Double-exponential function

A piecewise function with two exponential phases. The first exponential phase occurs between t1 and t2, and the second phase continues after t2 with a potentially different growth rate. The function ensures continuity at the transition point but not necessarily smoothness (in derivative).

Usage

fn_exp_exp(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 the two exponential phases. Must be greater than t1.

alpha

The exponential growth rate during the first phase (t1 to t2).

beta

The exponential growth rate after t2.

Value

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

Details

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

The function rises from 0 starting at t1 with exponential growth rate alpha, and transitions to a second exponential phase with rate beta at t2. The value at the transition point is preserved, ensuring continuity.

Examples

library(flexFitR)
plot_fn(
  fn = "fn_exp_exp",
  params = c(t1 = 35, t2 = 55, alpha = 1 / 20, beta = -1 / 30),
  interval = c(0, 108),
  n_points = 2000,
  auc_label_size = 3,
  y_auc_label = 0.2
)