A fuzzy opinion is a Gaussian fuzzy set with the center representing the opinion and the standard deviation representing the uncertainty about the opinion, and a fuzzy opinion network is a connection of a number of fuzzy opinions in a structured way. In this article, we propose a top-down hierarchical fuzzy opinion network to model how the opinion of a top leader is penetrated into the members in social organizations, and a bottom-up fuzzy opinion network to model how the opinions of a large number of agents are agglomerated layer-by-layer into a consensus or a few opinions in the social processes, such as an election. For the top-down hierarchical fuzzy opinion network, we prove that the opinions of all the agents converge to the leader's opinion but the uncertainties of the agents in different groups are generally converge to different values. We demonstrate that the speed of convergence is greatly improved by organizing the agents in a hierarchical structure of small groups. For the bottom-up hierarchical fuzzy opinion network, we simulate how a wide spectrum of opinions are negotiating and summarizing with each other, in a layer-by-layer fashion in some typical situations.

A deep convolutional fuzzy system (DCFS) on a high-dimensional input space is a multilayer connection of many low-dimensional fuzzy systems, where the input variables to the low-dimensional fuzzy systems are selected through a moving window across the input spaces of the layers. To design the DCFS based on an input-output data pairs, we propose a bottom-up layer-by-layer scheme. Specifically, by viewing each of the first-layer fuzzy systems as a weak estimator of the output based only on a very small portion of the input variables, we design these fuzzy systems using the Wang-Mendel method. After the first-layer fuzzy systems are designed, we pass the data through the first layer to form a new dataset and design the second-layer fuzzy systems based on this new dataset in the same way as designing the first-layer fuzzy systems. Repeating this process layer-by-layer, we design the whole DCFS. We also propose a DCFS with parameter sharing to save memory and computation. We apply the DCFS models to predict a synthetic chaotic plus random time-series and the real Hang Seng Index of the Hong Kong stock market.

We propose a mathematical model for the word-of-mouth communications among stock investors through social networks and explore how the changes of the investors' social networks influence the stock price dynamics and vice versa. An investor is modeled as a Gaussian fuzzy set (a fuzzy opinion) with the center and standard deviation as inputs and the fuzzy set itself as output. Investors are connected in the following fashion: the center input of an investor is taken as the average of the neighbors' outputs, where two investors are neighbors if their fuzzy opinions are close enough to each other, and the standard deviation (uncertainty) input is taken with local, global, or external reference schemes to model different scenarios of how investors define uncertainties. The centers and standard deviations of the fuzzy opinions are the expected prices and their uncertainties, respectively, that are used as inputs to the price dynamic equation. We prove that with the local reference scheme the investors converge to different groups in finite time, while with the global or external reference schemes all investors converge to a consensus within finite time and the consensus may change with time in the external reference case. We show how to model trend followers, contrarians, and manipulators within this mathematical framework and prove that the biggest enemy of a manipulator is the other manipulators. We perform Monte Carlo simulations to show how the model parameters influence the price dynamics, and we apply a modified version of the model to the daily closing prices of 15 top banking and real estate stocks in Hong Kong for the recent two years from December 5, 2013 to December 4, 2015 and discover that a sharp increase of the combined uncertainty is a reliable signal to predict the reversal of the current price trend.

We propose the concept of a conditional fuzzy set and prove that a type-2 fuzzy set is equivalent to a conditional fuzzy set. We show that both the conditional fuzzy sets and the type-2 fuzzy sets are fuzzy relations on the product space of the primary and secondary variables, and the difference is that the primary and secondary variables in the conditional fuzzy set framework are usually independent to each other, whereas in the type-2 fuzzy set framework, the secondary variable depends on the primary variable by definition. It is this dependence between the primary and secondary variables that makes the type-2 fuzzy sets a complex subject, while the conditional fuzzy sets do not have this built-in dependence and, thus, are much easier to analyze. With the fuzzy relation formulation, powerful tools in fuzzy set theory such as Zadeh's compositional rule of inference can be used to obtain the marginal fuzzy sets of the type-2 and conditional fuzzy sets, transforming the type-2 problems back to the conventional type-1 domain. With the help of the marginal fuzzy set concept, we show that a type-2 fuzzy logic system can be designed in the same way as designing a type-1 fuzzy logic system.

