Probability Density Under Birth–Death Models
These functions compute the probability density under some birth–death models, that is the probability of obtaining x species after a time t giving how speciation and extinction probabilities vary through time (these may be constant, or even equal to zero for extinction).
dyule(x, lambda = 0.1, t = 1, log = FALSE)
dbd(x, lambda, mu, t, conditional = FALSE, log = FALSE)
dbdTime(x, birth, death, t, conditional = FALSE,
BIRTH = NULL, DEATH = NULL, fast = FALSE)x |
a numeric vector of species numbers (see Details). |
lambda |
a numerical value giving the probability of speciation;
can be a vector with several values for |
mu |
id. for extinction. |
t |
id. for the time(s). |
log |
a logical value specifying whether the probabilities should
be returned log-transformed; the default is |
conditional |
a logical specifying whether the probabilities
should be computed conditional under the assumption of no extinction
after time |
birth, death |
a (vectorized) function specifying how the
speciation or extinction probability changes through time (see
|
BIRTH, DEATH |
a (vectorized) function giving the primitive
of |
fast |
a logical value specifying whether to use faster
integration (see |
These three functions compute the probabilities to observe x
species starting from a single one after time t (assumed to be
continuous). The first function is a short-cut for the second one with
mu = 0 and with default values for the two other arguments.
dbdTime is for time-varying lambda and mu
specified as R functions.
dyule is vectorized simultaneously on its three arguments
x, lambda, and t, according to R's rules of
recycling arguments. dbd is vectorized simultaneously x
and t (to make likelihood calculations easy), and
dbdTime is vectorized only on x; the other arguments are
eventually shortened with a warning if necessary.
The returned value is, logically, zero for values of x out of
range, i.e., negative or zero for dyule or if conditional
= TRUE. However, it is not checked if the values of x are
positive non-integers and the probabilities are computed and returned.
The details on the form of the arguments birth, death,
BIRTH, DEATH, and fast can be found in the links
below.
a numeric vector.
If you use these functions to calculate a likelihood function, it is
strongly recommended to compute the log-likelihood with, for instance
in the case of a Yule process, sum(dyule( , log = TRUE)) (see
examples).
Emmanuel Paradis
Kendall, D. G. (1948) On the generalized “birth-and-death” process. Annals of Mathematical Statistics, 19, 1–15.
x <- 0:10
plot(x, dyule(x), type = "h", main = "Density of the Yule process")
text(7, 0.85, expression(list(lambda == 0.1, t == 1)))
y <- dbd(x, 0.1, 0.05, 10)
z <- dbd(x, 0.1, 0.05, 10, conditional = TRUE)
d <- rbind(y, z)
colnames(d) <- x
barplot(d, beside = TRUE, ylab = "Density", xlab = "Number of species",
legend = c("unconditional", "conditional on\nno extinction"),
args.legend = list(bty = "n"))
title("Density of the birth-death process")
text(17, 0.4, expression(list(lambda == 0.1, mu == 0.05, t == 10)))
## Not run:
### generate 1000 values from a Yule process with lambda = 0.05
x <- replicate(1e3, Ntip(rlineage(0.05, 0)))
### the correct way to calculate the log-likelihood...:
sum(dyule(x, 0.05, 50, log = TRUE))
### ... and the wrong way:
log(prod(dyule(x, 0.05, 50)))
### a third, less preferred, way:
sum(log(dyule(x, 0.05, 50)))
## End(Not run)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.