Super-exponential linear function

A piecewise function that models an initial exponential growth phase based on a squared time difference, followed by a linear phase.

Usage

fn_exp2_lin(t, t1, t2, alpha, beta)

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 exponential and linear phases. Must be greater than t1.

alpha

The exponential growth rate controlling the curvature of the exponential phase.

beta

The slope of the linear phase after t2.

Value

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

Details

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

The exponential section rises gradually from 0 at t1 and accelerates as time increases. The linear section starts at t2 with a value matching the end of the exponential phase, ensuring continuity but not necessarily matching the derivative.

Examples

library(flexFitR)
plot_fn(
  fn = "fn_exp2_lin",
  params = c(t1 = 35, t2 = 55, alpha = 1 / 600, beta = -1 / 80),
  interval = c(0, 108),
  n_points = 2000,
  auc_label_size = 3
)