Non-Convex Penalized Estimation of Count Data Responses via Generalized Linear Model (GLM)
DOI:
https://doi.org/10.24203/ajfam.v8i3.6443Keywords:
Count Data, Minimax Concave Penalty (MCP), Non-convex penalization, Smoothed Clipped Absolute DeviationAbstract
This study provided a non-convex penalized estimation procedure via Smoothed Clipped Absolute Deviation (SCAD) and Minimax Concave Penalty (MCP) for count data responses to checkmate the problem of covariates exceeding the sample size . The Generalized Linear Model (GLM) approach was adopted in obtaining the penalized functions needed by the MCP and SCAD non-convex penalizations of Binomial, Poisson and Negative-Binomial related count responses regression. A case study of the colorectal cancer with six (6) covariates against sample size of five (5) was subjected to the non-convex penalized estimation of the three distributions. It was revealed that the non-convex penalization of Binomial regression via MCP and SCAD best explained four un-penalized covariates needed in determining whether surgical or therapy ideal for treating the turmoil.
References
Agarwal, A., Negahban, S. and Wainwright, M. J, “Fast global convergence of gradient methods for high-
dimensional statistical recovery” Annals of Statistics, vol. 40, no. 5, pp 2452–2482, 2012. doi:10.1214/12-AOS1032.
Chen, J. and Chen, Z., “Extended Bayesian information criteria for model selection with large model spaces”
Biometrika, vol. 95, no. 3, pp. 759–771, 2008.doi:10.1093/biomet/asn034.
Fan, J. and Li, R., “Variable selection via non-concave penalized likelihood and its oracle properties” Journal of the
American Statistical Association, vol. 96, no. 456, pp. 1348–1360, 2001. doi: 10.1198/016214501753882273.
Hirose, K and Yamamoto, M., “Penalized likelihood factor analysis via non-convex penalties”
th International Conference of the ERCIM WG on Computing & Statistics (ERCIM 2012), Conference
Centre, Oviedo, Spain, pp.1-33, December 2012.
Kim, Y., Kwon, S. and Choi, H, “Consistent model selection criteria on high dimensions” Journal of Machine
Learning Research, vol. 13, no. 1, pp. 1037-1057, 2012.
Liu, H.Y. Yao, T. and Li, R, “Global solutions to folded concave penalized non-convex learning” The Annals of
Statistics. Vol. 44, no. 2, pp. 629–659, 2016. doi:10.1214/15-AOS1380. © Institute of Mathematical Statistics.
Loh, P-L. and Wainwright, M. J, “Regularized M-estimators with non-convexity: Statistical and algorithmic theory
for local optima” Journal of Machine Learning Research, vol. 16, pp. 559–616, 2015.
Loh, P-L, “Local optima of non-convex regularized M-estimators” Electrical Engineering and Computer Sciences
University of California at Berkeley. Technical Report No. UCB/EECS-2013-54, 2013.
http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-54.html
Nesterov, Y, “Gradient methods for minimizing composite functions” Mathematical Programming, vol.140, no. 1, pp.
–161, 2013. Doi:10.1007/s10107-012-0629-5.
Wang, Z., Han, H. and Zhang, T, “Optimal computational and statistical rates of convergence for sparse non-convex
learning problems” The Annals of Statistics, vol. 42, no. 6, pp. 2164–2201, 2014. doi: 10.1214/14-AOS1238.
Wang, L., Kim, Y. and Li, R, “Calibrating non-convex penalized regression in ultra-high dimension” Annals of
Statistics, vol. 41, no. 5, pp. 2505–2536, 2013. doi: 10.1214/13-AOS1159 © Institute of Mathematical Statistics.
Xiao, L., Zhang, T., “A proximal-gradient homotopy method for the sparse least-squares problem” SIAM Journal on
Optimization, vol. 23, no. 2, pp. 1062–1091, 2013.
Yang, N. and Han, L., “A general theory of hypothesis tests and confidence regions for sparse high dimensional
models” The Annals of Statistics, vol. 45, no. 1, pp. 158-195, 2017. doi:10.1214/16AOS1448.
Yu, Y. and Feng, Y., “APPLE: approximate path for penalized likelihood estimators” Journal of Statistical
Computing, vol. 24, pp. 803–819, 2014. doi: 10.1007/s11222-013-9403-7.
Zhang, C.H., “Nearly unbiased variable selection under minimax concave penalty” Annals of Statistics, vol. 38,
no. 2, pp. 894–942, 2010. doi: 10.1214/09-aos729.
Zhang, C.-H and Zhang, T., “A general theory of concave regularization for high dimensional sparse estimation
problems” Statistical Science, vol. 27, no. 4, pp. 576-593, 2012.
Zhang, K., Yin, F. and Xiong, F., “Comparisons of penalized least squares methods by simulations” 1-16, 2014.
arXiv:1405.1796v1 [stat. Co].
Zhong, L.W. and Kwok, J.F., “Gradient descent with proximal average for non-convex and composite
regularization, Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, pp. 1 – 7, 2014.
Downloads
Published
Issue
Section
License
Copyright (c) 2020 Rasaki Olawale Olanrewaju, Johnson Funminiyi Ojo
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
- Papers must be submitted on the understanding that they have not been published elsewhere (except in the form of an abstract or as part of a published lecture, review, or thesis) and are not currently under consideration by another journal published by any other publisher.
- It is also the authors responsibility to ensure that the articles emanating from a particular source are submitted with the necessary approval.
- The authors warrant that the paper is original and that he/she is the author of the paper, except for material that is clearly identified as to its original source, with permission notices from the copyright owners where required.
- The authors ensure that all the references carefully and they are accurate in the text as well as in the list of references (and vice versa).
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Attribution-NonCommercial 4.0 International that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
- The journal/publisher is not responsible for subsequent uses of the work. It is the author's responsibility to bring an infringement action if so desired by the author.
