Abstract Function Point Analysis (FPA) is a method in software engineering that focuses on identifying the functions provided by a software system to users, such as data input, processing, output, and database management. These functions are classified according to complexity to quantify the system’s size in functional point units. In this paper, we propose two graph neural networks: a Graph-based Similarity Detection Neural Network (GSDNN) and a Prior-Structural Information Graph Neural Network (PSI-GNN) with a pre-trained layer using transfer learning, to define the best model for functional size prediction and uncover patterns and trends in data. Additionally, the NESMA (Netherlands Software Metrics Users Association) method, from the functional families approach, will be in focus, where the ISBSG (International Software Benchmarking Standards Group) dataset, which provides standardized and relevant data for comparing software performance, was used to analyze 1704 industrial software projects. The goal was to identify the graph architecture with the smallest number of experiments to be performed and the lowest Mean Magnitude Relative Error (MMRE) using orthogonal-array tuning optimization via Latin Square extraction. In the proposed approach, the number of experiments is fewer than 8 for each dataset, and a minimum MMRE value of 0.97% was obtained using PSI-GNN. Additionally, the impact of five input features on the change in MMRE value was analyzed with the top-performing model, employing the SHAP (SHapley Additive exPlanations) feature importance method, visualized through GraphExplainer. The frequency of user-initiated transactions, quantified technically, emerged as the most significant determinant within the NESMA framework.

The growing demand for eXplainable AI (XAI) has renewed interest in fuzzy cognitive maps (FCMs) due to their interpretability, causal transparency, and hybrid intelligence capabilities. However, current FCM explanation methods either overlook their dynamic behavior or limit themselves to feature attribution. Counterfactual explanations, which describe the minimal input changes required to alter outcomes, address this gap but remain largely unexplored in these models. The only existing FCM-specific approach relies on fuzzy discretization and predefined rules, producing causally invalid and overly conservative explanations. On the other hand, generic model-agnostic methods assume feature independence and suffer from instability, restrictive assumptions, and high computational costs. To overcome these limitations, this article presents the counterfactuals via the backpropagation (CF-BP) algorithm, a first backpropagation-based counterfactual explanation method for quasi-nonlinear FCMs (q-FCMs), which is a generalization of traditional FCMs that resolves convergence issues. CF-BP exploits the similarity between q-FCMs’ recurrent reasoning and neural network forward propagation, using exact analytical gradients to generate precise, causally consistent, and robust counterfactual explanations within the continuous state space of the model. Extensive evaluations, including hyperparameter sensitivity analysis and benchmarking against eight state-of-the-art (SOTA) model-agnostic methods, confirm the superior performance of the proposed method across key counterfactual quality metrics.

Fuzzy Cognitive Maps (FCMs) are knowledge-based recurrent neural networks that involve neural concepts and causal relationships. Despite being successful in several domains, classic FCMs often fall behind black-box models in terms of their approximation capabilities. However, the literature only reports a few studies devoted to understanding their theoretical foundations and the cause of their limited performance. In this paper, we prove that FCMs are not universal approximators and base our proof on recent theoretical findings and theorems related to the dynamic behavior of FCM-based models. Our results hold for activation functions that are bounded and monotonically increasing. These analytical findings and the empirical evidence (from the analysis of covering and proximity measures applied to synthetically generated FCMs) show that there are significant state space regions that are never produced for some problems. Consequently, classic FCM models cannot generally approximate these values, thus hindering their predictive capabilities in machine learning tasks. The same theoretical results that exposed the design weaknesses of FCMs can be used to overcome them. As the second contribution of our paper, we propose two enhanced FCM-based classifiers equipped with a quasi-nonlinear reasoning rule, together with a decision-making layer that uses derived analytical results. To fine-tune the classifiers' learnable parameters, we introduce a backpropagation-like algorithm that balances convergence and accuracy. Numerical simulations using realworld datasets indicate that our enhanced FCM-based classifiers significantly outperform the classical model.

Although deep learning models can solve complex prediction problems, they have been criticized for being ‘black boxes’. This implies that their decisions are difficult, if not impossible, to explain by simply inspecting their internal knowledge structures. Explainable Artificial Intelligence has attempted to open the black-box through model-specific and agnostic post-hoc methods that generate visualizations or derive associations between the problem features and the model predictions. This paper proposes a new method, termed REPROT, that explains the decisions of complex deep learning architectures based on local reducts of an image. A ‘reduct’ is a set of sufficiently descriptive features that can fully characterize the acquired knowledge. The created reducts are used to build a ‘prototype image’ that visually explains the inference obtained by a black-box model for an image. We focus on deep learning architectures whose complexity and internal particularities demand adapting existing model-specific explanation methods, making the explanation process more difficult. Experimental results show that the black-box model can detect an object using the prototype image generated from the reduct. Hence, the explanations will be given by “the minimum set of features sufficient for the neural model to detect an object”. The confidence scores obtained by architectures such as Inception, Yolo, and Mask R-CNN are higher for prototype images built from the reduct than those built from the most important superpixels according to the LIME method. Moreover, the target object is not detected on several occasions through the LIME output, thus supporting the superiority of the proposed explanation method.

