fit_sbm_const {castor}

Fit a phylogeographic Spherical Brownian Motion model.

Description

Given one or more rooted phylogenetic trees and geographic coordinates (latitudes & longitudes) for the tips of each tree, this function estimates the diffusivity of a Spherical Brownian Motion (SBM) model for the evolution of geographic location along lineages (Perrin 1928; Brillinger 2012). Estimation is done via maximum-likelihood and using independent contrasts between sister lineages.

Usage

fit_sbm_const(trees, 
        tip_latitudes, 
        tip_longitudes, 
        radius,
        phylodistance_matrixes  = NULL,
        clade_states            = NULL,
        planar_approximation    = FALSE,
        only_basal_tip_pairs    = FALSE,
        only_distant_tip_pairs  = FALSE,
        min_MRCA_time           = 0,
        max_MRCA_age            = Inf,
        max_phylodistance       = Inf,
        no_state_transitions    = FALSE,
        only_state              = NULL,
        min_diffusivity         = NULL,
        max_diffusivity         = NULL,
        Nbootstraps             = 0, 
        NQQ                     = 0,
        SBM_PD_functor          = NULL,
        focal_diffusivities     = NULL)

Arguments

trees

Either a single rooted tree or a list of rooted trees, of class "phylo". The root of each tree is assumed to be the unique node with no incoming edge. Edge lengths are assumed to represent time intervals or a similarly interpretable phylogenetic distance. When multiple trees are provided, it is either assumed that their roots coincide in time (if align_trees_at_root=TRUE) or that each tree's youngest tip was sampled at present day (if align_trees_at_root=FALSE).

tip_latitudes

Numeric vector of length Ntips, or a list of vectors, listing latitudes of tips in decimal degrees (from -90 to 90). If trees is a list of trees, then tip_latitudes should be a list of vectors of the same length as trees, listing tip latitudes for each of the input trees.

tip_longitudes

Numeric vector of length Ntips, or a list of vectors, listing longitudes of tips in decimal degrees (from -180 to 180). If trees is a list of trees, then tip_longitudes should be a list of vectors of the same length as trees, listing tip longitudes for each of the input trees.

radius

Strictly positive numeric, specifying the radius of the sphere. For Earth, the mean radius is 6371 km.

phylodistance_matrixes

Numeric matrix, or a list of numeric matrixes, listing phylogenetic distances between tips for each tree. If trees is a list of trees, then phylodistance_matrixes should be a list of the same length as trees, whose n-th element should be a numeric matrix comprising as many rows and columns as there are tips in the n-th tree; the entry phylodistance_matrixes[[n]][i,j] is the phylogenetic distance between tips i and j in tree n. If trees is a single tree, then phylodistance_matrixes can be a single numeric matrix. If NULL (default), phylogenetic distances between tips are calculated based on the provided trees, otherwise phylogenetic distances are taken from phylodistance_matrixes; in the latter case the trees are only used for the topology (determining tip pairs for independent contrasts), but not for calculating phylogenetic distances.

clade_states

Either NULL, or an integer vector of length Ntips+Nnodes, or a list of integer vectors, listing discrete states of every tip and node in the tree. If trees is a list of trees, then clade_states should be a list of vectors of the same length as trees, listing tip and node states for each of the input trees. For example, clade_states[[2]][10] specifies the state of the 10-th tip or node in the 2nd tree. States may be, for example, geographic regions, sub-types, discrete traits etc, and can be used to restrict independent contrasts to tip pairs within the same state (see option no_state_transitions).

planar_approximation

Logical, specifying whether to estimate the diffusivity based on a planar approximation of the SBM model, i.e. by assuming that geographic distances between tips are as if tips are distributed on a 2D cartesian plane. This approximation is only accurate if geographical distances between tips are small compared to the sphere's radius.

only_basal_tip_pairs

Logical, specifying whether to only compare immediate sister tips, i.e., tips connected through a single parental node.

only_distant_tip_pairs

Logical, specifying whether to only compare tips at distinct geographic locations.

min_MRCA_time

Numeric, specifying the minimum allowed time (distance from root) of the most recent common ancestor (MRCA) of sister tips considered in the fitting. In other words, an independent contrast is only considered if the two sister tips' MRCA has at least this distance from the root. Set min_MRCA_time<=0 to disable this filter.

max_MRCA_age

Numeric, specifying the maximum allowed age (distance from youngest tip) of the MRCA of sister tips considered in the fitting. In other words, an independent contrast is only considered if the two sister tips' MRCA has at most this age (time to present). Set max_MRCA_age=Inf to disable this filter.

max_phylodistance

Numeric, maximum allowed geodistance for an independent contrast to be included in the SBM fitting. Set max_phylodistance=Inf to disable this filter.

no_state_transitions

Logical, specifying whether to omit independent contrasts between tips whose shortest connecting paths include state transitions. If TRUE, only tips within the same state and with no transitions between them (as specified in clade_states) are compared. If TRUE, then clade_states must be provided.

only_state

Optional integer, specifying the state in which tip pairs (and their connecting ancestral nodes) must be in order to be considered. If specified, then clade_states must be provided.

min_diffusivity

Non-negative numeric, specifying the minimum possible diffusivity. If NULL, this is automatically chosen.

max_diffusivity

Non-negative numeric, specifying the maximum possible diffusivity. If NULL, this is automatically chosen.

