generalized additive models - feng
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Generalized Additive Models
Yufei Wang & Zhuoqun Cheng
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Contents
Generalized Additive Models
Piecewise Polynomials
Fitting Additive Models
Additive Logistic Models
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Generalized Additive Models
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Generalized Additive Models
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Generalized Additive Models
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Generalized Additive Models
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Generalized Additive Models
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Generalized Additive Models
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Piecewise Polynomials
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Piecewise Polynomials
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Then we add two constraint conditions:
Piecewise Polynomials
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Piecewise Polynomials
A more direct way to proceed in this case is to
use a basis that incorporates the constraints:
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Piecewise Polynomials
We often prefer smoother functions, and these can be
achieved by increasing the order of the local polynomial.
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Piecewise Polynomials
The function in this figure is continuous, and has
continuous first and second derivatives at the knots.
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Piecewise Polynomials
cubic spline
The function has two continuous derivatives at the knots.
It is known as a cubic spline.
It is not hard to show that the basis represents a cubic
spline with knots at 1 and 2:
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Piecewise Polynomials
More generally, an order M spline with knots j , j = 1, . . . ,K
is a piecewise-polynomial of order M, and has continuous
derivatives up to order M − 2.
Likewise the general form for the truncated-power basis set
would be:
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Fitting Additive Models
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Fitting Additive Models
The Backfitting Algorithm for Additive Models
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Additive Logistic Regression
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Additive Logistic Regression
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Additive Logistic Regression
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Additive Logistic Regression
Example : Predicting Email Spam
We apply a generalized additive model to the spam data. The data consists of information from 4601 email messages, in a study to screen email for
“ spam ” (i.e. junk email).
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The response variable is binary, with values email or spam, and there are 57 predictors as described below:
48 quantitative predictors—the percentage of words in the email that match a given word. Examples include business, address, internet, free and george. The idea was that these could be customized for Individual users.
6 quantitative predictors—the percentage of characters in the email That match a given character. The characters are ch;, ch(, ch[, ch!, ch$, and ch#.
The average length of uninterrupted sequences of capital letters:CAPAVE.
The length of the longest uninterrupted sequences of capital letters: CAPMAX.
The sum of the length of uninterrupted sequences of capital letters: CAPTOT.
Fitting Additive Models
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Additive Logistic Regression
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Additive Logistic Regression
Table 1
Table2 shows the predictors that are highly significant in
the additive model.
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Additive Logistic Regression
Table2
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Additive Logistic Regression
The figure shows the estimated functions for the significant predictors appearing in Table2.
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