• Previous Article
    A $ {BMAP/BMSP/1} $ queue with Markov dependent arrival and Markov dependent service batches
  • JIMO Home
  • This Issue
  • Next Article
    A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure
doi: 10.3934/jimo.2020072

Principal component analysis with drop rank covariance matrix

School of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, China

* Corresponding author: Bingo Wing-Kuen Ling

Received  November 2019 Revised  December 2019 Published  March 2020

This paper considers the principal component analysis when the covariance matrix of the input vectors drops rank. This case sometimes happens when the total number of the input vectors is very limited. First, it is found that the eigen decomposition of the covariance matrix is not uniquely defined. This implies that different transform matrices could be obtained for performing the principal component analysis. Hence, the generalized form of the eigen decomposition of the covariance matrix is given. Also, it is found that the matrix with its columns being the eigenvectors of the covariance matrix is not necessary to be unitary. This implies that the transform for performing the principal component analysis may not be energy preserved. To address this issue, the necessary and sufficient condition for the matrix with its columns being the eigenvectors of the covariance matrix to be unitary is derived. Moreover, since the design of the unitary transform matrix for performing the principal component analysis is usually formulated as an optimization problem, the necessary and sufficient condition for the first order derivative of the Lagrange function to be equal to the zero vector is derived. In fact, the unitary matrix with its columns being the eigenvectors of the covariance matrix is only a particular case of the condition. Furthermore, the necessary and sufficient condition for the second order derivative of the Lagrange function to be a positive definite function is derived. It is found that the unitary matrix with its columns being the eigenvectors of the covariance matrix does not satisfy this condition. Computer numerical simulation results are given to valid the results.

Citation: Yitong Guo, Bingo Wing-Kuen Ling. Principal component analysis with drop rank covariance matrix. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020072
References:
[1]

J. DehaeneM. Moonen and J. Vandewalle, An improved stochastic gradient algorithm for principal component analysis and subspace tracking, IEEE Transactions on Signal Processing, 45 (1997), 2582-2586.   Google Scholar

[2]

C. L. Fancourt and J. C. Principe, Competitive principal component analysis for locally stationary time series, IEEE Transactions on Signal Processing, 46 (1998), 3068-3081.  doi: 10.1109/78.726819.  Google Scholar

[3]

J. B. O. S. Filho and P. S. R. Diniz, A fixed-point online kernel principal component extraction algorithm, IEEE Transactions on Signal Processing, 65 (2017), 6244-6259.  doi: 10.1109/TSP.2017.2750119.  Google Scholar

[4]

I. A. Guimarães and A. C. Neto, Estimation in polytomous logistic model: Comparison of methods, Journal of Industrial and Management Optimization, Journal of Industrial and Management Optimization, 5 (2009), 239-252.  doi: 10.3934/jimo.2009.5.239.  Google Scholar

[5]

R. HeB. G. HuW. S. Zheng and X. W. Kong, Robust principal component analysis based on maximum correntropy criterion, IEEE Transactions on Image Processing, 20 (2011), 1485-1494.  doi: 10.1109/TIP.2010.2103949.  Google Scholar

[6]

S. M. Huang and J. F. Yang, Improved principal component regression for face recognition under illumination variations, IEEE Signal Processing Letters, 19 (2012), 179-182.  doi: 10.1109/LSP.2012.2185492.  Google Scholar

[7]

J. Kang, X. Lin and G. Yang, Research of multi-scale PCA algorithm for face recognition, International Conference on Information and Communications Technologies, ICT, (2015), 1–5. Google Scholar

[8]

M. S. KangJ. H. BaeB. S. Kang and K. T. Kim, ISAR cross-range scaling using iterative processing via principal component analysis and bisection algorithm, IEEE Transactions on Signal Processing, 64 (2016), 3909-3918.  doi: 10.1109/TSP.2016.2552511.  Google Scholar

[9]

