logff {VGAM} | R Documentation |
Estimating the (single) parameter of the logarithmic distribution.
logff(link = "logit", init.c = NULL, zero = NULL)
link |
Parameter link function for the parameter c,
which lies between 0 and 1.
See |
init.c |
Optional initial value for the c parameter. If given, it often pays to start with a larger value, e.g., 0.95. The default is to choose an initial value internally. |
zero |
Details at |
The logarithmic distribution is a generalized power series distribution that is based specifically on the logarithmic series (scaled to a probability function). Its probability function is f(y) = a * c^y / y, for y=1,2,3,..., where 0 < c < 1, and a = -1 / log(1-c). The mean is a*c/(1-c) (returned as the fitted values) and variance is a*c*(1-a*c)/(1-c)^2.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
The function log
computes the natural logarithm.
In the VGAM library, a link function with option
loge
corresponds to this.
Multiple responses are permitted.
The logarithmic distribution is sometimes confused with the log-series distribution. The latter was used by Fisher et al. for species abundance data and has two parameters.
T. W. Yee
Chapter 7 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.
Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011) Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.
rlog
,
log
,
loge
,
logoff
,
explogff
,
simulate.vlm
.
ldata <- data.frame(y = rlog(n = 1000, prob = logit(0.2, inverse = TRUE))) fit <- vglm(y ~ 1, logff, data = ldata, trace = TRUE, crit = "c") coef(fit, matrix = TRUE) Coef(fit) ## Not run: with(ldata, hist(y, prob = TRUE, breaks = seq(0.5, max(y) + 0.5, by = 1), border = "blue")) x <- seq(1, with(ldata, max(y)), by = 1) with(ldata, lines(x, dlog(x, Coef(fit)[1]), col = "orange", type = "h", lwd = 2)) ## End(Not run) # Example: Corbet (1943) butterfly Malaya data corbet <- data.frame(nindiv = 1:24, ofreq = c(118, 74, 44, 24, 29, 22, 20, 19, 20, 15, 12, 14, 6, 12, 6, 9, 9, 6, 10, 10, 11, 5, 3, 3)) fit <- vglm(nindiv ~ 1, logff, data = corbet, weights = ofreq) coef(fit, matrix = TRUE) chat <- Coef(fit)["c"] pdf2 <- dlog(x = with(corbet, nindiv), prob = chat) print(with(corbet, cbind(nindiv, ofreq, fitted = pdf2 * sum(ofreq))), digits = 1)