Phylogenetic information content of splitting leaves into two partitions

Calculate the phylogenetic information content (sensu Steel and Penny 2006 ) of a split, which reflects the probability that a uniformly selected random tree will contain# the split: a split that is consistent with a smaller number of trees will have a higher information content.

Usage

SplitInformation(A, B = A[1])

MultiSplitInformation(partitionSizes)

Arguments

A, B: Integer specifying the number of taxa in each partition.
partitionSizes: Integer vector specifying the number of taxa in each partition of a multi-partition split.

Value

SplitInformation() and MultiSplitInformation() return the phylogenetic information content, in bits, of a split that subdivides leaves into partitions of the specified sizes.

Details

SplitInformation() addresses bipartition splits, which correspond to edges in an unrooted phylogeny; MultiSplitInformation() supports splits that subdivide taxa into multiple partitions, which may correspond to multi-state characters in a phylogenetic matrix.

A simple way to characterise trees is to count the number of edges. (Edges are almost, but not quite, equivalent to nodes.) Counting edges (or nodes) provides a quick measure of a tree's resolution, and underpins the Robinson-Foulds tree distance measure. Not all edges, however, are created equal.

An edge splits the leaves of a tree into two subdivisions. The more equal these subdivisions are in size, the more instructive this edge is. Intuitively, the division of mammals from reptiles is a profound revelation that underpins much of zoology; recognizing that two species of bat are more closely related to each other than to any other mammal or reptile is still instructive, but somewhat less fundamental.

Formally, the phylogenetic (Shannon) information content of a split S, h(S), corresponds to the probability that a uniformly selected random tree will contain the split, P(S): h(S) = -log P(S). Base 2 logarithms are typically employed to yield an information content in bits.

As an example, the split AB|CDEF occurs in 15 of the 105 six-leaf trees; h(AB|CDEF) = -log P(AB|CDEF) = -log(15/105) ~ 2.81 bits. The split ABC|DEF subdivides the leaves more evenly, and is thus more instructive: it occurs in just nine of the 105 six-leaf trees, and h(ABC|DEF) = -log(9/105) ~ 3.54 bits.

As the number of leaves increases, a single even split may contain more information than multiple uneven splits – see the examples section below.

Summing the information content of all splits within a tree, perhaps using the 'TreeDist' function SplitwiseInfo(), arguably gives a more instructive picture of its resolution than simply counting the number of splits that are present – though with the caveat that splits within a tree are not independent of one another, so some information may be double counted. (This same charge applies to simply counting nodes, too.)

Alternatives would be to count the number of quartets that are resolved, perhaps using the 'Quartet' function QuartetStates(), or to use a different take on the information contained within a split, the clustering information: see the 'TreeDist' function ClusteringInfo() for details.

References

Steel MA, Penny D (2006). “Maximum parsimony and the phylogenetic information in multistate characters.” In Albert VA (ed.), Parsimony, Phylogeny, and Genomics, 163–178. Oxford University Press, Oxford.

Author

Martin R. Smith (martin.smith@durham.ac.uk)

Examples

# Eight leaves can be split evenly:
SplitInformation(4, 4)
#> [1] 5.529821

# or unevenly, which is less informative:
SplitInformation(2, 6)
#> [1] 3.459432

# A single split that evenly subdivides 50 leaves contains more information
# that seven maximally uneven splits on the same leaves:
SplitInformation(25, 25)
#> [1] 47.50376
7 * SplitInformation(2, 48)
#> [1] 45.98899
# Three ways to split eight leaves into multiple partitions:
MultiSplitInformation(c(2, 2, 4))
#> [1] 5.97728
MultiSplitInformation(c(2, 3, 3))
#> [1] 6.714246
MultiSplitInformation(rep(2, 4))
#> [1] 6.714246