castor (version 1.6.1)

loglikelihood_hbd: Galculate the log-likelihood of a homogenous birth-death model.

Description

Given a rooted ultrametric timetree, and a homogenous birth-death (HBD) model, i.e., with speciation rate \(\lambda\), extinction rate \(\mu\) and sampling fraction \(\rho\), calculate the likelihood of the tree under the model. The speciation and extinction rates may be time-dependent. ``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''). Alternatively to \(\lambda\) and \(\mu\), the likelihood may also be calculated based on the pulled diversification rate (PDR; Louca et al. 2018) and the product \(\rho(0)\cdot\lambda(0)\), or based on the pulled speciation rate (PSR). In either case, the time-profiles of \(\lambda\), \(\mu\), the PDR or the PSR are specified as piecewise polynomially functions (splines), defined on a discrete grid of ages.

Usage

loglikelihood_hbd(tree, 
                  oldest_age      = NULL,
                  age0            = 0,
                  rho0            = NULL,
                  rholambda0      = NULL,
                  age_grid        = NULL,
                  lambda          = NULL,
                  mu              = NULL,
                  PDR             = NULL,
                  PSR             = NULL,
                  splines_degree  = 1,
                  condition       = "auto",
                  relative_dt     = 1e-3)

Arguments

tree

A rooted ultrametric tree of class "phylo".

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 and rholambda0 are defined. If age0>0, then rho refers to the sampling fraction at age age0, and rholambda0 to the product between rho and the speciation rate at age age0. See below for more details.

rho0

Numeric between 0 (exclusive) and 1 (inclusive), specifying the sampling fraction of the tree at age0, i.e. the fraction of lineages extant at age0 that are included in the tree. Note that if \(rho0<1\), lineages extant at age0 are assumed to have been sampled randomly at equal probabilities. Can also be NULL, in which case rholambda0 and PDR (see below) must be provided.

rholambda0

Strictly positive numeric, specifying the product of the sampling fraction and the speciation rateat age0, units 1/time. Can be NULL, in which case rarefaction, lambda and mu must be provided.

age_grid

Numeric vector, listing discrete ages (time before present) on which either \(\lambda\) and \(\mu\), or the PDR, are specified. Listed ages must be strictly increasing, and must cover at least the full considered age interval (from age0 to oldest_age). Can also be NULL or a vector of size 1, in which case the speciation rate, extinction rate and PDR are assumed to be time-independent.

lambda

Numeric vector, of the same size as age_grid (or size 1 if age_grid==NULL), listing speciation rates (in units 1/time) at the ages listed in age_grid. Speciation rates should be non-negative, and are assumed to vary polynomially between grid points (see argument splines_degree). If NULL, then either PDR and rholambda0, or PSR alone, must be provided.

mu

Numeric vector, of the same size as age_grid (or size 1 if age_grid==NULL), listing extinction rates (in units 1/time)at the ages listed in age_grid. Extinction rates should be non-negative, and are assumed to vary polynomially between grid points (see argument splines_degree). If NULL, then PDR and rholambda0, or PSR alone, must be provided.

PDR

Numeric vector, of the same size as age_grid (or size 1 if age_grid==NULL), listing pulled diversification rates (in units 1/time) at the ages listed in age_grid. PDRs can be negative or positive, and are assumed to vary polynomially between grid points (see argument splines_degree). If NULL, then either lambda and mu, or PSR alone, must be provided.

PSR

Numeric vector, of the same size as age_grid (or size 1 if age_grid==NULL), listing pulled speciation rates (in units 1/time) at the ages listed in age_grid. PSRs should be non-negative, and are assumed to vary polynomially between grid points (see argument splines_degree). If NULL, then either lambda and mu, or PDR and rholambda0, must be provided.

splines_degree

Integer, either 0,1,2 or 3, specifying the polynomial degree of the provided lambda, mu, PDR and PSR (whichever applicable) between grid points in age_grid. For example, if splines_degree==1, then the provided lambda, mu, PDR and PSR are interpreted as piecewise-linear curves; if splines_degree==2 they are interpreted as quadratic splines; if splines_degree==3 they are interpreted as cubic splines. 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.

condition

Character, either "crown", "stem", "auto" or "none" (the last one is only available if lambda and mu are given), 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 and is only available when lambda and mu are provided. 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. Smaller values increase integration accuracy but increase computation time. Typical values are 0.0001-0.001. The default is usually sufficient.

