Inverse Fuzzy Research Articles

On Algebraic Properties of Inverse Fuzzy Languages

Aims/Objectives: In this paper, we investigate some of the algebraic properties of inverse fuzzy languages. We proved that a fuzzy automaton is inverse if and only if the transition monoid is an inverse monoid. A fuzzy language is an inverse fuzzy language if the minimal fuzzy automaton recognizing that fuzzy language is an inverse fuzzy automaton. We also discuss some more properties of an inverse fuzzy language based on the fact that an inverse monoid is one which is regular and idempotent commute.

Deriving Fuzzy Weights of the Fuzzy Analytic Network Process via Fuzzy Inverse Matrix

The analytic hierarchical process/network process (AHP/ANP) is a popular multi-criteria decision making approach for determining the optimal alternative or weights of criteria. Many papers have extended the AHP/ANP to consider the fuzzy environment to reflect the subjective uncertainty of decision-makers. However, the fuzzy ANP (FANP) is not as popular as the fuzzy AHP (FAHP), because the calculation of the fuzzy supermatrix results in the divergence of the steady-state. In this paper, we provide a novel mathematical programming model to calculate the limiting distribution of the fuzzy supermatrix by considering a fuzzy inverse matrix rather than directly calculate the fuzzy supermatrix by limiting powers. In addition, we use a numerical example to illustrate the proposed method and compare the results with the previous method. The numerical results indicate the proposed method has the least spread of the fuzzy weights, thus justifying the usefulness of the proposed method.

Working memory modeling using inverse fuzzy relational approach

The paper proposes a novel approach to fuzzy modeling of human working memory (WM) using electroencephalographic (EEG) signals, acquired during human face encoding and recall experiments in connection with a face recognition problem. The EEG signals acquired from the short term memory (STM) during memory encoding instances are considered as the input of the proposed working memory model. On the other hand, the EEG response of the WM to visual stimuli acquired during WM recall instances are considered as the output of the proposed working memory model. The entire experiment is primarily divided into two phases. In the first phase, the WM of a human subject is modeled by a fuzzy implication relation, describing a mapping from the STM response (during encoding) to the WM responses (during recall) to visual stimuli. During STM encoding, the subject is visually presented with the full face stimulus of a person. During WM recall, four partial face stimuli of the same person (made familiar during encoding) are used for the subject to recall the respective full face.The second phase is undertaken to validate the WM model by visually stimulating the subject again with randomly selected partial faces of people, being familiar in the first phase and the WM EEG responses are recorded. The WM responses along with the WM model, developed in the first phase, are used to retrieve the STM information by using an inverse fuzzy (implication) relation. Besides WM modeling, another important contribution of the paper lies in devising a solution to the inverse fuzzy relation computation in the settings of an optimization problem. An error metric is then defined to measure the discrepancy between the model-predicted STM encoding pattern and the actual pattern encoded by the STM (as captured by the EEG signal during encoding in the first phase). Apparently, smaller the error magnitude better is the accuracy of the proposed model to effectively differentiate people with memory failures. Experimentally it is observed that the proposed model yields a very small error, in the order of 10−4, thus showing a high level of similarity between actual and model predicted STM response for all the healthy subjects. An experiment undertaken using eLORETA software confirms that the orbito-frontal cortex of prefrontal lobe is responsible for STM encoding whereas dorsolateral prefrontal region is responsible for WM recall. An analysis undertaken reveals that the proposed WM model produces the best response in the theta frequency band of EEG spectra, thus assuring the association of the theta frequency range in the face recognition task. Comparative analysis performed also substantiates that the proposed technique of computing max–min inverse fuzzy relation outperforms the existing techniques for inverse fuzzy computation, with a successful retrieval accuracy of 87.92%. The proposed study would find interesting applications to diagnose memory failures for people with Pre-frontal lobe amnesia.

