# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2020010

## Modeling interactive components by coordinate kernel polynomial models

 1 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China 2 Division of Biostatistics, University of California, Berkeley, Berkeley, CA 94720, USA 3 Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN 37132, USA

* Corresponding author: Xin Guo

Received  October 2019 Published  June 2020

Fund Project: The work described in this paper is partially supported by FRCAC of Middle Tennessee State University, and is partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 25301115). All the three authors contributed equally to the paper

We proposed the use of coordinate kernel polynomials in kernel regression. This new approach, called coordinate kernel polynomial regression, can simultaneously identify active variables and effective interactive components. Reparametrization refinement is found critical to improve the modeling accuracy and prediction power. The post-training component selection allows one to identify effective interactive components. Generalization error bounds are used to explain the effectiveness of the algorithm from a learning theory perspective and simulation studies are used to show its empirical effectiveness.

Citation: Xin Guo, Lexin Li, Qiang Wu. Modeling interactive components by coordinate kernel polynomial models. Mathematical Foundations of Computing, doi: 10.3934/mfc.2020010
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Variable selection accuracy and average MSE for Example 1
 Algorithm TPR($x_1$) TPR($x_2$) FPR MSE CKPR-L 1.00 1.00 0.000 0.008 (0.000) CKPR-G 1.00 1.00 0.011 0.109 (0.015) LASSO 1.00 0.18 0.040 1.129 (0.015) COSSO 0.90 0.02 0.020 10.879 (8.345) SR-SIR (AIC) 1.00 0.89 0.460 - SR-SIR (BIC) 1.00 0.85 0.181 - SR-SIR (RIC) 1.00 0.75 0.053 -
 Algorithm TPR($x_1$) TPR($x_2$) FPR MSE CKPR-L 1.00 1.00 0.000 0.008 (0.000) CKPR-G 1.00 1.00 0.011 0.109 (0.015) LASSO 1.00 0.18 0.040 1.129 (0.015) COSSO 0.90 0.02 0.020 10.879 (8.345) SR-SIR (AIC) 1.00 0.89 0.460 - SR-SIR (BIC) 1.00 0.85 0.181 - SR-SIR (RIC) 1.00 0.75 0.053 -
Average and standard error of MSEs for Example 2
 $m=100$ $m=200$ $m=400$ CKPR-G 0.119 (0.003) 0.054 (0.001) 0.025 (0.0004) COSSO(GCV) 0.358 (0.009) 0.100 (0.003) 0.045 (0.001) COSSO(5CV) 0.378 (0.005) 0.094 (0.004) 0.043 (0.001) MARS 0.239 (0.008) 0.109 (0.003) 0.084 (0.001)
 $m=100$ $m=200$ $m=400$ CKPR-G 0.119 (0.003) 0.054 (0.001) 0.025 (0.0004) COSSO(GCV) 0.358 (0.009) 0.100 (0.003) 0.045 (0.001) COSSO(5CV) 0.378 (0.005) 0.094 (0.004) 0.043 (0.001) MARS 0.239 (0.008) 0.109 (0.003) 0.084 (0.001)
RMSE on three UCI data sets
 Ionosphere Sonar MR Wisc. BC $n$ 351 208 683 $p$ 33 60 9 CKPR-L $0.64 (0.04)$ $0.75 (0.06)$ $0.34 (0.02)$ CKPR-G $0.54 (0.03)$ $0.77 (0.06)$ $0.34 (0.02)$ Best in [5] $0.60 (0.05)$ $0.80 (0.04)$ $0.70 (0.01)$
 Ionosphere Sonar MR Wisc. BC $n$ 351 208 683 $p$ 33 60 9 CKPR-L $0.64 (0.04)$ $0.75 (0.06)$ $0.34 (0.02)$ CKPR-G $0.54 (0.03)$ $0.77 (0.06)$ $0.34 (0.02)$ Best in [5] $0.60 (0.05)$ $0.80 (0.04)$ $0.70 (0.01)$
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