Generalized additive model

In statistics, a generalized additive model (GAM) is a generalized linear model in which the linear predictor depends linearly on unknown smooth functions of some predictor variables, and interest focuses on inference about these smooth functions. GAMs were originally developed by Trevor Hastie and Robert Tibshirani to blend properties of generalized linear models with additive models.

The model relates a univariate response variable, Y, to some predictor variables, x_i. An exponential family distribution is specified for Y (for example normal, binomial or Poisson distributions) along with a link function g (for example the identity or log functions) relating the expected value of Y to the predictor variables via a structure such as

The functions f_i may be functions with a specified parametric form (for example a polynomial, or a spline depending on the levels of a factor variable) or may be specified non-parametrically, or semi-parametrically, simply as 'smooth functions', to be estimated by non-parametric means. So a typical GAM might use a scatterplot smoothing function, such as a locally weighted mean, for f₁(x₁), and then use a factor model for f₂(x₂). This flexibility to allow non-parametric fits with relaxed assumptions on the actual relationship between response and predictor, provides the potential for better fits to data than purely parametric models, but arguably with some loss of interpretability.

It had been known since the 1950s (via. the Kolmogorov–Arnold representation theorem) that any multivariate function could be represented as sums and compositions of univariate functions.

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