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Below are titles and abstracts of very recent research results. Some of these publications are in the review process and will not be available for distribution until they are published.

 

Type-2 Fuzzy Logic

""An Interval Approach to Fuzzistics for Interval Type-2 Fuzzy Sets," F. Liu and J. M. Mendel, FUZZ-IEEE 2007, London, UK, July 2007.

In this paper, a new and simple approach, called Interval Approach, to type-2 fuzzistics is presented, one that captures the strong points of both the person-MF and interval end-points approaches. It uses interval end-point data that are collected from a group of subjects, assumes a probability distribution for each person's data and maps the mean and standard deviation of that distribution into the parameters of an iteratively specified type-1 person MF. These type-1 person MFs are then aggregated using the union leading to the FOU for a word. Experiments show that this approach is easy to implement and the derived interval type-2 word models match our intuitions, i.e., the FOUs of the small-sounding words are located to the left, the FOUs of the medium-sounding words are located in the middle, and the FOUs of the large-sounding words are located to the right.

 

"Perceptual Reasoning: A New Computing With Words Engine," J. M. Mendel and D. Wu, Granular Computing Conference, San Jose, CA, Nov. 2007.

Zadeh proposed the paradigm of computing with words (CWW). We have proposed a CWW architecture for making subjective judgments, called a Perceptual Computer (Per-C). Because words mean different things to different people, the Per-C uses interval type-2 fuzzy sets (IT2 FSs). The encoder of the Per-C transforms words, in an application-dependent wordcodebook, into IT2 FSs. The central element of the Per-C is the CWW engine, which maps IT2 FSs to IT2 FSs. Several CWW engines have appeared in the literature, e.g., fuzzy IFTHEN rules to perform inference and/or reasoning based on Mamdani or TSK models, linguistic weighted averages (LWAs) to aggregate linguistic data, and linguistic summarization to perform human friendly data mining. In this paper a new CWW engine-Perceptual Reasoning (PR)-is proposed. It also uses fuzzy IF-THEN rules; however, unlike a traditional Mamdani or TSK model, in which fired rules are combined using the union, or addition, or during the defuzzification process, in PR a LWA is used to combine the fired rules. We prove that the output IT2 FSs of PR can only look like the IT2 FSs in the application codebook. This is very important for CWW, because the last component of the Per-C is a decoder which converts the CWW output IT2 FS back into a word, e.g. a word whose IT2 FS is most similar to it.

"Enhanced Karnik-Mendel Algorithms for Interval Type-2 Fuzzy Sets and Systems," D. Wu and J. M. Mendel, NAFIPS 2007, San Diego, CA, June 2007.

The Karnik-Mendel (KM) algorithms are iterative procedures widely used in fuzzy logic theory. They are known to converge monotonically and super-exponentially fast; however, several (usually two to six) iterations are still needed before convergence occurs. Methods to reduce their computational cost are proposed in this paper. Extensive simulations show that on average the enhanced KM algorithms can save about two iterations, which corresponds to more than a 39% reduction in computation time.

"Tutorial on the Uses of the Interval Type-2 Fuzzy Set's Wavy Slice Representation Theorem," J. M. Mendel, NAFIPS 2008, Paper #60103, New York City, NY, May 2008.

This tutorial paper demonstrates how the Embedded Sets Representation Theorem for a general T2 FS, when specialized to an IT2 FS, can be used as the starting point to solve many diverse problems that involve IT2 FSs. The problems considered are: Set theoretic operations, centroid, uncertainty measures, similarity, inference engine computations for Mamdani IT2 fuzzy logic systems, linguistic weighted average, person MF approach to type-2 fuzzistics, and Interval Approach to type-2 fuzzistics. Each solution obtained from the RT is a structural solution but is not a computational solution, however the latter are always found from the former. It is this author's recommendation that one should use the RT as a starting point whenever solving a new problem involving IT2 FSs, because it has had such great success in solving so many such problems in the past.

"Perceptual Reasoning Using Interval Type-2 Fuzzy Sets: Properties," D. Wu and J. M. Mendel, FUZZ-IEEE, Hong Kong, June 2008.

Perceptual Reasoning (PR) is an Approximate Reasoning mechanism that can be used as a Computing With Words (CWW) Engine, i.e., given input words, PR can infer the output from a rulebase. When the input words and the words in the ruelbase are modeled by interval type-2 fuzzy sets (IT2 FSs) the output of PR, Y_PR, is also an IT2 FS, and it will be mapped into a word in the codebook. For accurate mapping, we need we need to ensure that Y_PR resembles the IT2 FSs in the codebook. The concept of PR using IT2 FSs was originally proposed in [10]. In this paper, the procedures to compute PR are introduced, and the properties of PR are studied in more detail. More specifically, we show under what conditions Y_PR can be a shoulder or interior FOU.

"On New Quasi-Type-2 Fuzzy Logic Systems," J. M. Mendel and F. Liu, FUZZ-IEEE, Hong Kong, June 2008..

This paper provides an answer to the question that the type-2 fuzzy logic community is now asking: "What comes after interval type-2 fuzzy logic systems (IT2 FLSs)?" It demonstrates, through a geometrical understanding of the type-reduced set, that logical next steps in the progression from type-1 to interval type-2 to type-2 FLSs are quasi-T2 FLSs, either an interconnection of a T1 FLS and an IT2 FLS, or an interconnection of two IT2 FLSs, in which both FLSs are designed simultaneously. The quasi-T2 FLSs overcome the computational difficulties that are associated with set theoretic operations and type-reduction (TR) for general T2 FSs and FLSs, because all set theoretic operations can be performed as in existing T1 or IT2 FLSs, and because TR for an IT2 FLS can be performed using existing KM Algorithms.


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Smart Oil Field Technology

"Forecasting Injector-Producer Relationships from Production and Injection Rates Using an Extended Kalman Filter," F. Liu and J. M. Mendel, SPE Conference, Nov. 2007, Anaheim, CA.

This paper presents an adaptive method, using an extended Kalman filter (EKF), to forecast the injector-producer relationships (IPRs) between multiple injectors and a single producer based on measured production and injection rates. This EKF procedure provides the capability to adaptively infer and track the preferential transmissibility trends between injectors and producer.
Many methods have been previously used to infer IPRs by analyzing a producing well and its surrounding injectors. Most of these methods assume that the IPRs are stationary, i.e. that the parameters in the models are unchanged over the window during which the data are analyzed, so that when the IPRs change the analysis needs to be repeated for the new situation. This may not be practical because it may be very difficult to recognize when a change has occurred. To overcome this difficulty, we used an EKF to adaptively infer and track IPRs. For the EKF we used a very simple parametric model, one with two parameters per injector, so that if a producer depends upon n injectors our model contains exactly 2n parameters. The EKF adaptively estimates the 2n parameters, which then lets the IPR between each injector and a producer be estimated.
This approach was tested on both synthetic and real data. The former used Monte-Carlo simulations, whereas the latter used data from Section 5 of Chevron's Lost Hills Field. Test results on synthetic data demonstrate the feasibility of the EKF method, and test results on the real data match expert knowledge about the IPRs between injectors and a producer. All results confirm that this EKF method can provide a good way to infer and track the IPRs, and to provide better insight about the IPRs.
The EKF procedure has the following advantages: (1) the model used for modeling the IPRs is very simple and is extendible (i.e., each additional injector only adds two new parameters); (2) it can infer the IPRs on a continuous basis; and (3) it can track a change of the IPRs very rapidly (i.e., it is adaptive to such changes).

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