We propose a new mathematical framework for the evolution and propagation of opinions, called fuzzy opinion network, which is the connection of a number of Gaussian nodes, possibly through some weighted average, time delay, or logic operators, where a Gaussian node is a Gaussian fuzzy set with the center and the standard deviation being the node inputs and the fuzzy set itself being the node output. In this framework, an opinion is modeled as a Gaussian fuzzy set with the center representing the opinion itself and the standard deviation characterizing the uncertainty about the opinion. We study the basic connections of fuzzy opinion networks, including basic center, basic standard deviation (sdv), basic center-sdv, chain-in-center, and chain-in-sdv connections, and we analyze a number of dynamic connections to show how opinions and their uncertainties propagate and evolve across different network structures and scenarios. We explain what insights we might gain from these mathematical results about the formation and evolution of human opinions.

In this paper, we use fuzzy systems theory to convert the technical trading rules commonly used by stock practitioners into excess demand functions which are then used to drive the price dynamics. The technical trading rules are recorded in natural languages where fuzzy words and vague expressions abound. In Part I of this paper, we will show the details of how to transform the technical trading heuristics into nonlinear dynamic equations. First, we define fuzzy sets to represent the fuzzy terms in the technical trading rules; second, we translate each technical trading heuristic into a group of fuzzy IF-THEN rules; third, we combine the fuzzy IF-THEN rules in a group into a fuzzy system; and finally, the linear combination of these fuzzy systems is used as the excess demand function in the price dynamic equation. We transform a wide variety of technical trading rules into fuzzy systems, including moving average rules, support and resistance rules, trend line rules, big buyer, big seller, and manipulator rules, band and stop rules, and volume and relative strength rules. Simulation results show that the price dynamics driven by these technical trading rules are complex and chaotic, and some common phenomena in real stock prices such as jumps, trending, and self-fulfilling appear naturally.

In Part II of this study, we concentrate our analysis on the price dynamical model with the moving average rules developed in Part I. By decomposing the excessive demand function, we reveal that it is the interplay between trend-following and contrarian actions that generates the price chaos and gives parameter ranges for the price series to change from divergence to chaos and to oscillation. We prove that the price dynamical model has an infinite number of equilibriums, but all these equilibriums are unstable. We demonstrate the short-term predictability of the price volatility and derive the detailed formulas of the Lyapunov exponent as functions of the model parameters. We show that although the price is chaotic, the volatility converges to some constant very quickly at the rate of the Lyapunov exponent. We extract the formula relating the converged volatility to the model parameters based on Monte Carlo simulations. We explore the circumstances under which the returns are uncorrelated and illustrate in detail as to how the correlation index changes with the model parameters. Finally, we plot the strange attractor and the return distribution of the chaotic price series to illustrate the complex structure and the fat-tailed distribution of the returns.

In Part III of this study, we apply the price dynamical model with big buyers and big sellers developed in Part I of this paper to the daily closing prices of the top 20 banking and real estate stocks listed in the Hong Kong Stock Exchange. The basic idea is to estimate the strength parameters of the big buyers and the big sellers in the model and make buy/sell decisions based on these parameter estimates. We propose two trading strategies: 1) Follow-the-Big-Buyer, which buys when big buyer begins to appear and there is no sign of big sellers, holds the stock as long as the big buyer is still there, and sells the stock once the big buyer disappears; and 2) Ride-the-Mood, which buys as soon as the big buyer strength begins to surpass the big seller strength and sells the stock once the opposite happens. Based on the testing over 245 two-year intervals uniformly distributed across the seven years from July 3, 2007 to July 2, 2014, which includes a variety of scenarios, the net profits would increase 67% or 120% on average if an investor switched from the benchmark Buy-and-Hold strategy to the Follow-the-Big-Buyer or Ride-the-Mood strategies during this period, respectively.

