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

CHENYI HU

Professor and Chairman Computer S cience Department University of Central Arkansas, US A URL: www.cs .uca.edu

Interval Matrix Eigen/ Singular-Value Decomposition and an Application

January 4, 2008 1 RANMEP 2008,Taiwan Tsing Hua University

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

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

Interval data

S

• urce: Data collected from observation and computing

contain error inevitably Nature:Variability and uncertainty are the nature of real world phenomena

• Rangesvs. points

: qualitative indicators are often presented as ranges rather than points S

tream vs. s pot: segment and cross intersection

January 4, 2008 3 RANMEP 2008,Taiwan Tsing Hua University

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

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

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.

January 4, 2008 5 RANMEP 2008,Taiwan Tsing Hua University

Interval singular value

S imilarly, 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

January 4, 2008 6 RANMEP 2008,Taiwan Tsing Hua University

Interval computing

7

Moore proposed interval computing in later 1950’s Operations:

Arithmetic: + , -, •, ÷ S et : ∩ , ∪, ¬ 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

January 4, 2008 RANMEP 2008,Taiwan Tsing Hua University

Computational approaches

Interval representations:

Endpoint representation Midpoint-radius representation Computer representation Notations

Practical approach

An application

January 4, 2008 8 RANMEP 2008,Taiwan Tsing Hua University

An application in computational finance

Chen, Roll and Ross stock market forecasting (1986): changes in the stock market (S Pt) are linearly determined by the following five macroeconomics factors:

Growth rate variations of seasonally-adjusted Industrial Production Index (IP), Changes in expected inflation (DEIt) Changes in unexpected inflation (UIt), Default risk premiums (DEFt), and Unexpected changes in interest rates (TERMt)

January 4, 2008 9 RANMEP 2008,Taiwan Tsing Hua University

Point dataset

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S & P 500 interval forecasts

Figure 2. Out-of-sample 10-year rolling interval forecasts

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 1 9 4 1 9 4 3 1 9 4 6 1 9 4 9 1 9 5 2 1 9 5 5 1 9 5 8 1 9 6 1 1 9 6 4 1 9 6 7 1 9 7 1 9 7 3 1 9 7 6 1 9 7 9 1 9 8 2 1 9 8 5 1 9 8 8

• 1

9 9 1 1 9 9 4 1 9 9 7 2 2 3 Time Changes in stock market SPlow er bound Predicted low er SPupper bound Predicted upper

January 4, 2008 11 RANMEP 2008,Taiwan Tsing Hua University

Comparisons

Average Average error error Standard Standard deviation deviation Accuracy Accuracy ratio ratio Total # of Total # of missing missing OLS OLS 0.20572 0.20572 0.18996 0.18996

Confidence Interval based approaches Confidence Interval based approaches

std dev . (95%) std dev . (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 Inner Approx. Inner Approx. 0.073038 0.073038 0.038151 0.038151 0.385531 0.385531

Interval comp. Interval comp. 0.0516624 0.0516624 0.032238 0.032238 0.641877 0.641877

January 4, 2008 12 RANMEP 2008,Taiwan Tsing Hua University

Singular value decomposition

January 4, 2008 13 RANMEP 2008,Taiwan Tsing Hua University

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

January 4, 2008 14 RANMEP 2008,Taiwan Tsing Hua University

References

1.

Hu, C. et al, Knowledge proces

s ing with interval and s

• ft computing, S

pringer, expected in 2008.

2.

Collins, D. and Hu, C., Studying interval valued matrix games with fuzzy logic, J . S

• ft Computing,

12(2), 147-155, 2008.

3.

Hu, C. and He, L., An application of interval methods for s

tock market forecas ting, Reliable

Computing, 13(5), 2007

4.

He, L. and Hu, C., Impacts of interval meas urement on s tudies of economic variability: evidence from s tock market variability forecas ting, J

. Risk Finance, 8(5), 489-507, 2007

5.

Interval Computations , http:/ / www.cs .utep.edu/ interval-comp/ main.html

6.

Nooner, M. and Hu, C., A computational environment for interval matrices , the NS

F 2006 Workshop

• n Reliable Engineering Computing, pp. 65-74, 2006

7.

deKorvin, A., Hu, C. and Chen, P . , Generating and Applying Rule for Interval V

alued Fuzzy Obs ervations, Lecture Notes in Computer S

cience, Vol. 3177, pp. 279-284, S pringer-Verlag, 2004

8.

Hu, C., Xu, S ., and Yang, X., An Introduction to Interval Computation, J . Theory and Practice in S ystem S cience,Vol. 23, No. 4, pp. 59-62, in Chinese, 2003

January 4, 2008 15 RANMEP 2008,Taiwan Tsing Hua University