Linear  (model)

This ambiguous term has two different meanings.

    1) A model is said to be "linear in the variables" if the variables in it appear only in linear combinations.

• Multiple Linear Regression is a typical example of a predictive model that is linear in the variables.
• PCA is a typical example of a descriptive model that is linear in the variables.
• Note that a linear classifier (e.g., Discriminant Analysis) builds piecewise linear boundaries (straight lines, planes, hyperplanes).

    2) A model is said "linear in the parameters" if its  parameters appear only in linear expressions. A polynom of degree higher than one is linear in the parameters, but non linear in the variables.

A Neural network is linear neither in the variables, nor in the parameters.

In very general terms, models that are linear in the parameters are simpler, but cannot generalize  nearly as well as more sophisticated model (e.g. Neural networks) models that are not linear in the parameters.