fit_hbds_model_on_grid {castor}

Fit a homogenous birth-death-sampling model on a discrete time grid.

Description

Given a timetree (potentially sampled through time and not necessarily ultrametric), fit a homogenous birth-death-sampling (HBDS) model in which speciation, extinction and lineage sampling occurs at some continuous (Poissonian) rates \lambda, \mu and \psi, which are defined on a fixed grid of discrete time points and assumed to vary polynomially between grid points. Sampled lineages are kept in the pool of extant lineages at some “retention probability” \kappa, which may also depend on time. In addition, this model can include concentrated sampling attempts (CSAs) at a finite set of discrete time points t_1,..,t_m. “Homogenous” refers to the assumption that, at any given moment in time, all lineages exhibit the same speciation/extinction/sampling rates. Every HBDS model is thus defined based on the values that \lambda, \mu, \psi and \kappa take over time, as well as the sampling probabilities \rho_1,..,\rho_m and retention probabilities \kappa_1,..,\kappa_m during the concentrated sampling attempts. This function estimates the values of \lambda, \mu, \psi and \kappa on each grid point, as well as the \rho_1,..,\rho_m and \kappa_1,..,\kappa_m, by maximizing the corresponding likelihood of the timetree. Special cases of this model (when rates are piecewise constant through time) are sometimes known as “birth-death-skyline plots” in the literature (Stadler 2013). In epidemiology, these models are often used to describe the phylogenies of viral strains sampled over the course of the epidemic.

Usage

fit_hbds_model_on_grid( tree, 
                        root_age                = NULL,
                        oldest_age              = NULL,
                        age_grid                = NULL,
                        CSA_ages                = NULL,
                        min_lambda              = 0,
                        max_lambda              = +Inf,
                        min_mu                  = 0,
                        max_mu                  = +Inf,
                        min_psi                 = 0,
                        max_psi                 = +Inf,
                        min_kappa               = 0,
                        max_kappa               = 1,
                        min_CSA_probs           = 0,
                        max_CSA_probs           = 1,
                        min_CSA_kappas          = 0,
                        max_CSA_kappas          = 1,
                        guess_lambda            = NULL,
                        guess_mu                = NULL,
                        guess_psi               = NULL,
                        guess_kappa             = NULL,
                        guess_CSA_probs         = NULL,
                        guess_CSA_kappas        = NULL,
                        fixed_lambda            = NULL,
                        fixed_mu                = NULL,
                        fixed_psi               = NULL,
                        fixed_kappa             = NULL,
                        fixed_CSA_probs         = NULL,
                        fixed_CSA_kappas        = NULL,
                        fixed_age_grid          = NULL,
                        const_lambda            = FALSE,
                        const_mu                = FALSE,
                        const_psi               = FALSE,
                        const_kappa             = FALSE,
                        const_CSA_probs         = FALSE,
                        const_CSA_kappas        = FALSE,
                        splines_degree          = 1,
                        condition               = "auto",
                        ODE_relative_dt         = 0.001,
                        ODE_relative_dy         = 1e-3,
                        CSA_age_epsilon         = NULL,
                        Ntrials                 = 1,
                        max_start_attempts      = 1,
                        Nthreads                = 1,
                        max_model_runtime       = NULL,
                        Nbootstraps             = 0,
                        Ntrials_per_bootstrap   = NULL,
                        fit_control             = list(),
                        focal_param_values      = NULL,
                        verbose                 = FALSE,
                        diagnostics             = FALSE,
                        verbose_prefix          = "")

Arguments

tree

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.

root_age

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.

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 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.

age_grid

Numeric vector, listing ages in ascending order, on which \lambda, \mu, \psi and \kappa 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 \lambda, \mu, \psi and \kappa are assumed to be time-independent.

