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28 Pages

### Splines

Course: ISYE 7406, Fall 2008
School: Georgia Tech
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Word Count: 444

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Polynomials Piecewise and Splines Kwok-Leung Tsui Industrial &amp; Systems Engineering Georgia Institute of Technology 2/18/2009 1 General Models 2/18/2009 2 Examples of Piecewise Polynomials 2/18/2009 3 Piecewise Polynomials &amp; Splines 2/18/2009 4 Examples of Piecewise Polynomials 2/18/2009 5 Cubic Splines 2/18/2009 6 Model Fitting 2/18/2009 7 Number of Parameters in Splines...

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Polynomials Piecewise and Splines Kwok-Leung Tsui Industrial & Systems Engineering Georgia Institute of Technology 2/18/2009 1 General Models 2/18/2009 2 Examples of Piecewise Polynomials 2/18/2009 3 Piecewise Polynomials & Splines 2/18/2009 4 Examples of Piecewise Polynomials 2/18/2009 5 Cubic Splines 2/18/2009 6 Model Fitting 2/18/2009 7 Number of Parameters in Splines 2/18/2009 8 Natural Cubic Splines 2/18/2009 9 Dimension Reduction of Splines 2/18/2009 10 Filtering and Feature Extraction 2/18/2009 11 South African Heart Disease Example 2/18/2009 12 South African Heart Disease Example 2/18/2009 13 Smoothing Splines 2/18/2009 14 Smoothing Splines The over-parametrization problem is overcome by the penalty function. 2/18/2009 15 Fitting Smoothing Splines 2/18/2009 16 Degrees of Freedom of Smoothing Splines 2/18/2009 17 Example of Smoothing Splines 2/18/2009 18 Multi-dimensional Splines 2/18/2009 19 Other Smoothing Methods Generalized Additive Models (GAM) Multivariate Adaptive Regression Splines (MARS) Kernel Methods 2/18/2009 20 Generalized Additive Models Additive models (Stone, 1985). Sj : smooth nonparametric function The curse of dimensionality is avoided since each of the individual additive terms is estimated using a univariate smoother Estimates of the individual terms explain how the dependent variable changes with the corresponding independent variables. Generalized additives model (Hastie and Tibshirani, 1990). The mean of the dependent variable depends on an additive predictor through a nonlinear link function. Nonparametric log-linear models for Poisson data ij = n i + + j , 2/18/2009 log( ij ) = log(n ) + log( i + ) + log( + j ) 21 Generalized Additive Models Classical link function g g (E[Y|X]) = E [Y|X]; identity link. (E[Y|X]) g = logit [Y|X]; logit link. g (E[Y|X]) = probit [Y|X]; probit link. g (E[Y|X]) = log [Y|X]; log-linear or log-additive models for Poisson count data. example ( Additive logistic regression) log Pr (Y = 1 | X ) = + s1 ( X 1 ) + ... + s p (X p ) Pr (Y = 0 | X ) 2/18/2009 22 Generalized Additive Models Source: Snell,E.J. and Simpson,H.R (1991) A Handbook of Genstat Analysis A fit using a linear model A fit using a generalized additive model 2/18/2009 23 Generalized Additive Models (Summary) What it is ? A method of fitting a smooth relationship between two or more variables through a scatterplot of data points (cubic smoothing spline or kernel smoother) Provide an algorithm for simultaneously estimating all p functions. When to use ? The relationship between the variables is expected to be of a complex form, not easily fitted by standard linear or non-linear models. no a priori reason for using a particular model. Difference between generalized linear models and generalized additive models Smoothing functions are used instead of regression parameters. example: gen...

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