Value

A named list with the following elements:

success

Logical, indicating whether the calculation was successful. If FALSE, then the returned list includes an additional `error' element (character) containing a description of the error; all other return variables may be undefined.

loglikelihood

Numeric. If success==TRUE, this will be the natural logarithm of the likelihood of the tree under the given model.

Details

If age0>0, the input tree is essentially trimmed at age0 (omitting anything younger than age0), and the is likelihood calculated for the trimmed tree while shifting time appropriately. In that case, rho0 is interpreted as the sampling fraction at age0, i.e. the fraction of lineages extant at age0 that are repreented in the tree. Similarly, rholambda0 is the product of the sampling fraction and \(\lambda\) at age0.

This function supports three alternative parameterizations of HBD models, either using the speciation and extinction rates and sampling fraction (\(\lambda\), \(\mu\) and \(\rho(\tau_o)\) (for some arbitrary age \(\tau_o\)), or using the pulled diversification rate (PDR) and the product \(\rho(\tau_o)\cdot\lambda(\tau_o\) (sampling fraction times speciation rate at \(\tau_o\)), or using the pulled speciation rate (PSR). The latter two options should be interpreted as a parameterization of congruence classes, i.e. sets of models that have the same likelihood, rather than specific models, since multiple combinations of \(\lambda\), \(\mu\) and \(\rho(\tau_o)\) can have identical PDRs, \(\rho(\tau_o)\cdot\lambda(\tau_o)\) and PSRs (Louca and Pennell, in review).

For large trees the asymptotic time complexity of this function is O(Nips). The tree may include monofurcations as well as multifurcations, and the likelihood formula accounts for those (i.e., as if monofurcations were omitted and multifurcations were expanded into bifurcations).

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.

S. Louca and M. W. Pennell (in review as of 2019)

See Also

simulate_deterministic_hbd

fit_hbd_model_parametric

fit_hbd_model_on_grid

fit_hbd_pdr_on_grid

fit_hbd_pdr_parametric

Examples

# NOT RUN {
# generate a random tree with constant rates
Ntips  = 100
params = list(birth_rate_factor=1, death_rate_factor=0.2, rarefaction=0.5)
tree   = generate_random_tree(params, max_tips=Ntips, coalescent=TRUE)$tree

# get the loglikelihood for an HBD model with the same parameters that generated the tree
# in particular, assuming time-independent speciation & extinction rates
LL = loglikelihood_hbd( tree, 
                        rho0      = params$rarefaction, 
                        age_grid  = NULL, # assume time-independent rates
                        lambda    = params$birth_rate_factor,
                        mu        = params$death_rate_factor)
if(LL$success){
  cat(sprintf("Loglikelihood for constant-rates model = %g\n",LL$loglikelihood))
}

# get the likelihood for a model with exponentially decreasing (in forward time) lambda & mu
beta      = 0.01 # exponential decay rate of lambda over time
age_grid  = seq(from=0, to=100, by=0.1) # choose a sufficiently fine age grid
lambda    = 1*exp(beta*age_grid) # define lambda on the age grid
mu        = 0.2*lambda # assume similarly shaped but smaller mu
LL = loglikelihood_hbd( tree, 
                        rho0      = params$rarefaction, 
                        age_grid  = age_grid,
                        lambda    = lambda,
                        mu        = mu)
if(LL$success){
  cat(sprintf("Loglikelihood for exponential-rates model = %g\n",LL$loglikelihood))
}
# }