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May  2019, 2(2): 95-106. doi: 10.3934/mfc.2019008

Online learning for supervised dimension reduction

1. 

Computational Science PhD Program, Middle Tennessee State University, 1301 E Main Street, Murfreesboro, TN 37132, USA

2. 

Department of Mathematical Sciences, Middle Tennessee State University, 1301 E Main Street, Murfreesboro, TN 37132, USA

* Corresponding author: Qiang Wu

Received  March 2019 Published  May 2019

Online learning has attracted great attention due to the increasingdemand for systems that have the ability of learning and evolving. When thedata to be processed is also high dimensional and dimension reduction is necessary for visualization or prediction enhancement, online dimension reductionwill play an essential role. The purpose of this paper is to propose a new onlinelearning approach for supervised dimension reduction. Our algorithm is motivated by adapting the sliced inverse regression (SIR), a pioneer and effectivealgorithm for supervised dimension reduction, and making it implementable inan incremental manner. The new algorithm, called incremental sliced inverseregression (ISIR), is able to update the subspace of significant factors with intrinsic lower dimensionality fast and efficiently when new observations come in.We also refine the algorithm by using an overlapping technique and develop anincremental overlapping sliced inverse regression (IOSIR) algorithm. We verifythe effectiveness and efficiency of both algorithms by simulations and real dataapplications.

Citation: Ning Zhang, Qiang Wu. Online learning for supervised dimension reduction. Mathematical Foundations of Computing, 2019, 2 (2) : 95-106. doi: 10.3934/mfc.2019008
References:
[1]

A. AntoniadisS. Lambert-Lacroix and F. Leblanc, Effective dimension reduction methods for tumor classification using gene expression data, Bioinformatics, 19 (2003), 563-570. doi: 10.1093/bioinformatics/btg062. Google Scholar

[2]

Z. Bai and X. He, A chi-square test for dimensionality with non-Gaussian data, Journal of Multivariate Analysis, 88 (2004), 109-117. doi: 10.1016/S0047-259X(03)00056-3. Google Scholar

[3]

M. P. Barrios and S. Velilla, A bootstrap method for assessing the dimension of a general regression problem, Statistics & Probability Letters, 77 (2007), 247-255. doi: 10.1016/j.spl.2006.07.020. Google Scholar

[4]

C. Becker and R. Fried, Sliced inverse regression for high-dimensional time series, in Exploratory Data Analysis in Empirical Research, Springer, (2003), 3–11. Google Scholar

[5]

P. N. Belhumeur, J. P. Hespanha and D. J. Kriegman, Eigenfaces vs. Fisherfaces: recognition using class specific linear projection, European Conference on Computer Vision, (1996), 43–58. doi: 10.1007/BFb0015522. Google Scholar

[6]

E. Bura and R. D. Cook, Extending sliced inverse regression: the weighted chi-squared test, Journal of the American Statistical Association, 96 (2001), 996-1003. doi: 10.1198/016214501753208979. Google Scholar

[7]

S. ChandrasekaranB. S. ManjunathY.-F. WangJ. Winkeler and H. Zhang, An eigenspace update algorithm for image analysis, Graphical Models and Image Processing, 59 (1997), 321-332. Google Scholar

[8]

D. ChuL.-Z. ZhaoM. K.-P. Ng and X. Wang, Incremental linear discriminant analysis: A fast algorithm and comparisons, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 2716-2735. doi: 10.1109/TNNLS.2015.2391201. Google Scholar

[9]

R. D. Cook, Using dimension-reduction subspaces to identify important inputs in models of physical systems, in Proceedings of the section on Physical and Engineering Sciences, American Statistical Association Alexandria, VA, (1994), 18–25.Google Scholar

[10]

R. D. Cook and S. Weisberg, Sliced inverse regression for dimension reduction: Comment, Journal of the American Statistical Association, 86 (1991), 328-332.Google Scholar

[11]

R. D. Cook and and X. Zhang, Fused estimators of the central subspace in sufficient dimension reduction, Journal of the American Statistical Association, 109 (2014), 815-827. doi: 10.1080/01621459.2013.866563. Google Scholar

[12]

J. J. Dai, L. Lieu and D. Rocke, Dimension reduction for classification with gene expression microarray data, Statistical Applications in Genetics and Molecular Biology, 5 (2006), Art. 6, 21 pp. doi: 10.2202/1544-6115.1147. Google Scholar

[13]

