Modified Kostiakov Infiltration Model

These functions provide the Modified Kostiakov Infiltration Model equation (kostmod.fun()), as well as the self-starters for the nls() (SSkostmod()) and drc::drm() functions (DRCkostmod()).

Usage

kostmod.fun(x, alfa, beta, fc)

SSkostmod(x, alfa, beta, fc)

DRCkostmod()

Arguments

x

A numeric vector (e.g., time, distance).

alfa

The parameter controlling the shape of the curve. Must be non-zero.

beta

The exponent applied to x. Affects the curvature of the model.

fc

The value of the curve at the final observation.

Value

kostmod.fun() and SSkostmod() return a numeric value, while DRCkostmod() returns a list containing the nonlinear function and the self starter function.

Details

The model is defined as:

$$y = \alpha x^{\beta} + f_c$$

where alfa (\(\alpha\)) controls the shape of the curve, beta (\(\beta\)) is the exponent applied to x and affects curvature, and fc (\(f_c\)) is the value of the curve at the final observation (an additive constant).

See also

Other non-linear functions, self-starters: chaprich.fun(), exp3.fun(), expneg.fun(), expneg2.fun(), invpoly1.fun(), invpoly2.fun(), invpoly3.fun(), varexp.fun(), vargauss.fun(), varsph.fun()

Examples

x <- seq(0.5, 40)
y <- kostmod.fun(x, alfa = 0.3, beta = -0.8, fc = 0.1) + rnorm(40, sd = 0.01)
df <- data.frame(x = x, y = y)
mod = nls(y ~ SSkostmod(x, alfa, beta, fc), data = df)
summary(mod)
#> 
#> Formula: y ~ SSkostmod(x, alfa, beta, fc)
#> 
#> Parameters:
#>       Estimate Std. Error t value Pr(>|t|)    
#> alfa  0.307981   0.008137   37.85   <2e-16 ***
#> beta -0.791292   0.030040  -26.34   <2e-16 ***
#> fc    0.096380   0.003942   24.45   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.009563 on 37 degrees of freedom
#> 
#> Number of iterations to convergence: 4 
#> Achieved convergence tolerance: 4.869e-07
#> 
plot(x, y, cex = 0.8)
lines(x, predict(mod), col = 'blue')


if (FALSE) { # \dontrun{
mod = drc::drm(y ~ x, data = df, fct = DRCkostmod())
summary(mod)
plot(mod, log = "")
} # }