S. KaramizadehS. M. AbdullahA. A. Manaf and M. Zamani, An overview of principal component analysis, Journal of Signal and Information Processing, 4 (2013), 173-175.  doi: 10.4236/jsip.2013.43B031.  Google Scholar

[10]

H. C. LiuS. T. Li and L. Y. Fang, Robust object tracking based on principal component analysis and local sparse representation, IEEE Transactions on Instrumentation and Measurement, 64 (2015), 2863-2875.   Google Scholar

[11]

Z. Li, Q. L. Ye, Y. T. Guo, Z. K. Tian, B. W. K. Ling and R. W. K. Lam, Wearable non-invasive blood glucose estimation via empirical mode decomposition based hierarchical multiresolution analysis and random forest, 23rd IEEE International Conference on Digital Signal Processing, DSP, (2018), 1–5. doi: 10.1109/ICDSP.2018.8631545.  Google Scholar

[12]

A. MoradiJ. RazmiR. Babazadeh and A. Sabbaghnia, An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty, Journal of Industrial and Management Optimization, 15 (2019), 855-879.  doi: 10.3934/jimo.2018074.  Google Scholar

[13]

E. M. Moreno, Non-invasive estimate of blood glucose and blood pressure from a photoplethysmography by means of machine learning techniques, Artificial Intelligence in Medicine, 53 (2011), 127-138.   Google Scholar

[14]

P. P. MarkopulosG. N. Karystinos and D. A. Pados, Optimal algorithms for L1-subpace signal processing, IEEE Transactions on Signal Processing, 62 (2014), 5046-5058.  doi: 10.1109/TSP.2014.2338077.  Google Scholar

[15]

O. Ozgonenel and T. Yalcin, Principal component analysis (PCA) based neural network for motor protection, 10th IET International Conference on Developments in Power System Protection, DPSP, (2010), 1–5. doi: 10.1049/cp.2010.0252.  Google Scholar

[16]

S. Ouyang and Z. Bao, Fast principal component extraction by a weighted information criterion, IEEE Transactions on Signal Processing, 50 (2002), 1994-2002.  doi: 10.1109/TSP.2002.800395.  Google Scholar

[17]

Y. M. RenL. Liao and S. J. Maybank, Hyperspectral image spectral-spatial feature extraction via tensor principal component analysis, IEEE Geoscience and Remote Sensing Letters, 14 (2017), 1431-1435.  doi: 10.1109/LGRS.2017.2686878.  Google Scholar

[18]

C. E. Thomaz and G. A. Giraldi, A new ranking method for principal components analysis and its application to face image analysis, Image and Vision Computing, 28 (2009), 902-913.  doi: 10.1016/j.imavis.2009.11.005.  Google Scholar

[19]

G. Tang and A. Nehorai, Constrained Cramér-Rao bound on robust principal component analysis, IEEE Transactions on Signal Processing, 59 (2011), 5070-5076.  doi: 10.1109/TSP.2011.2161984.  Google Scholar

[20]

X. C XiuY. YangW.Q. LiuL.C. Kong and M.J. Shang, An improved total variation regularized RPCA for moving object detection with dynamic background, Journal of Industrial and Management Optimization, 13 (2017), 1-14.  doi: 10.3934/jimo.2019024.  Google Scholar

[21]

S. Y. YiZ. H. LaiZ. Y. HeY. Cheung and Y. Liu, Joint sparse principal component analysis, Pattern Recognition, 61 (2016), 524-536.  doi: 10.1016/j.patcog.2016.08.025.  Google Scholar

[22]

A. ZareA. OzdemirM. A. Iwen and S. Aviyente, Extension of PCA to higher order data structures: An introduction to tensors, tensor decompositions, and tensor PCA, Proceedings of the IEEE, 106 (2018), 1341-1358.  doi: 10.1109/JPROC.2018.2848209.  Google Scholar

[23]

