This paper 1) reviews the α-plane representation of a type-2 fuzzy set (T2 FS), which is a representation that is comparable to the α-cut representation of a type-1 FS (T1 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 α-plane computations that are the set theoretic operations for interval T2 (IT2) FSs; 3) reviews how the centroid of a T2 FS can be computed using α-plane computations that are also very simple because they can be performed using existing Karnik Mendel algorithms that are applied to each α-plane; 4) shows how many theoretically based geometrical properties can be obtained about the centroid, even before the centroid is computed; 5) provides examples that show that the mean value (defuzzified value) of the centroid can often be approximated by using the centroids of only 0 and 1 α-planes of a T2 FS; 6) examines a triangle quasi-T2 fuzzy logic system (Q-T2 FLS) whose secondary membership functions are triangles and for which all calculations use existing T1 or IT2 FS mathematics, and 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 to forecast noise-corrupted measurements of a chaotic Mackey-Glass time series.Index Terms-α-Plane, centroid, Mackey-Glass time series, quasi-type-2 fuzzy logic systems (Q-T2 FLSs), set theoretic operations, type-2 fuzzy sets (T2 FSs).
Removing Mixed Gaussian and Impulse Noise (MGIN) is considered to be one of the most essential topics in the domain of image restoration, and it is much more challenging than to remove pure Gaussian or impulse noise separately. Therefore, relatively fewer works have been published in this area. This paper proposes a new integrated approach for MGIN removal that is based on a Non-Singleton Interval Type-2 (NS-IT2) Fuzzy Logic System (FLS), and explains how to design such a NS-IT2 FLS using a Quantum-behaved Particle Swarm Optimization (QPSO) algorithm. Then the paper goes on to introduce two supplementary components, a Block-Matching 3-Dimensional Discrete Cosine Transformation (BM3D DCT) filter and a contrast scaling filter, which augment the overall performance of the NS-IT2 FLS. Finally, the paper shows that this proposed approach indeed provides both quantitatively and visually much better results compared to other often-used non-fuzzy techniques as well as its Type-1 (T1) and singleton IT2 (S-IT2) counterparts.
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 [5]. 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. I. Introduction One of the most important issues in water-flood management is the ability to detect preferential subsurface flow trends, i.e., where the water flows. Recently, one approach for doing this was to estimate the inter-well relationship of each injector and its contributing producers using only the production and injection rates. More specifically, the inter-well relationship, referred here as the "Injector-Producer-Relationship (IPR)", was evaluated by estimating parameters of a model that characterizes the reservoir. After obtaining the IPR values, one can ascertain whether an injector is contributing to a producer and, if so, by how much. In addition, one can often infer other features about the reservoir from the IPR values, e.g., the directional sweep efficiency of a given pattern and presence of directional fractures. [1], [2], [4], and [7]-[11] introduce many different methods, yet all aim to achieve the above goal. The most recent works are Albertoni and Lake [1] and Yousef, et al. [11], both of which model the reservoir as a continuous impulse response that converts input signals (injection rates) to output signals (production rates). Liu, et al [5] have pointed out the following difficulties that [1] and [11] have encountered:the parameters in the model and the IPRs are assumed stationary over the window of measurements for which processing occurs, and so when the IPRs change the processing needs to be redone for the new situation, and this may not be practical because the reservoir is dynamic and it may be very difficult to recognize when a change has occurred; and,the CM model is somewhat complex, and although it is characterized by two parameters, to use it, the primary bottom hole pressure impact also needs to be determined. Additionally, Albertoni and Lake [1] model the production rate as a weighted sum of delayed injection rates from the injectors assumed to contribute to a producer; but, in general, how many delayed injection rates to include in the model is unknown.
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