---
title: "Using secsse with complete phylogenies (with extinction)"
author: "Pedro Santos Neves"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Using secsse with complete phylogenies (with extinction)}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## Introduction

Most current studies of evolutionary dynamics make use of molecular phylogenies,
which, for most groups, contain only information on extant species. However,
when data on extinct species is available, usually through the presence of 
fossil data, we can use complete trees. Thus, we can leverage the data
from extinct lineages for maximum-likelihood estimation with secsse.

Note that here "complete tree" should not be taken as a complete sampling 
fraction, that is, all known species being present in the phylogeny and there 
being no missing data, but rather the assumption that all currently extinct 
species are included. This follows the nomenclature of Nee et al. (1994), 
who also coined the term "reconstructed tree" for phylogenies for which there
is no information on extinct lineages.

## Set-up

Like all ML analyses with secsse, we first need a few things to start with, 
starting with a dated phylogeny. For the purpose of this vignette, we are going
to simulate phylogenies with `secsse_sim()`. We will simulate a reconstructed
and a complete version of the same tree under the CR model. 

In order to simulate the trees, we need to specify the model and set starting 
parameters. Here we simulate a similar constant rate (CR) example to that of 
the _Simulating with secsse_ vignette. For more details on this model and full 
details of the functionality of `secsse_sim()`, see 
`vignette("sim_with_secsse", package = "secsse")`.

```{r sim_plot_tree}
library(secsse)

spec_matrix <- c()
spec_matrix <- rbind(spec_matrix, c(0, 0, 0, 1))
spec_matrix <- rbind(spec_matrix, c(1, 1, 1, 1))
lambda_list <- secsse::create_lambda_list(state_names = c(0, 1),
                                          num_concealed_states = 2,
                                          transition_matrix = spec_matrix,
                                          model = "CR")

mu_vector <- secsse::create_mu_vector(state_names = c(0, 1),
                                      num_concealed_states = 2,
                                      model = "CR",
                                      lambda_list = lambda_list)

shift_matrix <- c()
shift_matrix <- rbind(shift_matrix, c(0, 1, 3))
shift_matrix <- rbind(shift_matrix, c(1, 0, 4))

q_matrix <- secsse::create_q_matrix(state_names = c(0, 1),
                                    num_concealed_states = 2,
                                    shift_matrix = shift_matrix,
                                    diff.conceal = FALSE)

# Set-up starting parameters
speciation_rate <- 0.8
extinction_rate <- 0.2
q_01 <- 0.1
q_10 <- 0.1
used_params <- c(speciation_rate, extinction_rate, q_01, q_10)

sim_lambda_list <- secsse::fill_in(lambda_list, used_params)
sim_mu_vector   <- secsse::fill_in(mu_vector, used_params)
sim_q_matrix    <- secsse::fill_in(q_matrix, used_params)

# Simulate and plot the tree

sim_tree_complete <- secsse::secsse_sim(lambdas = sim_lambda_list,
                                        mus = sim_mu_vector,
                                        qs = sim_q_matrix,
                                        crown_age = 5,
                                        num_concealed_states = 2,
                                        seed = 40,
                                        drop_extinct = FALSE)

if (requireNamespace("diversitree")) {
  traits_for_plot_complete <- data.frame(
    trait = as.numeric(sim_tree_complete$obs_traits),
    row.names = sim_tree_complete$phy$tip.label
  )
  diversitree::trait.plot(tree = sim_tree_complete$phy,
                          dat = traits_for_plot_complete,
                          cols = list("trait" = c("blue", "red")),
                          type = "p")
} else {
  plot(sim_tree_complete$phy)
}

```

## Fitting the model

Finally, we run `secsse_ml()` on our complete tree, much in the same way as we 
would for one with extant species. However, this time we make sure to set the 
`is_complete_tree` argument to `TRUE` (defaults to `FALSE` if omitted). This
enables secsse to use the information present in extinct lineages.

```{r fitting_model_complete_tree}
idparsopt <- 1:4 # our maximum rate parameter was 4 -> We are keeping
# concealed and examined traits the same for the MLE.
idparsfix <- c(0) # we want to keep all zeros at zero
initparsopt <- rep(0.1, 4)
initparsfix <- c(0.0) # all zeros remain at zero.
sampling_fraction <- c(1, 1)

idparslist <- list()
idparslist[[1]] <- lambda_list
idparslist[[2]] <- mu_vector
idparslist[[3]] <- q_matrix

complete_tree_ml_CR <- secsse_ml(phy = sim_tree_complete$phy,
                                 traits = sim_tree_complete$obs_traits,
                                 num_concealed_states = 2,
                                 idparslist = idparslist,
                                 idparsopt = idparsopt,
                                 initparsopt = initparsopt,
                                 idparsfix = idparsfix,
                                 parsfix = initparsfix,
                                 sampling_fraction = sampling_fraction,
                                 verbose = FALSE)
```

