---
title: "Extending the Robinson-Foulds metric"
author: "[Martin R. Smith](https://smithlabdurham.github.io/)"
output: rmarkdown::html_vignette
bibliography: ../inst/REFERENCES.bib
csl: ../inst/apa-old-doi-prefix.csl
vignette: >
  %\VignetteIndexEntry{Extending the Robinson-Foulds metric}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r init, echo=FALSE, warning=FALSE, message=FALSE}
library('ape')
library('TreeDist')
standardMargin <- c(0.4, 0.4, 0.8, 0.4)
cbPalette8 <- c("#000000", "#E69F00", "#56B4E9", "#009E73", "#F0E442",
                "#0072B2", "#D55E00", "#CC79A7")
TwoTreePlot <- function() par(mfrow = c(1, 2), mar = standardMargin, cex = 0.9)

Plot <- function(tree, tree2 = NULL, highlight = character(0),
                  prune = if (is.null(tree2)) integer(0) 
                          else list(integer(0), integer(0))
                  ) {
  ABCDEFGHIJK <- LETTERS[1:11]
  if (is.null(tree2)) {
    if(!all(tree$tip.label %in% 1:11)) {
      tree$tip.label <- match(tree$tip.label, ABCDEFGHIJK)
    }
    
    highlightTip <- match(highlight, ABCDEFGHIJK)
    TreeDistPlot(tree, bold = highlightTip, prune = prune,
                 graft = which(tree$edge[, 2] == 
                                 match(highlightTip, tree$tip.label)))
  } else {
    par(mfrow = c(1, 2), mar = rep(0.4, 4))
    Plot(tree, highlight = highlight, prune = prune[[1]])
    Plot(tree2, highlight = highlight, prune = prune[[2]])
  }
}
```

# The Robinson–Foulds distance

An intuitive way to calculate the similarity between two trees is to calculate
the resolution of their strict consensus, which corresponds to the number
of splits that occur in both trees [@Schuh1980].
The corresponding distance measure, the Robinson–Foulds distance
[@Robinson1981], counts the number of splits that are unique to one of the 
two trees.

It is important to remember that counting splits is not the same as counting
clades, edges or nodes.  If a tree is drawn as rooted (even without a root 
edge),
then the number of splits is one less than the number of edges or clades,
and two less than the number of nodes.
For example, the trees below have one split, two edges and three nodes in 
common.

```{R, echo=FALSE, fig.width = 6, fig.align = 'center'}
startPar <- TwoTreePlot()
palette <- colorspace::qualitative_hcl(5, c=42, l=88)
TreeDistPlot(balancedTree <- 
               read.tree(text="(((1, 2), (3, 4)), ((5, 6), (7, 8)));"))
edgelabels(1:5, c(1, 2, 5, 9, 12), adj = c(1.5, 0.5), bg = palette[1]) # Splits
edgelabels(1:6, edge = c(1, 2, 5, 8, 9, 12), bg = palette[2],
           adj = c(-0.5, 0.5)) # Internal edges
nodelabels(1:7, bg = palette[3], adj=c(1, 0.5)) # Nodes
nodelabels(1:6, node = 10:15, bg = palette[4], adj=c(-1, 0.5)) # Clades
legend('bottomleft', col=palette[1:2], pch=15, bty='n',
       c('Splits (5)', 'Internal edges (6)'))
legend('bottomright', col=palette[3:4], pch=15, bty='n',
       c('Nodes (7)', 'Clades (6)'))

TreeDistPlot(read.tree(text="((1, 2, 3, 4), (5, 6, 7, 8));"))
edgelabels(1, 1, adj = c(1.5, 0.5), bg = palette[1])
edgelabels(1:2, edge = c(1, 6), adj = c(-0.5, 0.5), bg = palette[2])
nodelabels(1:3, bg = palette[3], adj = c(1, 0.5))
nodelabels(1:2, node=10:11, bg = palette[4], adj = c(-1, 0.5))

```

The simplicity of counting splits is appealing, but limited by the underlying
assumption that all splits are equivalent.

As an example, a split that separates eight leaves into two sets of four
(as in the right-hand tree above) has a $\frac{1}{`r choose(8, 4) / 2`}$
chance of being compatible with the reference tree.
In contrast, a split that separates two leaves from the other six has a
$\frac{1}{7}$ chance of matching the reference tree: the similarity observed 
is five times more likely to have arisen by chance.
In other words, failure to match an even split is less noteworthy than failure
to match an uneven one.

As a consequence, trees whose splits are less even will, on average, exhibit
higher Robinson–Foulds distances with comparison trees.
Compare a balanced and an unbalanced eight-taxon tree:

