Interval Matrix Eigen/ Singular-Value Decomposition and an - PowerPoint PPT Presentation

Interval Matrix Eigen/ Singular-Value Decomposition and an Application CHENYI HU Professor and Chairman Computer S cience Department University of Central Arkansas, US A URL: www.cs .uca.edu RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 1

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 RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 2

Interval data � S ource : 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 � S tream vs. s pot : segment and cross intersection RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 3

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 RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 4

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 λ x = Ax; and for any nonzero vector x ∈ X , ∃ λ ∈ Λ and A ∈ A such that λ x = Ax. RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 5

Interval singular value imilarly, the singular value set of an interval matrix of A � S consists of diagonal matrices Σ , ∋ ∀ A ∈ A, ∃ orthonormal , such that A = U Σ V matrices U and V � The challenge is to computationally find the eigenvalue and singular value sets of an interval matrix RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 6

Interval computing � Moore proposed interval computing in later 1950 ’ s � Operations: � Arithmetic: + , -, • , ÷ et : ∩ , ∪ , ¬ � S � Logic: < , = , > , ⊂ , ⊆ , ⊇ � Utility functions: midpoint(), width(), I/ O � Hardware and software: � Intel Itanium processor � S un S tudio, C+ + standard library, Interval BLAS � Ups and downs, past and the current RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 7

Computational approaches � Interval representations: � Endpoint representation � Midpoint-radius representation � Computer representation � Notations � Practical approach � An application RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 8

An application in computational finance � Chen, Roll and Ross stock market forecasting (1986): changes in the stock market (S P 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 ) RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 9

Point dataset RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 10

S & P 500 interval forecasts Figure 2. Out-of-sample 10-year rolling interval forecasts 0.2 0.15 0.1 Changes in stock market 0.05 0 0 3 6 9 2 5 8 1 4 7 0 3 6 9 2 5 8 1 4 7 0 3 4 4 4 4 5 5 5 6 6 6 7 7 7 7 8 8 8 9 9 9 0 0 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 -0.05 - -0.1 -0.15 -0.2 Time SPlow er bound Predicted low er SPupper bound Predicted upper RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 11

Comparisons Average Average Standard Standard Accuracy Accuracy Total # of Total # of error error deviation deviation ratio ratio missing missing OLS OLS 0.20572 0.20572 0.18996 0.18996 Confidence Interval based approaches Confidence Interval based approaches std dev std dev . (95%) . (95%) 0.723505 0.723505 0.311973 0.311973 0.125745 0.125745 5 (90%) (90%) 0.617097 0.617097 0.338792 0.338792 0.145145 0.145145 7 std error (95%) std error (95%) 0.36549 0.36549 0.431112 0.431112 0.1219 0.1219 35 35 (90%) (90%) 0.365712 0.365712 0.430969 0.430969 0.104936 0.104936 36 36 Low Low-up bounds up bounds 0.066643 0.066643 0.040998 0.040998 0.4617 0.4617 0 Inner Approx. Inner Approx. 0.073038 0.073038 0.038151 0.038151 0.385531 0.385531 0 Interval comp. Interval comp. 0.0516624 0.0516624 0.032238 0.032238 0.641877 0.641877 0 RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 12

Singular value decomposition RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 13

Conclusion and acknowledgements � Initial study on interval matrix singular value decomposition has discovered interesting results � Further studies are needed � Acknowledgements: � U. S . National S cience Foundation: CIS E/ CCF-0727798, and CIS E/ CCF-0202042 � Collaborators � NCTS RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 14

References Hu, C. et al, Knowledge proces oft computing , S pringer, expected in 2008. 1. s ing with interval and s Collins, D. and Hu, C., Studying interval valued matrix games with fuzzy logic, J . S oft Computing, 2. 12(2), 147-155, 2008. Hu, C. and He, L., An application of interval methods for s ting, Reliable 3. tock market forecas Computing, 13(5), 2007 He, L. and Hu, C., Impacts of interval meas urement on s tudies of economic variability: evidence from 4. ting, J . Risk Finance, 8(5), 489-507, 2007 s tock market variability forecas Interval Computations , http:/ / www.cs .utep.edu/ interval-comp/ main.html 5. , the NS F 2006 Workshop Nooner, M. and Hu, C., A computational environment for interval matrices 6. on Reliable Engineering Computing, pp. 65-74, 2006 deKorvin, A., Hu, C. and Chen, P . , Generating and Applying Rule for Interval V 7. alued Fuzzy ervations , Lecture Notes in Computer S cience, Vol. 3177, pp. 279-284, S pringer-Verlag, Obs 2004 Hu, C., Xu, S ., and Yang, X., An Introduction to Interval Computation , J . Theory and Practice in 8. S ystem S cience,Vol. 23, No. 4, pp. 59-62, in Chinese, 2003 RANMEP 2008,Taiwan Tsing Hua University January 4, 2008 15