Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance
Abstract
:1. Introduction
1.1. Alternating Approach
1.2. The Effect of Noise on SVD
2. Robustness Analysis
2.1. Robustness Analysis for Different Estimators
2.2. The Selection of K
3. Adjustable Robust SVD Algorithms
3.1. Myriad Robust SVD (MySVD)
Algorithm 1 Calculate the first eigentriple ${\lambda}_{1},{\mathbf{u}}_{1},{\mathbf{v}}_{1}$ 
Start with an initial guess of the leading left eigenvector ${\mathbf{u}}_{1}$ and a constant value p 
repeat 
for each column j do 
${K}_{c}\leftarrow \frac{1}{2}\left({M}_{{j}_{\left(\left(1p\right)N\right)}}{M}_{{j}_{\left(pN\right)}}\right)$ 
${a}_{j}\leftarrow arg{min}_{{a}_{j}}{\sum}_{i=1}^{n}log\left({K}_{c}^{2}+{\left({m}_{ij}{a}_{j}{u}_{i1}\right)}^{2}\right)$ 
end for 
${\lambda}_{1}\leftarrow {\left\right\mathbf{a}\left\right}_{2}$ 
${\mathbf{v}}_{1}\leftarrow \mathbf{a}/{\left\right\mathbf{a}\left\right}_{2}$ 
for each row i do 
${K}_{r}\leftarrow \frac{1}{2}\left({M}_{{i}_{\left(\left(1p\right)N\right)}}{M}_{{i}_{\left(pN\right)}}\right)$ 
${b}_{i}\leftarrow arg{min}_{{b}_{j}}{\sum}_{j=1}^{m}log\left({K}_{r}^{2}+{\left({m}_{ij}{b}_{i}{v}_{j1}\right)}^{2}\right)$ 
end for 
${\mathbf{u}}_{1}\leftarrow \mathbf{b}/{\left\right\mathbf{b}\left\right}_{2}$ 
until Convergence 
3.2. Sequential MySVD
Algorithm 2 Sequential MySVD 
Known: 
Original data $\mathbf{M}={\mathbf{USV}}^{T}$, new data $\mathbf{C}$ 
Process: 

4. Application
4.1. Model Set Up
4.2. Factor Extraction
4.3. Numerical Example
5. Conclusion and Future Research
Acknowledgments
Conflicts of Interest
Appendix A. A Simulation Example
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Estimator  Cost Function  Output, $\mathit{\theta}$ 

Linear  ${\sum}_{i=1}^{N}{({x}_{i}\theta )}^{2}$  $mean\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$ 
Median  ${\sum}_{i=1}^{N}{x}_{i}\theta $  $median\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$ 
Myriad  ${\sum}_{i=1}^{N}log({K}^{2}+{({x}_{i}\theta )}^{2})$  $myriad\{{x}_{1},{x}_{2},\dots ,{x}_{N}\}$ 
Factors  Conventional SVD  Myriad Robust SVD (MySVD) 

1  $23.7\%$  $34.8\%$ 
2  $10.2\%$  $14.6\%$ 
3  $9.4\%$  $8.4\%$ 
4  $6.2\%$  $6.3\%$ 
5  $2.3\%$  $2.0\%$ 
All Factors  $51.8\%$  $66.1\%$ 
${\mathit{\epsilon}}_{\mathbf{it}}$  hvalue  pvalue  ${\mathit{R}}^{2}$  

MySVD  $0.00078$  0  $0.3776$  $0.6959$ 
SVD  $0.0053$  1  $2.3\times {10}^{6}$  $0.6741$ 
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Wang, D. Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance. Data 2017, 2, 29. https://doi.org/10.3390/data2030029
Wang D. Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance. Data. 2017; 2(3):29. https://doi.org/10.3390/data2030029
Chicago/Turabian StyleWang, Deshen. 2017. "Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance" Data 2, no. 3: 29. https://doi.org/10.3390/data2030029