H. Zou and L. Xue, A selective overview of sparse principal component analysis, Proceedings of the IEEE, 106 (2018), 1311-1320.  doi: 10.1109/JPROC.2018.2846588.  Google Scholar

show all references

References:
[1]

J. DehaeneM. Moonen and J. Vandewalle, An improved stochastic gradient algorithm for principal component analysis and subspace tracking, IEEE Transactions on Signal Processing, 45 (1997), 2582-2586.   Google Scholar

[2]

C. L. Fancourt and J. C. Principe, Competitive principal component analysis for locally stationary time series, IEEE Transactions on Signal Processing, 46 (1998), 3068-3081.  doi: 10.1109/78.726819.  Google Scholar

[3]

J. B. O. S. Filho and P. S. R. Diniz, A fixed-point online kernel principal component extraction algorithm, IEEE Transactions on Signal Processing, 65 (2017), 6244-6259.  doi: 10.1109/TSP.2017.2750119.  Google Scholar

[4]

I. A. Guimarães and A. C. Neto, Estimation in polytomous logistic model: Comparison of methods, Journal of Industrial and Management Optimization, Journal of Industrial and Management Optimization, 5 (2009), 239-252.  doi: 10.3934/jimo.2009.5.239.  Google Scholar

[5]

R. HeB. G. HuW. S. Zheng and X. W. Kong, Robust principal component analysis based on maximum correntropy criterion, IEEE Transactions on Image Processing, 20 (2011), 1485-1494.  doi: 10.1109/TIP.2010.2103949.  Google Scholar

[6]

S. M. Huang and J. F. Yang, Improved principal component regression for face recognition under illumination variations, IEEE Signal Processing Letters, 19 (2012), 179-182.  doi: 10.1109/LSP.2012.2185492.  Google Scholar

[7]

J. Kang, X. Lin and G. Yang, Research of multi-scale PCA algorithm for face recognition, International Conference on Information and Communications Technologies, ICT, (2015), 1–5. Google Scholar

[8]

M. S. KangJ. H. BaeB. S. Kang and K. T. Kim, ISAR cross-range scaling using iterative processing via principal component analysis and bisection algorithm, IEEE Transactions on Signal Processing, 64 (2016), 3909-3918.  doi: 10.1109/TSP.2016.2552511.  Google Scholar

[9]

S. KaramizadehS. M. AbdullahA. A. Manaf and M. Zamani, An overview of principal component analysis, Journal of Signal and Information Processing, 4 (2013), 173-175.  doi: 10.4236/jsip.2013.43B031.  Google Scholar

[10]

H. C. LiuS. T. Li and L. Y. Fang, Robust object tracking based on principal component analysis and local sparse representation, IEEE Transactions on Instrumentation and Measurement, 64 (2015), 2863-2875.   Google Scholar

[11]

Z. Li, Q. L. Ye, Y. T. Guo, Z. K. Tian, B. W. K. Ling and R. W. K. Lam, Wearable non-invasive blood glucose estimation via empirical mode decomposition based hierarchical multiresolution analysis and random forest, 23rd IEEE International Conference on Digital Signal Processing, DSP, (2018), 1–5. doi: 10.1109/ICDSP.2018.8631545.  Google Scholar

[12]

A. MoradiJ. RazmiR. Babazadeh and A. Sabbaghnia, An integrated Principal Component Analysis and multi-objective mathematical programming approach to agile supply chain network design under uncertainty, Journal of Industrial and Management Optimization, 15 (2019), 855-879.  doi: 10.3934/jimo.2018074.  Google Scholar

[13]

E. M. Moreno, Non-invasive estimate of blood glucose and blood pressure from a photoplethysmography by means of machine learning techniques, Artificial Intelligence in Medicine, 53 (2011), 127-138.   Google Scholar

[14]

