Why Linear (and Piecewise Linear) Models Often Successfully Describe Complex Non-Linear Economic and Financial Phenomena: A~Fuzzy-Based Explanation
Subject Areas : Transactions on Fuzzy Sets and Systems
Hung Nguyen
1
,
Vladik Kreinovich
2
1 - Department of Mathematical Science, New Mexico State University, Las Cruce, New Mexico, USA and Faculty of Economics, Chiang Mai University, Chiang Mai, Thailand.
2 - Department of Computer Science, University of Texas at El Paso, El Paso, Texas, USA.
Keywords: Fuzzy Logic, Linear models, Piece-wise linear models, Economics and finance,
Abstract :
Economic and financial phenomena are highly complex and non-linear. However, surprisingly, in many cases, these phenomena are accurately described by linear models – or, sometimes, by piecewise linear ones. In this paper, we show that fuzzy techniques can explain the unexpected efficiency of linear and piecewise linear models: namely, we show that a natural fuzzy-based precisiation of imprecise (“fuzzy”) expert knowledge often leads to linear and piecewise linear models. We show this by applying invariance ideas to analyze which membership functions, which fuzzy “and”-operations (t-norms), and which fuzzy implication operations are most appropriate for applications to economics and finance. We also discuss which expert-motivated nonlinear models should be used to get a more accurate description of economic and financial phenomena: specifically, we show that a natural next step is to add cubic terms to the linear (and piece-wise linear) expressions, and, in general, to consider polynomial (and piece-wise polynomial) dependencies.
[1] J. Aczel, Lectures on Functional Equations and Their Applications, Dover Publ., New York, (2006).
[2] R. Belohlavek, J. W. Dauben and G. J. Klir, Fuzzy Logic and Mathematics: A Historical Perspective, Oxford University Press, New York, (2017).
[3] T. Bollerslev, Generalized Autoregressive Conditional Heteroskedasticitiy, Journal of Econometrics, 31(3) (1986), 307-327.
[4] P. J. Brockwell and R. A. Davis, Time Series: Theories and Methods, Springer Verlag, New York, (2009).
[5] W. Enders, Applied Econometric Time Series, Wiley, New York, (2014).
[6] G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall, Upper Saddle River, New Jersey, (1995).
[7] O. Kosheleva and M. Ceberio, Why polynomial formulas in soft computing, decision making, etc.?, Proceedings of the World Congress on Computational Intelligence WCCI’2010, July 18-23, 2010, Barcelona, Spain, (2010), 3310-3314.
[8] J. M. Mendel, Uncertain Rule-Based Fuzzy Systems: Introduction and New Directions, Springer, Cham, Switzerland, (2017).
[9] H. T. Nguyen, C. L. Walker and E. A. Walker, A First Course in Fuzzy Logic, Chapman and Hall/CRC, Boca Raton, Florida, (2019).
[10] V. Novák, I. Perfilieva and J. Moˇ ckoˇ r, Mathematical Principles of Fuzzy Logic, Kluwer, Boston, Dordrecht, (1999).
[11] L. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.