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Showing papers on "Neuro-fuzzy published in 2021"


Journal ArticleDOI
TL;DR: This article investigates the issue of neuro-fuzzy-based adaptive dynamic surface control (DSC) for uncertain fractional-order (FO) nonlinear systems in strict-feedback form where input constraint is considered in the systems.
Abstract: This article investigates the issue of neuro-fuzzy-based adaptive dynamic surface control (DSC) for uncertain fractional-order (FO) nonlinear systems in strict-feedback form where input constraint is considered in the systems. In the recursive steps, the neuro-fuzzy network systems are employed to deal with the unknown nonlinear terms existing in systems. Furthermore, based on a DSC scheme, a modified FO filter is constructed to overcome the problem of explosion of complexity caused by the traditional backstepping design. Moreover, according to the FO Lyapunov stability theory, a neuro-fuzzy-based adaptive controller is designed to guarantee all the signals of FO closed-loop systems tend to be bounded. Finally, the three examples are provided to verify the validity and superiority of the presented control scheme.

67 citations


Journal ArticleDOI
TL;DR: In the present review article, models created by several different machine learning approaches are comprehensively reviewed and it is concluded that artificial neural network is a very attractive approach for modeling both dynamic viscosity and thermal conductivity.

45 citations


Journal ArticleDOI
TL;DR: Experimental studies on popular medical data benchmarks demonstrate that the proposed work is able to learn compact rule bases involving simple rule antecedents, with statistically better or comparable performance to those achieved by state-of-the-art fuzzy classifiers.

39 citations


DOI
10 Nov 2021
TL;DR: In this article, the authors presented Neuro-Fuzzy system as a decision-making support in the selection of construction machines (the example of selecting a loader is provided), which makes the basis for selection, but also other elements of importance.
Abstract: In the paper is presented Neuro-Fuzzy System as a decision-making support in the selection of construction machines (the example of selecting a loader is provided). Construction characteristics of a loader make the basis for selection, but also other elements of importance. The data for Neuro-Fuzzy System modeling are prepared using the Multi-Criteria Decision Making (MCDM) methods: Logarithm Methodology of Additive Weights (LMAW), VIKOR, TOPSIS, MOORA and SAW. The paper also presents the method of aggregation of weights of rules premises (AWRP), which defines the key rules of Neuro-Fuzzy System. Finally, the training of the model is tested. The data for the selection of input variables and for model training are obtained by engaging experts

38 citations


Journal ArticleDOI
TL;DR: A review of different papers, reports, and other documents using ANN for MPPT control is presented in this paper, where the algorithms are based on ANN or in a hybrid combination with FL or a metaheuristic algorithm.
Abstract: The use of photovoltaic systems for clean electrical energy has increased. However, due to their low efficiency, researchers have looked for ways to increase their effectiveness and improve their efficiency. The Maximum Power Point Tracking (MPPT) inverters allow us to maximize the extraction of as much energy as possible from PV panels, and they require algorithms to extract the Maximum Power Point (MPP). Several intelligent algorithms show acceptable performance; however, few consider using Artificial Neural Networks (ANN). These have the advantage of giving a fast and accurate tracking of the MPP. The controller effectiveness depends on the algorithm used in the hidden layer and how well the neural network has been trained. Articles over the last six years were studied. A review of different papers, reports, and other documents using ANN for MPPT control is presented. The algorithms are based on ANN or in a hybrid combination with FL or a metaheuristic algorithm. ANN MPPT algorithms deliver an average performance of 98% in uniform conditions, exhibit a faster convergence speed, and have fewer oscillations around the MPP, according to this research.

35 citations


Journal ArticleDOI
TL;DR: The model developed in this work indicated that the ANFIS model can be used to accurately predict the solubility of drugs in supercritical solvents which can be consequently used for production of drugs with improved efficacy.

