonadThrow m+  ) =>+  -- | Dimensionalty after PCA transformation.+  Int ->+  -- | \(m \times n\) Feaute matrix \(\mathbf{X}\), with \(m\) different measurements (rows) in+  -- \(n\) different trials (columns).+  Matrix r Double ->+  m PCA+pca dim x = do+  -- Calculate the mean deviation form of the feature matrix and the covariance matrix from it.+  let x' = normalise . meanDeviation $ x+      cov = covariance x'++  -- Obtain eigenvalues and eigenvectors of the covariance matrix and sort them.+  (eigValsC :: Vector U (Complex Double), eigVecsC :: Matrix U (Complex Double)) <- eig cov+  let eigValsR = compute @U . Massiv.map realPart $ eigValsC+      eigVecsR = compute . Massiv.map realPart $ eigVecsC+  (eValS, eVecS) <- eigSort (eigValsR, eigVecsR)++  -- Use the subset of the eigenvectors with the largest eigenvalues to transform the features in+  -- mean deviation form into the result matrix Y.+  (pcaData, matA) <- transformToPCABasis dim eVecS x'++  -- Reconstuct the original data from lower dimensions and calculate the mean squared deviation.+  reconstructX <- (compute . transpose $ matA) .><. pcaData+  mse <- (/ fromIntegral n) . Massiv.sum . Massiv.map (** 2) <$> (x' .-. reconstructX)++  -- For output give the eigenvalues and eigenvectors that were kept.+  pcaEigenValues <- extractM 0 (Sz dim) eValS+  pcaEigenVecs <- extractM (0 :. 0) (Sz $ m :. dim) eVecS++  -- Calculate the amount of behaviour that could be kept.+  let remaining = (Massiv.sum pcaEigenValues / Massiv.sum eValS) * 100++  return+    PCA+      { x = compute x,+        x' = compute x',+        y = compute pcaData,+        a = compute matA,+        mse = mse,+        remaining = remaining,+        allEigenValues = eValS,+        pcaEigenValues = compute pcaEigenValues,+        allEigenVecs = compute eVecS,+        pcaEigenVecs = compute pcaEigenVecs+      }+  where+    Sz (m :. n) = size x++----------------------------------------------------------------------------------------------------+-- Variance++-- | Subtract the mean value of all columns from the feature matrix. Brings the feature matrix to+-- mean deviation form.+{-# SCC meanDeviation #-}+meanDeviation ::+  ( Source r Ix2 e,+    Fractional e,+    Unbox e,+    Numeric r e,+    Mutable r Ix2 e+  ) =>+  Matrix r e ->+  Matrix r e+meanDeviation mat = mat !-! compute meanMat+  where+    Sz (_ :. n) = Massiv.size mat+    featueMean = Massiv.foldlInner (+) 0 mat .* (1 / fromIntegral n)+    meanMat = expandInner (Sz n) const . compute @U $ featueMean++-- | Obtains the covariance matrix \(\mathbf{C_X}\) from the feature matrix \(\mathbf{X}\).+-- \[+--   \mathbf{C_X} \equiv \frac{1}{n - 1} \mathbf{X} \mathbf{X}^T+-- \]+-- where \(n\) is the number of columns in the matrix.+--+-- The feature matrix should be in mean deviation form, see 'meanDeviation'.+{-# SCC covariance #-}+covariance :: (Numeric r e, Mutable r Ix2 e, Fractional e) => Matrix r e -> Matrix r e+covariance x = (1 / (fromIntegral n - 1)) *. (x !><! (compute . transpose $ x))+  where+    Sz (_ :. n) = size x++-- | Normalise each value so that the maximum absolute value in each row becomes one.+normalise ::+  ( Ord e,+    Unbox e,+    Numeric r e,+    Fractional e,+    Source r Ix2 e,+    Mutable r Ix2 e+  ) =>+  Array r Ix2 e ->+  Array r Ix2 e+normalise mat =+  let absMat = Massiv.map abs mat+      maxPerRow = compute @U . foldlInner max 0 $ absMat+      divMat = compute . Massiv.map (1 /) . expandInner @Ix2 (Sz n) const $ maxPerRow+   in divMat !