An on-line learning algorithm for complex fuzzy logic

Aghakhani, Sara; Dick, Scott

doi:10.1109/fuzzy.2010.5584120

Cited by 25 publications

(20 citation statements)

References 31 publications

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“…Complex-valued fuzzy sets have been indeed successfully used in practical applications; see, e.g., [1,4,6,13,24] -and our analysis explains why. Let us show, on several examples, that they indeed help to reconstruct the membership values d i .…”

Section: How Good Are Complex-valued Degrees In Practice?mentioning

confidence: 97%

“…-if we have one quantity x 1 which is characterized by the tuple d (1) and -we have another quantity x 2 which is characterized by the tuple d (2) , -then we should be able to produce a tuple characterizing the sum x 1 + x 2 of these quantities, a tuple characterizing their difference…”

Section: We Face the Following Problemmentioning

confidence: 99%

“…We know the tuples d (1) and d (2) describing the two quantities x 1 and x 2 , and we want to find a tuple d corresponding to the value y = f (x 1 , x 2 ) for some known function f (x 1 , x 2 ). A natural idea to do it is as follows:…”

Section: Value X If One Of the Following Conditions Holdmentioning

confidence: 99%

“…Historically the first such transitions were observed in chemistry; as a result, the corresponding equations were first designed in chemistry and are thus known as equations of chemical kinetics. Let us use these equations to provide a numerical description of the transitions (1).…”

Section: How To Describe This Idea In Numerical Terms?mentioning

confidence: 99%

“…-first, we use the formula (1) to generate membership functions µ (1) (x 1 ) and µ (2) (x 2 ) corresponding to the tuples d (1) and d (2) ; -then, by applying Zadeh's extension principle to these membership functions, we compute the membership function µ(x) corresponding to y = f (x 1 , x 2 ); -finally, we need to generate the tuple d corresponding to the resulting membership function µ(x).…”

Section: Value X If One Of the Following Conditions Holdmentioning

confidence: 99%

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50 Years of fuzzy: from discrete to continuous to – Where?

Kreinovich

Nguyen

Kosheleva

et al. 2015

IFS

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Abstract. While many objects and processes in the real world are discrete, from the computational viewpoint, discrete objects and processes are much more difficult to handle than continuous ones. As a result, a continuous approximation is often a useful way to describe discrete objects and processes. We show that the need for such an approximation explains many features of fuzzy techniques, and we speculate on to which promising future directions of fuzzy research this need can lead us.Keywords: fuzzy techniques, discrete, continuous, interval-valued fuzzy, complex-valued fuzzy, computing with words, dynamical fuzzy logic, chemical kinetics, non-additive measures, symmetry Fuzzy Techniques as an Easier-to-Compute Continuous Approximation for Difficult-to-Compute Discrete Objects and ProcessesDiscrete objects and processes are ubiquitous. Many real-life objects are processes are discrete. On the macro level, there is an abrupt transition in space between physical bodies, there is an abrupt transition in time when, e.g., a glass breaks or a person changes his/her opinion. On the micro level, matter consists of discrete atoms and molecules, with abrupt transitions between different states of an atom.Continuous problems are easier to compute. While discrete objects and processes are ubiquitous in nature, from the computational viewpoint, it is often much easier to handle continuous problems. This may sound * Corresponding author. E-mail: vladik@utep.edu.counter-intuitive, since intuitively, if we restrict our search or optimization to only integer values, the problem would become easier -but it is not. For example, in the continuous case, it is relatively easy to find a solution x 1 , . . . , x n to a system of linearmany known feasible algorithms for that), the problem becomes NP-hard (computationally intractable) if we only allow discrete values of x i ; see, e.g., [9,28].Similarly, in the continuous case, it is relatively easy to find the values x 1 , . . . , x n that minimize a given quadratic function f (x 1 , . . . , x n ): it is sufficient to solve the corresponding system of linear equations ∂f ∂x i = 0. However, optimization of quadratic functions for discrete inputs, e.g., for x i ∈ {0, 1}, is NPhard [9,28]. Continuous approximations of discrete objects and processes are ubiquitous in physics. Because dealing with discrete objects and processes is often computationally complicated, physicists often approximate discrete objects with continuous ones. For example, it is not feasible to describe the changes in atmosphere by tracing all 10 23 molecules, but approximate equations that describe the atmosphere as a continuous field leads to many useful weather predictions. Similar, a solid body -in effect, a collection of atoms -is well described by a continuous density field, and an atomic nucleus -a collection of protons and neutrons -is well described by a continuous (liquid) model; see, e.g., [8].Such approximations are also useful in analyzing social phenomena. For example, in analyzing how epidemics spread, ...