L. ElnitskiR. C. HardisonJ. LiS. YangD. KolbeP. EswaraM. J. O'ConnorS. SchwartzW. Miller and F. Chiaromonte, Distinguishing regulatory DNA from neutral sites, Genome Research, 13 (2003), 64-72. doi: 10.1101/gr.817703. Google Scholar

[14]

L. Ferré, Determining the dimension in sliced inverse regression and related methods, Journal of the American Statistical Association, 93 (1998), 132-140. doi: 10.2307/2669610. Google Scholar

[15]

K. Fukumizu, F. R. Bach and M. I. Jordan, Kernel dimensionality reduction for supervised learning, in NIPS, (2003), 81–88.Google Scholar

[16]

A. GannounS. GirardC. Guinot and J. Saracco, Sliced inverse regression in reference curves estimation, Computational Statistics & Data Analysis, 46 (2004), 103-122. doi: 10.1016/S0167-9473(03)00141-5. Google Scholar

[17]

Y. A. Ghassabeh, A. Ghavami and H. A. Moghaddam, A new incremental face recognition system, in IEEE Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, (2007), 335–340. doi: 10.1109/IDAACS.2007.4488435. Google Scholar

[18]

Y. A. Ghassabeh and H. A. Moghaddam, A new incremental optimal feature extraction method for on-line applications, in International Conference Image Analysis and Recognition, Springer, (2007), 399–410. doi: 10.1007/978-3-540-74260-9_36. Google Scholar

[19]

B. GuV. S. ShengZ. WangD. HoS. Osman and S. Li, Incremental learning for v-support vector regression, Neural Networks, 67 (2015), 140-150. Google Scholar

[20]

P. M. HallD. A. Marshall and R. F. Martin, Incremental eigenanalysis for classification, BMVC, 98 (1998), 286-295. doi: 10.5244/C.12.29. Google Scholar

[21]

P. M. HallD. A. Marshall and R. F. Martin, Merging and splitting eigenspace models, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22 (2000), 1042-1049. doi: 10.1109/34.877525. Google Scholar

[22]

P. HeK.-T. Fang and C.-J. Xu, The classification tree combined with SIR and its applications to classification of mass spectra, Journal of Data Science, 1 (2003), 425-445. Google Scholar

[23]

T.-K. Kim, S.-F. Wong, B. Stenger, J. Kittler and R. Cipolla, Incremental linear discriminant analysis using sufficient spanning set approximations, in IEEE Conference on Computer Vision and Pattern Recognition, (2007), 1–8. doi: 10.1109/CVPR.2007.382985. Google Scholar

[24]

M. H. Law and A. K. Jain, Incremental nonlinear dimensionality reduction by manifold learning, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 377-391. doi: 10.1109/TPAMI.2006.56. Google Scholar

[25]

K.-C. Li, Sliced inverse regression for dimension reduction, Journal of the American Statistical Association, 86 (1991), 316-327. doi: 10.1080/01621459.1991.10475035. Google Scholar

[26]

K.-C. Li, On principal hessian directions for data visualization and dimension reduction: Another application of Stein's lemma, Journal of the American Statistical Association, 87 (1992), 1025-1039. doi: 10.1080/01621459.1992.10476258. Google Scholar

[27]

L. Li and H. Li, Dimension reduction methods for microarrays with application to censored survival data, Bioinformatics, 20 (2004), 3406-3412. doi: 10.1093/bioinformatics/bth415. Google Scholar

[28]

L. Li and X. Yin, Sliced inverse regression with regularizations, Biometrics, 64 (2008), 124-131. doi: 10.1111/j.1541-0420.2007.00836.x. Google Scholar

[29]

L.-P. Liu, Y. Jiang and Z.-H. Zhou, Least square incremental linear discriminant analysis, in 2009 Ninth IEEE International Conference on Data Mining, (2009), 298–306. doi: 10.1109/ICDM.2009.78. Google Scholar

[30]

G.-F. LuJ. Zou and Y. Wang, Incremental learning of complete linear discriminant analysis for face recognition, Knowledge-Based Systems, 31 (2012), 19-27. doi: 10.1016/j.knosys.2012.01.016. Google Scholar

[31]

G. M. Nkiet, Consistent estimation of the dimensionality in sliced inverse regression, Annals of the Institute of Statistical Mathematics, 60 (2008), 257-271. doi: 10.1007/s10463-006-0106-0. Google Scholar

[32]