A Fuzzy Inference System for Unsupervised Deblurring of Motion Blur in Electron Beam Calibration

This paper presents a novel method of restoring the electron beam (EB) measurements that are degraded by linear motion blur. This is based on a fuzzy inference system (FIS) and Wiener inverse filter, together providing autonomy, reliability, flexibility, and real-time execution. This system is capable of restoring highly degraded signals without requiring the exact knowledge of EB probe size. The FIS is formed of three inputs, eight fuzzy rules, and one output. The FIS is responsible for monitoring the restoration results, grading their validity, and choosing the one that yields to a better grade. These grades are produced autonomously by analyzing results of a Wiener inverse filter. To benchmark the performance of the system, ground truth signals obtained using an 18 μm wire probe were compared with the restorations. Main aims are therefore: (a) Provide unsupervised deblurring for device independent EB measurement; (b) improve the reliability of the process; and (c) apply deblurring without knowing the probe size. These further facilitate the deployment and manufacturing of EB probes as well as facilitate accurate and probe-independent EB characterization. This paper’s findings also makes restoration of previously collected EB measurements easier where the probe sizes are not known nor recorded.

Properties of fuzzy transform obtained from Lp minimization and a connection with Zadeh’s extension principle

In 2006 Perfilieva proved that the components Fk of the fuzzy transform are actually minimizers of certain weighted L2 type distances, obtained from a fuzzy partition. Inspired by this property, we consider and analyze the properties of a fuzzy transform when the components Fk are obtained as minimizers with respect to weighted Lp type distances, where p ≥ 1. In this way we can generate new approximation operators based on the fuzzy transform method. For these operators we obtain good rates of uniform convergence and what is more, they satisfy all the properties required for an effective approximation operator, such as additivity and homogeneity. If we consider the extended fuzzy transform and inverse fuzzy transform, respectively, in this setting of Lp type distances, then we get some nice supplementary properties like monotonicity of the operator and preservation of monotone functions. Finally, we consider fuzzy-valued versions of the L1 and L2 type fuzzy transforms, proposed recently as quantile and expectile smoothing tools, and we prove that the L1 type fuzzy-valued transform of a continuous monotonic function f coincides with the Zadeh’s fuzzy extension fEP evaluated at suitable fuzzy numbers, obtained from the basic functions of the used fuzzy partition.

Non-Probabilistic Inverse Fuzzy Model in Time Series Forecasting

Many models and techniques have been proposed by researchers to improve forecasting accuracy using fuzzy time series. However, very few studies have tackled problems that involve inverse fuzzy function into fuzzy time series forecasting. In this paper, we modify inverse fuzzy function by considering new factor value in establishing the forecasting model without any probabilistic approaches. The proposed model was evaluated by comparing its performance with inverse and noninverse fuzzy time series models in forecasting the yearly enrollment data of several universities, such as Alabama University, Universiti Teknologi Malaysia (UTM), and QiongZhou University; the yearly car accidents in Belgium; and the monthly Turkish spot gold price. The results suggest that the proposed model has potential to improve the forecasting accuracy compared to the existing inverse and non-inverse fuzzy time series models. This paper contributes to providing the better future forecast values using the systematic rules.

A Method Based on Extended Fuzzy Transforms to Approximate Fuzzy Numbers in Mamdani Fuzzy Rule-Based System

We propose a new Mamdani fuzzy rule-based system in which the fuzzy sets in the antecedents and consequents are assigned in a discrete set of points and approximated by using the extended inverse fuzzy transforms, whose components are calculated by verifying that the dataset is sufficiently dense with respect to the uniform fuzzy partition. We test our system in the problem of spatial analysis consisting in the evaluation of the livability of residential housings in all the municipalities of the district of Naples (Italy). Comparisons are done with the results obtained by using trapezoidal fuzzy numbers in the fuzzy rules.

Fuzzy Analytic Hierarchy Process: A performance analysis of various algorithms

Analytical Hierarchical Process (AHP) along with fuzzy set theory has been used extensively in the Multi-Criteria Decision Making (MCDM) process in which fuzzy numbers are utilized to represent human judgments more realistically. Over the past couple of decades, numerous articles have been published proposing algorithms through which priority vector (or weight vector) can be calculated from fuzzy comparison matrices. The aim of this study is to conduct a comprehensive performance analysis of the most popular algorithms proposed in this domain in terms of accuracy of weights calculated from fuzzy comparison matrices. Such an analysis is much needed by the researchers and practitioners. However none is available. An experimental analysis is conducted and the performance of various algorithms are evaluated with varying three parameters i.e., the size of the comparison matrix, the level of fuzziness and the level of inconsistency. We found that modified Logarithmic Least Squares Method and Fuzzy Inverse of Column Sum Method (FICSM) generally outperformed other algorithms, while Fuzzy Extent Analysis (FEA) which is the most frequently used algorithm in the literature provides the least accurate results. Furthermore, it was observed that a modified version of FEA method significantly improved its performance.

