Li-Xin Wang

Li-Xin Wang, Jerry M. Mendel

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.

Li-Xin Wang

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 fifteen top banking and real estate stocks in Hong Kong for the recent two years from Dec. 5, 2013 to Dec. 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.

Li-Xin Wang

A deep convolutional fuzzy system (DCFS) on a high-dimensional input space is a multi-layer 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 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 WM Method. After the first-layer fuzzy systems are designed, we pass the data through the first layer to form a new data set and design the second-layer fuzzy systems based on this new data set 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.

Li-Xin Wang

We propose a new heavy-tailed distribution --- Gaussian-Chain (GC) distribution, which is inspirited by the hierarchical structures prevailing in social organizations. We determine the mean, variance and kurtosis of the Gaussian-Chain distribution to show its heavy-tailed property, and compute the tail distribution table to give specific numbers showing how heavy is the heavy-tails. To filter out the heavy-tailed noise, we construct two filters --- 2nd and 3rd-order GC filters --- based on the maximum likelihood principle. Simulation results show that the GC filters perform much better than the benchmark least-squares algorithm when the noise is heavy-tail distributed. Using the GC filters, we propose a trading strategy, named Ride-the-Mood, to follow the mood of the market by detecting the actions of the big buyers and the big sellers in the market based on the noisy, heavy-tailed price data. Application of the Ride-the-Mood strategy to five blue-chip Hong Kong stocks over the recent two-year period from April 2, 2012 to March 31, 2014 shows that their returns are higher than the returns of the benchmark Buy-and-Hold strategy and the Hang Seng Index Fund.