sigmoid: Sigmoid Function

In pracma: Practical Numerical Math Functions

Sigmoid Function

Description

Sigmoid function (aka sigmoidal curve or logistic function).

Usage

sigmoid(x, a = 1, b = 0)
logit(x, a = 1, b = 0)

Arguments

x	numeric vector.
a, b	parameters.

Details

The sigmoidal function with parameters a,b is the function

y = 1/(1 + e^{-a (x-b)})

The sigmoid function is also the solution of the ordinary differentialequation

y' = y (1-y)

with y(0) = 1/2 and has an indefinite integral \ln(1 + e^x).

The logit function is the inverse of the sigmoid function and is (therefore) omly defined between 0 and 1. Its definition is

y = b + 1/a log(x/(1-x))

The parameters must be scalars; if they are vectors, only the first component will be taken.

Value

Numeric/complex scalar or vector.

Examples

x <- seq(-6, 6, length.out = 101)
y1 <- sigmoid(x)
y2 <- sigmoid(x, a = 2)
## Not run: 
plot(x, y1, type = "l", col = "darkblue", 
        xlab = "", ylab = "", main = "Sigmoid Function(s)")
lines(x, y2, col = "darkgreen")
grid()
## End(Not run)

# The slope in 0 (in x = b) is a/4
# sigmf with slope 1 and range [-1, 1].
sigmf <- function(x) 2 * sigmoid(x, a = 2) - 1

# logit is the inverse of the sigmoid function
x <- c(-0.75, -0.25, 0.25, 0.75)
y <- sigmoid(x)
logit(y)        #=> -0.75 -0.25  0.25  0.75

pracma documentation built on Nov. 10, 2023, 1:14 a.m.