Generalized Additive Models by Wood Simon N

Author:Wood, Simon N.
Language: eng
Format: epub
Publisher: CRC Press LLC

Figure 5.16 How the product of two marginal basis functions for smooth functions of x and z, separately, results in a basis function for a smooth function of x and z together. The two left panels show the 3rd and 5th basis functions for rank 8 cubic regression spline smooths of x and z, respectively. The right hand plot shows a3(x)d5(z), one of 64 similar basis functions of the tensor product smooth derived from these two marginal smooths.

where the αi, δl and βk are parameters, and the ai(x), dl(z) and bk(v) are known basis functions.

Now consider how fx, the smooth function of x, could be converted into a smooth function of x and z. What is required is for fx to vary smoothly with z, and this can be achieved by allowing its parameters, αi, to vary smoothly with z. Using the basis already available for representing smooth functions of z we could write:

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