We use fuzzy systems theory to convert the technical trading rules commonly used by stock practitioners into excess demand functions which are then used to drive the price dynamics. First, we define fuzzy sets to represent the fuzzy terms in the technical trading rules; second, we translate each technical trading heuristic into a group of fuzzy IF-THEN rules; third, we combine the fuzzy IF-THEN rules in a group into a fuzzy system; and finally, the linear combination of these fuzzy systems is used as the excess demand function in the price dynamic equation. We transform moving average rules, support and resistance rules, and trend line rules into fuzzy systems. Simulation results show that the price dynamics driven by these technical trading rules are complex and chaotic, and some common phenomena in real stock prices such as jumps, trending and self-fulfilling appear naturally.

We analyze some basic properties of the stock price dynamical model when trend-followers and contrarians interplay with each other. We prove that the price dynamical model has an infinite number of equilibriums, but all these equilibriums are unstable. We demonstrate the short-term predictability of the price volatility and derive the detailed formulas of the Lyapunov exponent as functions of the model parameters. We show that although the price is chaotic, the volatility converges to some constant very quickly at the rate of the Lyapunov exponent. We extract the formula relating the converged volatility to the model parameters based on Monte-Carlo simulations.

We apply the price dynamical model with big buyers and big sellers to the daily closing prices of the top 20 banking and real estate stocks listed in the Hong Kong Stock Exchange. The basic idea is to estimate the strength parameters of the big buyers and the big sellers in the model and make buy/sell decisions based on these parameter estimates. We propose two trading strategies: (i) Follow-the-Big-Buyer which buys when big buyer begins to appear and there is no sign of big sellers, holds the stock as long as the big buyer is still there, and sells the stock once the big buyer disappears; and (ii) Ride-the-Mood which buys as soon as the big buyer strength begins to surpass the big seller strength, and sells the stock once the opposite happens. Based on the testing over 245 two-year intervals uniformly distributed across the seven years from 03-July-2007 to 02-July-2014 which includes a variety of scenarios, the net profits would increase 67% or 120% on average if an investor switched from the benchmark Buy-and-Hold strategy to the Follow-the-Big-Buyer or Ride-the-Mood strategies during this period, respectively.

The process of pH process control is a challenging problem due to the strong nonlinearity and extreme sensitivity to disturbances of the process. This paper presents an application of one-step-ahead adaptive fuzzy control scheme for a strong acid-strong base neutralization process. The controller is designed based on a Mamdani type fuzzy system constructed to model the dynamics of the process. The fuzzy system model can take advantage of both a priori linguistic human knowledge through parameter initialization, and process measurements through online parameter adjustment using the least square algorithm with deadzone. In both setpoint tracking and disturbance rejection tasks, simulation experiments show satisfactory performances of the resulted pH control scheme as well as performance improvements owing to the linguistic information.

This paper discusses chaos control in discrete-time nonlinear systems using one-step-ahead adaptive fuzzy controller. The chaotic system in a general form is modeled by a Mamdani type fuzzy system with parameters tuned on-line using system input-output data by the least square algorithm with deadzone. Based on this fuzzy model, a one-step-ahead controller is designed to minimize the error between the desired and the real system outputs. Performance analysis and simulation results in control of the Sin and Lozi systems are given to show the effectiveness of the scheme.

In this paper, the so-called Wang-Mendel (WM) method for generating fuzzy rules from data is enhanced to make it a comprehensive and flexible fuzzy system approach to data description and prediction. In the description part, the core ideas of the WM method are used to develop three methods to extract fuzzy IF-THEN rules from data. The first method shows how to extract rules for the user-specified cases, the second method generates all the rules that can be generated directly from the data, and the third method extrapolates the rules generated by the second method over the entire domain of interest. In the prediction part, two fuzzy predictive models are constructed based on the fuzzy IF-THEN rules extracted by the methods of the description part. The first model gives a continuous output and is suitable for predicting continuous variables, and the second model gives a piecewise constant output and is suitable for predicting categorical variables. We show that by comparing the prediction accuracy of the fuzzy predictive models with different numbers of fuzzy sets covering the input variables, we can rank the importance of the input variables. We also propose an algorithm to optimize the fuzzy predictive models, and show how to use the models to solve pattern recognition problems. Throughout this paper, we use a set of real data from a steel rolling plant to demonstrate the ideas and test the models.