CSA_ages

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.

min_lambda

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

max_lambda

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted speciation rate \lambda 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_mu

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

max_mu

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted extinction rate \mu 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_psi

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

max_psi

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted Poissonian sampling rate \psi 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_kappa

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

max_kappa

Numeric vector of length Ngrid, or a single numeric, specifying upper bounds for the fitted retention probability \kappa 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_CSA_probs

Numeric vector of length NCSA (=length(CSA_ages)), or a single numeric, specifying lower bounds for the fitted sampling probabilities \rho_1,..,\rho_m at each concentrated sampling attempt. If a single numeric, the same lower bound applies at all CSAs. Note that, since \rho_1, \rho_2, ... are probabilities, min_CSA_probs should not be negative.

max_CSA_probs

Numeric vector of length NCSA, or a single numeric, specifying upper bounds for the fitted sampling probabilities \rho_1, \rho_2, ... at each concentrated sampling attempt. If a single numeric, the same upper bound applies at all CSAs. Note that, since \rho_1, \rho_2, ... are probabilities, max_CSA_probs should not be greater than 1.

min_CSA_kappas

Numeric vector of length NCSA, or a single numeric, specifying lower bounds for the fitted retention probabilities \kappa_1, \kappa_2, ... at each concentrated sampling attempt. If a single numeric, the same lower bound applies at all CSAs. Note that, since \kappa_1, \kappa_2, ... are probabilities, min_CSA_kappas should not be negative.

max_CSA_kappas

Numeric vector of length NCSA, or a single numeric, specifying upper bounds for the fitted sampling probabilities \kappa_1, \kappa_2, ... at each concentrated sampling attempt. If a single numeric, the same upper bound applies at all CSAs. Note that, since \kappa_1, \kappa_2, .. are probabilities, max_CSA_kappas should not be greater than 1.

guess_lambda

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

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

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

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

Initial guess for the \rho_1, \rho_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.

guess_CSA_kappas

Initial guess for the \kappa_1, \kappa_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.

fixed_lambda

Optional fixed (i.e. non-fitted) \lambda values on one or more age-grid points. Either NULL (\lambda is not fixed anywhere), or a single numeric (\lambda fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; \lambda 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).

fixed_mu

Optional fixed (i.e. non-fitted) \mu values on one or more age-grid points. Either NULL (\mu is not fixed anywhere), or a single numeric (\mu fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; \mu 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).

fixed_psi

Optional fixed (i.e. non-fitted) \psi values on one or more age-grid points. Either NULL (\psi is not fixed anywhere), or a single numeric (\psi fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; \psi 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).

fixed_kappa

Optional fixed (i.e. non-fitted) \kappa values on one or more age-grid points. Either NULL (\kappa is not fixed anywhere), or a single numeric (\kappa fixed to the same value at all grid points) or a numeric vector of size Ngrid (if fixed_age_grid=NULL; \kappa 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).

fixed_CSA_probs

Optional fixed (i.e. non-fitted) \rho_1, \rho_2, ... values on one or more age-grid points. Either NULL (none of the \rho_1, \rho_2,... are fixed), or a single numeric (\rho_1, \rho_2,... are fixed to the same value at all CSAs) or a numeric vector of size NCSA (one or more of the \rho_1, \rho_2, ... are fixed, use NA for non-fixed values).

fixed_CSA_kappas

Optional fixed (i.e. non-fitted) \kappa_1, \kappa_2, ... values on one or more age-grid points. Either NULL (none of the \kappa_1, \kappa_2,... are fixed), or a single numeric (\kappa_1, \kappa_2,... are fixed to the same value at all CSAs) or a numeric vector of size NCSA (one or more of the \kappa_1, \kappa_2, ... are fixed, use NA for non-fixed values).

fixed_age_grid

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.

const_lambda

Logical, specifying whether \lambda 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 \lambda 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.

const_mu

Logical, specifying whether \mu 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 \mu 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.

const_psi

Logical, specifying whether \psi 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 \psi 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.

const_kappa

Logical, specifying whether \kappa 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 \kappa 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.