*! mat+  where+    Sz (_ :. n) = size mat++----------------------------------------------------------------------------------------------------+-- Distance Measures++-- | Distance matrix generator functions.+type DistFn r e = Matrix r e -> Matrix r e++-- | Builds the distance measures in a permutation matrix/distance matrix.+buildDistMat ::+  (Mutable r Ix2 e) =>+  -- | Zip function to combine the elements of vectors \(\mathbf{a}\) \(\mathbf{b}\). Usually @(-)@.+  -- \( f(\mathbf{a}_i, \mathbf{b}_i) = \mathbf{c} \)+  (e -> e -> a) ->+  -- | Fold the vector \(\mathbf{c}\) elementwise to a distance \(d\).+  (a -> a -> a) ->+  -- | Accumulator of the fold function.+  a ->+  -- | \(m \times n\) matrix, with \(n\) \(m\)-dimensional points (column vectors of the matrix).+  Matrix r e ->+  -- | Resulting distance matrix.+  Matrix D a+buildDistMat zipFn foldFn acc mat =+  let a = transposeOuter . expandOuter @Ix3 (Sz n) const $ mat+      b = transposeInner a+      ab = Massiv.zipWith zipFn a b+      d = foldlInner foldFn acc ab+   in d+  where+    Sz (_ :. n) = size mat++-- | The \(L_p\) norm between two vectors. Generalisation of Manhattan and Euclidean distances.+-- \[+--   d(\mathbf{a}, \mathbf{b}) = \left( \sum \limits_{i=1}^n \lvert \mathbf{a}_i - \mathbf{b}_i \rvert ^p \right) ^ \frac{1}{p}+-- \]+{-# SCC lpNorm #-}+lpNorm :: (Mutable r Ix2 e, Floating e) => Int -> DistFn r e+lpNorm p = compute . buildDistMat zipFn foldFn 0+  where+    zipFn a b = (^ p) . abs $ a - b+    foldFn a b = (** (1 / fromIntegral p)) $ a + b++-- | The Manhattan distance between two vectors. Specialisation of the \(L_p\) norm for \(p = 1\).+-- \[+--   d(\mathbf{a}, \mathbf{b}) = \sum \limits_{i=1}^n \lvert \mathbf{a}_i - \mathbf{b}_i \rvert+-- \]+{-# SCC manhattan #-}+manhattan :: (Mutable r Ix2 e, Floating e) => DistFn r e+manhattan = lpNorm 1++-- | The Euclidean distance between two vectors. Specialisation of the \(L_p\) norm for \(p = 2\).+-- \[+--   d(\mathbf{a}, \mathbf{b}) = \sqrt{\sum \limits_{i=1}^n (\mathbf{a}_i - \mathbf{b}_i)^2}+-- \]+{-# SCC euclidean #-}+euclidean :: (Mutable r Ix2 e, Floating e) => DistFn r e+euclidean = lpNorm 2++-- | Mahalanobis distance between points. Suitable for non correlated axes.+-- \[+--   d(\mathbf{a}, \mathbf{b}) = \sqrt{(\mathbf{a} - \mathbf{b})^T \mathbf{S}^{-1} (\mathbf{a} - \mathbf{b})}+-- \]+-- where \(\mathbf{S}\) is the covariance matrix.+{-# SCC mahalanobis #-}+mahalanobis :: (Unbox e, Numeric r e, Mutable r Ix2 e, Mutable r Ix1 e, LA.Field e) => DistFn r e+mahalanobis points =+  let a = transposeOuter . expandOuter @Ix3 (Sz n) const $ points+      b = transposeInner a+      abDiff = compute @U $ a !-! b+      ixArray = makeArray @U @Ix2 @Ix2 Par (Sz $ n :. n) id+      distMat =+        Massiv.map+          ( \(x :. y) ->+              let ab = compute @U $ abDiff !> x !> y+               in ab ><! covInv !.! ab+          )+          ixArray+   in compute . Massiv.map sqrt $ distMat+  where+    Sz (_ :. n) = size points+    cov = covariance . meanDeviation $ points+    covInv = matH2M . LA.inv . matM2H $ cov++----------------------------------------------------------------------------------------------------+-- DBScan++-- | Exception for invalid search distances.