Why Linear (and Piecewise Linear) Models Often Successfully Describe Complex Non-Linear Economic and Financial Phenomena: A~Fuzzy-Based Explanation
محورهای موضوعی : 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.
کلید واژه: Fuzzy Logic, Linear models, Piece-wise linear models, Economics and finance,
چکیده مقاله :
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.
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.
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