fit_hbd_psr_on_grid: Fit pulled speciation rates of birth-death models on a time grid.

Description

Given an ultrametric timetree, estimate the pulled speciation rate of homogenous birth-death (HBD) models that best explains the tree via maximum likelihood. Every HBD model is defined by some speciation and extinction rates (\(\lambda\) and \(\mu\)) over time, as well as the sampling fraction \(\rho\) (fraction of extant species sampled). ``Homogenous'' refers to the assumption that, at any given moment in time, all lineages exhibit the same speciation/extinction rates. For any given HBD model there exists an infinite number of alternative HBD models that predict the same deterministic lineages-through-time curve and yield the same likelihood for any given reconstructed timetree; these ``congruent'' models cannot be distinguished from one another solely based on the tree.

Each congruence class is uniquely described by the ``pulled speciation rate'' (PSR), defined as the relative slope of the deterministic LTT over time, \(PSR=-M^{-1}dM/d\tau\) (where \(\tau\) is time before present). In other words, two HBD models are congruent if and only if they have the same PSR. This function is designed to estimate the generating congruence class for the tree, by fitting the PSR on a discrete time grid.

Usage

fit_hbd_psr_on_grid(  tree, 
                      oldest_age            = NULL,
                      age0                  = 0,
                      age_grid              = NULL,
                      min_PSR               = 0,
                      max_PSR               = +Inf,
                      guess_PSR             = NULL,
                      fixed_PSR             = NULL,
                      splines_degree        = 1,
                      condition             = "auto",
                      relative_dt           = 1e-3,
                      Ntrials               = 1,
                      Nbootstraps           = 0,
                      Ntrials_per_bootstrap = NULL,
                      Nthreads              = 1,
                      max_model_runtime     = NULL,
                      fit_control           = list(),
                      verbose               = FALSE,
                      verbose_prefix        = "")

Arguments

tree

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

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. If age0>0, the tree essentially is trimmed at age0, omitting anything younger than age0, and the PSR is fitted to the trimmed tree while shifting time appropriately.

age_grid

Numeric vector, listing ages in ascending order at which the PSR is allowed to vary independently. This grid must cover at least the age range from age0 to oldest_age. If NULL or of length <=1 (regardless of value), then the PSR is assumed to be time-independent.

min_PSR

Numeric vector of length Ngrid (=max(1,length(age_grid))), or a single numeric, specifying lower bounds for the fitted PSR at each point in the age grid. If a single numeric, the same lower bound applies at all ages. Note that the PSR is never negative.

max_PSR

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted PSR at each point in the age grid. If a single numeric, the same upper bound applies at all ages. Use +Inf to omit upper bounds.

guess_PSR

Initial guess for the PSR at each age-grid point. Either NULL (an initial guess will be computed automatically), or a single numeric (guessing the same PSR at all ages) or a numeric vector of size Ngrid specifying a separate guess at each age-grid point. To omit an initial guess for some but not all age-grid points, set their guess values to NA. Guess values are ignored for non-fitted (i.e., fixed) parameters.

fixed_PSR

Optional fixed (i.e. non-fitted) PSR values on one or more age-grid points. Either NULL (PSR is not fixed anywhere), or a single numeric (PSR fixed to the same value at all grid points) or a numeric vector of size Ngrid (PSR fixed at one or more age-grid points, use NA for non-fixed values).

splines_degree

Integer between 0 and 3 (inclusive), specifying the polynomial degree of the PSR between age-grid points. If 0, then the PSR is considered piecewise constant, if 1 then the PSR is considered piecewise linear, if 2 or 3 then the PSR is considered to be a spline of degree 2 or 3, respectively. The splines_degree influences the analytical properties of the curve, e.g. splines_degree==1 guarantees a continuous curve, splines_degree==2 guarantees a continuous curve and continuous derivative, and so on. A degree of 0 is generally not recommended.

condition

Character, either "crown", "stem" 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. 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.

Nbootstraps

Integer, specifying an optional number of bootstrap samplings to perform, for estimating standard errors and confidence intervals of maximum-likelihood fitted parameters. If 0, no bootstrapping is performed. Typical values are 10-100. At each bootstrap sampling, a random timetree is generated under the birth-death model according to the fitted PSR, the parameters are estimated anew based on the generated tree, and subsequently compared to the original fitted parameters. Each bootstrap sampling will use roughly the same information and similar computational resources as the original maximum-likelihood fit (e.g., same number of trials, same optimization parameters, same initial guess, etc).

