Interval Matrix Eigen/Singular-Value Decomposition and an Application

Transcription

1 Interval Matrix Eigen/Singular-Value Decomposition and an Application CHENYI HU Professor and Chairman Computer Science Department University of Central Arkansas, USA URL: 1 RANMEP 2008,Taiwan Tsing Hua University

2 Outline Why interval data and interval matrix? Eigenvalue/singular values of an interval matrix Interval computing Practical approach for interval matrix eigenvalue/singular value decomposition An application on computational finance 2 RANMEP 2008,Taiwan Tsing Hua University

3 Interval data Source: Data collected from observation and computing contain error inevitably Nature:Variability and uncertainty are the nature of real world phenomena Ranges vs. points: qualitative indicators are often presented as ranges rather than points Stream vs. spot: segment and cross intersection 3 RANMEP 2008,Taiwan Tsing Hua University

4 Interval matrix Interval matrix game Interval decision making matrix Interval coefficient matrix in interval linear systems of equations, Interval normal matrix in function least-squares approximation Interval Jacobean for nonlinear dynamic systems 4 RANMEP 2008,Taiwan Tsing Hua University

5 Interval eigenvalue/vectors Let A be an interval matrix. We call Λ and X the eigenvalue and eigenvector sets of A. If for any λ Λ, a nonzero vector x X and A A such that λx = Ax; and for any nonzero vector x X, λ Λ and A A such that λx = Ax. 5 RANMEP 2008,Taiwan Tsing Hua University

6 Interval singular value Similarly, the singular value set of an interval matrix of A consists of diagonal matrices Σ, A A, orthonormal matrices U and V, such that A =U ΣV The challenge is to computationally find the eigenvalue and singular value sets of an interval matrix 6 RANMEP 2008,Taiwan Tsing Hua University

7 Interval computing Moore proposed interval computing in later 1950 s Operations: Arithmetic: +, -,, Set :,, Logic: <, =, >,,, Utility functions: midpoint(), width(), I/O Hardware and software: Intel Itanium processor Sun Studio, C++ standard library, Interval BLAS Ups and downs, past and the current 7 RANMEP 2008,Taiwan Tsing Hua University

8 Computational approaches Interval representations: Endpoint representation Midpoint-radius representation Computer representation Notations Practical approach An application 8 RANMEP 2008,Taiwan Tsing Hua University

9 An application in computational finance Chen, Roll and Ross stock market forecasting (1986): changes in the stock market (SP t ) are linearly determined by the following five macroeconomics factors: Growth rate variations of seasonally-adjusted Industrial Production Index (IP), Changes in expected inflation (DEI t ) Changes in unexpected inflation (UI t ), Default risk premiums (DEF t ), and Unexpected changes in interest rates (TERM t ) 9 RANMEP 2008,Taiwan Tsing Hua University

10 Point dataset 10 RANMEP 2008,Taiwan Tsing Hua University

11 S & P 500 interval forecasts Figure 2. Out-of-sample 10-year rolling interval forecasts Changes in stock market Time SP low er bound Predicted low er SP upper bound Predicted upper 11 RANMEP 2008,Taiwan Tsing Hua University

12 Comparisons Average error Standard deviation OLS Accuracy ratio Total # of missing Confidence Interval based approaches std dev. (95%) (90%) std error (95%) (90%) Low-up bounds Inner Approx Interval comp RANMEP 2008,Taiwan Tsing Hua University

13 Singular value decomposition 13 RANMEP 2008,Taiwan Tsing Hua University

14 Conclusion and acknowledgements Initial study on interval matrix singular value decomposition has discovered interesting results Further studies are needed Acknowledgements: U. S. National Science Foundation: CISE/CCF , and CISE/CCF Collaborators NCTS 14 RANMEP 2008,Taiwan Tsing Hua University

15 References 1. Hu, C. et al, Knowledge processing with interval and soft computing, Springer, expected in Collins, D. and Hu, C., Studying interval valued matrix games with fuzzy logic, J. Soft Computing, 12(2), , Hu, C. and He, L., An application of interval methods for stock market forecasting, Reliable Computing, 13(5), He, L. and Hu, C., Impacts of interval measurement on studies of economic variability: evidence from stock market variability forecasting, J. Risk Finance, 8(5), , Interval Computations, 6. Nooner, M. and Hu, C., A computational environment for interval matrices, the NSF 2006 Workshop on Reliable Engineering Computing, pp , dekorvin, A., Hu, C. and Chen, P., Generating and Applying Rule for Interval Valued Fuzzy Observations, Lecture Notes in Computer Science, Vol. 3177, pp , Springer-Verlag, Hu, C., Xu, S., and Yang, X., An Introduction to Interval Computation, J. Theory and Practicein System Science,Vol. 23, No. 4, pp , in Chinese, RANMEP 2008,Taiwan Tsing Hua University