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 = 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 = 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 go xs (i-1) ds' dm