```{r two-trees, echo=FALSE, fig.height = 3, fig.width=6, fig.align = 'center'}
TwoTreePlot()
TreeDistPlot(balancedTree, edge.color = cbPalette8[3])
ape::edgelabels(1:5, c(1, 2, 5, 9, 12), bg = palette[1], adj=c(0.5, -0.25))
ape::edgelabels(c('4:4', rep('2:6', 4)), c(1, 2, 5, 9, 12),
                adj=c(0.5, 1.25), frame = 'n')
ape::edgelabels(c(5.53, rep(3.46, 4)), c(1, 2, 5, 9, 12),
                adj=c(0.5, 2.75), frame = 'n')
legend('topleft', 'Balanced tree', bty='n', inset=c(-0.05, -0.03))

TreeDistPlot(caterpillarTree <- 
               ape::read.tree(text="(1, (2, (3, (4, (5, (6, (7, 8)))))));"),
             edge.color = cbPalette8[2])
ape::edgelabels(1:5, bg = palette[1], c(4, 6, 8, 10, 12), adj = c(0.5, -0.25))
ape::edgelabels(c('2|6', '3|5', '4|4', '3|5', '2|6'), edge = c(4, 6, 8, 10, 12),
                frame = 'n', adj = c(0.5, 1.25))
ape::edgelabels(c(3.46, 5.04, 5.53, 5.04, 3.46), edge = c(4, 6, 8, 10, 12),
                frame = 'n', adj = c(0.5, 2.75))
legend('topleft', 'Asymmetric tree', bty = 'n', inset = c(-0.05, -0.03))
```

Each tree divides the eight taxa into five splits.
The [phylogenetic information content](information.html) of a split is a 
function of the probability that the split will match a uniformly chosen random
tree, i.e. the proportion of eight-leaf binary trees that contain the split in
question.
(Information content, in bits, is defined as $-\log_2(\textrm{probability})$.)
This, in turn, is a function of the evenness of the split:

```{r ic-of-splits, display='asis', echo=FALSE, warnings=FALSE}
splitSmall <- 2:4
splitLarge <- 8L - splitSmall

rootedTrees <- c(
  '1' = 1,
  '2' = 1 * 1,
  '3' = 3 * 1,
  '4' = 5 * 3 * 1,
  '5' = 7 * 5 * 3 * 1,
  '6' = 9 * 7 * 5 * 3 * 1
)

matchingTrees <- rootedTrees[splitSmall] * rootedTrees[splitLarge]
names(matchingTrees) <- paste0("Split size: ", splitSmall, ':', splitLarge)

matchingP <- matchingTrees / (11 * 9 * 7 * 5 * 3 * 1 * 1)

ic <- -log2(matchingP)

knitr::kable(cbind(
      'Matching trees' = paste(matchingTrees, "/ 10 395"),
      '_P_(Match in random tree)' = signif(matchingP, 3),
      'Phylogenetic information content' = paste(signif(ic, 3), 'bits')))
```

In the first tree, split 1 is even, dividing four taxa from four others (`4|4`);
splits 2--5 are maximally uneven (`2|6`).  If each split is treated as 
independent, then the total information content of the five splits is
`r signif(sum(ic[c(4, 2, 2, 2, 2) - 1]), 4)`&nbsp;bits,
whereas that of the five splits in the second tree, of sizes 
`2|6`, `3|5`, `4|4`, `3|5` and `2|6`, is
`r signif(sum(ic[c(2, 3, 4, 3, 2) - 1]), 4)`&nbsp;bits.
Put another way, a random tree will
on average share more splits with the balanced tree (whose splits are 
predominantly uneven and thus likely to be matched) than the asymmetric tree 
(which contains more even splits that are less likely to occur in a random 
tree).

Indeed, of the 10&nbsp;395 eight-leaf trees, many more bear at least one split 
in common with a balanced tree than with an asymmetric tree:

