pbd_ML demo

This document gives a demonstration how to use the package to obtain a maximum-likelihood estimate of the protracted birth-death speciation model.

First thing is to load the PBD package itself:

rm(list = ls())
library(PBD)

We will also need ape for branching.times:

library(ape)

Here we simulate a tree with known parameters:

seed <- 43
set.seed(seed)
b_1 <- 0.3 # speciation-initiation rate of good species
la_1 <- 0.2 # speciation-completion rate
b_2 <- b_1 # the speciation-initiation rate of incipient species
mu_1 <- 0.1 #  extinction rate of good species
mu_2 <- mu_1 # extinction rate of incipient species 
pars <- c(b_1, la_1, b_2, mu_1, mu_2)
age <- 15 # the age for the simulation 
phylogenies <- pbd_sim(pars = pars, age = age)
plot(phylogenies$recontree)

plot(phylogenies$igtree.extant)

## no colors provided. using the following legend:
##         g         i 
##   "black" "#DF536B"

plot(phylogenies$tree)

names(phylogenies)

##  [1] "tree"           "stree_random"   "stree_oldest"   "stree_youngest"
##  [5] "L"              "sL_random"      "sL_oldest"      "sL_youngest"   
##  [9] "igtree.extinct" "igtree.extant"  "recontree"      "reconL"        
## [13] "L0"

Now we try to recover the parameters by maximum likelihood estimation:

brts <- branching.times(phylogenies$recontree)  # branching times
init_b <- 0.2  # speciation-initiation rate
init_mu_1 <- 0.05  # extinction rate of good species
init_la_1 <- 0.3 # speciation-completion rate
#init_mu_2 <- 0.05  # extinction rate of incipient species  # not used

# The initial values of the parameters that must be optimized
initparsopt <- c(init_b, init_mu_1, init_la_1)

# The extinction rates between incipient and good species are equal
exteq <- TRUE

# The first element of the branching times is the crown age (and not the stem age)
soc <- 2

# Conditioning on non-extinction of the phylogeny
# as I actively selected for a nice phylogeny
cond <- 1

# Give the likelihood of the phylogeny (instead of the likelihood of the branching times)
btorph <- 1

Maximum likelihood estimation can now be performed:

r <- pbd_ML(
  brts = brts,
  initparsopt = initparsopt, 
  exteq = exteq,
  soc = soc, 
  cond = cond,
  btorph = btorph,
  verbose = FALSE
)

The ML parameter estimates are:

knitr::kable(r)

Comparing the known true value with the recovered values:

loglik_true <- PBD::pbd_loglik(pars, brts = brts)

## Parameters: 0.3, 0.2, 0.3, 0.1, 0.1, Loglikelihood: -36.539337

df <- as.data.frame(r)
df <- rbind(df, c(b_1, mu_1, la_1, mu_2, loglik_true, NA, NA))
row.names(df) <- c("ML", "true")
knitr::kable(df)

Ideally, all parameter columns should have the same values.

To test for the uncertainty of our ML estimate, we can do a parametric bootstrap.

The function pbd_bootstrap consists of a few steps:

1. Do an ML estimate
2. Run a simulation with those estimates
3. Perform ML estimation on the simulated data
4. Go to 2 depending on the setting of endmc.

endmc <- 10 # Sets the number of simulations for the bootstrap

b <- pbd_bootstrap(
  brts = brts,
  initparsopt = initparsopt, 
  exteq = exteq,
  soc = soc, 
  cond = cond,
  btorph = btorph,
  plotltt = FALSE,
  endmc = endmc,
  seed = seed
)