This article addresses parameter convergence problem in identification of nonlinear dynamic systems using fuzzy models. We first establish persistent excitation conditions and then propose several detailed algorithms to generate input signals that guarantee the convergence of the parameter estimates in the fuzzy system models to the true values in identifications of second-order nonlinear moving-average and auto-regressive-moving-average systems. Numerical example is given to illustrate the ideas and results.

Fuzzy systems have been proved to be universal approximators, yet a large number of fuzzy rules may be needed for high approximation accuracy. In this paper, we consider how to determine the minimum number of fuzzy rules required in a fuzzy system for approximation to achieve a given accuracy and how to construct this fuzzy system when there are only a limited number of input-output data pairs of the unknown system. The key point is to partition the input space nonuniformly. In particular, a tunnel algorithm is utilized for the single-input case. Numerical examples are given to demonstrate the proposed idea and algorithm.

In this paper, we propose a new fuzzy system architecture based on the idea of projection pursuit. This new model is suitable for the cases where the input variables are related to each other but their relationships cannot be precisely defined. The structure of the model is determined by projecting the sampling data onto lower and lower dimensional spaces in such a way that the rules automatically focus on the regions crowded with sampling data. The parameters in the model are estimated as the weighted average of the sampling data over the fuzzy region the parameter is responsible. The main advantages of this fuzzy projection pursuit model include its flexibility in searching relationships among fuzzily dependent input variables, its ability to deal with outliers in the data, and the ease of interpretation of its structure and parameters.

Fuzzy systems are proved to be universal approximators, yet they may need a large number of fuzzy rules for high approximation accuracy. For this rule-explosion problem, we introduce the idea on design of economical fuzzy systems. Given a specified architecture, a fuzzy system is called economical if it uses least fuzzy rules and at the same time it approximates the given function at a pre-required approximation accuracy. The essence of our solution is to partition the domain of the function nonuniformly. We investigate the approximation formulations of two commonly used fuzzy systems: the Mamdani and the TSK fuzzy systems with linear consequent. Then the economical fuzzy system design problem is formulated mathematically. Finally, an approximate algorithms is proposed toward designing Mamdani type economical fuzzy systems.

Many methods have been proposed in the literature for designing fuzzy systems from input-output data (the so-called neuro-fuzzy methods), but very little was done to analyze the performance of the methods from a rigorous mathematical point of view. In this paper, we establish approximation bounds for two of these methods - the table lookup scheme proposed by Wang et al. (1992) and the clustering method studied by Wang (1993, 1997). We derive detailed formulas of the error bounds between the nonlinear function to be approximated and the fuzzy systems designed using the methods based on input-output data. These error bounds show explicitly how the parameters in the two methods influence their approximation capability. We also propose modified versions for the two methods such that the designed fuzzy systems are well-defined over the whole input domain.

In this letter, the hierarchical fuzzy systems are analyzed and designed. In the analysis part, we prove that the hierarchical fuzzy systems are universal approximators and analyze the sensitivity of the fuzzy system output with respect to small perturbations in its inputs. In the design part, we derive a gradient descent algorithm for tuning the parameters of the hierarchical fuzzy system to match the input-output pairs. The algorithm is simulated for two examples and the results show that the algorithm is effective and the hierarchical structure gives good approximation accuracy.

We design a fuzzy system to compensate the delays due to backlash nonlinearity. The fuzzy compensator is constructed from some common-sense rules. We consider the general case of the unknown backlash parameter and develop an adaptation algorithm to estimate the backlash parameter online. We prove that under certain conditions the fuzzy compensator with the adaptation algorithm guarantees that the backlash output converges to the desired trajectory. Simulation and hardware implementation results show that the fuzzy compensator is robust to the estimation errors in the backlash parameters. Application to an industrial CNC machine tool is described.

