Piecewise Linear Functions

The Explicable Precision series comes to an end by introducing the formulae, geometry, and application of piecewise linear functions. It will be shown that such functions rival the precision of xgboost and random forest models while remaining explicable.

(0:00:00) Introduction
(0:03:24) Piecewise Linear Functions
(0:19:14) Modeling Strategy
(0:23:30) MARS Model On Transformed Variables
(0:38:09) Translating Back To Original Variables
(0:50:54) MARS Model On Original Variables
(1:05:43) Conclusion

References
1. Resident Evil 2. Capcom, 1998. Video game. Timestamp: 0.00.00
2. Pagan's Mind. "God's Equation." God's Equation, Limb Music, 2007. Timestamp: 0.00.10
3. Lundberg and Lee introduced their SHAP (SHapley Additive exPlanations) framework for interpreting predictions from models in 2017. Lundberg, Scott M., and Su-In Lee. "A Unified Approach to Interpreting Model Predictions." Advances in Neural Information Processing Systems, vol. 30, 2017, pp. 4765–4774. Timestamp: 0.03.13
4. Friedman introduced his MARS idea in his 1991 paper. Interestingly, he did not demonstrate the approximation properties of the MARS functions. Rather he introduced the functions as an alternative modeling approach. J. H. (1991). Multivariate Adaptive Regression Splines. The Annals Of Statistics 19 (1) 1 – 67. Timestamp: 0.15.13
5. “The Gang Buys a Boat.” It’s Always Sunny In Philadelphia, written by Charlie Day and Rob McElhenney, directed by Randall Einhorn, season 6, episode 3, FX, 30 Sept. 2010. Timestamp: 0.08.17
6. “Skytanic.” Archer, written by Adam Reed and Matt Thompson, directed by Mack Williams and Adam Reed, season 1, episode 7, FX, 18 Feb. 2010. Timestamp: 0.13.50
7. Salford Systems was acquired by Minitab in 2017. Its implementation of MARS may be found in Minitab’s Salford Predictive Modeler. Timestamp: 0.20.47
8. R Earth package documentation: https://www.rdocumentation.org/packag... . Timestamp: 0.20:53
9. My version of stepwise selection may be found here: https://github.com/IanDeters/Explicab.... Timestamp: 0.51.51
10. Free Guy. Directed by Shawn Levy, 20th Century Studios, 2021. Timestamp: 1.06.03
11. Linkin Park. "What I've Done." Minutes To Midnight, Warner Bros. Records, 2007. Timestamp: 1.07.20
12. Transformers. Directed by Michael Bay, Paramount Pictures, 2007. Timestamp: 1.07.58