# American Institute of Mathematical Sciences

• 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:

show all references

##### References:
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] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [2] Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045 [3] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435 [4] Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377 [5] Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164 [6] Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103 [7] Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $p$ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442 [8] Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076 [9] Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016 [10] Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247 [11] Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216 [12] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [13] Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011 [14] Mathew Gluck. Classification of solutions to a system of $n^{\rm th}$ order equations on $\mathbb R^n$. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246 [15] Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081 [16] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [17] Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012 [18] Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012 [19] M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014 [20] Qianqian Han, Xiao-Song Yang. Qualitative analysis of a generalized Nosé-Hoover oscillator. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020346

2019 Impact Factor: 1.366