A number of stable and optimal fuzzy controllers are developed for linear systems. Based on some classical results in control theory, we design the structure and parameters of fuzzy controllers such that the closed-loop fuzzy control systems are stable, provided that the process under control is linear and satisfies certain conditions. It turns out that if stability is the only requirement, there is much freedom in choosing the fuzzy controller parameters. Therefore, a performance criterion is set to optimalize the parameters. Using the Pontryagin minimum principle, we design an optimal fuzzy controller for linear systems with quadratic cost function. Finally, the optimal fuzzy controller is applied to a ball-and-beam system.

A deadzone compensator is designed for industrial positioning systems using a fuzzy logic (FL) controller. The classification property of FL systems makes them a natural candidate for the rejection of errors induced by the deadzone, which has regions in which it behaves differently. A tuning algorithm is given for the FL parameters, so that the deadzone compensation scheme becomes adaptive, guaranteeing small tracking errors and bounded parameter estimates. The adaptive FL deadzone compensator is implemented on an actual industrial CNC machine tool to show its efficacy.

An adaptive fuzzy controller is constructed from a set of fuzzy IF-THEN rules whose parameters are adjusted on-line according to some adaptation law for the purpose of controlling the plant to track a given-trajectory. In this paper, two adaptive fuzzy controllers are designed based on the Lyapunov synthesis approach. We require that the final closed-loop system must be globally stable in the sense that all signals involved (states, controls, parameters, etc.) must be uniformly bounded. Roughly speaking, the adaptive fuzzy controllers are designed through the following steps: first, construct an initial controller based on linguistic descriptions (in the form of fuzzy IF-THEN rules) about the unknown plant from human experts; then, develop an adaptation law to adjust the parameters of the fuzzy controller on-line. We prove, for both adaptive fuzzy controllers, that: (1) all signals in the closed-loop systems are uniformly bounded; and (2) the tracking errors converge to zero under mild conditions. We provide the specific formulas of the bounds so that controller designers can determine the bounds based on their requirements. Finally, the adaptive fuzzy controllers are used to control the inverted pendulum to track a given trajectory, and the simulation results show that: (1) the adaptive fuzzy controllers can perform successful tracking without using any linguistic information; and (2) after incorporating some linguistic fuzzy rules into the controllers, the adaptation speed becomes faster and the tracking error becomes smaller.

A direct adaptive fuzzy controller that does not require an accurate mathematical model of the system under control, is capable of incorporating fuzzy if-then control rules directly into the controllers, and guarantees the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded is developed. The specific formula for the bounds is provided, so that controller designers can determine the bounds based on their requirements. The direct adaptive fuzzy controller is used to regulate an unstable system to the origin and to control the Duffing chaotic system to track a trajectory. The simulation results show that the controller worked without using any fuzzy control rules, and that after fuzzy control rules were incorporated the adaptation speed became much faster. It is shown explicitly how the supervisory control forces the state to remain within the constraint set and how the adaptive fuzzy controller learns to regain control.>

Two fuzzy adaptive filters are developed: one uses a recursive-least-squares (RLS) adaptation algorithm, and the other uses a least-mean-square (LMS) adaptation algorithm. The RLS fuzzy adaptive filter is constructed through the following four steps: (1) define fuzzy sets in the filter input space Rn whose membership functions cover U; (2) construct a set of fuzzy IF-THEN rules which either come from human experts or are determined during the adaptation procedure by matching input-output data pairs; (3) construct a filter based on the set of rules; and (4) update the free parameters of the filter using the RLS algorithm. The design procedure for the LMS fuzzy adaptive filter is similar. The most important advantage of the fuzzy adaptive filters is that linguistic information (in the form of fuzzy IF-THEN rules) and numerical information (in the form of input-output pairs) can be combined in the filters in a uniform fashion. The filters are applied to nonlinear communication channel equalization problems.>

A novel approach based on fuzzy set theory is developed for recognizing handwritten rotated characters. This fuzzy approach consists of four steps: (1) generating crisp sets for reference characters rotated through different degrees; (2) fuzzifying these crisp sets; (3) determining the degrees of a given character to the fuzzy sets; and (4) classifying the given character based on an average rule or a maximum rule. Simulation results show that the fuzzy approach correctly classified 94% to 100% of a small test set of characters.>

