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Activation Functions

@statili/math ·v0.0.1-beta.0 ·1 export

# sigmoid function

packages/math/src/activation.ts:33
signature 
sigmoid(z: number): number
Sigmoid (logistic) activation function: σ(z) = 1 / (1 + e⁻ᶻ).

Maps any real-valued input to the open interval (0, 1), making it suitable for modelling probabilities. The function is smooth and differentiable everywhere, which is why it is used in gradient-based optimisation (e.g. logistic regression, neural networks).

Key properties:

  • σ(0) = 0.5
  • σ(z) → 1 as z → +∞
  • σ(z) → 0 as z → −∞
  • σ'(z) = σ(z) · (1 − σ(z)) (gradient used in back-propagation)

Usage in @statili/stats — the logistic regression implementation applies sigmoid to the linear predictor b₀ + b₁x₁ + … + bₖxₖ to obtain the estimated class probability P(y=1 | x).

Insights derivable by @statili/forge — the logit (inverse sigmoid) of a predicted probability reveals the log-odds: positive logit means the model favours class 1; negative means class 0.

Parameters

NameTypeDescription
z number Any real number (the linear predictor / logit).

Returns

number
A probability in the open interval (0, 1).

Example

sigmoid(0)    // 0.5
sigmoid(2)    // ≈ 0.8808
sigmoid(-2)   // ≈ 0.1192
sigmoid(100)  // ≈ 1  (numerically saturates to 1)