Definition

A model is considered linear when it is linear in its parameters, regardless of whether it’s linear in the inputs. Specifically, a model is linear if:

  • It can be expressed as a linear combination of the parameters
  • The output is a weighted sum of basis functions y(x) = Σ wⱼφⱼ(x)

Examples

  1. Simple linear regression: y(x) = w₀ + w₁x
    • Linear in both parameters and input
  2. Polynomial regression: y(x) = w₀ + w₁x + w₂x² + … + wₙxⁿ
    • Linear in parameters (w₀, w₁, w₂, …) but non-linear in input x