First-Order Inverse Polynomial Model

These functions provide First-Order Inverse Polynomial Model equation (invpoly1.fun()), as well as the self-starters for the nls() (SSinvpoly1()) and drc::drm() functions (DRCinvpoly1()).

Usage

invpoly1.fun(x, a, b)

SSinvpoly1(x, a, b)

DRCinvpoly1()

Arguments

x

A numeric vector of non-zero positive values (e.g., time, distance).

a

The coefficient of the \(1/x\) term in the linearised form. Controls the initial rise of the curve near the origin.

b

The coefficient of the \(x\) term in the denominator. The reciprocal \(1/b\) gives the asymptote of the response as \(x \to \infty\).

Value

invpoly1.fun() and SSinvpoly1() return a numeric value, while DRCinvpoly1() returns a list containing the nonlinear function and the self starter function.

Details

The model is defined as:

$$y = \frac{x}{a + b x}$$

where a is the coefficient of the \(1/x\) term in the linearised form (controls the initial rise near the origin) and b is the coefficient of the linear term in the denominator. The reciprocal \(1/b\) gives the asymptote as \(x \to \infty\).

The curve passes through the origin and approaches the asymptote \(1/b\) as \(x \to \infty\).

See also

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

Examples

x <- seq(1, 20, length.out = 50)
y <- invpoly1.fun(x, a = 2, b = 0.5) + rnorm(50, sd = 0.05)
df <- data.frame(x = x, y = y)
mod = nls(y ~ SSinvpoly1(x, a, b), data = df)
summary(mod)
#> 
#> Formula: y ~ SSinvpoly1(x, a, b)
#> 
#> Parameters:
#>   Estimate Std. Error t value Pr(>|t|)    
#> a 2.018758   0.065033   31.04   <2e-16 ***
#> b 0.500286   0.006379   78.43   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.04561 on 48 degrees of freedom
#> 
#> Number of iterations to convergence: 3 
#> Achieved convergence tolerance: 5.63e-08
#> 
plot(x, y, cex = 0.8)
lines(x, predict(mod), col = 'blue')


if (FALSE) { # \dontrun{
mod = drc::drm(y ~ x, data = df, fct = DRCinvpoly1())
summary(mod)
plot(mod, log = "")
} # }