Below are titles and abstracts of very recent research results.
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Type-2 Fuzzy Logic
"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.
J. M. Mendel, F. Liu and D. Zhai, "Alpha-plane representation for type-2 fuzzy sets: theory and applications," IEEE Trans. on Fuzzy Systems, vol. 17, pp. 1189-1207, October 2009.
This paper: (1) Reviews the alpha-plane representation of a type-2 fuzzy set (T2 FS), a representation that is comparable to the alpha-cut representation of a type-1 FS, and is useful for both theoretical and computational studies of and for T2 FSs; (2) Proves that set theoretic operations for T2 FSs can be computed using very simple alpha-plane computations that are the set theoretic operations for interval T2 FSs; (3) Reviews how the centroid of a T2 FS can be computed using alpha-plane computations that are also very simple because they can be performed using existing KM algorithms that are applied to each alpha-plane; (4) Shows how many theoretically-based geometrical properties can be obtained about the centroid, even before the centroid is computed; (5) Provides examples showing that the mean-value (defuzzified value) of the centroid can often be approximated by using the centroids of only the 0 and 1 alpha-planes of a T2 FS; (6) Examines a triangle quasi-T2 fuzzy logic system (FLS), whose secondary membership functions are triangles, and for which all calculations use existing T1 or interval T2 FS mathematics; hence, they may be a good next step in the hierarchy of FLSs, from T1 to IT2 to T2; and, (7) compares T1, IT2 and triangle Q-T2 FLSs for forecasting noise-corrupted measurements of a chaotic Mackey-Glass time series.
D. Zhai and J. M. Mendel, "Uncertainty measures for general type-2 fuzzy sets," submitted for publication, 2009.
Five uncertainty measures have previously been defined for Interval Type-2 Fuzzy Sets (IT2 FSs), namely centroid, cardinality, fuzziness, variance and skewness. Based on a recently developed _-plane representation for a general T2 FS, this paper generalizes these definitions to such T2 FSs and, more importantly, derives a unified strategy for computing all different uncertainty measures with low complexity. The uncertainty measures of T2 FSs with different shaped Footprints of Uncertainty and different kinds of secondary membership functions (MFs) are computed and are given as examples. Observations and summaries are made for these examples, and a Summary Interval Uncertainty Measure for a general T2 FS is proposed to simplify the interpretations. Comparative studies of uncertainty measures for Quasi-Type-2 (QT2), IT2 and T2 FSs are also performed to examine the feasibility of approximating T2 FSs using QT2 or even IT2 FSs.
D. Wu and J. M. Mendel, "Perceptual reasoning for perceptual computing: a similarity-based approach," accepted for publication in the IEEE Trans. on Fuzzy Systems, 2009.
Perceptual reasoning (PR) is an approximate reasoning method that can be used as a computing with words (CWW) engine in perceptual computing. There can be different approaches to implement PR, e.g., firing interval based PR (FI-PR) is proposed in , , , and similarity-based PR (S-PR) is proposed in this paper. Both approaches satisfy the requirement on a CWW engine that the result of combining fired rules should lead to a footprint of uncertainty (FOU) that resembles the three kinds of FOUs in a CWW codebook. A comparative study shows that S-PR leads to output FOUs that resembles word FOUs obtained from subject data much more closely than FI-PR; hence, S-PR is a better choice for a CWW engine than is FI-PR.
D. Wu and J. M. Mendel, "Interval type-2 fuzzy set subsethood measures as a decoder for perceptual computing," submitted for publication, 2009.
In some applications of computing with words, it is necessary to map an interval type-2 fuzzy set (IT2 FS) into one of several classes, which are also represented by IT2 FSs. This classifier can be implemented by a subsethood measure. Five existing subsethood measures for IT2 FSs are considered in this paper. Comparative studies show that Vlachos and Sergiadis's  IT2 FS subsethood measure gives the most reasonable outputs as a decoder in computing with words when the desired output is a class. The results in this paper will be useful in constructing a third kind of decoder (i.e., in addition to similarity measures and ranking methods) for perceptual computing.
J. Jhoo and J. M. Mendel, "Obtaining an FOU for a word form a single subject by an Individual Interval Approach," Proc. IEEE Systems, Man, and Cybernetics Conference, San Antonio, TX, 2009.
