R: Fit pulled diversification rates of birth-death models on a...

fit_hbd_pdr_on_grid {castor}

R Documentation

Fit pulled diversification rates of birth-death models on a time grid.

Description

Given an ultrametric timetree, estimate the pulled diversification 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 diversification rate” (PDR; Louca et al 2018), defined as PDR=\lambda-\mu+\lambda^{-1}d\lambda/d\tau (where \tau is time before present) as well as the product \rho\lambda_o (where \lambda_o is the present-day speciation rate). That is, two HBD models are congruent if and only if they have the same PDR and the same product \rho\lambda_o. This function is designed to estimate the generating congruence class for the tree, by fitting the PDR on a grid of discrete times as well as the product \rho\lambda_o.

Usage

fit_hbd_pdr_on_grid(  tree, 
                      oldest_age            = NULL,
                      age0                  = 0,
                      age_grid              = NULL,
                      min_PDR               = -Inf,
                      max_PDR               = +Inf,
                      min_rholambda0        = 1e-10,
                      max_rholambda0        = +Inf,
                      guess_PDR             = NULL,
                      guess_rholambda0      = NULL,
                      fixed_PDR             = NULL,
                      fixed_rholambda0      = 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`	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, and with respect to which `rholambda0` is defined. If `age0>0`, then `rholambda0` refers to the product of the sampling fraction at age `age0` and the speciation rate at age `age0`. See below for more details.
`age_grid`	Numeric vector, listing ages in ascending order at which the PDR 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 PDR is assumed to be time-independent.
`min_PDR`	Numeric vector of length Ngrid (=`max(1,length(age_grid))`), or a single numeric, specifying lower bounds for the fitted PDR at each point in the age grid. If a single numeric, the same lower bound applies at all ages. Use `-Inf` to omit lower bounds.
`max_PDR`	Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted PDR 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.
`min_rholambda0`	Strictly positive numeric, specifying the lower bound for the fitted `\rho\lambda_o` (sampling fraction times present-day extinction rate).
`max_rholambda0`	Strictly positive numeric, specifying the upper bound for the fitted `\rho\lambda_o`. Set to `+Inf` to omit this upper bound.
`guess_PDR`	Initial guess for the PDR at each age-grid point. Either `NULL` (an initial guess will be computed automatically), or a single numeric (guessing the same PDR 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.
`guess_rholambda0`	Numeric, specifying an initial guess for the product `\rho\lambda_o`. If `NULL`, a guess will be computed automatically.
`fixed_PDR`	Optional fixed (i.e. non-fitted) PDR values on one or more age-grid points. Either `NULL` (PDR is not fixed anywhere), or a single numeric (PDR fixed to the same value at all grid points) or a numeric vector of size Ngrid (PDR fixed at one or more age-grid points, use `NA` for non-fixed values).
`fixed_rholambda0`	Numeric, optionally specifying a fixed value for the product `\rho\lambda_o`. If `NULL` or `NA`, the product `\rho\lambda_o` is estimated.
`splines_degree`	Integer between 0 and 3 (inclusive), specifying the polynomial degree of the PDR between age-grid points. If 0, then the PDR is considered piecewise constant, if 1 then the PDR is considered piecewise linear, if 2 or 3 then the PDR 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 PDR and `\rho\lambda_o`, 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.

Details

If age0>0, the input tree is essentially trimmed at age0 (omitting anything younger than age0), and the PDR and rholambda0 are fitted to this new (shorter) tree, with time shifted appropriately. The fitted rholambda0 is thus the product of the sampling fraction at age0 and the speciation rate at age0. Note that the sampling fraction at age0 is simply the fraction of lineages extant at age0 that are represented in the timetree.

It is generally advised to provide as much information to the function fit_hbd_pdr_on_grid as possible, including reasonable lower and upper bounds (min_PDR, max_PDR, min_rholambda0 and max_rholambda0) and a reasonable parameter guess (guess_PDR and guess_rholambda0). It is also important that the age_grid is sufficiently fine to capture the expected major variations of the PDR 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_PDR`	Numeric vector of size Ngrid, listing fitted or fixed pulled diversification rates (PDR) at each age-grid point. Between grid points the fitted PDR 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`.
`fitted_rholambda0`	Numeric, specifying the fitted or fixed product `\rho\lambda(0)`.
`guess_PDR`	Numeric vector of size Ngrid, specifying the initial guess for the PDR at each age-grid point.
`guess_rholambda0`	Numeric, specifying the initial guess for `\rho\lambda(0)`.
`age_grid`	The age-grid on which the PDR 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 PDRs or `\rho\lambda0` were fixed, this will be equal to Ngrid+1.
`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 named list containing the elements `PDR` (numeric matrix of size `Nbootstraps` x Ngrid, listing the fitted PDR at each grid point and for each bootstrap) and `rholambda0` (a numeric vector of size `Nbootstraps`, listing the fitted `\rho\lambda_o` for each bootstrap).
`standard_errors`	If `Nbootstraps>0`, this will be a named list containing the elements `PDR` (numeric vector of size Ngrid, listing bootstrap-estimated standard errors for the fitted PDRs) and `rholambda0` (a single numeric, bootstrap-estimated standard error for the fitted `\rho\lambda_o`).
`medians`	If `Nbootstraps>0`, this will be a named list containing the elements `PDR` (numeric vector of size Ngrid, listing median fitted PDRs across bootstraps) and `rholambda0` (a single numeric, median fitted `\rho\lambda_o` across bootstraps).
`CI50lower`	If `Nbootstraps>0`, this will be a named list containing the elements `PDR` (numeric vector of size Ngrid, listing bootstrap-estimated lower bounds of the 50-percent confidence intervals for the fitted PDRs) and `rholambda0` (a single numeric, bootstrap-estimated lower bound of the 50-percent confidence intervals for the fitted `\rho\lambda_o`).
`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.

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))

# calculate true PDR
lambda_slopes = diff(lambdas)/diff(time_grid);
lambda_slopes = c(lambda_slopes[1],lambda_slopes)
PDRs = lambdas - mus - (lambda_slopes/lambdas)


# Fit PDR on grid
Ngrid     = 10
age_grid  = seq(from=0,to=root_age,length.out=Ngrid)
fit = fit_hbd_pdr_on_grid(tree, 
          age_grid    = age_grid,
          min_PDR     = -100,
          max_PDR     = +100,
          condition   = "crown",
          Ntrials     = 10,	# perform 10 fitting trials
          Nthreads    = 2,	# 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 & true PDR
  plot( x     = fit$age_grid,
        y     = fit$fitted_PDR,
        main  = 'Fitted & true PDR',
        xlab  = 'age',
        ylab  = 'PDR',
        type  = 'b',
        col   = 'red',
        xlim  = c(root_age,0)) 
  lines(x     = sim$final_time-time_grid,
        y     = PDRs,
        type  = 'l',
        col   = 'blue');
        
  # get fitted PDR as a function of age
  PDR_fun = approxfun(x=fit$age_grid, y=fit$fitted_PDR)
}

## End(Not run)

[Package castor version 1.8.0 Index]