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I'm trying to understand the basics of GAMs. Wood's book "Generalized Additive Models: an introduction with R" introduces GAMs via a cubic spline basis $b_j(x)$ (see p. 122), where $b_1(x)=1, b_2(x)=x$ and $b_{i+2}(x)=R(x, x_i^*)$ ($1 \leq i \leq q-2$, $x_i^*$ are knot locations), where $ R(x,z)=\left[ (z-1/2)^2 -1/12 \right]\left[ (x-1/2)^2 -1/12 \right]/4 - \left[(|x-z|-1/2)^4 - 1/2 (|x-z|-1/2)^2 +7/240 \right]/24. $

But there's definitely an error in the last expression since it's a spline of order 4. How should the correct formula look like?

Could anyone suggest another simple but mathematically rigorous (and relatively comprehensive) introduction to GAMs?

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GAM in "Introduction to statistical learning" (page 283) is defined as extension to multiple regression. Lets begin with a term representing simple multiple regression: $ \hat{y} = \beta_{0} + \beta_{1}x_{1} + ... +\beta_{n}x_{n} +\epsilon_i$. Now instead of just multiplying each variable $x_{1},x_{2},...,x_{n} $ with coefficients $ \beta_{1}, \beta_{2},...,\beta_{n} $, you could use some functions $ f_{1},...,f_{n} $ to model the relationship between predictors and response variable: $ \hat{y} = f_{1}(x_{1}) + ... + f_{n}(x_{n}) +\epsilon_i$. By doing this, you are adding possibility to model each predictor's contribution to the response variable with non-linear functions. This is the idea of GAMs.

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  • $\begingroup$ I think the OP knows that but the question is whether the formula quoted is correct for a spline of that order. $\endgroup$ – mdewey Dec 11 '18 at 13:12
  • $\begingroup$ OP clearly stated that he/she is trying to understand the basics, and asked for another explanation using math expressions. Sorry if the response is not suitable. $\endgroup$ – Nemanja Boskovic Dec 12 '18 at 12:36

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