Recently a simple and practical type-2-fuzzistics methodology called an Interval Approach (IA) was presented for obtaining interval type-2 fuzzy set (IT2 FS) models for words using data collected from a group of subjects. There may be times, however, when a group of subjects is not available. This paper proposes a way to obtain IT2 FS models from words collected from a single subject using an IA, and is called an Individual IA (IIA). Two methods are presented for doing this. Both use end-point and uncertainty data that are collected from an individual, assume a probability distribution on each interval, map them into pre-specified T1 membership functions (MF), interpret the MFs as nine embedded T1 FSs of an IT2 FS, and then aggregate the FSs using union to obtain the footprint of uncertainty (FOU) for the word. This approach not only captures the strong points of the previously developed IA but simplifies it. Experiments show that the IIA is easy to implement and the resulting FOUs match our intuition.
M. Biglarbegian, W. Melek and J. M. Mendel, "On the stability of interval type-2 TSK fuzzy logic control systems," IEEE Trans. on Systems, Man, and Cybernetics-Part B: Cybernetics, 2009.
Type-2 fuzzy logic systems have recently been utilized in many control processes due to their ability to model uncertainties. This paper proposes a novel inference mechanism for an interval type-2 TakagiSugenoKang fuzzy logic control system (IT2 TSK FLCS) when antecedents are type-2 fuzzy sets and consequents are crisp numbers (A2-C0). The proposed inference mechanism has a closed form, which makes it more feasible to analyze the stability of this FLCS. This paper focuses on control applications for the following cases: 1) Both plant and controller use A2-C0 TSK models, and 2) the plant uses type-1 TakagiSugeno (TS) and the controller uses IT2 TS models. In both cases, sufficient stability conditions for the stability of the closed-loop system are derived. Furthermore, novel linear-matrix inequality- based algorithms are developed for satisfying the stability conditions. Numerical analyses are included which validate the effectiveness of the new inference methods. Case studies reveal that an IT2 TS FLCS using the proposed inference engine clearly outperforms its type-1 TSK counter-part. Moreover, due to the simple nature of the proposed inference engine, it is easy to implement in real-time control systems. The methods presented in this paper lay the mathematical foundations for analyzing the stability and facilitating the design of stabilizing controllers of IT2 TSK FLCSs and IT2 TS FLCSs with significantly improved performance over type-1 approaches.
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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 Paper 110520, SPE Conference, Anaheim, CA, Nov. 2007.
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).
J. Jhoo, D. Wu, J. M. Mendel and A. Bugacov, "Forecasting the post-fracturing response of oil wells in a tight reservoir," SPE Paper 121394, presented at the 2009 SPE Western Regional Meeting, San Jose, CA, March 2009.
This paper proposes a two-stage multi-system architecture for forecasting post-fracturing responses in a tight oil reservoir using historical fracturing data. The first stage predicts the 180-day cumulative liquid (oil + water) production directly, and the second stage uses differential correction to predict the prediction error resulting from the first stage. The final prediction is a combination of the two stages. 5-fold cross-validation is used in each stage, resulting in five forecasters for each stage. The average of the five predictions is taken as the output of the corresponding stage. Each of the five forecasters in each stage consists of three independent subsystems (Location, Completion and Fracturing), whose inputs are subsets of the well properties. The Location subsystem is constructed by a weighted average, whereas Completion and Fracturing are constructed by fuzzy logic systems. The parameters of the three subsystems are optimized simultaneously using simulated annealing. The final design achieved over 70% prediction accuracy for more than 96% of the testing wells. The main advantages of our approach are that 1) it does not require a large training dataset; 2) it can cope well with incomplete data entries and uncertainties; and, 3) the redundancy in the input parameters is used to improve accuracy.
D. Zhai, J. M. Mendel and Fl Liu, "A new method for continual forecasting of interwell connectivity in waterfloods using an Extended Kalman Filter," SPE Paper 121393, presented at the 2009 SPE Western Regional Meeting, San Jose, CA, March 2009.
This paper is based on a relatively simple parametric model that characterizes the system function between a specific producer and each of its contributing injectors. The model has only two parameters for each producer-injector pair; so, if N injectors are assumed to contribute to a producer, there will be 2N unknown parameters. An adaptive strategy, using an Extended Kalman Filter (EKF), is used to estimate the 2N parameters, which are then used to generate N numeric Injector-Producer-Relationship (IPR) values for the N producer-injector pairs. The IPR values allow one to assess how well an injector influences the producer.
This same model and an EKF were first used in Liu, et al . The modified EKF used in this paper avoids problems that can arise when processing real data and provides additional information that is useful for future research. Our modified EKF is applied to real data from a section of an oil field. A validation strategy for the estimated IPR values is developed in terms of "prediction errors." A strategy is also presented for choosing an optimal set of injectors that affect a producer. Finally, a simple method is presented for converting producer-centric IPR values to injector-centric IPR values so that reservoir engineers can easily see which producers are being affected by a specific injector.
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