I just
pushed
some relatively small updates to the phytools function `fitMk`

,
which (aptly) fits the so-called M*k* discrete character evolution
model. The updates allow for more user control of optimization - specifically
by allowing the user to specify an optimization method (`opt.method`

),
which can be `"nlminb"`

or `"optim"`

, and for the user
to set initial values for the model parameters.

Note that `fitMk`

takes a lot of code from the ape function
`ace`

(for ancestral character estimation); however `fitMk`

allows the user to specify a fixed value of the transition matrix, `Q`

,
a non-flat prior probability density, `pi`

, ambiguous states at the
tips of the tree in the form of a matrix, and (now) an optimization method and
starting values for the optimization. The function `fitDiscrete`

in
geiger is quite powerful for fitting this model as well.

These updates are in part inspired by a recent post I wrote about an interesting failure of optimization in the function. Since we are doing numerical optimization, not exhaustive enumeration, there is always a chance we will fail to find the ML solution! Hopefully these updates will allow users to be more confident that they have in fact converged on the tree MLEs for their model.

`fitMk`

is also used internally by `make.simmap`

and
`rerootingMethod`

(and perhaps other functions that I'm not thinking of
now); however I have not yet migrated these options to these other methods.

Here's a demo with our dataset from before. This is a real empirical dataset, but one in which the taxon names have been obfuscated:

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

```
## [1] '0.5.52'
```

```
tree
```

```
##
## Phylogenetic tree with 271 tips and 270 internal nodes.
##
## Tip labels:
## t1, t2, t3, t4, t5, t6, ...
##
## Rooted; includes branch lengths.
```

```
head(x,n=100); cat("....\n")
```

```
## t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## t16 t17 t18 t19 t20 t21 t22 t23 t24 t25 t26 t27 t28 t29 t30
## 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0
## t31 t32 t33 t34 t35 t36 t37 t38 t39 t40 t41 t42 t43 t44 t45
## 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
## t46 t47 t48 t49 t50 t51 t52 t53 t54 t55 t56 t57 t58 t59 t60
## 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## t61 t62 t63 t64 t65 t66 t67 t68 t69 t70 t71 t72 t73 t74 t75
## 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## t76 t77 t78 t79 t80 t81 t82 t83 t84 t85 t86 t87 t88 t89 t90
## 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0
## t91 t92 t93 t94 t95 t96 t97 t98 t99 t100
## 0 0 0 0 0 0 0 0 1 0
```

```
## ....
```

```
plotTree(tree,type="fan",lwd=1,ftype="off")
tiplabels(pie=to.matrix(x[tree$tip.label],c(0,1)),
piecol=c("blue","red"),cex=0.3)
add.simmap.legend(colors=setNames(c("blue","red"),
c(0,1)),x=-176,y=169,prompt=FALSE,shape="circle")
```

OK, let's fit our model:

```
fit.nlminb<-fitMk(tree,x,opt.method="nlminb")
fit.nlminb
```

```
## Object of class "fitMk".
##
## Fitted (or set) value of Q:
## 0 1
## 0 -1.000148 1.000148
## 1 1.000148 -1.000148
##
## Fitted (or set) value of pi:
## 0 1
## 0.5 0.5
##
## Log-likelihood: -187.842905
##
## Optimization method used was "nlminb"
```

This was a rate that was about 200 × too high! Let's try the
`"optim"`

method for optimization. As you might guess, this substitutes
the `optim`

function for `nlminb`

internally, and is a
little bit slower as a consequence:

```
fit.optim<-fitMk(tree,x,opt.method="optim")
fit.optim
```

```
## Object of class "fitMk".
##
## Fitted (or set) value of Q:
## 0 1
## 0 -0.005093 0.005093
## 1 0.005093 -0.005093
##
## Fitted (or set) value of pi:
## 0 1
## 0.5 0.5
##
## Log-likelihood: -155.900603
##
## Optimization method used was "optim"
```

This gives us the correct answer this time; however it may not in general. For instance if we initial our transition rate with a value that is too high it will fail to converge on the correct solution:

```
fit.optim.q0.1<-fitMk(tree,x,opt.method="optim",q.init=0.1)
fit.optim.q0.1
```

```
## Object of class "fitMk".
##
## Fitted (or set) value of Q:
## 0 1
## 0 -1.389036 1.389036
## 1 1.389036 -1.389036
##
## Fitted (or set) value of pi:
## 0 1
## 0.5 0.5
##
## Log-likelihood: -187.842886
##
## Optimization method used was "optim"
```

Finally, as I showed
before
it is possible to visualize the likelihood surface for the transition rate. This
is also possible in two dimensions for different forward & backward rates,
*q*_{0,1} & *q*_{1,0}. For instance:

```
## one dimension
q<-seq(0.001,0.1,by=0.001)
logL<-sapply(q,function(q,tree,x)
logLik(fitMk(tree,x,fixedQ=matrix(c(-q,q,q,-q),2,2))),
tree=tree,x=x)
plot(q,logL,type="l",xlab="q[0,1] & q[1,0]",ylab="log(L)")
```

```
## two dimensions
logL<-sapply(q,function(q1,q2,tree,x) sapply(q2,
function(q2,q1,tree,x) logLik(fitMk(tree,x,fixedQ=
matrix(c(-q1,q1,q2,-q2),2,2))),tree=tree,x=x,q1=q1),
tree=tree,x=x,q2=q)
contour(logL,x=q,y=q,nlevels=100,xlab="q[0,1]",ylab="q[1,0]",
main="log(L)")
fitARD<-fitMk(tree,x,model="ARD")
points(fitARD$rates[2],fitARD$rates[1],col="red",pch=19)
```

Cool.

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