A. ÖztaşM. PalaE. ÖzbayE. KancaN. Caglar and M. A. Bhatti, Predicting the compressive strength and slump of high strength concrete using neural network, Construction and Building Materials, 20 (2006), 769-775. Google Scholar

[33]

S. PangS. Ozawa and N. Kasabov, Incremental linear discriminant analysis for classification of data streams, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35 (2005), 905-914. doi: 10.1109/TSMCB.2005.847744. Google Scholar

[34]

C.-X. Ren and D.-Q. Dai, Incremental learning of bidirectional principal components for face recognition, Pattern Recognition, 43 (2010), 318-330. doi: 10.1016/j.patcog.2009.05.020. Google Scholar

[35]

P. Rodriguez and B. Wohlberg, A matlab implementation of a fast incremental principal component pursuit algorithm for video background modeling, in 2014 IEEE International Conference on Image Processing (ICIP), (2014), 3414–3416. doi: 10.1109/ICIP.2014.7025692. Google Scholar

[36]

J. R. Schott, Determining the dimensionality in sliced inverse regression, Journal of the American Statistical Association, 89 (1994), 141-148. doi: 10.1080/01621459.1994.10476455. Google Scholar

[37]

C. M. Setodji and R. D. Cook, K-means inverse regression, Technometrics, 46 (2004), 421-429. doi: 10.1198/004017004000000437. Google Scholar

[38]

F. X. Song, D. Zhang, Q. Chen and J. Yang, A novel supervised dimensionality reduction algorithm for online image recognition, in Pacific-Rim Symposium on Image and Video Technology, Springer, (2006), 198–207. doi: 10.1007/11949534_20. Google Scholar

[39]

J.-G. WangE. Sung and W.-Y. Yau, Incremental two-dimensional linear discriminant analysis with applications to face recognition, Journal of Network and Computer Applications, 33 (2010), 314-322. doi: 10.1016/j.jnca.2009.12.014. Google Scholar

[40]

X. Wang, Incremental and Regularized Linear Discriminant Analysis, Ph.D thesis, National University of Singapore, 2012.Google Scholar

[41]

J. WengY. Zhang and W.-S. Hwang, Candid covariance-free incremental principal component analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 1034-1040. Google Scholar

[42]

H.-M. Wu, Kernel sliced inverse regression with applications to classification, Journal of Computational and Graphical Statistics, 17 (2008), 590-610. doi: 10.1198/106186008X345161. Google Scholar

[43]

Q. WuS. Mukherjee and F. Liang, Localized sliced inverse regression, Journal of Computational and Graphical Statistics, 19 (2010), 843-860. doi: 10.1198/jcgs.2010.08080. Google Scholar

[44]

Y. XiaH. TongW. Li and L.-X. Zhu, An adaptive estimation of dimension reduction space, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64 (2002), 363-410. doi: 10.1111/1467-9868.03411. Google Scholar

[45]

J. Yan, Z. Lei, D. Yi and S. Z. Li, Towards incremental and large scale face recognition, in 2011 International Joint Conference on Biometrics (IJCB), IEEE, (2011), 1–6.Google Scholar

[46]

J. Ye, Q. Li, H. Xiong, H. Park, R. Janardan and V. Kumar, IDR/QR: An incremental dimension reduction algorithm via QR decomposition, KDD '04 Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2004), 364–373. doi: 10.1145/1014052.1014093. Google Scholar

[47]

I.-C. Yeh, Modeling of strength of high-performance concrete using artificial neural networks, Cement and Concrete Research, 28 (1998), 1797-1808. doi: 10.1016/S0008-8846(98)00165-3. Google Scholar

[48]

I.-C. Yeh, Design of high-performance concrete mixture using neural networks and nonlinear programming, Journal of Computing in Civil Engineering, 13 (1999), 36-42. doi: 10.1061/(ASCE)0887-3801(1999)13:1(36). Google Scholar

[49]

I.-C. Yeh, Modeling slump flow of concrete using second-order regressions and artificial neural networks, Cement and Concrete Composites, 29 (2007), 474-480. doi: 10.1016/j.cemconcomp.2007.02.001. Google Scholar

[50]

N. Zhang, Z. Yu and Q. Wu, Overlapping sliced inverse regression for dimension reduction, preprint, arXiv: 1806.08911.Google Scholar

[51]

T. ZhangW. Ye and Y. Shan, Application of sliced inverse regression with fuzzy clustering for thermal error modeling of CNC machine tool, The International Journal of Advanced Manufacturing Technology, 85 (2016), 2761-2771. doi: 10.1007/s00170-015-8135-6. Google Scholar

