LNRE {zipfR} | R Documentation |
Type density g (tdlnre
), type distribution G
(tplnre
), type quantiles G^{-1} (tqlnre
),
probability density f (dlnre
), distribution function
F (plnre
), quantile function F^{-1} (qlnre
),
logarithmic type and probability densities (ltdlnre
and
ldlnre
), and random sample generation (rlnre
) for LNRE
models.
tdlnre(model, x, ...) tplnre(model, q, lower.tail=FALSE, ...) tqlnre(model, p, lower.tail=FALSE, ...) dlnre(model, x, ...) plnre(model, q, lower.tail=TRUE, ...) qlnre(model, p, lower.tail=TRUE, ...) ltdlnre(model, x, base=10, log.x=FALSE, ...) ldlnre(model, x, base=10, log.x=FALSE, ...) rlnre(model, n, ...)
model |
an object belonging to a subclass of lnre ,
representing a LNRE model |
x |
vector of type probabilities pi for which the density function is evaluated |
q |
vector of type probability quantiles, i.e. threshold values \rho on the type probability axis |
p |
vector of tail probabilities |
lower.tail |
if TRUE , lower tail probabilities or type
counts are returned / expected in the p argument. Note that
the defaults differ for distribution function and type distribution,
and see "Details" below. |
base |
positive number, the base with respect to which the log-transformation is peformed (see "Details" below) |
log.x |
if TRUE , the values passed in the argument
x are assumed to be logarithmic, i.e. \log_a \pi |
n |
size of random sample to generate. If length(n) > 1 ,
the length is taken to be the number required. |
... |
further arguments are passed through to the method implementations (currently unused) |
Note that the order in which arguments are specified differs from the
analogous functions for common statistical distributions in the R
standard library. In particular, the LNRE model model
always
has to be given as the first parameter so that R can dispatch the
function call to an appropriate method implementation for the chosen
LNRE model.
Some of the functions may not be available for certain types of LNRE
models. In particular, no analytical solutions are known for the
distribution and quantiles of GIGP models, so the functions
tplnre
, tqlnre
, plnre
, qlnre
and
rlnre
(which depends on qlnre
and tplnre
) are not
implemented for objects of class lnre.gigp
.
The default tails differ for the distribution function (plnre
,
qlnre
) and the type distribution (tplnre
,
tqlnre
), in order to match the definitions of F(\rho) and
G(\rho). While the distribution function defaults to lower
tails (lower.tail=TRUE
, corresponding to F and
F^{-1}), the type distribution defaults to upper tails
(lower.tail=FALSE
, corresponding to G and G^{-1}).
Unlike for standard distriutions, logarithmic tail probabilities
(log.p=TRUE
) are not provided for the LNRE models, since here
the focus is usually on the bulk of the distribution rather than on
the extreme tails.
The log-transformed density functions f* and g*
returned by ldlnre
and ltdlnre
, respectively, can be
understood as probability and type densities for \log_a \pi
instead of \pi, and are useful for visualization of LNRE
populations (with a logarithmic scale for the parameter \pi on
the x-axis). For example,
G(log_a rho) = integral_{log_a rho}^{0} g*(t) dt
For rnlre
, a factor of length n
, representing a random
sample from the population described by the specified LNRE model.
For all other functions, a vector of non-negative numbers of the same
length as the second argument (x
, p
or q
).
tdlnre
returns the type density g(\pi) for the values of
\pi specified in the vector x
. tplnre
returns the
type distribution G(\rho) (default) or its complement
1-G(\rho) (if lower.tail=TRUE
), for the values of
\rho specified in the vector q
. tqlnre
returns
type quantiles, i.e. the inverse G^{-1}(x) (default) or
G^{-1}(S-x) (if lower.tail=TRUE
) of the type
distribution, for the type counts x specified in the vector
p
.
dlnre
returns the probability density f(\pi) for the
values of \pi specified in the vector x
. plnre
returns the distribution function F(\rho) (default) or its
complement 1-F(\rho) (if lower.tail=FALSE
), for the
values of \rho specified in the vector q
. qlnre
returns quantiles, i.e. the inverse F^{-1}(p) (default) or
F^{-1}(1-p) (if lower.tail=FALSE
) of the distribution
function, for the probabilities p specified in the vector
p
.
ldlnre
and ltdlnre
compute logarithmically transformed
versions of the probability and type density functions, respectively,
taking logarithms with respect to the base a specified in the
base
argument (default: a=10). See "Details" above for
more information.
lnre
for more information about LNRE models and how to
initialize them
random samples generated with rnlre
can be further processed
with the functions vec2tfl
, vec2spc
and
vec2vgc
## define ZM and fZM LNRE models ZM <- lnre("zm", alpha=.8, B=1e-3) FZM <- lnre("fzm", alpha=.8, A=1e-5, B=.05) ## random samples from the two models head(table(rlnre(ZM, 10000))) head(table(rlnre(FZM, 10000))) ## plot logarithmic type density functions x <- 10^seq(-6, 1, by=.01) # pi = 10^(-6) .. 10^(-1) y.zm <- ltdlnre(ZM, x) y.fzm <- ltdlnre(FZM, x) ## Not run: plot(x, y.zm, type="l", lwd=2, col="red", log="x", ylim=c(0,14000)) ## Not run: lines(x, y.fzm, lwd=2, col="blue") ## Not run: zipfR.legend(2, legend=c("ZM", "fZM"), lwd=3, col=c("red", "blue")) ## probability pi_k of k-th type according to FZM model k <- 10 plnre(FZM, tqlnre(FZM, k-1)) - plnre(FZM, tqlnre(FZM, k)) ## number of types with pi >= 1e-6 tplnre(ZM, 1e-6) ## lower tail fails for infinite population size ## Not run: tplnre(ZM, 1e-3, lower=TRUE) ## total probability mass assigned to types with pi <= 1e-6 plnre(ZM, 1e-6)