+newtype DistanceInvalidException e = DistanceInvalidException e deriving (Show, Eq)++instance (Typeable e, Show e) => Exception (DistanceInvalidException e)++-- | Representation of clusters.+type Clusters = Vector B IntSet++-- | DBScan algorithm.+{-# SCC dbscan #-}+dbscan ::+  ( MonadThrow m,+    Ord e,+    Num e,+    Typeable e,+    Show e,+    Source r Ix2 e+  ) =>+  -- | Distance measure to build the distance matrix of all points.+  DistFn r e ->+  -- | Minimal number of members in a cluster.+  Int ->+  -- | Search radius \(\epsilon\)+  e ->+  -- | \(n\) \(m\)-dimensional data points as column vectors of a \(m \times n\) matrix.+  Matrix r e ->+  -- | Resulting clusters.+  m Clusters+dbscan distFn nPoints epsilon points+  | isEmpty points = throwM $ SizeEmptyException (Sz 0 :: Sz1)+  | nPoints < 1 = throwM $ SizeNegativeException (Sz nPoints)+  | epsilon <= 0 = throwM $ DistanceInvalidException epsilon+  | otherwise =+    let pointNeighbours = ifoldlInner collectNeighbours mempty distMat+        allClusters = joinOverlapping . compute @B $ pointNeighbours+        largeClusters = sfilter (\s -> IntSet.size s >= nPoints) allClusters+     in return $ compute largeClusters+  where+    -- The distance matrix in the measure chosen by the distance function.+    distMat = distFn points++    -- Function to collect the neighbours of a point within the search radius epsilon.+    {-# SCC collectNeighbours #-}+    collectNeighbours (_ :. n) acc d = if d <= epsilon then IntSet.insert n acc else acc++    -- Construct the overlap matrix of all clusters.+    compareSets :: (IntSet -> IntSet -> Bool) -> Vector B IntSet -> Matrix D Bool+    compareSets fn clVec =+      let a = expandOuter @Ix2 sz const clVec+          b = transpose a+          compareMat = Massiv.zipWith fn a b+       in compareMat+      where+        sz = size clVec++    -- Overlap matrix. Checks if two sets have any overlap. Sets do overlap with themself.+    overlap :: Vector B IntSet -> Matrix D Bool+    overlap = compareSets (\a b -> not $ IntSet.disjoint a b)++    -- Check if any set overlaps wiht **any** other set.+    anyOtherOverlap :: Vector B IntSet -> Bool+    anyOtherOverlap = Massiv.any (== True) . imap (\(m :. n) v -> if m == n then False else v) . overlap++    -- Check if two sets are identical. Sets are identical to themself.+    same :: Vector B IntSet -> Matrix D Bool+    same = compareSets (==)++    -- Join all overlapping clusters recursively.+    {-# SCC joinOverlapping #-}+    joinOverlapping :: Vector B IntSet -> Vector B IntSet+    joinOverlapping clVec =+      let -- The overlap matrix of the clusters.+          ovlpMat = compute @U . overlap $ clVec+          anyOvlp = anyOtherOverlap clVec++          -- Join all sets that have overlap but keep them redundantly (no reduction of the amount+          -- of clusters).+          joined =+            ifoldlInner+              (\(_ :. n) acc ovlp -> if ovlp then (clVec ! n) <> acc else acc)+              mempty+              ovlpMat++          -- Find all sets at different indices that are the same. This is an upper triangular+          -- matrix with the main diagonal being False.