const_CSA_probs

Logical, specifying whether the \rho_1, \rho_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 \rho_1, \rho_2, ... are fixed (i.e. via fixed_CSA_probs), then only the non-fixed CSA_probs are assumed to be identical to one another.

const_CSA_kappas

Logical, specifying whether the \kappa_1, \kappa_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 \kappa_1, \kappa_2, ... are fixed (i.e. via fixed_CSA_kappas), then only the non-fixed CSA_kappas are assumed to be identical to one another.

splines_degree

Integer between 0 and 3 (inclusive), specifying the polynomial degree of \lambda, \mu, \psi and \kappa between age-grid points. If 0, then \lambda, \mu, \psi and \kappa 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.

condition

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.

ODE_relative_dt

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

ODE_relative_dy

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).

CSA_age_epsilon

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.

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.

max_start_attempts

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.

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).

Nbootstraps

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.

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.

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.

focal_param_values

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.

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

Warning: In the absence of concentrated sampling attempts (NCSA=0), and without well-justified a priori constraints on either \lambda, \mu, \psi and/or \kappa, it is generally impossible to reliably estimate \lambda, \mu, \psi and \kappa from timetrees alone. This routine (and any other software that claims to estimate \lambda, \mu, \psi and \kappa solely from timetrees) should thus be treated with great suspicion. Many epidemiological models make the (often reasonable assumption) that \kappa=0; note that even in this case, one generally can't co-estimate \lambda, \mu and \psi from the timetree alone.

It is advised to provide as much information to the function fit_hbds_model_on_grid as possible, including reasonable lower and upper bounds (min_lambda, max_lambda, min_mu, max_mu, min_psi, max_psi, min_kappa, max_kappa) and reasonable parameter guesses. It is also important that the age_grid is sufficiently fine to capture the expected major variations of \lambda, \mu, \psi and \kappa over time, but keep in mind the serious risk of overfitting when age_grid is too fine and/or the tree is too small. The age_grid does not need to be uniform, i.e., you may want to use a finer grid in regions where there's more data (tips) available. If strong lower and upper bounds are not available and fitting takes a long time to run, consider using the option max_model_runtime to limit how much time the fitting allows for each evaluation of the likelihood.

Note that here "age" refers to time before present, i.e., age increases from tips to root and age 0 is present-day. CSAs are enumerated in the order of increasing age, i.e., from the present to the past. Similarly, the age grid specifies time points from the present towards the past.

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.

guess_loglikelihood

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

param_fitted

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 \lambda,\mu, \psi and \kappa at each age-grid point) as well as the elements CSA_probs and CSA_kappas (each being a numeric vector of size NCSA).

param_guess

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 \lambda 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 \mu, \psi and \kappa.

age_grid

Numeric vector of size NG, the age-grid on which \lambda, \mu, \psi and \kappa are defined. This will be the same as the provided age_grid, unless the latter was NULL or of length <=1.

CSA_ages

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

NFP

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

Ndata

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

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.

condition

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

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.

standard_errors

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.

CI50lower

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.

CI50upper

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 \lambda at age age_grid[1] will be between CI50lower$lambda[1] and CI50upper$lambda[1]. Only included if Nbootstraps>0.

CI95lower

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.

CI95upper

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.

consistency

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.

Author(s)

Stilianos Louca

References

T. Stadler, D. Kuehnert, S. Bonhoeffer, A. J. Drummond (2013). Birth-death skyline plot reveals temporal changes of epidemic spread in HIV and hepatitis C virus (HCV). PNAS. 110:228-233.

A. Lindholm, D. Zachariah, P. Stoica, T. B. Schoen (2019). Data consistency approach to model validation. IEEE Access. 7:59788-59796.

