Last week I posted on visualizing the mean number of changes (or changes per unit of edge length) per interval of time from the root of the tree to the present day. This can create a nice plot; however it is more or less meaningless unless we can compare it to what we might expect under some kind of neutral process.

Consequently, today I have added some new function & methods that:

1) Automate calculation of this CTT or 'changes-through-time' plot based on stochastic mapped trees.

2) Simulate CTT plots for a given transition matrix, ** Q** and (optionally) ancestral state,
by first simulating a discrete character & then sampling stochastic character maps consistent with this
character. This can be done a large number of times to generate a null distribution of the accumulation
of changes through time under the model.

3) Visualize our empirical CTT plot and a 100×(1-α)% distribution of CTTs under the a null model.

Here's what that looks like:

```
library(phytools)
packageVersion("phytools")
```

```
## [1] '0.6.49'
```

```
tree
```

```
##
## Phylogenetic tree with 200 tips and 199 internal nodes.
##
## Tip labels:
## t5, t29, t68, t199, t200, t11, ...
##
## Rooted; includes branch lengths.
```

```
y
```

```
## t5 t29 t68 t199 t200 t11 t18 t107 t108 t72 t3 t24 t85 t86 t44
## b c c c c c c c c c c c c c b
## t48 t51 t52 t186 t195 t196 t115 t116 t91 t134 t135 t113 t42 t90 t97
## b b b b b b c c c c c c c a a
## t98 t158 t159 t50 t144 t145 t32 t43 t151 t152 t112 t30 t53 t54 t47
## b a a a a a a a a a a a a a a
## t82 t83 t57 t58 t157 t176 t177 t76 t75 t92 t93 t95 t139 t140 t153
## a a a a a a a a a a a a a a a
## t154 t36 t37 t80 t101 t102 t73 t74 t31 t124 t125 t33 t162 t163 t96
## a a a a a a a a a a a a a a a
## t114 t122 t123 t132 t133 t78 t79 t71 t17 t21 t22 t14 t25 t26 t89
## a a a a a b a a a a a a a a a
## t187 t188 t170 t171 t23 t172 t173 t84 t148 t149 t156 t182 t183 t81 t197
## a a a a a c c a a a c c c c c
## t198 t141 t49 t189 t190 t19 t20 t184 t185 t155 t110 t111 t106 t117 t118
## c c c c c c c c c c c c c c c
## t13 t4 t38 t39 t12 t150 t180 t181 t45 t46 t8 t142 t143 t40 t164
## b c c c c c c c c c c c c c c
## t165 t178 t179 t55 t56 t41 t59 t87 t88 t35 t193 t194 t138 t174 t175
## c c c c c c a a a a a a c c c
## t15 t9 t128 t129 t160 t161 t119 t62 t63 t10 t2 t191 t192 t69 t70
## c a a a a a a a a a a b b b b
## t146 t147 t103 t77 t109 t168 t169 t136 t137 t130 t131 t120 t121 t126 t127
## b b b b b b b b b b b b b b b
## t166 t167 t7 t64 t65 t34 t16 t27 t66 t67 t99 t100 t60 t61 t104
## b b b a a a b b b b b b b b b
## t105 t94 t28 t6 t1
## b b b b b
## Levels: a b c
```

```
trees<-make.simmap(tree,y,nsim=100,model="ER")
```

```
## make.simmap is sampling character histories conditioned on the transition matrix
##
## Q =
## a b c
## a -0.003042964 0.001521482 0.001521482
## b 0.001521482 -0.003042964 0.001521482
## c 0.001521482 0.001521482 -0.003042964
## (estimated using likelihood);
## and (mean) root node prior probabilities
## pi =
## a b c
## 0.3333333 0.3333333 0.3333333
```

```
## Done.
```

```
object<-ctt(trees)
object
```

```
## Object of class "ctt" consisting of:
## (1) a matrix (segments) with the beginning & ending time of each segment.
## (2) a vector (nchanges) with the mean number of changes in each segment.
## (3) a vector (edge.length) containing the total edge length of each segement.
## (4) an object of class "phylo".
```

```
plot(object,type="number")
```

More often it seems more likely that we'd be interested in tracking the number of changes per unit of edge length rather than the total number of changes - because in all reconstructed phylogeny of extant taxa there is more edge length towards the tips of the tree than towards the root.

```
plot(object,type="rate")
```

Of course, as I mentioned in the preamble, it probably doesn't make much sense to generate this type
of visualization unless we have a sense of what might be expected under some kind of reasonable
null hypothesis - such as a constant rate of character evolution through time & among lineages. Luckily,
we can simulate this too. The function `sim.ctt`

will simulate a CTT plot for a given
transition matrix, ** Q**, by first simulating a discrete character and then by sampling stochastic
character maps for that character. This is still somewhat slow, so it may take a few minutes to run.

I'll use the function `sim.multiCtt`

to simulate various rather than a single CTT:

```
Q<-trees[[1]]$Q
Q
```

```
## a b c
## a -0.003042964 0.001521482 0.001521482
## b 0.001521482 -0.003042964 0.001521482
## c 0.001521482 0.001521482 -0.003042964
```

```
nulo<-sim.multiCtt(tree,Q,nsim=100)
```

```
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
## Starting stochastic mapping with simulated data vector.... Done.
```

```
nulo
```

```
## 100 objects of class "ctt" in a list.
```

```
plot(nulo,type="number")
plot(object,add=TRUE,type="number")
```

That's kind of neat. Now, for the rate. Here, I will change the α level for our confidence interval.

```
plot(nulo,alpha=0.2,ylim=c(0,0.025))
plot(object,add=TRUE)
```

In this case, the data were simulated under a process of declining rate towards the present. Perhaps that was captured by our plots?

```
tree<-pbtree(n=200,scale=100)
Q<-matrix(c(-0.004,0.002,0.002,
0.002,-0.004,0.002,
0.002,0.002,-0.004),3,3)
rownames(Q)<-colnames(Q)<-letters[1:3]
EB<-phytools:::ebTree(tree,-0.06)
EB$edge.length<-EB$edge.length/max(nodeHeights(EB))*100
y<-as.factor(sim.history(EB,10*Q)$states)
```

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