## 7.5 Other Potentially Useful Tools

There are a few other models that have the ability to uncover predictive interactions that are worthy of mentioning here. Multivariate adaptive regression splines (MARS) is a nonlinear modeling technique for a continuous response that searches through individual predictors and individual values of each predictor to find the predictor and sample value for creating a hinge function that describes the relationship between the predictor and response (Friedman 1991). The hinge function was described previously in Section 6.2.1 and enables the model to search for nonlinear relationships. In addition to being able to search through individual predictors, MARS can be used to search for products of predictors in order to make nonlinear interactions that isolate portions of the predictor space. The MARS technique also has built-in feature selection. The primary disadvantage of this approach is that it is computationally taxing. MARS has also been extended to classification outcomes, and this method is called flexible discriminant analysis (FDA).

Cubist (Kuhn and Johnson 2013) is a rule-based regression model that builds an initial tree and decomposes it into set of rules that are pruned and perhaps eliminated. For each rule, a separate linear model is defined. One example of a rule for the Ames housing data might be:

```
if
Year_Built <= 1952
Central_Air = No
Longitude <= -93.6255
then
log10 Sale Price = 261.38176 + 0.355 Gr_Liv_Area +
2.88 Longitude +
0.26 Latitude
```

This structure creates a set of disjoint interactions; the set of rules may not cover every combination of values for the predictors that are used in the rules. New data points being predicted may belong to multiple rules and, in this case, the associated linear model predictions are averaged. This model structure is very flexible and has a strong chance of finding local interactions within the entire predictor space. For example, when a property has a small total living area but has a large number of bedrooms, there is usually a *negative* slope in regression models associated with the number of bedrooms. This is certainly not true in general but can be when applied only to properties that are nearly “all bedroom.”

### References

Friedman, J. 1991. “Multivariate Adaptive Regression Splines.” *The Annals of Statistics* 19 (1):1–141.

Kuhn, M, and K Johnson. 2013. *Applied Predictive Modeling*. Springer.