module Data.Clustering.Hierarchical
(-- * Dendrogram data type
Dendrogram(..)
,elements
,cutAt
-- * Linkage data type
,Linkage(..)
-- * Generic clustering function
,dendrogram
-- * Functions for specific linkages
,singleLinkage
,completeLinkage
,upgma
,fakeAverageLinkage
) where
import qualified Data.IntMap as IM
import Control.Applicative ((<$>), (<*>))
import Control.Monad.ST (runST)
import Data.Array (listArray, (!))
import Data.Foldable (Foldable (..))
import Data.Function (on)
import Data.Monoid (mappend)
import Data.Traversable (Traversable(..))
import Data.Clustering.Hierarchical.Internal.DistanceMatrix
-- | Data structure for storing hierarchical clusters. The
-- distance between clusters is stored on the branches.
-- Distances between leafs are the distances between the elements
-- on those leafs, while distances between branches are defined
-- by the linkage used (see 'Linkage').
data Dendrogram d a =
Leaf a
-- ^ The leaf contains the item @a@ itself.
| Branch d (Dendrogram d a) (Dendrogram d a)
-- ^ Each branch connects two clusters/dendrograms that are
-- @d@ distance apart.
deriving (Eq, Ord, Show)
-- | List of elements in a dendrogram.
elements :: Dendrogram d a -> [a]
elements = go []
where
go acc (Leaf x) = x : acc
go acc (Branch _ l r) = go (go acc r) l
-- | @dendro \`cutAt\` threshold@ cuts the dendrogram @dendro@ at
-- all branches which have distances strictly greater than
-- @threshold@.
--
-- For example, suppose we have
--
-- @
-- dendro = Branch 0.8
-- (Branch 0.5
-- (Branch 0.2
-- (Leaf \'A\')
-- (Leaf \'B\'))
-- (Leaf \'C\'))
-- (Leaf \'D\')
-- @
--
-- Then:
--
-- @
-- dendro \`cutAt\` 0.9 == dendro \`cutAt\` 0.8 == [dendro] -- no changes
-- dendro \`cutAt\` 0.7 == dendro \`cutAt\` 0.5 == [Branch 0.5 (Branch 0.2 (Leaf \'A\') (Leaf \'B\')) (Leaf \'C\'), Leaf \'D\']
-- dendro \`cutAt\` 0.4 == dendro \`cutAt\` 0.2 == [Branch 0.2 (Leaf \'A\') (Leaf \'B\'), Leaf \'C\', Leaf \'D\']
-- dendro \`cutAt\` 0.1 == [Leaf \'A\', Leaf \'B\', Leaf \'C\', Leaf \'D\'] -- no branches at all
-- @
cutAt :: Ord d => Dendrogram d a -> d -> [Dendrogram d a]
cutAt dendro threshold = go [] dendro
where
go acc x@(Leaf _) = x : acc
go acc x@(Branch d l r) | d <= threshold = x : acc
| otherwise = go (go acc r) l -- cut!
-- | Does not recalculate the distances!
instance Functor (Dendrogram d) where
fmap f (Leaf d) = Leaf (f d)
fmap f (Branch s c1 c2) = Branch s (fmap f c1) (fmap f c2)
instance Foldable (Dendrogram d) where
foldMap f (Leaf d) = f d
foldMap f (Branch _ c1 c2) = foldMap f c1 `mappend` foldMap f c2
instance Traversable (Dendrogram d) where
traverse f (Leaf d) = Leaf <$> f d
traverse f (Branch s c1 c2) = Branch s <$> traverse f c1 <*> traverse f c2
-- | The linkage type determines how the distance between
-- clusters will be calculated. These are the linkage types
-- currently available on this library.
data Linkage =
SingleLinkage
-- ^ The distance between two clusters @a@ and @b@ is the
-- /minimum/ distance between an element of @a@ and an element
-- of @b@.
| CompleteLinkage
-- ^ The distance between two clusters @a@ and @b@ is the
-- /maximum/ distance between an element of @a@ and an element
-- of @b@.
| UPGMA
-- ^ Unweighted Pair Group Method with Arithmetic mean, also
-- called \"average linkage\". The distance between two
-- clusters @a@ and @b@ is the /arithmetic average/ between the
-- distances of all elements in @a@ to all elements in @b@.
| FakeAverageLinkage
-- ^ This method is usually wrongly called \"average linkage\".