Optimal fuzzy inverse dynamics control of a parallelogram mechanism based on a new multi-objective PSO

This work presents a multi-objective optimization method based on high exploration particle swarm optimization, called MOHEPSO, for optimization problems with multiple objectives. In order to conve...

Time Series Seasonal Analysis Based on Fuzzy Transforms

We define a new seasonal forecasting method based on fuzzy transforms. We use the best interpolating polynomial for extracting the trend of the time series and generate the inverse fuzzy transform on each seasonal subset of the universe of discourse for predicting the value of an assigned output. In the first example, we use the daily weather dataset of the municipality of Naples (Italy) starting from data collected from 2003 to 2015 making predictions on mean temperature, max temperature and min temperature, all considered daily. In the second example, we use the daily mean temperature measured at the weather station “Chiavari Caperana” in the Liguria Italian Region. We compare the results with our method, the average seasonal variation, Auto Regressive Integrated Moving Average (ARIMA) and the usual fuzzy transforms concluding that the best results are obtained under our approach in both examples. In addition, the comparison results show that, for seasonal time series that have no consistent irregular variations, the performance obtained with our method is comparable with the ones obtained using Support Vector Machine- and Artificial Neural Networks-based models.

An adaptive image filter based on the fuzzy transform for impulse noise reduction

Impulse noise, also known as impulsive noise, is one of the most common types of noise occurring in digital images. The median filter and morphological filters are often used to remove impulsive noise, but these filters do not preserve image details. In this paper, we first apply the median filter in order to reduce the amount of impulsive noise in a corrupted image. After an application of the direct fuzzy transform (FT) to the resulting image, we restored the pixel values corresponding to locations flagged by the fuzzy rule-based noise detector by means of the inverse fuzzy transform. Finally, we obtained the output of our proposed image filter by combining the restored pixels with the ones marked as noiseless by the aforementioned noise fuzzy detector. We compared the results of our approach that we called adaptive FT-based image filter (AFT-IF) with the ones obtained by a number of other image filters for impulsive noise reduction.

Fuzzy Transform-Based Fusion of Multiple Images

Extensive development has taken place in the field of image fusion and various algorithms of image fusion have attracted the attention of many researchers in the recent past. Various algorithms of image fusion are used to combine information from multiple source images into a single fused image. In this paper, fusion of multiple images using fuzzy transform is proposed. Images to be fused are initially decomposed into same size blocks. These blocks are then fuzzy transformed and fused using maxima coefficient value-based fusion rule. Finally, the fused image is obtained by performing inverse fuzzy transform. The performance of the proposed algorithm is evaluated by performing experiments on multifocus, medical and visible/infrared images. Further, the performance of the proposed algorithm is compared with the state-of-the-art image fusion algorithms, both subjectively and objectively. Experimental results and comparative study show that the proposed fusion algorithm fuses the multiple images effectively and produces better fusion results for medical and visible/infrared images.

A new representation for inverse fuzzy transform and its application

In this study, a new representation formula for basic functions of fuzzy transform is introduced and some new approximating properties of the inverse fuzzy transform are described. In particular, using block pulse functions, we present properties of sinusoidal basic functions. As an application, we present a new fuzzy-based method for numerical solution of nonlinear Fredholm integral equations.

Multi-dimensional fuzzy transform and projection

Fuzzy and inverse fuzzy transforms, introduced by I. Perfilieva, is an important tool for signal and image fuzzy representation. It led to several variants, as the natural least-squares minimization named fuzzy projection. We deal with this transform, proposing first to show its simplicity and its easy numerical implementation in any dimension of space. Secondly, we point out that a key parameter permits to control its numerical robustness. Thirdly, we show that this parameter also governs the stability of the representation by fuzzy projection, when choosing different partitions of the space. We conclude in discussing linear representation; especially triangular representation for signals and pyramidal representation for images.