+          sameMat =+            compute @U+              . imap (\(m :. n) v -> if m >= n then False else v)+              . same+              . compute @B+              $ joined++          -- Remove all sets that are redundant. Redundancy is checked by two criteria:+          --   1. Is this cluster the same set of points as any other cluster? If yes, it is+          --      redundant.+          --   2. Is this cluster isolated it is not redundant.+          nonRed =+            compute @B+              . sifilter+                ( \ix _ ->+                    let sameAsAnyOther = Massiv.any (== True) $ sameMat !> ix+                     in not sameAsAnyOther+                )+              $ joined+       in if anyOvlp then joinOverlapping nonRed else clVec++----------------------------------------------------------------------------------------------------+-- Hierarchical Cluster Analysis++-- | Nodes of a dendrogram.+data DendroNode e = DendroNode+  { distance :: e,+    cluster :: IntSet+  }+  deriving (Eq, Show, Generic)++instance (FromJSON e) => FromJSON (DendroNode e)++instance (ToJSON e) => ToJSON (DendroNode e)++-- | A dendrogram as a binary tree.+newtype Dendrogram e = Dendrogram {unDendro :: BinTree (DendroNode e)}+  deriving (Show, Eq, Generic)++instance ToJSON e => ToJSON (Dendrogram e)++instance FromJSON e => FromJSON (Dendrogram e)++-- | An accumulator to finally build a dendrogram by a bottom-up algorithm. Not to be exposed in the+-- API.+type DendroAcc e = Vector B (Dendrogram e)++-- | Mutable version of the dendrogram accumulator.+type DendroAccM m e = MArray (PrimState m) B Ix1 (Dendrogram e)++-- | Cut a 'Dendrogram' at a given distance and obtain all clusters from it.+cutDendroAt :: Ord e => Dendrogram e -> e -> Clusters+cutDendroAt dendro dist =+  let nodes = takeLeafyBranchesWhile (\DendroNode {distance} -> distance >= dist) . unDendro $ dendro+      clusters = compute @B . Massiv.map cluster . compute @B $ nodes+   in clusters++-- | A strategy/distance measure for clusters.+data JoinStrat e+  = SingleLinkage+  | CompleteLinkage+  | Median+  | UPGMA+  | WPGMA+  | Centroid+  | Ward+  | LWFB e+  | LW e e e e+  deriving (Eq, Show)++-- | Lance Williams formula to update distances.+{-# SCC lanceWilliams #-}+lanceWilliams ::+  Fractional e =>+  -- | How to calculate distance between clusters of points.+  JoinStrat e ->+  -- | Number of points in cluster \(A\).+  Int ->+  -- | Number of points in cluster \(B\)+  Int ->+  -- | Number of points in cluster \(C\)+  Int ->+  -- | \(d(A, B)\)+  e ->+  -- | \(d(A, C)\)+  e ->+  -- | \(d(B, C)\)+  e ->+  -- | Updated distance \(D \(A \cup B, C\)+  e+lanceWilliams js nA nB nC dAB dAC dBC = alpha1 * dAC + alpha2 * dBC + beta * dAB + gamma * abs (dAC - dBC)+  where+    nA' = fromIntegral nA+    nB' = fromIntegral nB+    nC' = fromIntegral nC+    (alpha1, alpha2, beta, gamma) = case js of+      SingleLinkage -> (1 / 2, 1 / 2, 0, - 1 / 2)+      CompleteLinkage -> (1 / 2, 1 / 2, 0, 1 / 2)+      Median -> (1 / 2, 1 / 2, - 1 / 4, 0)+      UPGMA -> (nA' / (nA' + nB'), nB' / (nA' + nB'), 0, 0)+      WPGMA -> (1 / 2, 1 / 2, 0, 0)+      Centroid -> (nA' / (nA' + nB'), nB' / (nA' + nB'), - (nA' * nB') / ((nA' + nB') ^ (2 :: Int)), 0)+      Ward -> ((nA' + nC') / (nA' + nB' + nC'), (nA' + nC') / (nA' + nB' + nC'), - (nA' + nC') / (nA' + nB' + nC'), 0)+      LWFB b -> ((1 - b) / 2, (1 - b) / 2, b, 0)+      LW a b c d -> (a, b, c, d)++----------------------------------------------------------------------------------------------------+-- Müllner Generic Hierarchical Clustering++-- | A neighbourlist. At index @i@ of the vector it contains a tuple with the minimal distance of+-- this cluster to any other cluster and the index of the other cluster.