Basis expansions and regularization
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Basis Expansions and Regularization. Based on Chapter 5 of Hastie, Tibshirani and Friedman (Prepared by David Madigan). Basis Expansions for Linear Models. Here the h m ’s might be:. h m ( X )= X m , m =1,…, p recovers the original model h m ( X )= X j 2 or h m ( X )= X j X k

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Basis expansions and regularization l.jpg
Basis Expansions and Regularization

Based on Chapter 5 of

Hastie, Tibshirani and Friedman

(Prepared by David Madigan)

Basis expansions for linear models l.jpg
Basis Expansions for Linear Models

Here the hm’s might be:

  • hm(X)=Xm, m=1,…,p recovers the original model

  • hm(X)=Xj2 orhm(X)= XjXk

  • hm(X)=I(LmXk Um),

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Regression Splines

Bottom left panel uses:

Number of parameters = (3 regions) X (2 params per region)

- (2 knots X 1 constraint per knot)

= 4

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Cubic Spline

continuous first and second derivatives

Number of parameters = (3 regions) X (4 params per region)

- (2 knots X 3 constraints per knot)

= 6Knot discontinuity essentially invisible to the human eye

Natural cubic spline l.jpg
Natural Cubic Spline

Adds a further constraint that the fitted function is linear beyond the boundary knots

A natural cubic spline model with K knots is represented by K basis functions:

Each of these basis functions has zero 2nd and 3rd derivative outside the boundary knots

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Natural Cubic Spline Models

Can use these ideas in, for example, regression models.

For example, if you use 4 knots and hence 4 basis functions per predictor variable, then simply fit logistic regression model with four times the number of predictor variables…

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Smoothing Splines

Consider this problem: among all functions f(x) with two continuous derivatives, find the one that minimizes the penalized residual sum of squares:

smoothing parameter

=0 : f can be any function that interpolates the data

=infinity : least squares line

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Smoothing Splines

Theorem: The unique minimizer of this penalized RSS is a natural cubic spline with knots at the unique values of xi , i=1,…,N

Seems like there will be N features and presumably overfitting of the data. But,… the smoothing term shrinks the model towards the linear fit

This is a generalized ridge regression

Can show that where K does not depend on 

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Nonparametric Logistic Regression

Consider logistic regression with a single x:

and a penalized log-likelihood criterion:

Again can show that the optimal f is a natural spline with knots at the datapoint

Can use Newton-Raphson to do the fitting.

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Thin-Plate Splines

The discussion up to this point has been one-dimensional. The higher-dimensional analogue of smoothing splines are “thin-plate splines.” In 2-D, instead of minimizing:


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Thin-Plate Splines

The solution has the form:

a type of “radial basis function”