## Finding the maximum likelihood estimates ...
## 
## You are optimizing b mu_1 lambda_1 
## You are fixing nothing 
## Extinction rate of incipient species is exactly the same as for good species.
## The likelihood for the initial parameter values is -35.1429 
## Optimizing the likelihood - this may take a while. 
## 
##  Maximum likelihood parameter estimates: b: 0.264410, mu_1: 0.000001, lambda_1: 0.042447, mu_2: 0.000001 
##  Maximum loglikelihood: -33.066326 
##  The expected duration of speciation for these parameters is: 7.481272 
##  The median duration of speciation for these parameters is: 6.868607 
## Bootstrapping ...
## 
## Simulated data set 1 out of 10 
## Simulated data set 2 out of 10 
## Simulated data set 3 out of 10 
## Simulated data set 4 out of 10 
## Simulated data set 5 out of 10 
## Simulated data set 6 out of 10 
## Simulated data set 7 out of 10 
## Simulated data set 8 out of 10 
## Simulated data set 9 out of 10 
## Simulated data set 10 out of 10

knitr::kable(b[[3]])

From the bootstrap analysis, we get

• Again the ML estimate
• The ML estimates for simulations run with those estimates

Putting this in a table:

dg <- rbind(df, 
  list(
    b = b[[1]]$b, 
    mu_1 = b[[1]]$mu_1, 
    lambda_1 = b[[1]]$lambda_1, 
    mu_2 = b[[1]]$mu_2,
    loglik = b[[1]]$loglik,
    df = b[[1]]$df,
    conv = b[[1]]$conv
  ),
  list(
    b = b[[3]]$b, 
    mu_1 = b[[3]]$mu_1, 
    lambda_1 = b[[3]]$lambda_1, 
    mu_2 = b[[3]]$mu_2,
    loglik = b[[3]]$loglik,
    df = b[[3]]$df,
    conv = b[[3]]$conv
  )
)
dg

##              b         mu_1    lambda_1         mu_2    loglik df conv
## ML   0.2642439 6.424792e-10 0.042509306 6.424792e-10 -33.06632  3    0
## true 0.3000000 1.000000e-01 0.200000000 1.000000e-01 -36.53934 NA   NA
## 1    0.2644103 1.293611e-06 0.042447108 1.293611e-06 -33.06633  3    0
## 11   0.1422727 6.178418e-03 0.132053688 6.178418e-03 -16.77821  3    0
## 2    0.1933750 4.387956e-02 0.173796013 4.387956e-02 -28.24429  3    0
## 3    0.1924405 2.544596e-07 0.121625361 2.544596e-07 -27.91703  3    0
## 4    3.3308024 3.058947e+00 0.008703876 3.058947e+00 -49.97862  3    0
## 5    0.4307373 1.746037e-01 0.010257024 1.746037e-01 -16.89841  3    0
## 6    6.5081980 6.212549e+00 0.003041870 6.212549e+00 -22.91142  3    0
## 7    0.3076034 8.936738e-04 0.014070605 8.936738e-04 -28.04804  3    0
## 8    0.2731540 4.576958e-06 0.050923109 4.576958e-06 -38.97566  3    0
## 9    0.4716201 2.005959e-06 0.003967112 2.005959e-06 -41.75817  3    0
## 10   6.8221865 6.478270e+00 0.007963620 6.478270e+00 -94.97368  3    0

row.names(dg) <- c("ML", "true", "ML2", paste("BS", 1:endmc, sep = ""))
knitr::kable(dg)

We expect rows ML and ML2 to be identical. Their values are indeed very similar.

We can calculate the loglikelihood for

ml_b <- b[[1]]$b
ml_mu_1 <- b[[1]]$mu_1
ml_la_1 <- b[[1]]$lambda_1
ml_mu_2 <- b[[1]]$mu_2
ml_pars1 <- c(ml_b, ml_mu_1, ml_la_1, ml_mu_2)
ml_pars2 <- c(cond, btorph, soc, 0, "lsoda")

l <- pbd_loglik(
  pars1 = ml_pars1,
  pars2 = ml_pars2,
  brts = brts
)
print(l)

## [1] -33.06633

# Create .md, .html, and .pdf files
setwd(paste(getwd(), "vignettes", sep = "/"))
knit("PBD_ML_demo.Rmd")
markdown::markdownToHTML('PBD_ML_demo.md', 'PBD_ML_demo.html', options=c("use_xhml"))
system("pandoc -s PBD_ML_demo.html -o PBD_ML_demo.pdf")