February 17, 2011

Additive Models (Advanced Data Analysis from an Elementary Point of View)

The curse of dimensionality limits the usefulness of fully non-parametric regression in problems with many variables: bias remains under control, but variance grows rapidly with dimensionality. (The number of points required to pin down a [hyper-]surface to within a given tolerance grows exponentially in the number of dimensions.) Parametric models do not have this problem, but have bias and do not let us discover anything about the true function. Structured or constrained non-parametric regression compromises, by adding some bias so as to reduce variance. Additive models are an example, where each input variable has a "partial response function", which add together to get the total regression function; the partial response functions are otherwise arbitrary. Additive models include linear models as a special case, but still evade the curse of dimensionality. Visualization and interpretation of additive models by display of the partial response functions. Fitting additive models is done iteratively, starting with some initial guess about each partial response function and then doing one-dimensional smoothing, so that the guesses correct each other until a self-consistent solution is reached. Incorporation of parametric terms, and interactions by joint smoothing of subsets of variables. Examples in R using the California house-price data. Conclusion: there is hardly ever any reason to prefer linear models to additive ones, and the continued thoughtless use of linear regression is a scandal.

PDF notes, incorporating R examples

Advanced Data Analysis from an Elementary Point of View

Posted at February 17, 2011 21:30 | permanent link

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