Ntrials_per_bootstrap

Integer, specifying the number of fitting trials to perform for each bootstrap sampling. If NULL, this is set equal to max(1,Ntrials). Decreasing Ntrials_per_bootstrap will reduce computation time, at the expense of potentially inflating the estimated confidence intervals; in some cases (e.g., for very large trees) this may be useful if fitting takes a long time and confidence intervals are very narrow anyway. Only relevant if Nbootstraps>0.

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.

verbose

Logical, specifying whether to print progress reports and warnings to the screen. Note that errors always cause a return of the function (see return values success and error).

verbose_prefix

Character, specifying the line prefix for printing progress reports to the screen.

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''.

loglikelihood

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

fitted_PSR

Numeric vector of size Ngrid, listing fitted or fixed pulled speciation rates (PSR) at each age-grid point. Between grid points the fitted PSR should be interpreted as a piecewise polynomial function (natural spline) of degree splines_degree; to evaluate this function at arbitrary ages use the castor routine evaluate_spline.

guess_PSR

Numeric vector of size Ngrid, specifying the initial guess for the PSR at each age-grid point.

age_grid

The age-grid on which the PSR is defined. This will be the same as the provided age_grid, unless the latter was NULL or of length <=1.

NFP

Integer, number of fitted (i.e., non-fixed) parameters. If none of the PSRs were fixed, this will be equal to Ngrid.

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.

bootstrap_estimates

If Nbootstraps>0, this will be a numeric matrix of size Nbootstraps x Ngrid, listing the fitted PSR at each grid point and for each bootstrap.

standard_errors

If Nbootstraps>0, this will be a numeric vector of size NGrid, listing bootstrap-estimated standard errors for the fitted PSR at each grid point.

CI50lower

If Nbootstraps>0, this will be a numeric vector of size Ngrid, listing bootstrap-estimated lower bounds of the 50-percent confidence intervals for the fitted PSR at each grid point.

CI50upper

Similar to CI50lower, listing upper bounds of 50-percentile confidence intervals.

CI95lower

Similar to CI50lower, listing lower bounds of 95-percentile confidence intervals.

CI95upper

Similar to CI95lower, listing upper bounds of 95-percentile confidence intervals.

Details

It is generally advised to provide as much information to the function fit_hbd_psr_on_grid as possible, including reasonable lower and upper bounds (min_PSR and max_PSR) and a reasonable parameter guess (guess_PSR). It is also important that the age_grid is sufficiently fine to capture the expected major variations of the PSR over time, but keep in mind the serious risk of overfitting when age_grid is too fine and/or the tree is too small.

References

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

Examples

Run this code

# 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)
sim       = 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 = sim$tree
root_age = castor::get_tree_span(tree)$max_distance
cat(sprintf("Tree has %d tips, spans %g Myr\n",length(tree$tip.label),root_age))

# Fit PSR on grid
oldest_age=root_age/2 # only consider recent times when fitting
Ngrid     = 10
age_grid  = seq(from=0,to=oldest_age,length.out=Ngrid)
fit = fit_hbd_psr_on_grid(tree, 
          oldest_age  = oldest_age,
          age_grid    = age_grid,
          min_PSR     = 0,
          max_PSR     = +100,
          condition   = "crown",
          Ntrials     = 10,	# perform 10 fitting trials
          Nthreads    = 10,	# use two CPUs
          max_model_runtime = 1) 	# limit model evaluation to 1 second
if(!fit$success){
  cat(sprintf("ERROR: Fitting failed: %s\n",fit$error))
}else{
  cat(sprintf("Fitting succeeded:\nLoglikelihood=%g\n",fit$loglikelihood))
  # plot fitted PSR
  plot( x     = fit$age_grid,
        y     = fit$fitted_PSR,
        main  = 'Fitted PSR',
        xlab  = 'age',
        ylab  = 'PSR',
        type  = 'b',
        xlim  = c(root_age,0)) 
        
 # plot deterministic LTT of fitted model
  plot( x     = fit$age_grid,
        y     = fit$fitted_LTT,
        main  = 'Fitted dLTT',
        xlab  = 'age',
        ylab  = 'lineages',
        type  = 'b',
        log   = 'y',
        xlim  = c(root_age,0)) 

  # get fitted PSR as a function of age
  PSR_fun = approxfun(x=fit$age_grid, y=fit$fitted_PSR)
}
# }

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