flexFitR (version 1.2.2)

fn_exp2_exp: Super-exponential exponential function

Description

A piecewise function that models an initial exponential phase with quadratic time dependence, followed by a second exponential phase with a different growth rate.

Usage

fn_exp2_exp(t, t1, t2, alpha, beta)

Value

A numeric vector of the same length as t, representing the function values.

Arguments

t

A numeric vector of input values (e.g., time).

t1

The onset time of the response. The function is 0 for all values less than t1.

t2

The transition time between the two exponential phases. Must be greater than t1.

alpha

The curvature-controlled exponential rate during the first phase (t1 to t2).

beta

The exponential growth rate after t2.

Details

$$ f(t; t_1, t_2, \alpha, \beta) = \begin{cases} 0 & \text{if } t < t_1 \\ e^{\alpha \cdot (t - t_1)^2} - 1 & \text{if } t_1 \leq t \leq t_2 \\ \left(e^{\alpha \cdot (t_2 - t_1)^2} - 1\right) \cdot e^{\beta \cdot (t - t_2)} & \text{if } t > t_2 \end{cases} $$

Examples

library(flexFitR)
plot_fn(
  fn = "fn_exp2_exp",
  params = c(t1 = 35, t2 = 55, alpha = 1 / 600, beta = -1 / 30),
  interval = c(0, 108),
  n_points = 2000,
  auc_label_size = 3,
  y_auc_label = 0.15
)