[52]

H. Zhao and P. C. Yuen, Incremental linear discriminant analysis for face recognition, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 210-221. doi: 10.1109/TSMCB.2007.908870. Google Scholar

[53]

H. ZhaoP. C. Yuen and J. T. Kwok, A novel incremental principal component analysis and its application for face recognition, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 36 (2006), 873-886. Google Scholar

show all references

References:
[1]

A. AntoniadisS. Lambert-Lacroix and F. Leblanc, Effective dimension reduction methods for tumor classification using gene expression data, Bioinformatics, 19 (2003), 563-570. doi: 10.1093/bioinformatics/btg062. Google Scholar

[2]

Z. Bai and X. He, A chi-square test for dimensionality with non-Gaussian data, Journal of Multivariate Analysis, 88 (2004), 109-117. doi: 10.1016/S0047-259X(03)00056-3. Google Scholar

[3]

M. P. Barrios and S. Velilla, A bootstrap method for assessing the dimension of a general regression problem, Statistics & Probability Letters, 77 (2007), 247-255. doi: 10.1016/j.spl.2006.07.020. Google Scholar

[4]

C. Becker and R. Fried, Sliced inverse regression for high-dimensional time series, in Exploratory Data Analysis in Empirical Research, Springer, (2003), 3–11. Google Scholar

[5]

P. N. Belhumeur, J. P. Hespanha and D. J. Kriegman, Eigenfaces vs. Fisherfaces: recognition using class specific linear projection, European Conference on Computer Vision, (1996), 43–58. doi: 10.1007/BFb0015522. Google Scholar

[6]

E. Bura and R. D. Cook, Extending sliced inverse regression: the weighted chi-squared test, Journal of the American Statistical Association, 96 (2001), 996-1003. doi: 10.1198/016214501753208979. Google Scholar

[7]

S. ChandrasekaranB. S. ManjunathY.-F. WangJ. Winkeler and H. Zhang, An eigenspace update algorithm for image analysis, Graphical Models and Image Processing, 59 (1997), 321-332. Google Scholar

[8]

D. ChuL.-Z. ZhaoM. K.-P. Ng and X. Wang, Incremental linear discriminant analysis: A fast algorithm and comparisons, IEEE Transactions on Neural Networks and Learning Systems, 26 (2015), 2716-2735. doi: 10.1109/TNNLS.2015.2391201. Google Scholar

[9]

R. D. Cook, Using dimension-reduction subspaces to identify important inputs in models of physical systems, in Proceedings of the section on Physical and Engineering Sciences, American Statistical Association Alexandria, VA, (1994), 18–25.Google Scholar

[10]

R. D. Cook and S. Weisberg, Sliced inverse regression for dimension reduction: Comment, Journal of the American Statistical Association, 86 (1991), 328-332.Google Scholar

[11]

R. D. Cook and and X. Zhang, Fused estimators of the central subspace in sufficient dimension reduction, Journal of the American Statistical Association, 109 (2014), 815-827. doi: 10.1080/01621459.2013.866563. Google Scholar

[12]

J. J. Dai, L. Lieu and D. Rocke, Dimension reduction for classification with gene expression microarray data, Statistical Applications in Genetics and Molecular Biology, 5 (2006), Art. 6, 21 pp. doi: 10.2202/1544-6115.1147. Google Scholar

[13]

L. ElnitskiR. C. HardisonJ. LiS. YangD. KolbeP. EswaraM. J. O'ConnorS. SchwartzW. Miller and F. Chiaromonte, Distinguishing regulatory DNA from neutral sites, Genome Research, 13 (2003), 64-72. doi: 10.1101/gr.817703. Google Scholar

[14]

L. Ferré, Determining the dimension in sliced inverse regression and related methods, Journal of the American Statistical Association, 93 (1998), 132-140. doi: 10.2307/2669610. Google Scholar

[15]

K. Fukumizu, F. R. Bach and M. I. Jordan, Kernel dimensionality reduction for supervised learning, in NIPS, (2003), 81–88.Google Scholar

[16]

A. GannounS. GirardC. Guinot and J. Saracco, Sliced inverse regression in reference curves estimation, Computational Statistics & Data Analysis, 46 (2004), 103-122. doi: 10.1016/S0167-9473(03)00141-5. Google Scholar

[17]

