Exponential-linear function

A piecewise function that models a response with an initial exponential growth phase followed by a linear phase. Commonly used to describe processes with rapid early increases that slow into a linear trend, while maintaining continuity.

Usage

fn_exp_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 during 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

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

The exponential segment starts from 0 at t1, and the linear segment continues smoothly from the end of the exponential part. This ensures value continuity at t2, but not necessarily smoothness in slope.

Examples

library(flexFitR)
plot_fn(
  fn = "fn_exp_lin",
  params = c(t1 = 35, t2 = 55, alpha = 1 / 20, beta = -1 / 40),
  interval = c(0, 108),
  n_points = 2000,
  auc_label_size = 3
)