+type Neighbourlist r e = Vector r (e, Ix1)++-- | A distance matrix.+type DistanceMatrix r e = Matrix r e++-- | Performance improved hierarchical clustering algorithm. @GENERIC_LINKAGE@ from figure 3,+-- <https://arxiv.org/pdf/1109.2378.pdf>.+{-# SCC hca #-}+hca ::+  ( MonadThrow m,+    Mutable r Ix1 e,+    Mutable r Ix2 e,+    Mutable r Ix1 (e, Ix1),+    Manifest (R r) Ix1 e,+    OuterSlice r Ix2 e,+    Ord e,+    Unbox e,+    Fractional e+  ) =>+  DistFn r e ->+  JoinStrat e ->+  Matrix r e ->+  m (Dendrogram e)+hca distFn joinStrat points+  | Massiv.isEmpty points = throw $ SizeEmptyException (Sz nPoints)+  | otherwise = do+    let -- The distance matrix from the points.+        distMat = distFn points++    -- Initial vector of nearest neighbour to each point.+    nNghbr <- nearestNeighbours distMat++    let -- Initial priority queue of points. Has the minimum distance of all points.+        pq = PQ.fromList . Massiv.toList . Massiv.imap (\k (d, _) -> k PQ.:-> d) $ nNghbr+        -- Set of points not joined yet. Initially all points.+        s = IntSet.fromDistinctAscList [0 .. nPoints - 1]+        -- Initial dendrogram accumulator. The vector of all points as their own cluster.+        dendroAcc =+          makeArray @B @Ix1+            Par+            (Sz nPoints)+            (\p -> Dendrogram . Leaf $ DendroNode {distance = 0, cluster = IntSet.singleton p})++    distMatM <- return . unsafePerformIO . thaw $ distMat+    nNghbrM <- return . unsafePerformIO . thaw $ nNghbr+    dendroAccM <- return . unsafePerformIO . thaw $ dendroAcc++    return . unsafePerformIO $ agglomerate joinStrat distMatM nNghbrM pq s dendroAccM+  where+    Sz (_mFeatures :. nPoints) = size points++-- | Agglomerative clustering by the improved generic linkage algorithm. This is the main loop+-- recursion L 10-43.+{-# SCC agglomerate #-}+agglomerate ::+  ( MonadThrow m,+    PrimMonad m,+    MonadUnliftIO m,+    PrimState m ~ RealWorld,+    Mutable r Ix2 e,+    OuterSlice r Ix2 e,+    Manifest (R r) Ix1 e,+    Mutable r Ix1 (e, Ix1),+    Fractional e,+    Ord e+  ) =>+  -- | Join strategy for clusters and therefore how to calculate cluster-cluster distances.+  JoinStrat e ->+  -- | Distance matrix.+  MArray (PrimState m) r Ix2 e ->+  -- | List of nearest neighbours for each point.+  MArray (PrimState m) r Ix1 (e, Ix1) ->+  -- | Priority queue with the distances as priorities and the cluster index as keys.+  PQ.PSQ Ix1 e ->+  -- | A set \(S\), that keeps track which clusters have already been joined.+  IntSet ->+  -- | Accumulator of the dendrogram. Should collapse to a singleton vector.+  DendroAccM m e ->+  -- | The final dendrogram, after all clusters have been joined.+  m (Dendrogram e)+agglomerate joinStrat distMat nNghbr pq s dendroAcc+  | IntSet.null s = throw $ IndexException "No clusters left. This must never happen."+  | otherwise = do+    -- Obtain candidates for the two clusters to join and the minimal distance in the priority queue.