castor (version 1.6.1)

fit_hbd_model_parametric: Fit a parametric homogenous birth-death model to a timetree.

Description

Given an ultrametric timetree, fit a homogenous birth-death (HBD) model in which speciation and extinction rates (\(\lambda\) and \(\mu\)) are given as parameterized functions of time before present. ``Homogenous'' refers to the assumption that, at any given moment in time, all lineages exhibit the same speciation/extinction rates (in the literature this is sometimes referred to simply as ``birth-death model''). Every HBD model is defined based on the values that \(\lambda\) and \(\mu\) take over time as well as the sampling fraction \(\rho\) (fraction of extant species sampled); in turn, \(\lambda\), \(\mu\) and \(\rho\) can be parameterized by a finite set of parameters. This function estimates these parameters by maximizing the likelihood (Morlon et al. 2011) of the timetree under the resulting HBD model.

Usage

fit_hbd_model_parametric( tree, 
                          param_values,
                          param_guess        = NULL,
                          param_min          = -Inf,
                          param_max          = +Inf,
                          param_scale        = NULL,
                          oldest_age         = NULL,
                          age0               = 0,
                          lambda,
                          mu                 = 0,
                          rho0               = 1,
                          age_grid           = NULL,
                          condition          = "auto",
                          relative_dt        = 1e-3,
                          Ntrials            = 1,
                          max_start_attempts = 1,
                          Nthreads           = 1,
                          max_model_runtime  = NULL,
                          fit_control        = list())

Arguments

tree

An ultrametric timetree of class "phylo", representing the time-calibrated reconstructed phylogeny of a set of extant species.

param_values

Numeric vector, specifying fixed values for a some or all model parameters. For fitted (i.e., non-fixed) parameters, use NaN or NA. For example, the vector c(1.5,NA,40) specifies that the 1st and 3rd model parameters are fixed at the values 1.5 and 40, respectively, while the 2nd parameter is to be fitted. The length of this vector defines the total number of model parameters. If entries in this vector are named, the names are taken as parameter names. Names should be included if you'd like returned parameter vectors to have named entries, or if the functions lambda, mu or rho query parameter values by name (as opposed to numeric index).

param_guess

Numeric vector of size NP, specifying a first guess for the value of each model parameter. For fixed parameters, guess values are ignored. Can be NULL only if all model parameters are fixed.

param_min

Optional numeric vector of size NP, specifying lower bounds for model parameters. If of size 1, the same lower bound is applied to all parameters. Use -Inf to omit a lower bound for a parameter. If NULL, no lower bounds are applied. For fixed parameters, lower bounds are ignored.

param_max

Optional numeric vector of size NP, specifying upper bounds for model parameters. If of size 1, the same upper bound is applied to all parameters. Use +Inf to omit an upper bound for a parameter. If NULL, no upper bounds are applied. For fixed parameters, upper bounds are ignored.

param_scale

Optional numeric vector of size NP, specifying typical scales for model parameters. If of size 1, the same scale is assumed for all parameters. If NULL, scales are determined automatically. For fixed parameters, scales are ignored. It is strongly advised to provide reasonable scales, as this facilitates the numeric optimization algorithm.

oldest_age

Strictly positive numeric, specifying the oldest time before present (``age'') to consider when calculating the likelihood. If this is equal to or greater than the root age, then oldest_age is taken as the stem age, and the classical formula by Morlon et al. (2011) is used. If oldest_age is less than the root age, the tree is split into multiple subtrees at that age by treating every edge crossing that age as the stem of a subtree, and each subtree is considered an independent realization of the HBD model stemming at that age. This can be useful for avoiding points in the tree close to the root, where estimation uncertainty is generally higher. If oldest_age==NULL, it is automatically set to the root age.

age0

Non-negative numeric, specifying the youngest age (time before present) to consider for fitting, and with respect to which rho is defined. If age0>0, then rho0 refers to the sampling fraction at age age0, i.e. the fraction of lineages extant at age0 that are included in the tree. See below for more details.

lambda

Function specifying the speciation rate at any given age (time before present) and for any given parameter values. This function must take exactly two arguments, the 1st one being a numeric vector (one or more ages) and the 2nd one being a numeric vector of size NP (parameter values), and return a numeric vector of the same size as the 1st argument with strictly positive entries. Can also be a single number (i.e., lambda is fixed).

mu

Function specifying the extinction rate at any given age and for any given parameter values. This function must take exactly two arguments, the 1st one being a numeric vector (one or more ages) and the 2nd one being a numeric vector of size NP (parameter values), and return a numeric vector of the same size as the 1st argument with non-negative entries. Can also be a single number (i.e., mu is fixed).