32 citations


Journal ArticleDOI
TL;DR: This field is an emerging research direction and it is increasingly paying much more attention, and fuzzy systems make great effects on deep learning models in the aspect of classification, prediction, natural language processing, auto-control, etc., and the fusion is applied into different fields.
Abstract: Deep learning presents excellent learning ability in constructing learning model and greatly promotes the development of artificial intelligence, but its conventional models cannot handle uncertain or imprecise circumstances. Fuzzy systems, can not only depict uncertain and vague concepts widely existing in the real world, but also improve the prediction accuracy in deep learning models. Thus, it is important and necessary to go through the recent contributions about the fusion of deep learning and fuzzy systems. At first, we introduce the deep learning into fuzzy community from two perspectives: statistical results of relevant publications and conventional deep learning algorithms. Then, the fusing framework and graphic form of deep learning and fuzzy systems are constructed. Followed by, are the current situations of several types of fuzzy techniques used in deep learning, some reasons why use fuzzy techniques in deep learning, and the application fields of the fusion, respectively. Finally, some discussions and future challenges are provided regarding the fusion technology of deep learning and fuzzy systems, the application scenarios of fusing deep learning and fuzzy systems, and some limitations of the current fusion, respectively. After summarizing the recent contributions, we have found that this field is an emerging research direction and it is increasingly paying much more attention. Especially, fuzzy systems make great effects on deep learning models in the aspect of classification, prediction, natural language processing, auto-control, etc., and the fusion is applied into different fields, like but not limited to computer science, natural language, medical system, smart energy management systems and machinery industry.

26 citations


Journal ArticleDOI
TL;DR: Results show that the ANFIS SC framework with cluster radius 0.6 can remarkably diagnose the PV system faults with high accuracy.

26 citations


Journal ArticleDOI
TL;DR: The outcome of the present study reveals that the ANFIS modeling, aided with data pre-processing and input optimization, is a suitable technique for simulating the quality of surface water.
Abstract: Modeling surface water quality using soft computing techniques is essential for the effective management of scarce water resources and environmental protection. The development of accurate predictive models with significant input parameters and inconsistent datasets is still a challenge. Therefore, further research is needed to improve the performance of the predictive models. This study presents a methodology for dataset pre-processing and input optimization for reducing the modeling complexity. The objective of this study was achieved by employing a two-sided detection approach for outlier removal and an exhaustive search method for selecting essential modeling inputs. Thereafter, the adaptive neuro-fuzzy inference system (ANFIS) was applied for modeling electrical conductivity (EC) and total dissolved solids (TDS) in the upper Indus River. A larger dataset of a 30-year historical period, measured monthly, was utilized in the modeling process. The prediction capacity of the developed models was estimated by statistical assessment indicators. Moreover, the 10-fold cross-validation method was carried out to address the modeling overfitting issue. The results of the input optimization indicate that Ca2+, Na+, and Cl− are the most relevant inputs to be used for EC. Meanwhile, Mg2+, HCO3−, and SO42− were selected to model TDS levels. The optimum ANFIS models for the EC and TDS data showed R values of 0.91 and 0.92, and the root mean squared error (RMSE) results of 30.6 µS/cm and 16.7 ppm, respectively. The optimum ANFIS structure comprises a hybrid training algorithm with 27 fuzzy rules of triangular fuzzy membership functions for EC and a Gaussian curve for TDS modeling, respectively. Evidently, the outcome of the present study reveals that the ANFIS modeling, aided with data pre-processing and input optimization, is a suitable technique for simulating the quality of surface water. It could be an effective approach in minimizing modeling complexity and elaborating proper management and mitigation measures.

26 citations


Journal ArticleDOI
TL;DR: This paper aims to evaluate whether it is possible to obtain reasonable accuracy using a hybrid model for electric power generation forecasting considering solar trackers and presents a model and evaluates the solar prediction capacity using the Wavelet Neuro-Fuzzy algorithm.
Abstract: The generation of electric energy by photovoltaic (PV) panels depends on many parameters, one of them is the sun’s angle of incidence. By using solar active trackers, it is possible to maximize generation capacity through real-time positioning. However, if the engines that update the position of the panels use more energy than the difference in efficiency, the solar tracker system becomes ineffective. In this way, a time series forecasting method can be assumed to determine the generation capacity in a pre-established horizon prediction to evaluate if a position update would provide efficient results. Among a wide range of algorithms that can be used in forecasting, this work considered a Neuro-Fuzzy Inference System due to its combined advantages such as smoothness property from Fuzzy systems and adaptability property from neural networks structures. Focusing on time series forecasting, this article presents a model and evaluates the solar prediction capacity using the Wavelet Neuro-Fuzzy algorithm, where Wavelets were included in the model for feature extraction. In this sense, this paper aims to evaluate whether it is possible to obtain reasonable accuracy using a hybrid model for electric power generation forecasting considering solar trackers. The main contributions of this work are related to the efficiency improvement of PV panels. By assuming a hybrid computational model, it is possible to make a forecast and determine if the use of solar tracking is interesting during certain periods. Finally, the proposed model showed promising results when compared to traditional Nonlinear autoregressive model structures.

