# American Institute of Mathematical Sciences

May  2019, 2(2): 169-181. doi: 10.3934/mfc.2019012

## An RKHS approach to estimate individualized treatment rules based on functional predictors

 1 Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong, China 2 School of Mathematical Sciences, Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

* Corresponding author: Lei Shi

Published  July 2019

In recent years there has been massive interest in precision medicine, which aims to tailor treatment plans to the individual characteristics of each patient. This paper studies the estimation of individualized treatment rules (ITR) based on functional predictors such as images or spectra. We consider a reproducing kernel Hilbert space (RKHS) approach to learn the optimal ITR which maximizes the expected clinical outcome. The algorithm can be conveniently implemented although it involves infinite-dimensional functional data. We provide convergence rate for prediction under mild conditions, which is jointly determined by both the covariance kernel and the reproducing kernel.

Citation: Jun Fan, Fusheng Lv, Lei Shi. An RKHS approach to estimate individualized treatment rules based on functional predictors. Mathematical Foundations of Computing, 2019, 2 (2) : 169-181. doi: 10.3934/mfc.2019012
##### References:
 [1] T. T. Cai and M. Yuan, Minimax and adaptive prediction for functional linear regression, Journal of the American Statistical Association, 107 (2012), 1201–1216. doi: 10.1080/01621459.2012.716337. [2] A. Ciarleglio, E. Petkova, T. Tarpey and R. T. Ogden, Flexible functional regression methods for estimating individualized treatment regimes, 5 (2016), 185–199. doi: 10.1002/sta4.114. [3] J. Fan, T. Hu, Q. Wu and D. X. Zhou, Consistency analysis of an empirical minimum error entropy algorithm, Applied and Computational Harmonic Analysis, 41 (2016), 164-189.  doi: 10.1016/j.acha.2014.12.005. [4] X. Guo, J. Fan and D. X. Zhou, Sparsity and error analysis of empirical feature-based regularization schemes, Journal of Machine Learning Research, 17 (2016), 3058-3091. [5] Z. C. Guo, S. B. Lin and D. X. Zhou, Learning theory of distributed spectral algorithms, Inverse Problems, 33 (2017), 074009, 29pp. doi: 10.1088/1361-6420/aa72b2. [6] T. Hu, J. Fan, Q. Wu and D. X. Zhou, Regularization schemes for minimum error entropy principle, Analysis and Applications, 13 (2015), 437-455.  doi: 10.1142/S0219530514500110. [7] S. B. Lin, X. Guo and D. X. Zhou, Distributed learning with regularized least squares, Journal of Machine Learning Research, 18 (2017), Paper No. 92, 31 pp. [8] I. McKeague and M. Qian, Estimation of treatment policies based on functional predictors, Statistica Sinica, 24 (2014), 1461–1485. [9] S. A. Murphy, Optimal dynamic treatment regimes, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65 (2003), 331–366. doi: 10.1111/1467-9868.00389. [10] S. A. Murphy, An experimental design for the development of adaptive treatment strategies, Statistics in Medicine, 24 (2005), 1455–1481. doi: 10.1002/sim.2022. [11] Z. L. Qi and Y. F. Liu, D-learning to estimate optimal individual treatment rules, Electronic Journal of Statistics, 12 (2018), 3601–3638. doi: 10.1214/18-EJS1480. [12] J. O. Ramsay and B. W. Silverman, Applied Functional Data Analysis, Springer, New York, 2002. doi: 10.1007/b98886. [13] L. Shi, Distributed Learning with Indefinite Kernels, Analysis and Applications, 2019. doi: 10.1142/S021953051850032X. [14] R. Song, W. Wang, D. Zeng and M. R. Kosorok, Penalized q-learning for dynamic treatment regimes, Statistica Sinica, 25 (2015), 901–920. [15] M. Yuan and T. T. Cai, A reproducing kernel Hilbert space approach to functional linear regression, The Annals of Statistics, 38 (2010), 3412–3444. doi: 10.1214/09-AOS772. [16] T. Zhang, Learning bounds for kernel regression using effective data dimensionality, Neural Computation, 17 (2005), 2077-2098.  doi: 10.1162/0899766054323008. [17] Y. Zhao, D. Zeng, A. J. Rush and M. R. Kosorok, Estimating individualized treatment rules using outcome weighted learning, Journal of the American Statistical Association, 107 (2012), 1106–1118. doi: 10.1080/01621459.2012.695674.