A commonly used routine in seismic signal processing is deconvolution, which comprises two operations: reflectivity detection and magnitude estimation. Existing statistical detectors are computationally expensive. In the paper, a Hopfield neural network is constructed to perform the reflectivity detection operation. The basic idea is to represent the reflectivity detection problem by an equivalent optimization problem and then construct a Hopfield neural network to solve this optimization problem. The neural detector is applied to a synthetic seismic trace and 30 real seismic traces. The processing results show that the accuracy of the neural detector is about the same as that of the existing detectors, but the speed of the neural detector is much faster.

Fuzzy systems are represented as series expansions of fuzzy basis functions which are algebraic superpositions of fuzzy membership functions. Using the Stone-Weierstrass theorem, it is proved that linear combinations of the fuzzy basis functions are capable of uniformly approximating any real continuous function on a compact set to arbitrary accuracy. Based on the fuzzy basis function representations, an orthogonal least-squares (OLS) learning algorithm is developed for designing fuzzy systems based on given input-output pairs; then, the OLS algorithm is used to select significant fuzzy basis functions which are used to construct the final fuzzy system. The fuzzy basis function expansion is used to approximate a controller for the nonlinear ball and beam system, and the simulation results show that the control performance is improved by incorporating some common-sense fuzzy control rules.

A general method is developed to generate fuzzy rules from numerical data. The method consists of five steps: divide the input and output spaces of the given numerical data into fuzzy regions; generate fuzzy rules from the given data; assign a degree of each of the generated rules for the purpose of resolving conflicts among the generated rules; create a combined fuzzy rule base based on both the generated rules and linguistic rules of human experts; and determine a mapping from input space to output space based on the combined fuzzy rule base using a defuzzifying procedure. The mapping is proved to be capable of approximating any real continuous function on a compact set to arbitrary accuracy. Applications to truck backer-upper control and time series prediction problems are presented.>

Three Hopfield (1984, 1985) neural networks are developed to realize a new adaptive minimum prediction-error deconvolution procedure. The first neural network is developed to detect the reflectivity sequence. The second neural network is developed to determine the magnitudes of the detected reflections. The third neural network is developed to estimate the seismic wavelet. A block-component method is proposed for simultaneous reflectivity estimation and wavelet extraction based on these three neural networks. These three neural networks and the block-component method are simulated for a narrowband wavelet. Real seismic data are processed using the block-component method, and the results are compared with those using the minimum variance deconvolution (MVD) filter and the maximum-likelihood based SMLR detector.>

Two three-dimensional structured networks are developed for solving linear equations and the Lyapunov equation. The basic idea of the structured network approaches is to first represent a given equation-solving problem by a 3-D structured network so that if the network matches a desired pattern array, the weights of the linear neurons give the solution to the problem: then, train the 3-D structured network to match the desired pattern array using some training algorithms; and finally, obtain the solution to the specific problem from the converged weights of the network. The training algorithms for the two 3-D structured networks are proved to converge exponentially fast to the correct solutions. Simulations were performed to show the detailed convergence behaviors of the 3-D structured networks.>

A two-level three-layer structured network is developed to estimate the moving-average model parameters based on second-order and third-order cumulant matching. The structured network is a multilayer feedforward network composed of linear summers in which the weights of these summers have a clear physical meaning. The first level is composed of random access memory units, which are used to control the connectivities of the second-level summers. The second level is composed of three layers of linear summers in which the weight of any summer represents the moving-average parameter to be estimated. The connectivities among these summers are controlled by the first-level memory units in such a way that the outputs of the second-level structured network equal the desired second-order or third-order statistics if the summer weights equal their corresponding true moving-average parameter values. Each second-order and third-order cumulant is viewed as a pattern which the structured network needs to learn, and a steepest-descent algorithm is proposed for training the structured network. The author also presents extensions to particular sorts of estimation, and results of simulations.