24 citations


Journal ArticleDOI
TL;DR: In this article, the authors analyzed pupils' knowledge in mathematics by adaptive neuro fuzzy inference system (ANFIS) after implementation of distance learning application or e-learning (electronic learning).

Journal ArticleDOI
TL;DR: The proposed ENFS-Uni0 model shows an outperformance of the related works in terms of higher accuracy trend lines over time, while offering a high degree of interpretability through coherent neuro-fuzzy rules to solve the classification problems.

Journal ArticleDOI
22 Jan 2021
TL;DR: The proposed methods of prediction would be useful for practitioners to simplify the procedures during software development in large datasets for reducing failures of software.
Abstract: Reliability of software is an essential concern for users for a long time. Software reliability is mainly obtained through modeling and estimating. There are numerous methods for reducing the failure rate. However, the existing methods are nonlinear. Hence the parameter estimation of these methods is difficult. This paper concerns on estimation and prediction of software reliability through different soft computing methods for improving the reliability of software. For estimation and prediction, the authors of this paper take two soft computing methodologies, including fuzzy logic and neural network. The outcomes seem to give satisfactory results on large datasets. For experiments, this paper is using two different large datasets of Apache server and MyLyn application software for showing the effectiveness of the results. The proposed methods of prediction would be useful for practitioners to simplify the procedures during software development in large datasets for reducing failures of software.

Journal ArticleDOI
TL;DR: Investigating the relationship among porosity, permeability, and pore throat radii using three methods such as multiple regression analysis, artificial neural network (ANN), and adaptive neuro-fuzzy inference system (ANFIS) for application in transition zone permeability modeling finds both the ANN and ANFIS are later found to be more effective and efficient and thus recommended as compared with the multiple regression technique commonly used in the industry.

Journal ArticleDOI
TL;DR: In this paper, a three-stage parameter learning method of the neuro-fuzzy based Hammerstein-Wiener model corrupted by process noise using combined signals is developed, which completely separate the parameter learning issues of the static input nonlinear block, the linear dynamic block and the static output nonlinear blocks, which effectively simplifies the process of parameter learning.
Abstract: The Hammerstein–Wiener model is a nonlinear system with three blocks where a dynamic linear block is sandwiched between two static nonlinear blocks. For parameter learning of the Hammerstein–Wiener model, the synchronous parameter learning methods are proposed to learn the model parameters by constructing hybrid model of the three series block, such as over parameterization method, subspace method and maximum likelihood method. It should be pointed out that the aforementioned methods appeared the product term of model parameters in the process of parameter learning, and parameter separation method is further adopted to separate hybrid parameters, which increases the complexity of parameter learning. To address this issue, a novel three-stage parameter learning method of the neuro-fuzzy based Hammerstein–Wiener model corrupted by process noise using combined signals is developed in this paper. The combined signals are designed to completely separate the parameter learning issues of the static input nonlinear block, the linear dynamic block and the static output nonlinear block, which effectively simplifies the process of parameter learning of the Hammerstein–Wiener model. Parameter learning of the Hammerstein–Wiener model are summarized into the following three aspects: The first one is to learn the output static nonlinear block parameters using two sets of separable signals with different sizes. The second one is to estimate the linear dynamic block parameters by means of the correlation analysis method, the unmeasurable intermediate variable information problem is effectively handled. The final one is to determine the parameters of the static input nonlinear block and the moving average noise model using recursive extended least square scheme. The simulation results are presented to illustrate that the proposed learning approach yields high learning accuracy and good robustness for the Hammerstein–Wiener model corrupted by process noise.

Journal ArticleDOI
TL;DR: This paper proposes and examines alternative variants for consequent parameter updates, namely multi-innovation RFWLS, recursive correntropy and especially recursive weighted total least squares (RWTLS) and shows thatRFWLS can be largely outperformed by the proposed alternative variants, and this with even lower sensitivity on various data noise levels.

