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. 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, xi. 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 fi may be functions with a specified parametric form (for example a polynomial, or an un-penalized regression spline of a 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. Unfortunately, though the Kolmogorov–Arnold representation theorem asserts the existence of a function of this form, it gives no mechanism whereby one could be constructed. Certain constructive proofs exist, but they tend to require highly complicated (i.e. fractal) functions, and thus are not suitable for modeling approaches. Therefore, the Generalized Additive Model drops the outer sum, and demands instead that the function belong to a simpler class. However the success of the neural networks and their universal approximation properties have shown that more general alternatives exist to GAM. where Φ {displaystyle Phi } is a smooth monotonic function. Writing g {displaystyle g} for the inverse of Φ {displaystyle Phi } , this is traditionally written as