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

In castor: Efficient Phylogenetics on Large Trees

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,
                      diagnostics           = FALSE,
                      verbose_prefix        = "")

Arguments

tree	A rooted ultrametric timetree of class "phylo", representing the time-calibrated phylogeny of a set of extant sampled 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", "auto", "stemN" or "crownN" (where N is an integer >=2), 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 at that time. If "stem", the likelihood is conditioned on the survival of the stem lineage, with the process having started at oldest_age. Note that "crown" and "crownN"" really only make sense when oldest_age is equal to the root age, while "stem" is recommended if oldest_age differs from the root age. If "stem2", the condition is that the process yielded at least two sampled tips, and similarly for "stem3" etc. If "crown3", the condition is that a splitting occurred at the root age, both child clades survived, and in total yielded at least 3 sampled tips (and similarly for "crown4" etc). 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).
diagnostics	Logical, specifying whether to print detailed information (such as model likelihoods) at every iteration of the fitting routine. For debugging purposes mainly.
verbose_prefix	Character, specifying the line prefix for printing progress reports to the screen.

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.

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.

Author(s)

Stilianos Louca

References

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

S. Louca and M. W. Pennell (2020). Extant timetrees are consistent with a myriad of diversification histories. Nature. 580:502-505.

See Also

simulate_deterministic_hbd

loglikelihood_hbd

fit_hbd_model_parametric

fit_hbd_model_on_grid

fit_hbd_pdr_parametric

fit_hbd_pdr_on_grid

fit_hbd_psr_on_best_grid_size

model_adequacy_hbd

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)
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,
          Nthreads    = 4,
          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)
}

## End(Not run)

castor documentation built on Aug. 18, 2023, 1:07 a.m.