Mathematical Methods in Engineering by Unknown

Author:Unknown
Language: eng
Format: epub
ISBN: 9783319909721
Publisher: Springer International Publishing

(9.8)

where K m is the number of truncated linear functions multiplied in the mth basis function, is the input variable corresponding to the jth truncated linear function in the mth basis function, is the knot value corresponding to the variable and is the selected sign +1 or −1. A lack-of-fit criterion is used to compare the possible basis functions. The search for new basis functions can be restricted to interactions of a maximum order. For example, if only up to two-factor interactions are permitted, then K m ≤ 2 would be restricted in Eq. (9.8). The MARS algorithm to estimate the model function f(x) consists of two sub-algorithms [2], the forward stepwise and backward stepwise. The forward stepwise is used for searching the basis function. It starts with the constant basis function, the only one present initially. The process is terminated when a user-pointed out value M max is attained. The goal of the backward stepwise algorithm is to hinder to over-fitting by reducing the complexity of the model without degrading the fit to the data and contains removing from the model basis functions that contribute to the smallest increase in the residual squared error at each stage, producing an optimally estimated model with respect to each number of terms, called α. For estimating the optimal value of α, generalized cross-validation can be used which indicates the lack of fit when using MARS. This criterion is defined by

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