+    candidates <- getJoinCandidates nNghbr pq++    -- If the distance between a b is not the minimal distance that the priority queue has found, the+    -- neighbour list must be wrong and recalculated.+    (a, b, delta, nNghbrU1, pqU1) <- recalculateNghbr candidates s distMat nNghbr pq++    -- Remove the minimal element from the priority queue and join clusters a and b. The cluster+    -- accumulator is reduced in its size: a is removed and b is updated with the joined cluster.+    (newS, pqU2, newAcc) <- joinClusters a b delta s pqU1 dendroAcc++    -- Update the distance matrix in the row and column of b but not at (b,b) and not at (a,b) and+    -- (b,a).+    newDistMat <- updateDistMat joinStrat a b newS distMat newAcc++    -- Redirect neighbours to b, if they previously pointed to a.+    nNghbrU2 <- redirectNeighbours a b newS newDistMat nNghbrU1++    -- Update the neighbourlist and priority queue with the new distances to b.+    (newNNghbr, newPQ) <- updateBNeighbour b s newDistMat nNghbrU2 pqU2++    -- If the problem has been reduced to a single cluster the algorithm is done and the final+    -- dendrogram can be obtained from the accumulator at index b. Otherwise join further.+    if IntSet.size newS == 1+      then newAcc `readM` b+      else agglomerate joinStrat newDistMat newNNghbr newPQ newS newAcc++-- | Obtain candidates for the clusters to join by looking at the minimal distance in the priority+-- queue and the neighbourlist. L 11-13+{-# SCC getJoinCandidates #-}+getJoinCandidates ::+  ( MonadThrow m,+    PrimMonad m,+    Mutable r Ix1 (e, Ix1),+    Ord e+  ) =>+  MArray (PrimState m) r Ix1 (e, Ix1) ->+  PQ.PSQ Ix1 e ->+  m (Ix1, Ix1, e)+getJoinCandidates nNghbr pq = do+  (a PQ.:-> d) <- case PQ.findMin pq of+    Nothing -> throwM $ IndexException "Empty priority queue"+    Just v -> return v+  (_, b) <- nNghbr `readM` a+  return (a, b, d)++-- | If the minimal distance @d@ found is not the distance between @a@ and @b@ recalculate the+-- neighbour list, update the priority queue and obtain a new set of a,b and a distance between them.+-- L 14-20.+{-# SCC recalculateNghbr #-}+recalculateNghbr ::+  ( MonadThrow m,+    PrimMonad m,+    MonadUnliftIO m,+    PrimState m ~ RealWorld,+    OuterSlice r Ix2 e,+    Manifest (R r) Ix1 e,+    Mutable r Ix1 (e, Ix1),+    Mutable r Ix2 e,+    Ord e+  ) =>+  (Ix1, Ix1, e) ->+  IntSet ->+  MArray (PrimState m) r Ix2 e ->+  MArray (PrimState m) r Ix1 (e, Ix1) ->+  PQ.PSQ Ix1 e ->+  m (Ix1, Ix1, e, MArray (PrimState m) r Ix1 (e, Ix1), PQ.PSQ Ix1 e)+recalculateNghbr (cA, cB, d) s distMat nNghbr pq = do+  dAB <- distMat `readM` (cA :. cB)+  if d == dAB+    then return (cA, cB, d, nNghbr, pq)+    else do+      -- Recalculate the nearest neighbours just on index cA. Consider only clusters, that were not+      -- merged yet.+      dmRowA <- searchRow cA s distMat >>= unsafeFreeze Par+      newNeighbourA@(minDistA, _) <- minimumM dmRowA+      writeM nNghbr cA newNeighbourA++      -- Update the priority queue at key cA with the new distance.+      let newPQ = PQ.adjust (const minDistA) cA pq++      -- Determine new a, b and d from the updated neighbour list and priority queue.