Logistic Functions

Functions to compute the logistic sigmoid function and its inverse, the logit function.

These functions are fundamental in statistics, machine learning, and probability theory, particularly in logistic regression and neural networks.

Logistic Sigmoid Function:

• logistic_sigmoid(x): \(\sigma(x) = \frac{1}{1 + e^{-x}} = \frac{e^x}{1 + e^x}\)

Logit Function:

• logit(x): $$logit(x) = \log\left(\frac{x}{1 - x}\right)$$

Usage

logistic_sigmoid(x)

logit(x)

Arguments

x

Numeric value for which to compute the functions

Value

A single numeric value with the computed logit or logistic sigmoid function.

See also

Boost Documentation for more details on the mathematical background.

Examples

# Logistic Sigmoid Function
logistic_sigmoid(0)     # Returns 0.5
#> [1] 0.5
logistic_sigmoid(2)     # Returns ~0.881
#> [1] 0.8807971
logistic_sigmoid(-2)    # Returns ~0.119
#> [1] 0.1192029

# Logit Function (inverse of sigmoid)
logit(0.5)              # Returns 0
#> [1] 0
logit(0.7)              # Returns ~0.847
#> [1] 0.8472979
logit(0.881)            # Returns ~2 (inverse of sigmoid(2))
#> [1] 2.001934