Y. A. Ghassabeh, A. Ghavami and H. A. Moghaddam, A new incremental face recognition system, in IEEE Workshop on Intelligent Data Acquisition and Advanced Computing Systems: Technology and Applications, (2007), 335–340. doi: 10.1109/IDAACS.2007.4488435. Google Scholar

[18]

Y. A. Ghassabeh and H. A. Moghaddam, A new incremental optimal feature extraction method for on-line applications, in International Conference Image Analysis and Recognition, Springer, (2007), 399–410. doi: 10.1007/978-3-540-74260-9_36. Google Scholar

[19]

B. GuV. S. ShengZ. WangD. HoS. Osman and S. Li, Incremental learning for v-support vector regression, Neural Networks, 67 (2015), 140-150. Google Scholar

[20]

P. M. HallD. A. Marshall and R. F. Martin, Incremental eigenanalysis for classification, BMVC, 98 (1998), 286-295. doi: 10.5244/C.12.29. Google Scholar

[21]

P. M. HallD. A. Marshall and R. F. Martin, Merging and splitting eigenspace models, IEEE Transactions on Pattern Analysis and Machine Intelligence, 22 (2000), 1042-1049. doi: 10.1109/34.877525. Google Scholar

[22]

P. HeK.-T. Fang and C.-J. Xu, The classification tree combined with SIR and its applications to classification of mass spectra, Journal of Data Science, 1 (2003), 425-445. Google Scholar

[23]

T.-K. Kim, S.-F. Wong, B. Stenger, J. Kittler and R. Cipolla, Incremental linear discriminant analysis using sufficient spanning set approximations, in IEEE Conference on Computer Vision and Pattern Recognition, (2007), 1–8. doi: 10.1109/CVPR.2007.382985. Google Scholar

[24]

M. H. Law and A. K. Jain, Incremental nonlinear dimensionality reduction by manifold learning, IEEE Transactions on Pattern Analysis and Machine Intelligence, 28 (2006), 377-391. doi: 10.1109/TPAMI.2006.56. Google Scholar

[25]

K.-C. Li, Sliced inverse regression for dimension reduction, Journal of the American Statistical Association, 86 (1991), 316-327. doi: 10.1080/01621459.1991.10475035. Google Scholar

[26]

K.-C. Li, On principal hessian directions for data visualization and dimension reduction: Another application of Stein's lemma, Journal of the American Statistical Association, 87 (1992), 1025-1039. doi: 10.1080/01621459.1992.10476258. Google Scholar

[27]

L. Li and H. Li, Dimension reduction methods for microarrays with application to censored survival data, Bioinformatics, 20 (2004), 3406-3412. doi: 10.1093/bioinformatics/bth415. Google Scholar

[28]

L. Li and X. Yin, Sliced inverse regression with regularizations, Biometrics, 64 (2008), 124-131. doi: 10.1111/j.1541-0420.2007.00836.x. Google Scholar

[29]

L.-P. Liu, Y. Jiang and Z.-H. Zhou, Least square incremental linear discriminant analysis, in 2009 Ninth IEEE International Conference on Data Mining, (2009), 298–306. doi: 10.1109/ICDM.2009.78. Google Scholar

[30]

G.-F. LuJ. Zou and Y. Wang, Incremental learning of complete linear discriminant analysis for face recognition, Knowledge-Based Systems, 31 (2012), 19-27. doi: 10.1016/j.knosys.2012.01.016. Google Scholar

[31]

G. M. Nkiet, Consistent estimation of the dimensionality in sliced inverse regression, Annals of the Institute of Statistical Mathematics, 60 (2008), 257-271. doi: 10.1007/s10463-006-0106-0. Google Scholar

[32]

A. ÖztaşM. PalaE. ÖzbayE. KancaN. Caglar and M. A. Bhatti, Predicting the compressive strength and slump of high strength concrete using neural network, Construction and Building Materials, 20 (2006), 769-775. Google Scholar

[33]

S. PangS. Ozawa and N. Kasabov, Incremental linear discriminant analysis for classification of data streams, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 35 (2005), 905-914. doi: 10.1109/TSMCB.2005.847744. Google Scholar

[34]

C.-X. Ren and D.-Q. Dai, Incremental learning of bidirectional principal components for face recognition, Pattern Recognition, 43 (2010), 318-330. doi: 10.1016/j.patcog.2009.05.020. Google Scholar

[35]

P. Rodriguez and B. Wohlberg, A matlab implementation of a fast incremental principal component pursuit algorithm for video background modeling, in 2014 IEEE International Conference on Image Processing (ICIP), (2014), 3414–3416. doi: 10.1109/ICIP.2014.7025692. Google Scholar