Journal ArticleDOI
TL;DR: This paper proposes the use of a neuro‐fuzzy machine learning technique for predicting the most complex hypertensive disorder in pregnancy called HELLP syndrome, and serves as an inference mechanism for cloud‐based mobile applications, for effective monitoring through the analysis of symptoms presented by pregnant women.
Abstract: The exchange of information among health professionals is a common practice among clinics, laboratories, and hospitals. Cloud‐based clinical data exchange platforms enable valuable information to be available in real time and in a secure and private manner. The increasing availability of data in health information systems allows specialists to extract knowledge using pattern recognition techniques for the identification and prediction of risk situations that could lead to severe complications for a patient. Hence, this paper proposes the use of a neuro‐fuzzy machine learning technique for predicting the most complex hypertensive disorder in pregnancy called HELLP syndrome. This classifier serves as an inference mechanism for cloud‐based mobile applications, for effective monitoring through the analysis of symptoms presented by pregnant women. Results show that the proposed model achieves excellent results regarding several indicators, such as precision (0.685), recall (0.756), the F‐measure (0.705), and the area under the receiver operating characteristic curve (0.829). This technique can accurately predict situations that could lead to the death of both a mother and fetus, at any location and time.

Journal ArticleDOI
TL;DR: The study examines various factors of ANFIS to come up with an effective time series prediction model and indicates that Spain and Italy data can strengthen the predictive power of COVID-19 cases in the UK.

Journal ArticleDOI
25 Oct 2021-Energy
TL;DR: In this paper, a hybrid optimized model of Adaptive Neuro-Fuzzy Inference System (ANFIS), Recurrent Kalman Filter (RKF), and Neuro-Wavelet (WNN) for wind power forecasting driven by doubly fed induction generator (DFIG).

Journal ArticleDOI
TL;DR: A robust intelligent control problem for a class of power-electronic converters via a neuro-fuzzy learning mechanism using type-2 fuzzy neural network to approximate the developed TSMC and the corresponding adaptive T2FNN controller with online parameter adjustment is established.
Abstract: This article considers a robust intelligent control problem for a class of power-electronic converters via a neuro-fuzzy learning mechanism. First, a terminal sliding-mode control (TSMC) is designed to ensure finite-time error convergence and further enhance the system performance. Meanwhile, a saturation function is utilized in the proposed TSMC. Then, by using type-2 fuzzy neural network (T2FNN) to approximate the developed TSMC, the corresponding adaptive T2FNN controller with online parameter adjustment is established. To enhance the generalization ability for the uncertainties, the recurrent feature-selection algorithm is added into T2FNN. Moreover, the existence of adaptive compensator comprised by upper bound updated law can avoid the impact of the approximation error. Finally, to show the superiorities of the T2FNN controller, it is applied to active power filter, and the simulation and experimental results are compared with the existing literature.

Journal ArticleDOI
TL;DR: In this paper, a Mamdani-type fuzzy logic model and a Neuro-fuzzy system (FIS) model are developed to elucidate the operational parameters of supercritical CO2 extractor.
Abstract: Catalyst regeneration is highly important in chemical processes in terms of process performance and economic efficiency. In this study, deactivated rib ring nickel alumina (Ni/Al2O3) catalysts are regenerated in a pilot plant supercritical CO2 extractor. The catalysts are collected from the Iron Midrex reformer unit and are provided by Khouzestan Steel Company (KSC). A Mamdani-type fuzzy logic model and a Neuro-fuzzy system (FIS) model are developed to elucidate the operational parameters i.e. temperature, pressure, and flow rate of supercritical carbon dioxide. The models may define and predict non-experienced data for the output variable, which is the mass concentration of removed deactivating substances from the spent catalyst. Triangular architecture fuzzy models with seventeen rules for the first model, and fourteen rules as the training data for the latter one, are considered to develop the models. The proposed FIS model is validated by using the root mean square error (RMSE) and considering three of the experiments as the checking data. Moreover, the response surface methodology (RSM) is used to find the optimal output value and the related input parameters values. The results show that the proposed neuro-fuzzy model is valid with the RMSE of less than 1.3 %, and the optimum output value found by the RSM is 0.0767, when the temperature, pressure, and flow rate are 57.7, 182.5, and is 0.538, respectively.

