A timetree of class "phylo", representing the time-calibrated reconstructed phylogeny of a set of extant and/or extinct species. Tips of the tree are interpreted as terminally sampled lineages, while monofurcating nodes are interpreted as non-terminally sampled lineages, i.e., lineages sampled at some past time point and with subsequently sampled descendants.

Positive numeric, specifying the age of the tree's root. Can be used to define a time offset, e.g. if the last tip was not actually sampled at the present. If NULL, this will be calculated from the tree and it will be assumed that the last tip was sampled at the present.

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 interpreted as the stem age. 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 HBDS 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.

Numeric vector, listing ages in ascending order, on which λ, μ, ψ and κ are fitted and allowed to vary independently. This grid must cover at least the age range from the present (age 0) to oldest_age. If NULL or of length <=1 (regardless of value), then λ, μ, ψ and κ are assumed to be time-independent.

Optional numeric vector, listing ages (in ascending order) at which concentrated sampling attempts (CSAs) occurred. If NULL, it is assumed that no concentrated sampling attempts took place and that all tips were sampled according to the continuous sampling rate psi.

Numeric vector of length Ngrid (=max(1,length(age_grid))), or a single numeric, specifying lower bounds for the fitted speciation rate λ at each point in the age grid. If a single numeric, the same lower bound applies at all ages.

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted speciation rate λ 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.

Numeric vector of length Ngrid, or a single numeric, specifying lower bounds for the fitted extinction rate μ at each point in the age grid. If a single numeric, the same lower bound applies at all ages.

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted extinction rate μ 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.

Numeric vector of length Ngrid, or a single numeric, specifying lower bounds for the fitted Poissonian sampling rate ψ at each point in the age grid. If a single numeric, the same lower bound applies at all ages.

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted Poissonian sampling rate ψ 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.

Numeric vector of length Ngrid, or a single numeric, specifying lower bounds for the fitted retention probability κ at each point in the age grid. If a single numeric, the same lower bound applies at all ages.

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted retention probability κ 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.

Numeric vector of length NCSA (=length(CSA_ages)), or a single numeric, specifying lower bounds for the fitted sampling probabilities ρ_1,..,ρ_m at each concentrated sampling attempt. If a single numeric, the same lower bound applies at all CSAs. Note that, since ρ_1, ρ_2, ... are probabilities, min_CSA_probs should not be negative.

Numeric vector of length NCSA, or a single numeric, specifying upper bounds for the fitted sampling probabilities ρ_1, ρ_2, ... at each concentrated sampling attempt. If a single numeric, the same upper bound applies at all CSAs. Note that, since ρ_1, ρ_2, ... are probabilities, max_CSA_probs should not be greater than 1.

Numeric vector of length NCSA, or a single numeric, specifying lower bounds for the fitted retention probabilities κ_1, κ_2, ... at each concentrated sampling attempt. If a single numeric, the same lower bound applies at all CSAs. Note that, since κ_1, κ_2, ... are probabilities, min_CSA_kappas should not be negative.

Numeric vector of length NCSA, or a single numeric, specifying upper bounds for the fitted sampling probabilities κ_1, κ_2, ... at each concentrated sampling attempt. If a single numeric, the same upper bound applies at all CSAs. Note that, since κ_1, κ_2, .. are probabilities, max_CSA_kappas should not be greater than 1.

Initial guess for λ at each age-grid point. Either NULL (an initial guess will be computed automatically), or a single numeric (guessing the same λ at all ages) or a numeric vector of size Ngrid specifying a separate guess for λ 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.

Initial guess for μ at each age-grid point. Either NULL (an initial guess will be computed automatically), or a single numeric (guessing the same μ at all ages) or a numeric vector of size Ngrid specifying a separate guess for μ 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.

Initial guess for ψ at each age-grid point. Either NULL (an initial guess will be computed automatically), or a single numeric (guessing the same ψ at all ages) or a numeric vector of size Ngrid specifying a separate guess for ψ 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.

Initial guess for κ at each age-grid point. Either NULL (an initial guess will be computed automatically), or a single numeric (guessing the same κ at all ages) or a numeric vector of size Ngrid specifying a separate guess for κ 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.

Initial guess for the ρ_1, ρ_2, ... at each concentrated sampling attempt. Either NULL (an initial guess will be computed automatically), or a single numeric (guessing the same value at every CSA) or a numeric vector of size NCSA specifying a separate guess at each CSA. To omit an initial guess for some but not all CSAs, set their guess values to NA. Guess values are ignored for non-fitted (i.e., fixed) parameters.

Initial guess for the κ_1, κ_2, ... at each concentrated sampling attempt. Either NULL (an initial guess will be computed automatically), or a single numeric (guessing the same value at every CSA) or a numeric vector of size NCSA specifying a separate guess at each CSA. To omit an initial guess for some but not all CSAs, set their guess values to NA. Guess values are ignored for non-fitted (i.e., fixed) parameters.

