weibull4: Four-Parametric Weibull Function In cardidates: Identification of Cardinal Dates in Ecological Time Series

Description

Four-parametric Weibull function and its definite integral.

Usage

 1 2 3 fweibull4(x, p) aweibull4(p, lower, upper)

Arguments

 x vector of function arguments, p vector of function parameters with: p vertical shift (zero offset), p vertical scaling, p shape parameter a of dweibull, and p scale parameter b of dweibull, lower lower limit of the cumulative (integrated) function, upper upper limit of the cumulative (integrated) function.

Details

The four-parametric Weibull function is essentially based on the Weibull density function dweibull and its integral by the Weibull distribution function pweibull with two additional parameters for y scaling and zero offset. It can be given by:

f(x) = p1 + p2 (p3/p4) (x/p4)^(p3-1) exp(- (x/p4)^p3)

for x ≥ 0.

Value

fweibull4 gives the Weibull function and aweibull4 its definite integral (cumulative sum or area under curve).

Examples

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 x <- seq(0, 5, 0.02) plot(x, fweibull4(x, c(0, 1, 2, 1)), type = "l", ylim = c(0, 2)) points(x, dweibull(x, 2, 1), pch = "+") ## identical to former ## shape lines(x, fweibull4(x, c(0, 2, 1.5, 1)), type = "l", col = "orange") ## horizontal scaling lines(x, fweibull4(x, c(0, 2, 2, 2)), type = "l", col = "green") ## shifting lines(x, fweibull4(x, c(1, 1, 2, 1)), type = "l", col = "blue") ## vertical scaling lines(x, fweibull4(x, c(0, 2, 2, 1)), type = "l", col = "red") ## definite integral p <- c(0, 1, 2, 2) plot(x, aweibull4(p, lower = 0, upper = x)) p <- c(0.1, 1, 2, 2) plot(x, aweibull4(p, lower = 0, upper = x))

Example output   