Journal ArticleDOI
TL;DR: In this article, a Self-Identification Neuro-Fuzzy Inference Model (SINFIM) was proposed for modeling the relationship between rainfall and runoff on a Chilean watershed.

Journal ArticleDOI
TL;DR: The result shows that LSRGNFM‐LDC technique improves the accuracy and minimizes the time consumption as well as error rate than existing works during COVID prediction.
Abstract: Coronavirus disease (COVID-19) is a harmful disease caused by the new SARS-CoV-2 virus. COVID-19 disease comprises symptoms such as cold, cough, fever, and difficulty in breathing. COVID-19 has affected many countries and their spread in the world has put humanity at risk. Due to the increasing number of cases and their stress on administration as well as health professionals, different prediction techniques were introduced to predict the coronavirus disease existence in patients. However, the accuracy was not improved, and time consumption was not minimized during the disease prediction. To address these problems, least square regressive Gaussian neuro-fuzzy multi-layered data classification (LSRGNFM-LDC) technique is introduced in this article. LSRGNFM-LDC technique performs efficient COVID prediction with better accuracy and lesser time consumption through feature selection and classification. The preprocessing is used to eliminate the unwanted data in input features. Preprocessing is applied to reduce the time complexity. Next, Deming Least Square Regressive Feature Selection process is carried out for selecting the most relevant features through identifying the line of best fit. After the feature selection process, Gaussian neuro-fuzzy classifier in LSRGNFM-LDC technique performs the data classification process with help of fuzzy if-then rules for performing prediction process. Finally, the fuzzy if-then rule classifies the patient data as lower risk level, medium risk level and higher risk level with higher accuracy and lesser time consumption. Experimental evaluation is performed by Novel Corona Virus 2019 Dataset using different metrics like prediction accuracy, prediction time, and error rate. The result shows that LSRGNFM-LDC technique improves the accuracy and minimizes the time consumption as well as error rate than existing works during COVID prediction.

Journal ArticleDOI
TL;DR: ANFIS scheme provides appropriate results with less computational burden and simple structure with optimised responses in challenging conditions, and the capability of the proposed method is validated for different operating conditions through simulation and experimental results.
Abstract: Adaptive neuro-fuzzy inference system (ANFIS) technique is a significant alternative of research which is structured with a combination of two soft-computing strategies of fuzzy logic and artificial neural network. The design of ANFIS controller for a single-phase full-bridge inverter with pulse width modulation is demonstrated here in the presence of different disturbances. Moreover, an LC filter is designed to decrease the disturbing harmonics which the stability of the filter can be noted as an important issue. Based on the fuzzy C-mean clustering method used for decreasing fuzzy rules, the computational burden has been improved resulting in faster dynamic performance. This method considers the system as a black-box structure which omits the need for an exact model of system and can be an appropriate technique for ill-defined systems. Additionally, to deal with the variations of supply DC voltage, a fractional-order proportional-integral-derivative controller is designed which is tuned by particle swarm optimiser algorithm and can generate a sinusoidal reference for the system input. This double-loop control technique is known as cascade control strategy. It can be seen that ANFIS scheme provides appropriate results with less computational burden and simple structure with optimised responses in challenging conditions. The capability of the proposed method is validated for different operating conditions through simulation and experimental results.

Journal ArticleDOI
TL;DR: The results show that both ANFIS and SAC can deal very successfully with modeling complexities associated with full three-dimensional models subjected to multi-excitations and both methods have shown a great potential in enhancing the structural performance under seismic activities.

Journal ArticleDOI
TL;DR: In this article, a Deep Convolutional Neuro-Fuzzy Method (DCNFM) was proposed to detect and quantify different kinds of infections affecting the paddy plant crop, such as brown spots, bacterial blight, and leaf blasts.
Abstract: A significant number of the world’s population is dependent on rice for survival. In addition to sugarcane and corn, rice is said to be the third most growing staple food in the world. As a consequence of intensive usage of man-made fertilizers, paddy plant diseases have also risen at a faster pace in current history. Exploring the possible disease spread and classifying to detect the consequent impact at an early stage will prevent the loss and improve rice production. The core task of this research is to recognize and quantify different kinds of infections (disease) affecting the paddy plant crop, such as brown spots, bacterial blight, and leaf blasts. Both detection and recognition are carried out based on the risk analysis of paddy crop leaf images. We suggest a Deep Convolutional Neuro-Fuzzy Method (DCNFM) that combines one of the advanced machine learning variant, namely deep convolutional neural networks (DCNNs) and uncertainty handler called fuzzy logic. The synthesis has the benefits of both fuzzy logic and DCNNs when dealing with unstructured data, extracting essential features from imprecise and ambiguous datasets. From the crop field, continuous image data are captured through image sensors and fed as a primary input to the proposed model to analyze the risk and then later to classify them for precise recognition/detection of the disease. The detection/recognition rate of the DCNFM is found to be 98.17% which is comparatively found to be effective in comparison with the traditional CNN model.