+      (a PQ.:-> newD) <- case PQ.findMin newPQ of+        Nothing -> throwM $ IndexException "Empty priority queue"+        Just v -> return v+      (_, b) <- nNghbr `readM` a+      recalculateNghbr (a, b, newD) s distMat nNghbr newPQ++-- | Joins the selected clusters \(A\) and \(B\) and updates the dendrogram accumulator at index b.+-- A will not be removed so that the accumulator never shrinks.+-- L 21-24+{-# SCC joinClusters #-}+joinClusters ::+  ( MonadThrow m,+    PrimMonad m,+    Ord e+  ) =>+  Ix1 ->+  Ix1 ->+  e ->+  IntSet ->+  PQ.PSQ Ix1 e ->+  DendroAccM m e ->+  m (IntSet, PQ.PSQ Ix1 e, DendroAccM m e)+joinClusters a b d s pq acc = do+  clA <- acc `readM` a+  let newPQ = PQ.deleteMin pq+  modifyM_+    acc+    ( \clB ->+        return+          . Dendrogram+          $ Node+            ( DendroNode+                { distance = d,+                  cluster = (cluster . root . unDendro $ clA) <> (cluster . root . unDendro $ clB)+                }+            )+            (unDendro clA)+            (unDendro clB)+    )+    b+  let newS = IntSet.delete a s+  return (newS, newPQ, acc)++-- | Update the distance matrix with a Lance-Williams update in the rows and columns of cluster b.+-- L 25-27+{-# SCC updateDistMat #-}+updateDistMat ::+  ( MonadThrow m,+    PrimMonad m,+    MonadUnliftIO m,+    Mutable r Ix2 e,+    Fractional e+  ) =>+  JoinStrat e ->+  Ix1 ->+  Ix1 ->+  IntSet ->+  MArray (PrimState m) r Ix2 e ->+  DendroAccM m e ->+  m (MArray (PrimState m) r Ix2 e)+updateDistMat js a b s distMat dendroAcc+  | nDM /= nDM = throwM $ SizeMismatchException (Sz nDM) (Sz nCl)+  | mDM /= nDM = throwM $ SizeMismatchException (Sz mDM) (Sz nDM)+  | otherwise = do+    dAB <- distMat `readM` (a :. b)+    nA <- clSize a+    nB <- clSize b+    forIO_ ixV $ \ix -> do+      dAX <- distMat `readM` (a :. ix)+      nX <- clSize ix+      modifyM_ distMat (\dBX -> return $ lanceWilliams js nA nB nX dAB dAX dBX) (ix :. b)+      modifyM_ distMat (\dBX -> return $ lanceWilliams js nA nB nX dAB dAX dBX) (b :. ix)+    return distMat+  where+    Sz (mDM :. nDM) = msize distMat+    Sz nCl = msize dendroAcc+    ixV = Massiv.fromList @U Par . IntSet.toAscList . IntSet.delete b $ s+    clSize i = IntSet.size . cluster . root . unDendro <$> dendroAcc `readM` i++-- | Updates the neighbourlist. All elements with a smaller index than a, that had a as a nearest+-- neighbour are blindly redirected to the union of a and b, now at index b.+-- L 28-32+{-# SCC redirectNeighbours #-}+redirectNeighbours ::+  ( MonadThrow m,+    PrimMonad m,+    MonadUnliftIO m,+    Mutable r Ix1 (e, Ix1),+    Mutable r Ix2 e+  ) =>+  Ix1 ->+  Ix1 ->+  IntSet ->+  MArray (PrimState m) r Ix2 e ->+  MArray (PrimState m) r Ix1 (e, Ix1) ->+  m (MArray (PrimState m) r Ix1 (e, Ix1))+redirectNeighbours a b s distMat nNghbr = do+  forIO_ ixV $ \ix ->+    modifyM_+      nNghbr+      ( \old@(_, nghbrX) ->+          if nghbrX == a+            then distMat `readM` (ix :. b) >>= \dXB -> return (dXB, b)+            else return old+      )+      ix+  return nNghbr+  where+    ixV = compute @U . sfilter (< a) . Massiv.fromList @U Par . IntSet.toAscList $ s++-- | Updates the list of nearest neighbours for all combinations that might have changed by+-- recalculation with the joined cluster AB at index b.