Optional fixed (i.e. non-fitted) λ values on one or more age-grid points. Either NULL (λ is not fixed anywhere), or a single numeric (λ fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; λ fixed on one or more age-grid points, use NA for non-fixed values) or a numeric vector of the same size as fixed_age_grid (if fixed_age_grid!=NULL, in which case all entries in fixed_lambda must be finite numbers).

Optional fixed (i.e. non-fitted) μ values on one or more age-grid points. Either NULL (μ is not fixed anywhere), or a single numeric (μ fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; μ fixed on one or more age-grid points, use NA for non-fixed values) or a numeric vector of the same size as fixed_age_grid (if fixed_age_grid!=NULL, in which case all entries in fixed_mu must be finite numbers).

Optional fixed (i.e. non-fitted) ψ values on one or more age-grid points. Either NULL (ψ is not fixed anywhere), or a single numeric (ψ fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; ψ fixed on one or more age-grid points, use NA for non-fixed values) or a numeric vector of the same size as fixed_age_grid (if fixed_age_grid!=NULL, in which case all entries in fixed_psi must be finite numbers).

Optional fixed (i.e. non-fitted) κ values on one or more age-grid points. Either NULL (κ is not fixed anywhere), or a single numeric (κ fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; κ fixed on one or more age-grid points, use NA for non-fixed values) or a numeric vector of the same size as fixed_age_grid (if fixed_age_grid!=NULL, in which case all entries in fixed_kappa must be finite numbers).

Optional fixed (i.e. non-fitted) ρ_1, ρ_2, ... values on one or more age-grid points. Either NULL (none of the ρ_1, ρ_2,... are fixed), or a single numeric (ρ_1, ρ_2,... are fixed to the same value at all CSAs) or a numeric vector of size NCSA (one or more of the ρ_1, ρ_2, ... are fixed, use NA for non-fixed values).

Optional fixed (i.e. non-fitted) κ_1, κ_2, ... values on one or more age-grid points. Either NULL (none of the κ_1, κ_2,... are fixed), or a single numeric (κ_1, κ_2,... are fixed to the same value at all CSAs) or a numeric vector of size NCSA (one or more of the κ_1, κ_2, ... are fixed, use NA for non-fixed values).

Optional numeric vector, specifying an age grid on which fixed_lambda, fixed_mu, fixed_psi and fixed_kappa (whichever is provided) are defined instead of on the age_grid. If fixed_age_grid is provided, then each of fixed_lambda, fixed_mu, fixed_psi and fixed_kappa must be defined (i.e. have a finite non-negative value) on every point in fixed_age_grid. Entries in fixed_age_grid must be in ascending order and must cover at least the ages 0 to oldest_age.

This option may be useful if you want to fit some parameters on a coarse grid, but want to specify (fix) some other parameters on a much finer grid. Also note that if fixed_age_grid is used, all parameters lambda, mu, psi and kappa are internally re-interpolated onto fixed_age_grid when evaluating the likelihood; hence, in general fixed_age_grid should be much finer than age_grid. In most situations you would probably want to keep the default fixed_age_grid=NULL.

Logical, specifying whether λ should be assumed constant across the grid, i.e. time-independent. Setting const_lambda=TRUE reduces the number of free (i.e., independently fitted) parameters. If λ is fixed on some grid points (i.e. via fixed_lambda), then only the non-fixed lambdas are assumed to be identical to one another.

Logical, specifying whether μ should be assumed constant across the grid, i.e. time-independent. Setting const_mu=TRUE reduces the number of free (i.e., independently fitted) parameters. If μ is fixed on some grid points (i.e. via fixed_mu), then only the non-fixed mus are assumed to be identical to one another.

Logical, specifying whether ψ should be assumed constant across the grid, i.e. time-independent. Setting const_psi=TRUE reduces the number of free (i.e., independently fitted) parameters. If ψ is fixed on some grid points (i.e. via fixed_psi), then only the non-fixed psis are assumed to be identical to one another.

Logical, specifying whether κ should be assumed constant across the grid, i.e. time-independent. Setting const_kappa=TRUE reduces the number of free (i.e., independently fitted) parameters. If κ is fixed on some grid points (i.e. via fixed_kappa), then only the non-fixed kappas are assumed to be identical to one another.

Logical, specifying whether the ρ_1, ρ_2, ... should be the same across all CSAs. Setting const_CSA_probs=TRUE reduces the number of free (i.e., independently fitted) parameters. If some of the ρ_1, ρ_2, ... are fixed (i.e. via fixed_CSA_probs), then only the non-fixed CSA_probs are assumed to be identical to one another.

Logical, specifying whether the κ_1, κ_2, ... should be the same across all CSAs. Setting const_CSA_kappas=TRUE reduces the number of free (i.e., independently fitted) parameters. If some of the κ_1, κ_2, ... are fixed (i.e. via fixed_CSA_kappas), then only the non-fixed CSA_kappas are assumed to be identical to one another.