Journal ArticleDOI
TL;DR: A hybrid neuro-fuzzy-wavelet model with mother wavelet db7 is proposed to modelling the rainfall-runoff transformation for forecasting daily flow rates in the Sebaou basin located in Tizi Ouzou region.
Abstract: Heavy rainfall over a short period or slowly during long periods can significantly increase the amount of water. Where it results in floods that can pose a direct threat, capable of causing human and material losses. Over the past two decades, artificial intelligence has been widely applied in the field of hydrology as well as in many other areas of hydraulic engineering. The wavelet transform is a very popular technique in the analysis of non-stationary time series and particularly effective for hydrological series. Currently, the application of intelligent hybrid systems in different fields has shown good performance and unparalleled efficiency. As such, in this work, we propose a hybrid neuro-fuzzy-wavelet model to modelling the rainfall-runoff transformation for forecasting daily flow rates in the Sebaou basin located in Tizi Ouzou region. The results obtained are very encouraging and better than those obtained by the models used for comparison in this research. According to the results, the hybrid neuro-fuzzy-wavelet model with mother wavelet db7 gave us better performance for daily flow rates forecasting.

Journal ArticleDOI
TL;DR: This paper focuses on the ADS schemes which have applied fuzzy logic in combination with other machine learning and data mining techniques to deal with the inherent uncertainty in the intrusion detection process.
Abstract: In the data security context, anomaly detection is a branch of intrusion detection that can detect emerging intrusions and security attacks. A number of anomaly detection systems (ADSs) have been proposed in the literature that using various algorithms and techniques try to detect the intrusions and anomalies. This paper focuses on the ADS schemes which have applied fuzzy logic in combination with other machine learning and data mining techniques to deal with the inherent uncertainty in the intrusion detection process. For this purpose, it first presents the key knowledge about intrusion detection systems and then classifies the fuzzy ADS approaches regarding their utilized fuzzy algorithm. Afterward, it summarizes their major contributions and illuminates their advantages and limitations. Finally, concluding issues and directions for future researches in the fuzzy ADS context are highlighted.

Journal ArticleDOI
TL;DR: The simulations results showed that the mobile robot could navigate successfully into maze environment with both proposed approaches but ANFIS controller provided better results in comparison to fuzzy controller.
Abstract: The navigation of autonomous mobile robots has in recent times gained interest from many researchers in different areas such as the industrial, agricultural, and military sectors. This pap...

Journal ArticleDOI
TL;DR: Experiments show that CANFIS-ART approach not only outperforms these state-of-the-art techniques but also demonstrates a higher level of generalization capability, increasing the data quality contained in databases with missing values.
Abstract: Data imputation aims to solve missing values problem which is common in nowadays applications. Many techniques have been proposed to solve this problem from statistical methods such as Mean/Mode to machine learning models. In this paper, an approach based on Co-active Neuro-Fuzzy Inference System named CANFIS-ART is proposed to automate data imputation procedure. This model is constructed from the Neural Network adaptative capabilities and fuzzy logic qualitative approach using the Fuzzy-ART algorithm. Performance of CANFIS-ART model is compared to other state-of-the-art imputation techniques such as Multilayer Perceptron or Hot-Deck, among others, using a total of eighteen databases exposed to a perturbation procedure based on the random generation of non-monotone missing values pattern. The data sets cover a wide range of fields, types of variables and sizes. A comparison of databases imputed by these models using a set of three classifiers has been conducted. A statistical analysis of these results employing Wilcoxon signed-ranked test has been included. Experiments show that CANFIS-ART approach not only outperforms these state-of-the-art techniques but also demonstrates a higher level of generalization capability, increasing the data quality contained in databases with missing values.