Now we can see what our results look like.

```{r complete_tree_res}
CR_par_complete <- secsse::extract_par_vals(idparslist, complete_tree_ml_CR$MLpars)
complete_tree_ml_CR
CR_par_complete
spec_rates_complete <- CR_par_complete[1]
ext_rates_complete <- CR_par_complete[2]
Q_01_complete <- CR_par_complete[3]
Q_10_complete <- CR_par_complete[4]
spec_rates_complete
ext_rates_complete
Q_01_complete
Q_10_complete
```

## Comparing with reconstructed trees

It would be interesting to see how they compare with the same tree without any
extant species. Let's follow the standard procedure using a similar phylogeny -
the same tree we used before - but where all the extinct lineages have been 
removed. We'll keep all other model specification the same.

```{r fitting_ml_reconstructed_tree}

sim_tree_reconstructed <- secsse::secsse_sim(lambdas = sim_lambda_list,
                                             mus = sim_mu_vector,
                                             qs = sim_q_matrix,
                                             crown_age = 5,
                                             num_concealed_states = 2,
                                             seed = 40,
                                             drop_extinct = TRUE)

if (requireNamespace("diversitree")) {
  traits_for_plot_reconstructed <- data.frame(
    trait = as.numeric(sim_tree_reconstructed$obs_traits),
    row.names = sim_tree_reconstructed$phy$tip.label
  )
  diversitree::trait.plot(tree = sim_tree_reconstructed$phy,
                          dat = traits_for_plot_reconstructed,
                          cols = list("trait" = c("blue", "red")),
                          type = "p")
} else {
  plot(sim_tree_reconstructed$phy)
}

reconstructed_tree_ml <- secsse_ml(phy = sim_tree_reconstructed$phy,
                                   traits = sim_tree_reconstructed$obs_traits,
                                   num_concealed_states = 2,
                                   idparslist = idparslist,
                                   idparsopt = idparsopt,
                                   initparsopt = initparsopt,
                                   idparsfix = idparsfix,
                                   parsfix = initparsfix,
                                   sampling_fraction = sampling_fraction,
                                   verbose = FALSE,
                                   is_complete_tree = FALSE)

```


```{r reconstructed_tree_res_comparison}
reconstructed_tree_ml_CR <- reconstructed_tree_ml$ML
CR_par_reconstructed <- secsse::extract_par_vals(
  idparslist,
  reconstructed_tree_ml$MLpars
)
reconstructed_tree_ml
CR_par_reconstructed
spec_rates_reconstructed <- CR_par_reconstructed[1]
ext_rates_reconstructed <- CR_par_reconstructed[2]
Q_01_reconstructed <- CR_par_reconstructed[3]
Q_10_reconstructed <- CR_par_reconstructed[4]

knitr::kable(
  data.frame(
    Reconstructed_tree = c(
      spec_rates_reconstructed,
      ext_rates_reconstructed,
      Q_01_reconstructed,
      Q_10_reconstructed
    ),  
    Complete_tree = c(
      spec_rates_complete,
      ext_rates_complete,
      Q_01_complete,
      Q_10_complete
    ),
    Generating_parameters = c(
      speciation_rate,
      extinction_rate,
      q_01,
      q_10
    ),
    row.names = c(
      "Speciation rate",
      "Extinction rate",
      "Transition rate 01",
      "Transition rate 10"
    )
  )
)
```

We see that including extinct species results in a better esimation 
particularly of the extinction rate. This effect is especially noticeable if
there are many extinct species present in the tree.
Additionally, we see that the estimation of the transition rate from state 1 to 
0 also improved.

As a final note, do note that this is just a simple simulation example and
care should be exercised with model selection and specification when fitting
secsse to empirical datasets to make predictions about evolutionary patterns.

## References

Nee S, May RM, Harvey PH. The reconstructed evolutionary process. Philos Trans 
R Soc Lond B Biol Sci. 1994 May 28;344(1309):305-11. 
https://doi.org/10.1098/rstb.1994.0068.