[36]

J. R. Schott, Determining the dimensionality in sliced inverse regression, Journal of the American Statistical Association, 89 (1994), 141-148. doi: 10.1080/01621459.1994.10476455. Google Scholar

[37]

C. M. Setodji and R. D. Cook, K-means inverse regression, Technometrics, 46 (2004), 421-429. doi: 10.1198/004017004000000437. Google Scholar

[38]

F. X. Song, D. Zhang, Q. Chen and J. Yang, A novel supervised dimensionality reduction algorithm for online image recognition, in Pacific-Rim Symposium on Image and Video Technology, Springer, (2006), 198–207. doi: 10.1007/11949534_20. Google Scholar

[39]

J.-G. WangE. Sung and W.-Y. Yau, Incremental two-dimensional linear discriminant analysis with applications to face recognition, Journal of Network and Computer Applications, 33 (2010), 314-322. doi: 10.1016/j.jnca.2009.12.014. Google Scholar

[40]

X. Wang, Incremental and Regularized Linear Discriminant Analysis, Ph.D thesis, National University of Singapore, 2012.Google Scholar

[41]

J. WengY. Zhang and W.-S. Hwang, Candid covariance-free incremental principal component analysis, IEEE Transactions on Pattern Analysis and Machine Intelligence, 25 (2003), 1034-1040. Google Scholar

[42]

H.-M. Wu, Kernel sliced inverse regression with applications to classification, Journal of Computational and Graphical Statistics, 17 (2008), 590-610. doi: 10.1198/106186008X345161. Google Scholar

[43]

Q. WuS. Mukherjee and F. Liang, Localized sliced inverse regression, Journal of Computational and Graphical Statistics, 19 (2010), 843-860. doi: 10.1198/jcgs.2010.08080. Google Scholar

[44]

Y. XiaH. TongW. Li and L.-X. Zhu, An adaptive estimation of dimension reduction space, Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64 (2002), 363-410. doi: 10.1111/1467-9868.03411. Google Scholar

[45]

J. Yan, Z. Lei, D. Yi and S. Z. Li, Towards incremental and large scale face recognition, in 2011 International Joint Conference on Biometrics (IJCB), IEEE, (2011), 1–6.Google Scholar

[46]

J. Ye, Q. Li, H. Xiong, H. Park, R. Janardan and V. Kumar, IDR/QR: An incremental dimension reduction algorithm via QR decomposition, KDD '04 Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, (2004), 364–373. doi: 10.1145/1014052.1014093. Google Scholar

[47]

I.-C. Yeh, Modeling of strength of high-performance concrete using artificial neural networks, Cement and Concrete Research, 28 (1998), 1797-1808. doi: 10.1016/S0008-8846(98)00165-3. Google Scholar

[48]

I.-C. Yeh, Design of high-performance concrete mixture using neural networks and nonlinear programming, Journal of Computing in Civil Engineering, 13 (1999), 36-42. doi: 10.1061/(ASCE)0887-3801(1999)13:1(36). Google Scholar

[49]

I.-C. Yeh, Modeling slump flow of concrete using second-order regressions and artificial neural networks, Cement and Concrete Composites, 29 (2007), 474-480. doi: 10.1016/j.cemconcomp.2007.02.001. Google Scholar

[50]

N. Zhang, Z. Yu and Q. Wu, Overlapping sliced inverse regression for dimension reduction, preprint, arXiv: 1806.08911.Google Scholar

[51]

T. ZhangW. Ye and Y. Shan, Application of sliced inverse regression with fuzzy clustering for thermal error modeling of CNC machine tool, The International Journal of Advanced Manufacturing Technology, 85 (2016), 2761-2771. doi: 10.1007/s00170-015-8135-6. Google Scholar

[52]

H. Zhao and P. C. Yuen, Incremental linear discriminant analysis for face recognition, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 38 (2008), 210-221. doi: 10.1109/TSMCB.2007.908870. Google Scholar

[53]

H. ZhaoP. C. Yuen and J. T. Kwok, A novel incremental principal component analysis and its application for face recognition, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), 36 (2006), 873-886. Google Scholar

Figure 1.  Simulation results for model (9). (a) Trace correlation and (b) cumulative calculation time by SIR, ISIR, and IOSIR
Figure 2.  Mean square errors (MSE) for two real data applications: (a) for Concrete Compressive Strength data and (b) for Cpusmall data
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