See Also

simulate_deterministic_hbds, fit_hbds_model_parametric

Examples

## Not run: 
# define lambda & mu & psi as functions of time
# Assuming an exponentially varying lambda & mu, and a constant psi
time2lambda = function(times){ 2*exp(0.1*times) }
time2mu     = function(times){ 0.1*exp(0.09*times) }
time2psi    = function(times){ rep(0.2, times=length(times)) }

# define concentrated sampling attempts
CSA_times   = c(3,4)
CSA_probs   = c(0.1, 0.2)

# generate random tree based on lambda, mu & psi
# assume that all sampled lineages are removed from the pool (i.e. kappa=0)
time_grid = seq(from=0, to=100, by=0.01)
simul = generate_tree_hbds( max_time    = 5,
                            time_grid   = time_grid,
                            lambda      = time2lambda(time_grid),
                            mu          = time2mu(time_grid),
                            psi         = time2psi(time_grid),
                            kappa       = 0,
                            CSA_times   = CSA_times,
                            CSA_probs   = CSA_probs,
                            CSA_kappas  = 0)
tree     = simul$tree
root_age = simul$root_age
cat(sprintf("Tree has %d tips\n",length(tree$tip.label)))

# Define an age grid on which lambda_function & mu_function shall be fitted
fit_age_grid = seq(from=0,to=root_age,length.out=3)

# Fit an HBDS model on a grid
# Assume that psi is known and that sampled lineages are removed from the pool
# Hence, we only fit lambda & mu & CSA_probs
cat(sprintf("Fitting model to tree..\n"))
fit = fit_hbds_model_on_grid(tree, 
                             root_age           = root_age,
                             age_grid           = fit_age_grid,
                             CSA_ages           = rev(simul$final_time - CSA_times),
                             fixed_psi          = time2psi(simul$final_time-fit_age_grid),
                             fixed_kappa        = 0,
                             fixed_CSA_kappas   = 0,
                             Ntrials            = 4,
                             Nthreads           = 4,
                             Nbootstraps        = 0,
                             verbose            = TRUE,
                             verbose_prefix     = "  ")
if(!fit$success){
    cat(sprintf("ERROR: Fitting failed: %s\n",fit$error))
}else{
    # compare fitted lambda to true lambda
    plot(x=fit$age_grid, 
         y=fit$param_fitted$lambda, 
         type='l', 
         col='#000000', 
         xlim=c(root_age,0),
         xlab='age', 
         ylab='lambda')
    lines(x=simul$final_time-time_grid, 
          y=time2lambda(time_grid), 
          type='l', 
          col='#0000AA')
}


# compare true and fitted model in terms of their LTTs
LTT      = castor::count_lineages_through_time(tree, Ntimes=100, include_slopes=TRUE)
LTT$ages = root_age - LTT$times

cat(sprintf("Simulating deterministic HBDS (true model)..\n"))
age0 = 0.5 # reference age at which to equate LTTs
LTT0 = approx(x=LTT$ages, y=LTT$lineages, xout=age0)$y # tree LTT at age0
fsim = simulate_deterministic_hbds( age_grid        = fit$age_grid,
                                    lambda          = fit$param_fitted$lambda,
                                    mu              = fit$param_fitted$mu,
                                    psi             = fit$param_fitted$psi,
                                    kappa           = fit$param_fitted$kappa,
                                    CSA_ages        = fit$CSA_ages,
                                    CSA_probs       = fit$param_fitted$CSA_probs,
                                    CSA_kappas      = fit$param_fitted$CSA_kappas,
                                    requested_ages  = seq(0,root_age,length.out=200),
                                    age0            = age0,
                                    LTT0            = LTT0,
                                    splines_degree  = 1)
if(!fsim$success){
    cat(sprintf("ERROR: Could not simulate fitted model: %s\n",fsim$error))
    stop()
}
plot(x=LTT$ages, y=LTT$lineages, type='l', col='#0000AA', lwd=2, xlim=c(root_age,0))
lines(x=fsim$ages, y=fsim$LTT, type='l', col='#000000', lwd=2)

## End(Not run)