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Section: How Good Are Complex-valued Degrees In Practice?mentioning

confidence: 97%

Section: We Face the Following Problemmentioning

confidence: 99%

Section: Value X If One Of the Following Conditions Holdmentioning

confidence: 99%

Section: How To Describe This Idea In Numerical Terms?mentioning

confidence: 99%

Section: Value X If One Of the Following Conditions Holdmentioning

confidence: 99%

See 3 more Smart Citations

50 Years of fuzzy: from discrete to continuous to – Where?

Kreinovich

Nguyen

Kosheleva

et al. 2015

IFS

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Time-Series Forecasting via Complex Fuzzy Logic

Yazdanbakhsh

Dick

2014

Frontiers of Higher Order Fuzzy Sets

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Adaptive neuro-complex-fuzzy inference system (ANCFIS) is a neurofuzzy system that employs complex fuzzy sets for time-series forecasting. One of the particular advantages of this architecture is that each input to the network is a windowed segment of the time series, rather than a single lag as in most other neural networks. This allows ANCFIS to predict even chaotic time series very accurately, using a small number of rules. Some recent findings, however, indicate that published results on ANCFIS are suboptimal; they could be improved by changing how the length of an input window is determined, and/or subsampling the input window.We compare the performance of ANCFIS using three different approaches to defining an input window, across six time-series datasets. These include chaotic datasets and time series up to 20,000 observations in length. We found that the optimal choice of input formats was dataset dependent, and may be influenced by the size of the dataset. We finally develop a recommended approach to determining input windows that balances the twin concerns of accuracy and computation time. IntroductionTime-series forecasting has emerged as the first major application of complex fuzzy sets and logic, which were first described by Ramot in [1]. Beginning in 2007, complex-valued neuro-fuzzy systems were developed to inductively learn forecasting models; these include the adaptive neuro-complex-fuzzy inference system (ANCFIS) architecture [2], and the family of complex neuro-fuzzy system (CNFS) architectures [3]. Both ANCFIS and CNFS are modifications of the well-known ANFIS architecture, in which complex fuzzy sets and complex-valued network signals are used. These architectures showed that complex fuzzy sets were naturally useful in creating very accurate forecasting models. ANCFIS in particular is also very parsimonious;

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A Natural Formalization of Changing-One’s-Mind Leads to Square Root of “Not” and to Complex-Valued Fuzzy Logic

Kosheleva

Kreinovich

2021

Explainable AI and Other Applications of Fuzzy Techniques

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We show that a natural formalization of the process of changing one's mind leads to such seemingly non-intuitive ideas as square root of "not" and complex-valued fuzzy degrees. Formulation of the ProblemEverything is a Matter of Degree -At Least Our Opinions. It is rare that we immediately become convinced in the truth of some statement. Ok, this happens in mathematics: once we hear (or read) and understand the proof, we are 100% convinced. However, outside mathematical proofs, our degree of belief in a statement changes gradually. We may eventually reach the stage when we are absolutely convinced that a given statement is true (or when we are absolutely convinced that a given statement is false), but before we reach this stage, we go through intermediate stages, in which we only have a partial degree of belief in the given statement.This idea of intermediate degrees of belief is one of the main ideas behind Zadeh's fuzzy logic (see, e.g., [2,6,10,12,14,18]) -that everything is a matter of degree.

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An on-line learning algorithm for complex fuzzy logic

Cited by 25 publications

References 31 publications

50 Years of fuzzy: from discrete to continuous to – Where?

50 Years of fuzzy: from discrete to continuous to – Where?

Time-Series Forecasting via Complex Fuzzy Logic

A Natural Formalization of Changing-One’s-Mind Leads to Square Root of “Not” and to Complex-Valued Fuzzy Logic

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