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\name{yule.time}
\alias{yule.time}
\title{Fits the Time-Dependent Yule Model}
\usage{
yule.time(phy, birth, BIRTH = NULL, root.time = 0, opti = "nlm", start = 0.01)
}
\arguments{
\item{phy}{an object of class \code{"phylo"}.}
\item{birth}{a (vectorized) function specifying how the birth
(speciation) probability changes through time (see details).}
\item{BIRTH}{a (vectorized) function giving the primitive of
\code{birth}.}
\item{root.time}{a numeric value giving the time of the root node
(time is measured from the past towards the present).}
\item{opti}{a character string giving the function used for
optimisation of the likelihood function. Three choices are possible:
\code{"nlm"}, \code{"nlminb"}, or \code{"optim"}, or any unambiguous
abbreviation of these.}
\item{start}{the initial values used in the optimisation.}
}
\description{
This function fits by maximum likelihood the time-dependent Yule
model. The time is measured from the past (\code{root.time}) to the
present.
}
\details{
The model fitted is a straightforward extension of the Yule model with
covariates (see \code{\link{yule.cov}}). Rather than having
heterogeneity among lineages, the speciation probability is the same
for all lineages at a given time, but can change through time.
The function \code{birth} \emph{must} meet these two requirements: (i)
the parameters to be estimated are the formal arguments; (ii) time is
named \code{t} in the body of the function. However, this is the
opposite for the primitive \code{BIRTH}: \code{t} is the formal
argument, and the parameters are used in its body. See the examples.
It is recommended to use \code{BIRTH} if possible, and required if
speciation probability is constant on some time interval. If this
primitive cannot be provided, a numerical integration is done with
\code{\link[stats]{integrate}}.
The standard-errors of the parameters are computed with the Hessian of
the log-likelihood function.
}
\value{
An object of class \code{"yule"} (see \code{\link{yule}}).
}
\author{Emmanuel Paradis}
\references{
Hubert, N., Paradis, E., Bruggemann, H. and Planes, S. (2011) Community
assembly and diversification in Indo-Pacific coral reef
fishes. \emph{Ecology and Evolution}, \bold{1}, 229--277.
}
\seealso{
\code{\link{branching.times}}, \code{\link{ltt.plot}},
\code{\link{birthdeath}}, \code{\link{yule}}, \code{\link{yule.cov}},
\code{\link{bd.time}}
}
\examples{
### define two models...
birth.logis <- function(a, b) 1/(1 + exp(-a*t - b)) # logistic
birth.step <- function(l1, l2, Tcl) { # 2 rates with one break-point
ans <- rep(l1, length(t))
ans[t > Tcl] <- l2
ans
}
### ... and their primitives:
BIRTH.logis <- function(t) log(exp(-a*t) + exp(b))/a + t
BIRTH.step <- function(t)
{
out <- numeric(length(t))
sel <- t <= Tcl
if (any(sel)) out[sel] <- t[sel] * l1
if (any(!sel)) out[!sel] <- Tcl * l1 + (t[!sel] - Tcl) * l2
out
}
data(bird.families)
### fit both models:
yule.time(bird.families, birth.logis)
yule.time(bird.families, birth.logis, BIRTH.logis) # same but faster
\dontrun{yule.time(bird.families, birth.step)} # fails
yule.time(bird.families, birth.step, BIRTH.step,
opti = "nlminb", start = c(.01, .01, 100))
}
\keyword{models}
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