-- The distance between cluster @a = a1 U a2@ (that is, cluster
-- @a@ was formed by the linkage of clusters @a1@ and @a2@) and
-- an old cluster @b@ is @(d(a1,b) + d(a2,b)) / 2@. So when
-- clustering two elements to create a cluster, this method is
-- the same as UPGMA. However, in general when joining two
-- clusters this method assigns equal weights to @a1@ and @a2@,
-- while UPGMA assigns weights proportional to the number of
-- elements in each cluster. See, for example:
--
-- *
-- <http://www.cs.tau.ac.il/~rshamir/algmb/00/scribe00/html/lec08/node21.html>,
-- which defines the real UPGMA and gives the equation to
-- calculate the distance between an old and a new cluster.
--
-- *
-- <http://github.com/JadeFerret/ai4r/blob/master/lib/ai4r/clusterers/average_linkage.rb>,
-- code for \"average linkage\" on ai4r library implementing
-- what we call here @FakeAverageLinkage@ and not UPGMA.
deriving (Eq, Ord, Show, Enum)
-- Some cluster distances
cdistSingleLinkage :: Ord d => ClusterDistance d
cdistSingleLinkage = \(_, d1) (_, d2) -> d1 `min` d2
cdistCompleteLinkage :: Ord d => ClusterDistance d
cdistCompleteLinkage = \(_, d1) (_, d2) -> d1 `max` d2
cdistUPGMA :: Fractional d => ClusterDistance d
cdistUPGMA = \(b1,d1) (b2,d2) ->
let n1 = fromIntegral (size b1)
n2 = fromIntegral (size b2)
in (n1 * d1 + n2 * d2) / (n1 + n2)
cdistFakeAverageLinkage :: Fractional d => ClusterDistance d
cdistFakeAverageLinkage = \(_, d1) (_, d2) -> (d1 + d2) / 2
-- | /O(n^3)/ Calculates a complete, rooted dendrogram for a list
-- of items and a linkage type. If your distance type has an
-- 'Ord' instance but not a 'Fractional' one, then please use
-- specific functions 'singleLinkage' or 'completeLinkage' that
-- have less restrictive types.
dendrogram :: (Ord d, Fractional d)
=> Linkage -- ^ Linkage type to be used.
-> [a] -- ^ Items to be clustered.
-> (a -> a -> d) -- ^ Distance function between items.
-> Dendrogram d a -- ^ Complete dendrogram.
dendrogram linkage = dendrogram' cdist
where
cdist = case linkage of
SingleLinkage -> cdistSingleLinkage
CompleteLinkage -> cdistCompleteLinkage
FakeAverageLinkage -> cdistFakeAverageLinkage
UPGMA -> cdistUPGMA
-- | /O(n^3)/ Like 'dendrogram', but specialized to single
-- linkage (see 'SingleLinkage') which does not require
-- 'Fractional'.
singleLinkage :: Ord d => [a] -> (a -> a -> d) -> Dendrogram d a
singleLinkage = dendrogram' cdistSingleLinkage
-- | /O(n^3)/ Like 'dendrogram', but specialized to complete
-- linkage (see 'CompleteLinkage') which does not require
-- 'Fractional'.
completeLinkage :: Ord d => [a] -> (a -> a -> d) -> Dendrogram d a
completeLinkage = dendrogram' cdistCompleteLinkage
-- | /O(n^3)/ Like 'dendrogram', but specialized to 'UPGMA'.
upgma :: (Fractional d, Ord d) => [a] -> (a -> a -> d) -> Dendrogram d a
upgma = dendrogram' cdistUPGMA
-- | /O(n^3)/ Like 'dendrogram', but specialized to fake average
-- linkage (see 'FakeAverageLinkage').
fakeAverageLinkage :: (Fractional d, Ord d) => [a]
-> (a -> a -> d) -> Dendrogram d a
fakeAverageLinkage = dendrogram' cdistFakeAverageLinkage
-- | Worker function to create dendrograms based on a
-- 'ClusterDistance' (and not a 'Linkage').
dendrogram' :: Ord d => ClusterDistance d
-> [a] -> (a -> a -> d) -> Dendrogram d a
dendrogram' cdist items dist = runST (act ())
where
n = length items
act _noMonomorphismRestrictionPlease = do
let xs = listArray (1, n) items
fromDistance (dist `on` (xs !)) n >>= go xs (n-1) IM.empty
go xs i ds dm = xs `seq` i `seq` ds `seq` dm `seq` do
((c1,c2), distance) <- findMin dm
cu <- mergeClusters cdist dm (c1,c2)
let dendro c = case size c of
1 -> Leaf $! xs ! key c
_ -> ds IM.! key c
d1 = dendro c1
d2 = dendro c2
du = d1 `seq` d2 `seq` Branch distance d1 d2
case i of
1 -> return du
_ -> let ds' = IM.insert (key cu) du $
IM.delete (key c1) $
IM.delete (key c2) ds
in du `seq` go xs (i-1) ds' dm