P. P. MarkopulosG. N. Karystinos and D. A. Pados, Optimal algorithms for L1-subpace signal processing, IEEE Transactions on Signal Processing, 62 (2014), 5046-5058.  doi: 10.1109/TSP.2014.2338077.  Google Scholar

[15]

O. Ozgonenel and T. Yalcin, Principal component analysis (PCA) based neural network for motor protection, 10th IET International Conference on Developments in Power System Protection, DPSP, (2010), 1–5. doi: 10.1049/cp.2010.0252.  Google Scholar

[16]

S. Ouyang and Z. Bao, Fast principal component extraction by a weighted information criterion, IEEE Transactions on Signal Processing, 50 (2002), 1994-2002.  doi: 10.1109/TSP.2002.800395.  Google Scholar

[17]

Y. M. RenL. Liao and S. J. Maybank, Hyperspectral image spectral-spatial feature extraction via tensor principal component analysis, IEEE Geoscience and Remote Sensing Letters, 14 (2017), 1431-1435.  doi: 10.1109/LGRS.2017.2686878.  Google Scholar

[18]

C. E. Thomaz and G. A. Giraldi, A new ranking method for principal components analysis and its application to face image analysis, Image and Vision Computing, 28 (2009), 902-913.  doi: 10.1016/j.imavis.2009.11.005.  Google Scholar

[19]

G. Tang and A. Nehorai, Constrained Cramér-Rao bound on robust principal component analysis, IEEE Transactions on Signal Processing, 59 (2011), 5070-5076.  doi: 10.1109/TSP.2011.2161984.  Google Scholar

[20]

X. C XiuY. YangW.Q. LiuL.C. Kong and M.J. Shang, An improved total variation regularized RPCA for moving object detection with dynamic background, Journal of Industrial and Management Optimization, 13 (2017), 1-14.  doi: 10.3934/jimo.2019024.  Google Scholar

[21]

S. Y. YiZ. H. LaiZ. Y. HeY. Cheung and Y. Liu, Joint sparse principal component analysis, Pattern Recognition, 61 (2016), 524-536.  doi: 10.1016/j.patcog.2016.08.025.  Google Scholar

[22]

A. ZareA. OzdemirM. A. Iwen and S. Aviyente, Extension of PCA to higher order data structures: An introduction to tensors, tensor decompositions, and tensor PCA, Proceedings of the IEEE, 106 (2018), 1341-1358.  doi: 10.1109/JPROC.2018.2848209.  Google Scholar

[23]

H. Zou and L. Xue, A selective overview of sparse principal component analysis, Proceedings of the IEEE, 106 (2018), 1311-1320.  doi: 10.1109/JPROC.2018.2846588.  Google Scholar

Figure 1.  Plots of the feature values. (a) Sample I.D.: 1-5. (b) Sample I.D.: 6-10. (c) Sample I.D.: 11-15. (d) Sample I.D.: 16-20
[1]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

[2]

Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233

[3]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017

[4]

Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397

[5]

Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051

[6]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[7]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136

[8]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[9]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184

[10]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[11]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709

[12]

Tao Wu, Yu Lei, Jiao Shi, Maoguo Gong. An evolutionary multiobjective method for low-rank and sparse matrix decomposition. Big Data & Information Analytics, 2017, 2 (1) : 23-37. doi: 10.3934/bdia.2017006

[13]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133

[14]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597

[15]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[16]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[17]

Meiqiao Ai, Zhimin Zhang, Wenguang Yu. First passage problems of refracted jump diffusion processes and their applications in valuing equity-linked death benefits. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021039

[18]

Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a one-sided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247-257. doi: 10.3934/proc.2013.2013.247

[19]

Jan Prüss, Laurent Pujo-Menjouet, G.F. Webb, Rico Zacher. Analysis of a model for the dynamics of prions. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 225-235. doi: 10.3934/dcdsb.2006.6.225

[20]

Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (123)
  • HTML views (395)
  • Cited by (0)

Other articles
by authors

[Back to Top]