+-- L+{-# SCC updateWithNewBDists #-}+updateWithNewBDists ::+  ( MonadThrow m,+    MonadUnliftIO m,+    PrimMonad m,+    Mutable r Ix2 e,+    Mutable r Ix1 (e, Ix1),+    Ord e+  ) =>+  Ix1 ->+  IntSet ->+  MArray (PrimState m) r Ix2 e ->+  MArray (PrimState m) r Ix1 (e, Ix1) ->+  PQ.PSQ Ix1 e ->+  m (MArray (PrimState m) r Ix1 (e, Ix1), PQ.PSQ Ix1 e)+updateWithNewBDists b s distMat nNghbr pq = do+  pqT <- newTVarIO pq+  forIO_ ixV $ \ix -> do+    dBX <- distMat `readM` (ix :. b)+    currentPQ <- readTVarIO pqT+    minDistX <- case PQ.lookup ix currentPQ of+      Nothing -> throwM $ IndexException "Empty priority queue."+      Just v -> return v+    if dBX < minDistX+      then do+        writeM nNghbr ix (dBX, b)+        atomically . writeTVar pqT . PQ.adjust (const dBX) ix $ currentPQ+      else atomically . writeTVar pqT $ currentPQ++  newPQ <- readTVarIO pqT+  return (nNghbr, newPQ)+  where+    ixV = compute @U . Massiv.sfilter (< b) . Massiv.fromList @U Par . IntSet.toAscList $ s++--  | Updates the list of nearest neighbours and the priority queue at key b.+-- L 39-40+{-# SCC updateBNeighbour #-}+updateBNeighbour ::+  ( MonadThrow m,+    PrimMonad m,+    MonadUnliftIO m,+    Mutable r Ix1 (e, Ix1),+    Mutable r Ix2 e,+    Ord e+  ) =>+  Ix1 ->+  IntSet ->+  MArray (PrimState m) r Ix2 e ->+  MArray (PrimState m) r Ix1 (e, Ix1) ->+  PQ.PSQ Ix1 e ->+  m (MArray (PrimState m) r Ix1 (e, Ix1), PQ.PSQ Ix1 e)+updateBNeighbour b s distMat nNghbr pq =+  if b >= nNeighbours+    then return (nNghbr, pq)+    else do+      rowAB <- searchRow b s distMat >>= unsafeFreeze Par+      newNeighbourB@(distB, neighbourB) <- minimumM rowAB+      writeM nNghbr b newNeighbourB+      let newPQ = PQ.adjust (const distB) neighbourB pq+      return (nNghbr, newPQ)+  where+    Sz nNeighbours = msize nNghbr++-- | Find the nearest neighbour for each point from a distance matrix. For each point it stores the+-- minimum distance and the index of the other point, that is the nearest neighbour but at a higher+-- index.+{-# SCC nearestNeighbours #-}+nearestNeighbours ::+  ( MonadThrow m,+    Mutable r Ix1 e,+    Mutable r Ix1 (e, Ix1),+    OuterSlice r Ix2 e,+    Source (R r) Ix1 e,+    Ord e,+    Unbox e+  ) =>+  Matrix r e ->+  m (Vector r (e, Ix1))+nearestNeighbours distMat+  | m /= n = throwM $ IndexException "Distance matrix is not square"+  | m == 0 = throwM $ IndexException "Distance matrix is empty"+  | otherwise =+    let rows = compute @B . outerSlices $ distMat+        minDistIx =+          Massiv.imap (\i v -> unsafePerformIO . minDistAtVec i . compute @U $ v) . init $ rows+     in return . compute $ minDistIx+  where+    Sz (m :. n) = size distMat++-- Make a search row for distances. Takes row x from a distance matrix and zips them with their+-- column index. Then keeps only the valid elements of the row, that are still part of the available+-- points. A minimum or maximum search can be performed on the resulting vector and a valid pair of+-- distance and index can be obtained.+searchRow ::+  ( PrimMonad m,+    MonadThrow m,+    MonadUnliftIO m,+    Mutable r Ix2 e,+    Mutable r Ix1 (e, Ix1)+  ) =>+  Ix1 ->+  IntSet ->+  MArray (PrimState m) r Ix2 e ->+  m (MArray (PrimState m) r Ix1 (e, Ix1))+searchRow x s dm =+  makeMArray Par (size ixV) $ \ix -> do+    dmIx <- ixV !? ix+    (dm `readM` (x :. dmIx)) >>= \dist -> return (dist, dmIx)+  where+    ixV :: Vector U Ix1+    ixV = compute @U . sfilter (> x) . Massiv.fromList @U Par . IntSet.toAscList $ s