Exponential Model with Inverse Predictor (Type 3)

These functions provide Exponential Model with Inverse Predictor (Type 3) equation (exp3.fun()), as well as the self-starters for the nls() (SSexp3()) and drc::drm() functions (DRCexp3()).

Usage

exp3.fun(x, a, b)

SSexp3(x, a, b)

DRCexp3()

Arguments

x: A numeric vector of non-zero positive values (e.g., time, distance).
a: A scaling parameter representing the value of the response when 1/x approaches zero (i.e., as x tends to infinity). Must be positive.
b: The rate parameter in the exponent. Controls the curvature of the response; negative values yield a decreasing curve, positive values an increasing curve.

Value

exp3.fun() and SSexp3() return a numeric value, while DRCexp3() returns a list containing the nonlinear function and the self starter function.

Details

The model is defined as:

$$y = a \, e^{b / x}$$

where a is a scaling parameter representing the response as x tends to infinity, and b is the rate parameter in the exponent. Negative values of b yield a decreasing curve; positive values yield an increasing curve.

Examples

x <- seq(1, 20, length.out = 50)
y <- exp3.fun(x, a = 5, b = -3) + rnorm(50, sd = 0.1)
df <- data.frame(x = x, y = y)
mod = nls(y ~ SSexp3(x, a, b), data = df)
summary(mod)
#> 
#> Formula: y ~ SSexp3(x, a, b)
#> 
#> Parameters:
#>   Estimate Std. Error t value Pr(>|t|)    
#> a  5.03009    0.03456  145.54   <2e-16 ***
#> b -3.02623    0.06047  -50.05   <2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> 
#> Residual standard error: 0.09231 on 48 degrees of freedom
#> 
#> Number of iterations to convergence: 3 
#> Achieved convergence tolerance: 6.714e-06
#> 
plot(x, y, cex = 0.8)
lines(x, predict(mod), col = 'blue')


if (FALSE) { # \dontrun{
mod = drc::drm(y ~ x, data = df, fct = DRCexp3())
summary(mod)
plot(mod, log = "")
} # }