```{r all-8-tip-trees, echo=FALSE, cache=TRUE, fig.width=4, fig.height=4, fig.align='center'}
calculate <- FALSE
if (calculate) {
  all8 <- phangorn::allTrees(8, tip.label = 1:8, rooted = FALSE)
  inBalanced <- RobinsonFoulds(all8, balancedTree, similarity = TRUE)
  inCaterpillar <- RobinsonFoulds(all8, caterpillarTree, similarity = TRUE)
} else {
  inBalanced <- rep(0:5, c(7088, 2708, 512, 76, 10, 1))
  inCaterpillar <- rep(0:5, c(8162, 1808, 350, 64, 10, 1))
}

par(cex = 0.8)
sch <- hist(inCaterpillar + 0.7, breaks = 0:18 / 3 - (1/6),
            main = '8-leaf trees with N common splits', cex.main = 1,
            xlim = c(0, 6), axes = FALSE,
            xlab = 'Splits in common', ylab = 'Number of trees')
sbh <- hist(inBalanced + 0.4, breaks = 0:18 / 3 - (1/6), plot = FALSE)
plot(sch, col = paste0(cbPalette8[2], '44'), add = TRUE)
plot(sbh, col = paste0(cbPalette8[3], '44'), add = TRUE)
text(1/6, 100, paste0('Balanced: ', sum(inBalanced == 0)), 
     pos = 4, srt = 90, cex = 0.7)
text(0.5, 100, paste0('Asymmetric: ', sum(inCaterpillar == 0)), 
     pos = 4, srt = 90, cex = 0.7)


legend('topright', pch = 22,
       pt.cex = 2, col = 'black',
       pt.bg = paste0(cbPalette8[2:3], '44'), bty = 'n',
       c('Asymmetric', 'Balanced'))

axis(1, at = 0:5 + 0.5, labels = 0:5)
axis(2)
```


# Information-corrected Robinson–Foulds distance

This differing information content can be accommodated by weighting each
split according to the amount of phylogenetic information it contains
[@SmithDist].
The two tree pairs below both have a Robinson–Foulds distance of two,
but the first pair differ with regard to 
an uneven split (`ABCDEF|GH`), so obtain a total difference
of 22.54 &minus; (3.46 + 5.04 + 5.53 + 5.04) = 3.46&nbsp;bits:

```{r, fig.height = 3, fig.width=6, fig.align = 'center'}
tree1 <- ape::read.tree(text='(1, (2, (3, (4, (5, (6, (7, 8)))))));')
tree2 <- ape::read.tree(text='(1, (2, (3, (4, (5, (7, (6, 8)))))));')
tree3 <- ape::read.tree(text='(1, (2, (3, (5, (4, (6, (7, 8)))))));')

VisualizeMatching(InfoRobinsonFoulds, tree1, tree2, 
                  Plot = TreeDistPlot, prune = 12)
```

whereas the second pair differ in the resolution of a more even, and thus more
information-rich, split (`ABCD|EFGH`), and so receive a distance score of 
5.53&nbsp;bits:

```{r, fig.height = 3, fig.width=6, fig.align = 'center'}
VisualizeMatching(InfoRobinsonFoulds, tree1, tree3, 
                  Plot = TreeDistPlot, prune = 8)
```

# Generalized Robinson–Foulds distances

Even when accounting for the information content of splits in this way,
the Robinson–Foulds distance is readily saturated: the maximum value can be 
obtained by moving a single leaf.

```{r, fig.height = 3, fig.width=6, fig.align = 'center'}
tree1 <- ape::read.tree(text='(1, (2, (3, (4, (5, (6, (7, 8)))))));')
tree2 <- ape::read.tree(text='(8, (1, (2, (3, (4, (5, (6, 7)))))));')

