Insight into Fuzzy Modeling Techniques for Data Analysis

This tutorial aims to get the audience acquainted with two unique and potent techniques of computational intelligence: Fuzzy Transform (shortly F-transform) and few methods based on the theory of Fuzzy Natural Logic. From many possible applications of these techniques, we have chosen applications in the analysis and forecasting of time series, and mining information from them, and also applications in pattern recognition and image processing.

The fuzzy transform (F-transform for short) is a unique and very general mathematical technique that has now profoundly developed theory, and that has applications in many areas, for example in time series analysis and forecasting, image processing, mining information from numerical data, numerical solution of differential and integral equations, and elsewhere.

In general, the F-transform establishes a correspondence between a set of continuous (discrete) functions (functional objects like a signal, images, time series, etc.) on a bounded domain (of the real line, Euclidean space) and a set of n-dimensional (real) vectors, called F-transform components. The core idea consists of two steps: granulation of the domain (called a fuzzy partition) and feature extraction - representation of functional objects by features (F-transform components).

In detail, a fuzzy partition of a domain consists of a (usually) finite number of fuzzy sets (factors, clusters, granules, etc.) and can be learned from given data. Each fuzzy set in the partition is assigned a particular value called the F-transform component (feature). A finite number of computed components represents the given function object. This step is called direct F-transform. An inversion formula defines the inverse F-transform, which converts the n-dimensional vector of components into a continuous function, which approximates the original one.

There are many advantages of this method: we obtain a simple (reduced) approximate representation of the original functional object in a feature space, we can set the approximation to have desirable properties, and we can use the inversion formula instead of the precise representation of the original function. Moreover, when solving various problems (pattern recognition, numerical solution of integral or differential equations, etc.), we may operate with the F-transform components (instead of original objects) so that the problem is transformed into a particular problem in the n-dimensional vector space and solved using methods of linear algebra. An approximation theorem stating that the sequence of inverse F-transforms uniformly converges to the original function justifies this method. Besides many other excellent properties, the F-transform has exceptional filtering properties, and its computational complexity is polynomial.

In the tutorial, we will explain the principles of the F-transform methodology and demonstrate applications in pattern recognition, image processing, and time series processing and forecasting. The methodology employs dimensionality reduction and feature extraction, namely: we first project a given object on a feature space (comprised of the F-transform components) and then reconstruct it using the inverse step. By this, we obtain a decomposition into a robust part and its residual (respectively, a trend-cycle and a seasonal component in case of time series) with the rest in the form of random disturbances. Unlike the traditional approach, which assumes the trend-cycle to be an a priori given function on the entire time domain, the F-transform makes it possible to find a data-driven shape of the trend-cycle and to express it using an analytic formula. It has been proved that the F-transform can significantly reduce noise. Therefore, the estimation of the trend cycle is very credible.

In forecasting, we employ techniques of Fuzzy Natural Logic (FNL). We will explain the learning of the linguistic description (a set of fuzzy/linguistic rules), with which we characterize the predicted trend cycle, and the Perception-based Logical Deduction (PbLD), using which we can evaluate the values of the predicted components. Finally, using the inverse F-transform, we obtain the functional representation of the predicted trend-cycle. We can easily interpret the learned linguistic description because of using (a fragment of) natural language.

Another useful outcome of the combination of F-transform and FNL is mining meaningful information about time series in natural language expressions. We generate linguistic comments to the general trend (tendency) of a time series in a given time slice, find intervals of monotonous behavior, and also detect structural breaks in the time series (such as sudden changes of its course, sudden changes in its volatility, or detection of unexpectedly distant outliers). These methods are based on the application of the first-degree F-transform, which provides an estimation of the average tangent even in cases when the course of the time series is by no means clear.

Outline

The tutorial will be accompanied by a demonstration of the mentioned above methodology using experimental software LFL Forecaster and many real-world applications to image processing and pattern recognition. The LFL Forecaster is fully comparable with commercial forecasting systems (such as ForecastPro^TM) and additionally provides us with easily interpretable information.

Expected length of the tutorial: 4h

The level of the tutorial: Introductory

Session	Duration
Irina Perfilieva: Introduction to F-transform and higher-degree F-transform. Real-world applications in image processing and pattern recognition (car license plates, on-line tracking).	2 hrs
Vilém Novák: Introduction to fuzzy natural logic: evaluative linguistic expressions, perception-based logical deduction. Analysis and forecasting of time series, and mining information from them: detection and evaluation of periods with monotonous behavior, detection of structural breaks.	2 hrs

Biography of Presenters

Professor Saeid Nahavandi received a Ph.D. from Durham University, U.K. in 1991. He is an Alfred Deakin Professor, Pro Vice-Chancellor, Chair of Engineering, and the Founding Director of the Institute for Intelligent Systems Research and Innovation at Deakin University. His research interests include modelling of complex systems, robotics and haptics. Professor Nahavandi is Editor-In-Chief: IEEE SMC Magazine, the Senior Associate Editor: IEEE Systems Journal, Associate Editor of IEEE Transactions on Systems, Man and Cybernetics: Systems, and IEEE Press Editorial Board member. Professor Nahavandi is a Fellow of IEEE (FIEEE), Engineers Australia (FIEAust), the Institution of Engineering and Technology (FIET). Saeid is a Fellow of the Australian Academy of Technology and Engineering (ATSE). He has published more than 900 journal and conference papers.

Assoc. Prof. Abbas Khosravi received his PhD from Deakin University in 2010. He is currently an associate professor with the Institute for Intelligent Systems Research and Innovation at Deakin University. His current research interests include machine learning, artificial intelligence, and their applications for data mining, computer vision, optimization, and operation planning. He has received several prestigious grants to conduct fundamental and applied research in the field of AI-based uncertainty quantification. He has published more than 200 journal and conference papers and his h-index is 37 based on Google Scholar.