weibull4: Four-Parametric Weibull Function

In cardidates: Identification of Cardinal Dates in Ecological Time Series

Four-Parametric Weibull Function

Description

Four-parametric Weibull function and its definite integral.

Usage

fweibull4(x, p)

aweibull4(p, lower, upper)

Arguments

x	vector of function arguments,
p	vector of function parameters with: p[1] vertical shift (zero offset), p[2] vertical scaling, p[3] shape parameter a of dweibull, and p[4] 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) = p_1 + p_2 (p_3/p_4) {(x/p_4)}^{p_3-1} \exp (-{(x/b)}^{p_3})

for x \ge 0.

Value

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

See Also

dweibull, weibull6, fitweibull, peakwindow, CDW, cardidates

Examples

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))

cardidates documentation built on Oct. 8, 2023, 1:06 a.m.