Modelling and Smoothing Parameter Estimation With Multiple Quadratic Penalties
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2000
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Abstract
Summary Penalized likelihood methods provide a range of practical modelling tools, including spline smoothing, generalized additive models and variants of ridge regression. Selecting the correct weights for penalties is a critical part of using these methods and in the single-penalty case the analyst has several well-founded techniques to choose from. However, many modelling problems suggest a formulation employing multiple penalties, and here general methodology is lacking. A wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimize ||W 1/2 (Xp−y) ||2ρ+Σi=1 m θi p′S i p (p is a parameter vector, X a design matrix, S i a non-negative definite coefficient matrix defining the ith penalty with associated smoothing parameter θi, W a diagonal weight matrix, y a vector of data or pseudodata and ρ an ‘overall’ smoothing parameter included for computational efficiency). This paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross-validation to this problem and how this allows non-linear, generalized linear and linear models to be fitted using multiple penalties, substantially increasing the scope of penalized modelling methods. Examples of non-linear modelling, generalized additive modelling and anisotropic smoothing are given.
| Reference Key |
openalex_W2116394790
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|---|---|
| Authors | Simon N. Wood |
| Journal | Journal of the Royal Statistical Society Series B (Statistical Methodology) |
| Year | 2000 |
| DOI |
10.1111/1467-9868.00240
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| URL | |
| Keywords | Keywords not found |
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