---
title: "Comparing sets of trees from different analyses"
author: "[Martin R. Smith](https://smithlabdurham.github.io/)"
output: 
  rmarkdown::html_vignette:
    fig_width: 6
    fig_height: 4
bibliography: ../inst/REFERENCES.bib
csl: https://raw.githubusercontent.com/citation-style-language/styles/master/apa-old-doi-prefix.csl
vignette: >
  %\VignetteIndexEntry{Comparing sets of trees from different analyses}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

A common application of tree space analysis is to compare the outputs of
different analyses – for instance, trees obtained from different gene sequences,
or results obtained using different models or methods 
(e.g. Bayesian, maximum likelihood, or parsimony).

## Shiny app

This can be accomplished quickly using the `MapTrees()` graphical user interface:

- <span style="border: solid 2px #e69f00">Load trees</span> from file: Select first tree file

- Select an appropriate sample size

- Select <span style="border: solid 2px #009e73">Replace existing</span>

- Load each additional set of trees from file using <span style="border: solid 2px #d55e00">Add batch to existing</span>

![Load tree batches](Batch1.svg){ width="245px" }


- On the <span style="border: solid 2px #e69f00">Display</span> tab, 
  select <span style="border: solid 2px #009e73">Point symbols: One per batch</span>,
  or <span style="border: solid 2px #d55e00">Colour points by: Batch</span>


![Tree batch styles](Batch2.svg){ width=100% }


## Scripting at the R command line

More control over the mapping can be obtained at the command line:

```{r generate-trees}
# Load trees
library("TreeTools", quietly = TRUE)
batch1 <- as.phylo(1:60, 8) # Generate 60 similar trees 
batch2 <- as.phylo(seq(200, 800, length.out = 30), 8) # A separate batch of 30 trees
styles <- c(1, 2) # Select plotting colours / symbols
treeStyle <- rep(styles, c(length(batch1), length(batch2)))

# Calculate distances
library("TreeDist")
distances <- ClusteringInfoDistance(c(batch1, batch2))

# Construct over-simple 2D PCoA mapping
mapping <- cmdscale(distances, k = 2)
```

```{r plot-mapping, fig.align = "center"}
# Plot mapping
par(mar = rep(0, 4))
plot(mapping,
     asp = 1, # Preserve aspect ratio - do not distort distances
     ann = FALSE, axes = FALSE, # Don't label axes: dimensions are meaningless
     col = treeStyle, # Colour
     pch = treeStyle # Plotting symbol
     )
legend("left", c("Batch 1", "Batch 2"), col = styles, pch = styles)
```

For more robust analyses than the (potentially misleading!) 2D plot above,
consult the companion [vignette](treespace.html).
Note also that mapped areas and their regions of overlap may not correspond
to reality; see the warnings and recommendations in @SmithSpace.

## Comparing trees' dispersal / hypervolume

Interpreting and comparing the areas of tree space from a projection can be
misleading -- the expanded apparent area of Greenland under the Mercator projection
being a familiar example.

![Mapping can introduce distortion](https://i0.wp.com/geoffboeing.com/wp-content/uploads/2015/08/greenland-vs-africa-mercator-projection.jpg)

As such, it is always best to work with original distances when interpreting
whether sets of trees occupy larger or smaller regions of tree space.

### Distances from median

One approach is to plot distances from a median tree:

```{r compare-dist-from-median, fig.align = "center"}
# Calculate median trees
median1 <- median(batch1)
median2 <- median(batch2)

# Compute distance from each tree to the median of its batch
dist1 <- ClusteringInfoDist(batch1, median1)
dist2 <- ClusteringInfoDist(batch2, median2)

# Set resolution of histogram
nBreaks <- 10
breaks <- seq(0, max(dist1, dist2), length.out = nBreaks)

# Plot first distance set
hist(dist1, col = "#00000022", breaks = breaks,
     main = "Distance from median of batch",
     xlab = "Clustering information distance",
     ylim = c(0, 25) # Omit this line to infer Y axis limit from first batch.
     )

# Add second distance set
hist(dist2, col = "#ff000022", breaks = breaks, add = TRUE)

# Add legend
legend("topleft", c("Batch 1", "Batch 2"),
       fill = c("#00000022",  "#ff000022"))
```

In the plotted example, distances to the median tree are greater for batch 2
than batch 1, indicating a more dispersed set of trees that occupies a greater
hypervolume.  Note that the increased frequency at higher distances is expected:
the outer shell of a sphere contains more volume than a layer of equivalent
thickness closer to the centre, and this phenomenon becomes more pronounced as
the dimensionality of tree space increases.

### Consensus resolution

A complementary approach is to identify the resolution of the consensus of each
batch of trees.  This approach shares many of the 
[problems with the Robinson--Foulds distance](
https://ms609.github.io/TreeDist/articles/Robinson-Foulds.html): in particular,
resolution can be decimated by a single "rogue" taxon whose position is poorly
defined [@SmithRogue].
Detecting and removing rogue taxa can provide a more meaningful point of
comparison.

```{r rogue-detection}
# Create tree set with a rogue taxon
batch3 <- AddTipEverywhere(as.phylo(7, 7), "t8")

# Set up plotting area
par(mfrow = c(2, 2), mar = rep(0.4, 4))

# Plot naive strict consensus
plot(consensus(batch1, p = 1))
plot(consensus(batch3, p = 1))

if (requireNamespace("Rogue", quietly = TRUE)) {
  cons1 <- ConsensusWithout(batch1, p = 1,
                            Rogue::QuickRogue(batch1, p = 1)[-1, "taxon"])
  cons3 <- ConsensusWithout(batch3, p = 1,
                            Rogue::QuickRogue(batch3, p = 1)[-1, "taxon"])
  
  # The information content of each tree gives a measure of its resolution,
  # accounting for omitted rogue leaves
  SplitwiseInfo(cons1) # 8.5 bits
  SplitwiseInfo(cons3) # 15.1 bits: higher resolution indicates that these
  # trees are more similar, notwithstanding rogue taxa.
  
  # Plot the trees
  plot(cons1)
  plot(cons3)
} else {
  message("The package 'Rogue' is required to run this example.")
}
```

Whereas a direct interpretation of this analysis is not straightforward, it can
provide a complementary way of understanding the distribution of trees
across tree space.

## References