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##### References:
 [1] T. T. Cai and M. Yuan, Minimax and adaptive prediction for functional linear regression, Journal of the American Statistical Association, 107 (2012), 1201–1216. doi: 10.1080/01621459.2012.716337. [2] A. Ciarleglio, E. Petkova, T. Tarpey and R. T. Ogden, Flexible functional regression methods for estimating individualized treatment regimes, 5 (2016), 185–199. doi: 10.1002/sta4.114. [3] J. Fan, T. Hu, Q. Wu and D. X. Zhou, Consistency analysis of an empirical minimum error entropy algorithm, Applied and Computational Harmonic Analysis, 41 (2016), 164-189.  doi: 10.1016/j.acha.2014.12.005. [4] X. Guo, J. Fan and D. X. Zhou, Sparsity and error analysis of empirical feature-based regularization schemes, Journal of Machine Learning Research, 17 (2016), 3058-3091. [5] Z. C. Guo, S. B. Lin and D. X. Zhou, Learning theory of distributed spectral algorithms, Inverse Problems, 33 (2017), 074009, 29pp. doi: 10.1088/1361-6420/aa72b2. [6] T. Hu, J. Fan, Q. Wu and D. X. Zhou, Regularization schemes for minimum error entropy principle, Analysis and Applications, 13 (2015), 437-455.  doi: 10.1142/S0219530514500110. [7] S. B. Lin, X. Guo and D. X. Zhou, Distributed learning with regularized least squares, Journal of Machine Learning Research, 18 (2017), Paper No. 92, 31 pp. [8] I. McKeague and M. Qian, Estimation of treatment policies based on functional predictors, Statistica Sinica, 24 (2014), 1461–1485. [9] S. A. Murphy, Optimal dynamic treatment regimes, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 65 (2003), 331–366. doi: 10.1111/1467-9868.00389. [10] S. A. Murphy, An experimental design for the development of adaptive treatment strategies, Statistics in Medicine, 24 (2005), 1455–1481. doi: 10.1002/sim.2022. [11] Z. L. Qi and Y. F. Liu, D-learning to estimate optimal individual treatment rules, Electronic Journal of Statistics, 12 (2018), 3601–3638. doi: 10.1214/18-EJS1480. [12] J. O. Ramsay and B. W. Silverman, Applied Functional Data Analysis, Springer, New York, 2002. doi: 10.1007/b98886. [13] L. Shi, Distributed Learning with Indefinite Kernels, Analysis and Applications, 2019. doi: 10.1142/S021953051850032X. [14] R. Song, W. Wang, D. Zeng and M. R. Kosorok, Penalized q-learning for dynamic treatment regimes, Statistica Sinica, 25 (2015), 901–920. [15] M. Yuan and T. T. Cai, A reproducing kernel Hilbert space approach to functional linear regression, The Annals of Statistics, 38 (2010), 3412–3444. doi: 10.1214/09-AOS772. [16] T. Zhang, Learning bounds for kernel regression using effective data dimensionality, Neural Computation, 17 (2005), 2077-2098.  doi: 10.1162/0899766054323008. [17] Y. Zhao, D. Zeng, A. J. Rush and M. R. Kosorok, Estimating individualized treatment rules using outcome weighted learning, Journal of the American Statistical Association, 107 (2012), 1106–1118. doi: 10.1080/01621459.2012.695674.
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