VisualizeMatching(RobinsonFouldsMatching, tree1, tree2, Plot = TreeDistPlot)
```

Generalized Robinson–Foulds distances [@Nye2006;@Bocker2013] seek to address
this issue. 
This category of metrics aim to acknowledge semblances between similar-but-not-quite-identical pairs of splits, which would contribute 
zero to tree similarity under the standard Robinson–Foulds measure.

Generalized RF distances work by finding a _matching_ that pairs splits from
one tree with splits in the other.
Each pairing is scored according to the similarity of the paired splits;
the sum of these scores is the score of the matching.
The tree distance is given by the score of the optimal matching.

## Constructing a matching

Let's consider two trees that differ in the position of one wildcard leaf,
and in the resolution of a clade:

```{r, fig.height = 3, fig.width=6, fig.align = 'center'}
tree1 <- ape::read.tree(text='((A, B), ((C, (D, E)), (F, (G, (H, I)))));')
tree2 <- ape::read.tree(text='((A, B), ((C, D, (E, I)), (F, (G, H))));')

Plot(tree1, tree2, highlight = 'I', prune = list(8, integer(0)))
```

These trees obtain a Robinson–Foulds distance of nine: a large distances, as
the maximum possible for trees of this resolution is eleven.
`AB|CDEFGHI` is the only split in common between the two trees:

```{r, fig.height = 3, fig.width=6, fig.align = 'center'}
TwoTreePlot()
VisualizeMatching(RobinsonFouldsMatching, tree1, tree2)
```

This distance score is higher than might be expected, given how much the trees 
have in common;
removing the single leaf 'I' results in two trees that differ only 
in the resolution of a single node:

```{r, fig.height = 3, fig.width=6, fig.align = 'center'}
TwoTreePlot()
VisualizeMatching(RobinsonFouldsMatching,
                  drop.tip(tree1, 'I'),
                  drop.tip(tree2, 'I'))
```

This hidden similarity can be better reflected if similar, but non-identical,
splits are assigned non-zero similarity scores.

There are various ways to score the similarity between two splits. 
One is to build on the idea introduced above, where identical splits are
scored according to their phylogenetic information content.
Non-matching splits can be scored according to the amount of phylogenetic 
information that they hold in common, which is a function of the proportion of 
trees that are consistent with both splits.
(A full explanation is provided in the discussion of
[Generalized Robinson–Foulds distances](Generalized-RF.html).)

```{r, fig.height = 3, fig.width=6, fig.align = 'center'}
TwoTreePlot()
VisualizeMatching(SharedPhylogeneticInfo, tree1, tree2)
```

Here, the split `AB|CDEFGHI` occurs in both trees,
and, as it happens, makes the largest contribution to the tree similarity 
score (3.70) for this particular pair of trees.
This is the same contribution it would
have made to the information-corrected Robinson–Foulds similarity.

The split `ABCDEF|GHI` in the left-hand tree is paired with the split
`ABCDEFI|GH` in the right-hand tree.  Had `ABCDEF|GHI` been available in the 
right-hand tree, then this perfect match would have been assigned a similarity 
of `SplitInformation(3, 6)` = 5.57&nbsp;bits.  The partial match is instead 
allocated a lower score of 2.12&nbsp;bits.  Pairings of incompatible splits,
i.e. those that cannot co-exist on a tree, such as `ABCDEFG|HI` - `ABCDFGH|EI`,
have no phylogenetic information in common.
([clustering information](information.html) is an alternative way to think
about split similarity that recognizes similarity even between incompatible
splits.)

The matching depicted above is one of many.
It happens to be optimal: an optimal matching can be found by considering the
similarity score of each possible pairing, and solving a linear assignment
problem to find the optimal set of pairings.

We can view the splits in each tree, named according to the number of their
associated node:

```{r}
summary(TreeTools::as.Splits(tree1, LETTERS[1:9]))
summary(TreeTools::as.Splits(tree2, LETTERS[1:9]))
```

We can then see the similarity scores for each pair of splits, along with the 
optimal matching:

```{r}
attributes(SharedPhylogeneticInfo(tree1, tree2, reportMatching = TRUE))
```

`..` denotes that the fifth matching contributes zero to similarity score; an 
alternative optimal matching would leave these splits unpaired.

## What next?

- Alternatives measures of split similarity, such as 
[mutual clustering information](information.html), give rise to other
[Generalized Robinson–Foulds distances](Generalized-RF.html), and can be used
to generate meaningful [tree spaces](treespace.html).

```{r echo=FALSE}
par(startPar)
```

## References