rho0

Function specifying the sampling fraction (fraction of extant species sampled at age0) for any given parameter values. This function must take exactly one argument, a numeric vector of size NP (parameter values), and return a numeric between 0 (exclusive) and 1 (inclusive). Can also be a single number (i.e., rho0 is fixed).

age_grid

Numeric vector, specifying ages at which the lambda and mu functionals should be evaluated. This age grid must be fine enough to capture the possible variation in \(\lambda\) and \(\mu\) over time, within the permissible parameter range. If of size 1, then lambda & mu are assumed to be time-independent. Listed ages must be strictly increasing, and must cover at least the full considered age interval (from 0 to oldest_age). Can also be NULL or a vector of size 1, in which case the speciation rate and extinction rate is assumed to be time-independent.

condition

Character, either "crown", "stem", "none" or "auto", specifying on what to condition the likelihood. If "crown", the likelihood is conditioned on the survival of the two daughter lineages branching off at the root. If "stem", the likelihood is conditioned on the survival of the stem lineage. Note that "crown" really only makes sense when oldest_age is equal to the root age, while "stem" is recommended if oldest_age differs from the root age. "none" is usually not recommended. If "auto", the condition is chosen according to the recommendations mentioned earlier.

relative_dt

Strictly positive numeric (unitless), specifying the maximum relative time step allowed for integration over time, when calculating the likelihood. Smaller values increase integration accuracy but increase computation time. Typical values are 0.0001-0.001. The default is usually sufficient.

Ntrials

Integer, specifying the number of independent fitting trials to perform, each starting from a random choice of model parameters. Increasing Ntrials reduces the risk of reaching a non-global local maximum in the fitting objective.

max_start_attempts

Integer, specifying the number of times to attempt finding a valid start point (per trial) before giving up on that trial. Randomly choosen extreme start parameters may occasionally result in Inf/undefined likelihoods, so this option allows the algorithm to keep looking for valid starting points.

Nthreads

Integer, specifying the number of parallel threads to use for performing multiple fitting trials simultaneously. This should generally not exceed the number of available CPUs on your machine. Parallel computing is not available on the Windows platform.

max_model_runtime

Optional numeric, specifying the maximum number of seconds to allow for each evaluation of the likelihood function. Use this to abort fitting trials leading to parameter regions where the likelihood takes a long time to evaluate (these are often unlikely parameter regions).

fit_control

Named list containing options for the nlminb optimization routine, such as iter.max, eval.max or rel.tol. For a complete list of options and default values see the documentation of nlminb in the stats package.

Value

A list with the following elements:

success

Logical, indicating whether model fitting succeeded. If FALSE, the returned list will include an additional ``error'' element (character) providing a description of the error; in that case all other return variables may be undefined.

objective_value

The maximized fitting objective. Currently, only maximum-likelihood estimation is implemented, and hence this will always be the maximized log-likelihood.

objective_name

The name of the objective that was maximized during fitting. Currently, only maximum-likelihood estimation is implemented, and hence this will always be ``loglikelihood''.

param_fitted

Numeric vector of size NP (number of model parameters), listing all fitted or fixed model parameters in their standard order (see details above). If param_names was provided, elements in fitted_params will be named.

param_guess

Numeric vector of size NP, listing guessed or fixed values for all model parameters in their standard order. If param_names was provided, elements in param_guess will be named.

loglikelihood

The log-likelihood of the fitted model for the given timetree.

NFP

Integer, number of fitted (i.e., non-fixed) model parameters.

AIC

The Akaike Information Criterion for the fitted model, defined as \(2k-2\log(L)\), where \(k\) is the number of fitted parameters and \(L\) is the maximized likelihood.

BIC

The Bayesian information criterion for the fitted model, defined as \(\log(n)k-2\log(L)\), where \(k\) is the number of fitted parameters, \(n\) is the number of data points (number of branching times), and \(L\) is the maximized likelihood.

converged

Logical, specifying whether the maximum likelihood was reached after convergence of the optimization algorithm. Note that in some cases the maximum likelihood may have been achieved by an optimization path that did not yet converge (in which case it's advisable to increase iter.max and/or eval.max).

Niterations

Integer, specifying the number of iterations performed during the optimization path that yielded the maximum likelihood.

Nevaluations

Integer, specifying the number of likelihood evaluations performed during the optimization path that yielded the maximum likelihood.

trial_start_objectives

Numeric vector of size Ntrials, listing the initial objective values (e.g., loglikelihoods) for each fitting trial, i.e. at the start parameter values.

trial_objective_values

Numeric vector of size Ntrials, listing the final maximized objective values (e.g., loglikelihoods) for each fitting trial.

trial_Nstart_attempts

Integer vector of size Ntrials, listing the number of start attempts for each fitting trial, until a starting point with valid likelihood was found.

trial_Niterations

Integer vector of size Ntrials, listing the number of iterations needed for each fitting trial.

trial_Nevaluations

Integer vector of size Ntrials, listing the number of likelihood evaluations needed for each fitting trial.