Nbootstraps

Non-negative integer, specifying an optional number of parametric bootstraps to performs for estimating standard errors and confidence intervals.

NQQ

Integer, optional number of simulations to perform for creating QQ plots of the theoretically expected distribution of geodistances vs. the empirical distribution of geodistances (across independent contrasts). The resolution of the returned QQ plot will be equal to the number of independent contrasts used for fitting. If <=0, no QQ plots will be calculated.

SBM_PD_functor

SBM probability density functor object. Used internally for efficiency and for debugging purposes, and should be kept at its default value NULL.

focal_diffusivities

Optional numeric vector, listing diffusivities of particular interest and for which the log-likelihoods should be returned. This may be used e.g. for diagnostic purposes, e.g. to see how "sharp" the likelihood peak is at the maximum-likelihood estimate.

Details

For short expected transition distances this function uses the approximation formula by Ghosh et al. (2012). For longer expected transition distances the function uses a truncated approximation of the series representation of SBM transition densities (Perrin 1928). It is assumed that tips are sampled randomly without any biases for certain geographic regions. If you suspect strong geographic sampling biases, consider using the function fit_sbm_geobiased_const.

This function can use multiple trees to fit the diffusivity under the assumption that each tree is an independent realization of the same SBM process, i.e. all lineages in all trees dispersed with the same diffusivity.

If edge.length is missing from one of the input trees, each edge in the tree is assumed to have length 1. The tree may include multifurcations as well as monofurcations, however multifurcations are internally expanded into bifurcations by adding dummy nodes.

Value

A list with the following elements:

success

Logical, indicating whether the fitting was successful. If FALSE, then an additional return variable, error, will contain a description of the error; in that case all other return variables may be undefined.

diffusivity

Numeric, the estimated diffusivity, in units distance^2/time. Distance units are the same as used for the radius, and time units are the same as the tree's edge lengths. For example, if the radius was specified in km and edge lengths are in Myr, then the estimated diffusivity will be in km^2/Myr.

loglikelihood

Numeric, the log-likelihood of the data at the estimated diffusivity.

Ncontrasts

Integer, number of independent contrasts (i.e., tip pairs) used to estimate the diffusivity. This is the number of independent data points used.

phylodistances

Numeric vector of length Ncontrasts, listing the phylogenetic distances of the independent contrasts used in the fitting.

geodistances

Numeric vector of length Ncontrasts, listing the geographical distances of the independent contrasts used in the fitting.

focal_loglikelihoods

Numeric vector of the same length as focal_diffusivities, listing the log-likelihoods for the diffusivities provided in focal_diffusivities.

standard_error

Numeric, estimated standard error of the estimated diffusivity, based on parametric bootstrapping. Only returned if Nbootstraps>0.

CI50lower

Numeric, lower bound of the 50% confidence interval for the estimated diffusivity (25-75% percentile), based on parametric bootstrapping. Only returned if Nbootstraps>0.

CI50upper

Numeric, upper bound of the 50% confidence interval for the estimated diffusivity, based on parametric bootstrapping. Only returned if Nbootstraps>0.

CI95lower

Numeric, lower bound of the 95% confidence interval for the estimated diffusivity (2.5-97.5% percentile), based on parametric bootstrapping. Only returned if Nbootstraps>0.

CI95upper

Numeric, upper bound of the 95% confidence interval for the estimated diffusivity, based on parametric bootstrapping. Only returned 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.

QQplot

Numeric matrix of size Ncontrasts x 2, listing the computed QQ-plot. The first column lists quantiles of geodistances in the original dataset, the 2nd column lists quantiles of hypothetical geodistances simulated based on the fitted model.

SBM_PD_functor

SBM probability density functor object. Used internally for efficiency and for debugging purposes.

Author(s)

Stilianos Louca

References

F. Perrin (1928). Etude mathematique du mouvement Brownien de rotation. 45:1-51.

D. R. Brillinger (2012). A particle migrating randomly on a sphere. in Selected Works of David Brillinger. Springer.

A. Ghosh, J. Samuel, S. Sinha (2012). A Gaussian for diffusion on the sphere. Europhysics Letters. 98:30003.

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

S. Louca (2021). Phylogeographic estimation and simulation of global diffusive dispersal. Systematic Biology. 70:340-359.

See Also

fit_sbm_geobiased_const, simulate_sbm, fit_sbm_parametric, fit_sbm_linear, fit_sbm_on_grid

Examples

## Not run: 
# generate a random tree
tree = generate_random_tree(list(birth_rate_intercept=1),max_tips=500)$tree

# simulate SBM on the tree
D = 1e4
simulation = simulate_sbm(tree, radius=6371, diffusivity=D)

# fit SBM on the tree
fit = fit_sbm_const(tree,simulation$tip_latitudes,simulation$tip_longitudes,radius=6371)
cat(sprintf('True D=%g, fitted D=%g\n',D,fit$diffusivity))

## End(Not run)