logistic: Generalised logistic function

In mnirs: Muscle Near-Infrared Spectroscopy Processing and Analysis

Generalised logistic function

Description

Calculate a 4- or 5-parameter logistic (sigmoidal) curve. The 4-parameter symmetric form is fit by analyse_kinetics() with method = "sigmoidal" and shape = "symmetric" (default), and by stats::nls() via the self-starting wrapper SSlogistic().

Usage

logistic(t, A, B, xmid, slope, asym = NULL)

Arguments

t	A numeric vector of the predictor variable (time).
A	A numeric parameter for the starting asymptote of the response variable.
B	A numeric parameter for the ending asymptote of the response variable.
xmid	A numeric parameter for the time at the inflection point (the steepest point) of the curve, in units of the predictor variable t.
slope	A numeric parameter for the response rate dx/dt at the inflection xmid.
asym	A numeric parameter for the asymmetry index of the curve; the fraction of the amplitude (y(xmid) - A) / (B - A) at which the inflection xmid occurs, in ⁠(0, 1)⁠. asym = 0.5 is symmetric and equivalent to the 4-parameter form. If NULL (default), a symmetric 4-parameter model is used.

Details

The 5-parameter Richards form is exported for advanced use directly with stats::nls() but is not used by analyse_kinetics() due to convergence instability. For asymmetric responses, prefer gompertz() / gompertz_left(), which are more stable.

Model equations

Both forms are re-parameterised from the Richards generalised logistic model so xmid is the time at inflection and slope is the response rate dx/dt at the inflection.

• 4-parameter (symmetric): A + (B - A) / (1 + exp(-4 * slope * (t - xmid) / (B - A)))

• 5-parameter (asymmetric): A + (B - A) / (1 + exp(-k * (t - xmid)))^(1 / v) with v = -log(2) / log(asym) and k = 2 * slope * v / ((B - A) * asym).

The inflection is at t = xmid with dx/dt = slope and y(xmid) = A + (B - A) * asym for any asym in ⁠(0, 1)⁠:

• asym = 0.5 (v = 1) collapses to the 4-parameter form.

• asym -> 0 gives an early-acceleration curve (inflection near A).

• asym -> 1 gives a late-acceleration curve (inflection near B).

• asym = 0.368 (1/e) approximates a right-inflection gompertz() curve.

• asym = 0.632 (1 - 1/e) approximates a left-inflection gompertz_left() curve.

Value

A numeric vector of predicted values the same length as the predictor variable t.

See Also

analyse_kinetics(), SSlogistic(), gompertz(), gompertz_left(), sigmoidal_drift(), monoexponential()

Examples

## create an asymmetric logistic curve with random noise
set.seed(15)
t <- 1:60
x <- logistic(t, A = 10, B = 100, xmid = 30, slope = 4, asym = 0.3) +
    rnorm(length(t), 0, 2)
data <- data.frame(t, x)

## 5-parameter fit with the self-starting wrapper
model <- nls(x ~ SSlogistic(t, A, B, xmid, slope, asym), data = data)
summary(model)

y <- predict(model, data)


    if (requireNamespace("ggplot2", quietly = TRUE)) {
        ggplot2::ggplot(data, ggplot2::aes(t, x)) +
            theme_mnirs() +
            ggplot2::geom_point() +
            ggplot2::geom_line(ggplot2::aes(y = y))
    }

mnirs documentation built on Sept. 13, 2026, 1:06 a.m.