Details

This function is designed to estimate a finite set of scalar parameters (\(p_1,..,p_n\in\R\)) that determine the speciation rate \(\lambda\), the extinction rate \(\mu\) and the sampling fraction \(\rho\), by maximizing the likelihood of observing a given timetree under the HBD model. For example, the investigator may assume that both \(\lambda\) and \(\mu\) vary exponentially over time, i.e. they can be described by \(\lambda(t)=\lambda_o\cdot e^{-\alpha t}\) and \(\mu(t)=\mu_o\cdot e^{-\beta t}\) (where \(\lambda_o\), \(\mu_o\) are unknown present-day rates and \(\alpha\), \(\beta\) are unknown factors, and \(t\) is time before present), and that the sampling fraction \(\rho\) is known. In this case the model has 4 free parameters, \(p_1=\lambda_o\), \(p_2=\mu_o\), \(p_3=\alpha\) and \(p_4=\beta\), each of which may be fitted to the tree.

It is generally advised to provide as much information to the function fit_hbd_model_parametric as possible, including reasonable lower and upper bounds (param_min and param_max), a reasonable parameter guess (param_guess) and reasonable parameter scales param_scale. If some model parameters can vary over multiple orders of magnitude, it is advised to transform them so that they vary across fewer orders of magnitude (e.g., via log-transformation). It is also important that the age_grid is sufficiently fine to capture the variation of lambda and mu over time, since the likelihood is calculated under the assumption that both vary linearly between grid points.

References

H. Morlon, T. L. Parsons, J. B. Plotkin (2011). Reconciling molecular phylogenies with the fossil record. Proceedings of the National Academy of Sciences. 108:16327-16332.

S. Louca et al. (2018). Bacterial diversification through geological time. Nature Ecology & Evolution. 2:1458-1467.

See Also

simulate_deterministic_hbd

loglikelihood_hbd

fit_hbd_model_on_grid

fit_hbd_pdr_on_grid

fit_hbd_pdr_parametric

Examples

# NOT RUN {
# Generate a random tree with exponentially varying lambda & mu
Ntips     = 10000
rho       = 0.5 # sampling fraction
time_grid = seq(from=0, to=100, by=0.01)
lambdas   = 2*exp(0.1*time_grid)
mus       = 1.5*exp(0.09*time_grid)
tree      = generate_random_tree( parameters  = list(rarefaction=rho), 
                                  max_tips    = Ntips/rho,
                                  coalescent  = TRUE,
                                  added_rates_times     = time_grid,
                                  added_birth_rates_pc  = lambdas,
                                  added_death_rates_pc  = mus)$tree
root_age = castor::get_tree_span(tree)$max_distance
cat(sprintf("Tree has %d tips, spans %g Myr\n",length(tree$tip.label),root_age))

# Define a parametric HBD model, with exponentially varying lambda & mu
# Assume that the sampling fraction is known
# The model thus has 4 parameters: lambda0, mu0, alpha, beta
lambda_function = function(ages,params){
	return(params['lambda0']*exp(-params['alpha']*ages));
}
mu_function = function(ages,params){
	return(params['mu0']*exp(-params['beta']*ages));
}
rho_function = function(params){
	return(rho) # rho does not depend on any of the parameters
}

# Define an age grid on which lambda_function & mu_function shall be evaluated
# Should be sufficiently fine to capture the variation in lambda & mu
age_grid = seq(from=0,to=100,by=0.01)

# Perform fitting
# Lets suppose extinction rates are already known
cat(sprintf("Fitting model to tree..\n"))
fit = fit_hbd_model_parametric(	tree, 
                      param_values  = c(lambda0=NA, mu0=3, alpha=NA, beta=-0.09),
                      param_guess   = c(1,1,0,0),
                      param_min     = c(0,0,-1,-1),
                      param_max     = c(10,10,1,1),
                      param_scale   = 1, # all params are in the order of 1
                      lambda        = lambda_function,
                      mu            = mu_function,
                      rho0          = rho_function,
                      age_grid      = age_grid,
                      Ntrials       = 10,    # perform 10 fitting trials
                      Nthreads      = 2,     # use 2 CPUs
                      max_model_runtime = 1, # limit model evaluation to 1 second
                      fit_control       = list(rel.tol=1e-6))
if(!fit$success){
	cat(sprintf("ERROR: Fitting failed: %s\n",fit$error))
}else{
	cat(sprintf("Fitting succeeded:\nLoglikelihood=%g\n",fit$loglikelihood))
	print(fit)
}
# }