Forecasting and Diagnosing Cardiovascular Disease Based on Inverse Fuzzy Models

Advance formalization of a generalized fuzzy-logic mathematical model for forecasting and diagnosing cardiovascular complications is described. The model improves time of decision-making of anesthetist regarding the causes of complications and optimizes the perioperative period by prenosological prevention of occurring precursors of complications, reducing their total number and severity of clinical manifestations.

Embedded Indirect Adaptive Fuzzy Controller Based on T–S Fuzzy Inverse Model

The present paper is a trial to shed further light on the Indirect Adaptive Fuzzy Logic Controller (IDAFLC). In this concern, the proposed technique is predestined from two levels, where the lower level is based on Mamdani fuzzy controller. On the other hand, the upper level is an inverse model based on a Takagi–Sugeno method, in which its output is used to adapt the parameters of the fuzzy controller in the lower level. Moreover, the upper level contains learning mechanism to adapt model identification parameters. The proposed IDAFLC is implemented using an Arduino DUE kit. From the practical results, it is proved that the proposed adaptive controller has the ability to adapt the model identification parameters and improves, successfully, both the performance response and the disturbance due to the load and also measurement error of sensor in the speed control of the DC motor.

Fuzzy Time Series Prediction Model and Application based on Fuzzy Inverse

This paper presents a forecast model of fuzzy time series prediction to study the prediction of enrollment. It takes the percentage of enrollment changes as the domain, constructs the inverse fuzzy number and predicts the enrollment of Qiong Zhou University from 2005 to 2013. Compared with the existing models, the mean square error and prediction error of the ameliorated model are smaller and the precision is higher. The authors get the better method to solve the prediction problem based on the inverse fuzzy prediction model.

Lattice fuzzy transforms from the perspective of mathematical morphology

The compositions of direct and inverse fuzzy transforms constitute powerful tools in knowledge extraction and representation that have been applied to a large variety of problems in computational intelligence as well as in image processing and computer vision. Fuzzy transforms (FTs) have linear as well as lattice-based versions. In this paper, we extend the latter FTs, known as lattice FTs, and relate these operators and their underlying mathematical structures to the ones of mathematical morphology (MM), in particular to the ones of MM on complete lattices and L-fuzzy MM.

An interval-valued inversion of the non-additive interval-valued F-transform: Use for upsampling a signal

As shown in a recent paper, the fuzzy transform can be reinterpreted as a sampling process with an ill-known sampling kernel (or point spread function). This reinterpretation leads to an interval-valued direct fuzzy transform: the non-additive fuzzy transform. It provides the convex set of all sampled values that should have been obtained by sampling a continuous signal with a convex set of sampling kernels. Its dual transform, the non-additive inverse fuzzy transform, allows computation of an interval-valued reconstruction of a sampled signal, which is the set of all values that would have been obtained by interpolating or approximating this sampled signal when using a convex set of reconstruction kernels. In this paper, we consider using a new interval-valued inversion process within this new framework to upsample a signal, i.e. reconstruct a high resolution signal with a low resolution signal, in a semi-blind context. As illustrated in the experimental part of the paper, the main advantage of using this method, compared to previous techniques, is its inherent robustness with respect to modeling the sampling process.

General approximation of fuzzy numbers by F-transform

In this paper we will prove that in most of the cases the extended inverse fuzzy transform preserves the quasi-concavity of a fuzzy number and hence it can be used to generate fuzzy numbers by approximating the restriction of the membership function to its support. In the case of continuous fuzzy numbers with cores containing more than one element, the rate of uniform convergence is of linear type and the same holds when we approximate the important characteristics of a fuzzy number such as the value or the ambiguity. Moreover we have the preservation of the support and the convergence of the core which in addition can be determined precisely. In the case of continuous fuzzy numbers with one-element core, it is in general necessary to normalize the approximation, but the support is preserved again and the core can be determined exactly in this case too. Moreover, the approximations have again linear rate of uniform convergence.