# 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[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) = 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).

`dweibull`, `weibull6`, `fitweibull`, `peakwindow`, `CDW`, `cardidates`

## 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

```Loading required package: boot