Dendrogram data type

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).

Constructors

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.

Instances

Functor (Dendrogram d)	Does not recalculate the distances!
Foldable (Dendrogram d)
Traversable (Dendrogram d)
(Eq d, Eq a) => Eq (Dendrogram d a)
(Ord d, Ord a) => Ord (Dendrogram d a)
(Show d, Show a) => Show (Dendrogram d a)

elements :: Dendrogram d a -> [a]Source

List of elements in a dendrogram.

cutAt :: Ord d => Dendrogram d a -> d -> [Dendrogram d a]Source

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

Linkage data type

data Linkage Source

The linkage type determines how the distance between clusters will be calculated. These are the linkage types currently available on this library.

Constructors

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.

Instances

Enum Linkage
Eq Linkage
Ord Linkage
Show Linkage

Generic clustering function

dendrogram Source

Arguments

:: (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.

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.

Functions for specific linkages

singleLinkage :: Ord d => [a] -> (a -> a -> d) -> Dendrogram d aSource

O(n^3) Like dendrogram, but specialized to single linkage (see SingleLinkage) which does not require Fractional.

completeLinkage :: Ord d => [a] -> (a -> a -> d) -> Dendrogram d aSource

O(n^3) Like dendrogram, but specialized to complete linkage (see CompleteLinkage) which does not require Fractional.

upgma :: (Fractional d, Ord d) => [a] -> (a -> a -> d) -> Dendrogram d aSource

O(n^3) Like dendrogram, but specialized to UPGMA.

fakeAverageLinkage :: (Fractional d, Ord d) => [a] -> (a -> a -> d) -> Dendrogram d aSource

O(n^3) Like dendrogram, but specialized to fake average linkage (see FakeAverageLinkage).