Integer between 0 and 3 (inclusive), specifying the polynomial degree of λ, μ, ψ and κ between age-grid points. If 0, then λ, μ, ψ and κ are considered piecewise constant, if 1 they are considered piecewise linear, if 2 or 3 they are considered to be splines 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. The case splines_degree=0 is also known as “skyline” model.

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 generally not recommended. If "auto", the condition is chosen according to the above recommendations.

Positive unitless number, specifying the default relative time step for the ordinary differential equation solvers. Typical values are 0.01-0.001.

Positive unitless number, specifying the relative difference between subsequent simulated and interpolated values, in internally used ODE solvers. Typical values are 1e-2 to 1e-5. A smaller ODE_relative_dy increases interpolation accuracy, but also increases memory requirements and adds runtime (scaling with the tree's age span, not with Ntips).

Non-negative numeric, in units of time, specfying the age radius around a concentrated sampling attempt, within which to assume that sampling events were due to that concentrated sampling attempt. If NULL, this is chosen automatically based on the anticipated scale of numerical rounding errors. Only relevant if concentrated sampling attempts are included.

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.

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.

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.

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).

Integer, specifying the number of parametric bootstraps to perform for estimating standard errors and confidence intervals of estimated parameters. Set to 0 for no bootstrapping.

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.

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.

Optional list, listing combinations of parameter values of particular interest and for which the log-likelihoods should be returned. Every element of this list should itself be a named list, containing the elements lambda, mu, psi and kappa (each being a numeric vector of size NG) as well as the elements CSA_probs and CSA_kappas (each being a numeric vector of size NCSA). This may be used e.g. for diagnostic purposes, e.g. to examine the shape of the likelihood function.

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).

Logical, specifying whether to print detailed information (such as model likelihoods) at every iteration of the fitting routine. For debugging purposes mainly.

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

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.

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

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

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

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

Named list, specifying the fixed and fitted model parameters. This list will contain the elements lambda, mu, psi and kappa (each being a numeric vector of size NG, listing λ,μ, ψ and κ at each age-grid point) as well as the elements CSA_probs and CSA_kappas (each being a numeric vector of size NCSA).

Named list, specifying the guessed model parameters. This list will contain the elements lambda, mu, psi and kappa (each being a numeric vector of size NG) as well as the elements CSA_probs and CSA_kappas (each being a numeric vector of size NCSA). Between grid points λ 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. The same also applies to μ, ψ and κ.

Numeric vector of size NG, the age-grid on which λ, μ, ψ and κ are defined. This will be the same as the provided age_grid, unless the latter was NULL or of length <=1.

Numeric vector of size NCSA, ting listhe ages at which concentrated sampling attempts occurred. This is the same as provided to the function.

Integer, number of free (i.e., independently) fitted parameters. If none of the λ, μ and ρ were fixed, and const_lambda=FALSE and const_mu=FALSE, then NFP will be equal to 2*Ngrid+1.

Integer, the number of data points (sampling and branching events) used for fitting.

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.

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.

Character, specifying what conditioning was root for the likelihood (e.g. "crown" or "stem").

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).

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

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

Named list specifying the standard errors of the parameters, based on parametric bootstrapping. This list will contain the elements lambda, mu, psi and kappa (each being a numeric vector of size NG) as well as the elements CSA_probs and CSA_kappas (each being a numeric vector of size NCSA). Only included if Nbootstraps>0. Note that the standard errors of non-fitted (i.e., fixed) parameters will be zero.

Named list specifying the lower end of the 50% confidence interval (i.e. the 25% quantile) for each parameter, based on parametric bootstrapping. This list will contain the elements lambda, mu, psi and kappa (each being a numeric vector of size NG) as well as the elements CSA_probs and CSA_kappas (each being a numeric vector of size NCSA). Only included if Nbootstraps>0.

Similar to CI50lower, but listing the upper end of the 50% confidence interval (i.e. the 75% quantile) for each parameter. For example, the confidence interval for λ at age age_grid[1] will be between CI50lower$lambda[1] and CI50upper$lambda[1]. Only included if Nbootstraps>0.

Similar to CI50lower, but listing the lower end of the 95% confidence interval (i.e. the 2.5% quantile) for each parameter. Only included if Nbootstraps>0.

Similar to CI50upper, but listing the upper end of the 95% confidence interval (i.e. the 97.5% quantile) for each parameter. Only included if Nbootstraps>0.

Numeric between 0 and 1, estimated consistency of the data with the fitted model. If L denotes the loglikelihood of new data generated by the fitted model (under the same model) and M denotes the expectation of L, then consistency is the probability that |L-M| will be greater or equal to |X-M|, where X is the loglikelihood of the original data under the fitted model. Only returned if Nbootstraps>0. A low consistency (e.g., <0.05) indicates that the fitted model is a poor description of the data. See Lindholm et al. (2019) for background.