Fuzzy Cognitive Mapping (FCM) and the extensive family of models derived from it have firmly established their strong position in the landscape of machine learning algorithms. Specifically designed for pattern classification and multi-output regression, the recently introduced Recurrence-aware Long-term Cognitive Network (r-LTCN) model is one of these FCM-inspired extensions. On the one hand, this recurrent neural network connects all temporal states generated during the reasoning process with the decision-making layer. On the other hand, it uses a quasi-nonlinear reasoning rule devoted to avoiding convergence issues caused by unique fixed points, which typically emerge in other FCM models. In the original paper, the authors employed a combination of unsupervised and supervised learning to compute the r-LTCNs’ learnable parameters. Despite r-LTCNs’ astounding performance for a wide variety of pattern classification problems, the literature reports no attempt to train these recurrent neural systems in a fully supervised manner nor provide insights into their performance in other machine learning settings. This paper brings forward a modified Backpropagation Through Time learning (BPTT) algorithm devoted to training r-LTCN models used for multi-output regressions tasks rather than pattern classification. The proposed BPPT includes a simple yet effective mechanism to deal with the vanishing gradient within the recurrent layer that operates as a closed system while being tailored to the quasi-nonlinear reasoning mechanism. Empirical evaluation of the proposed BPTT algorithm using 20 multi-output regression problems reveals that it produces lower prediction errors compared with other state-of-the-art learning approaches.

This paper presents a Prolog-based reasoning module to generate counterfactual explanations given the predictions computed by a black-box classifier. Our approach comprises four well-defined stages that can be applied to any structured pattern classification problem. Firstly, we pre-process the given dataset by imputing missing values and normalizing the numerical features. Secondly, we transform numerical features into symbolic ones using fuzzy clustering such that extracted fuzzy clusters are mapped to an ordered set of predefined symbols. Thirdly, we encode instances as a Prolog rule using the nominal values, the predefined symbols, the decision classes, and the confidence values. Fourthly, we compute the overall confidence of each Prolog rule using fuzzy-rough set theory to handle the uncertainty caused by transforming numerical quantities into symbols. This step comes with an additional theoretical contribution to a new similarity function to compare the previously defined Prolog rules involving confidence values. Finally, we implement a chatbot as a proxy between humans and the Prolog-based reasoning module to resolve natural language queries and generate counterfactual explanations. During the numerical simulations using synthetic datasets, we study the performance of our system when using different fuzzy operators and similarity functions.

This paper proposes a post-hoc explanation method for computing concept attribution in Fuzzy Cognitive Map (FCM) models used for scenario analysis, based on SHapley Additive exPlanations (SHAP) values. The proposal is inspired by the lack of approaches to exploit the often-claimed intrinsic interpretability of FCM models while considering their dynamic properties. Our method uses the initial activation values of concepts as input features, while the outputs are considered as the hidden states produced by the FCM model during the recurrent reasoning process. Hence, the relevance of neural concepts is computed taking into account the model’s dynamic properties and hidden states, which result from the interaction among the initial conditions, the weight matrix, the activation function, and the selected reasoning rule. The proposed post-hoc method can handle situations where the FCM model might not converge or converge to a unique fixed-point attractor where the final activation values of neural concepts are invariant. The effectiveness of the proposed approach is demonstrated through experiments conducted on real-world case studies.

The paper presents the presumably correct decision sets as a tool to analyze uncertainty in the form of inconsistency in decision systems. As a first step, problem instances are gathered into three regions containing weak members, borderline members, and strong members. This is accomplished by using the membership degrees of instances to their neighborhoods while neglecting their actual labels. As a second step, we derive the presumably correct and incorrect sets by contrasting the decision classes determined by a neighborhood function with the actual decision classes. We extract these sets from either the regions containing strong members or the whole universe, which defines the strict and relaxed versions of our theoretical formalism. These sets allow isolating the instances difficult to handle by machine learning algorithms as they are responsible for inconsistent patterns. The simulations using synthetic and real-world datasets illustrate the advantages of our model compared to rough sets, which is deemed a solid state-of-the-art approach to cope with inconsistency. In particular, it is shown that we can increase the accuracy of selected classifiers up to 36% by weighting the presumably correct and incorrect instances during the training process.

This article presents a comprehensive approach for time-series classification. The proposed model employs a fuzzy cognitive map (FCM) as a classification engine. Preprocessed input data feed the employed FCM. Map responses, after a postprocessing procedure, are used in the calculation of the final classification decision. The time-series data are staged using the moving-window technique to capture the time flow in the training procedure. We use a backward error propagation algorithm to compute the required model hyperparameters. Four model hyperparameters require tuning. Two are crucial for the model construction: 1) FCM size (number of concepts) and 2) window size (for the moving-window technique). Other two are important for training the model: 1) the number of epochs and 2) the learning rate (for training). Two distinguishing aspects of the proposed model are worth noting: 1) the separation of the classification engine from pre- and post-processing and 2) the time flow capture for data from concept space. The proposed classifier joins the key advantage of the FCM model, which is the interpretability of the model, with the superior classification performance attributed to the specially designed pre- and postprocessing stages. This article presents the experiments performed, demonstrating that the proposed model performs well against a wide range of state-of-the-art time-series classification algorithms.

Machine-learning solutions for pattern classification problems are nowadays widely deployed in society and industry. However, the lack of transparency and accountability of most accurate models often hinders their safe use. Thus, there is a clear need for developing explainable artificial intelligence mechanisms. There exist model-agnostic methods that summarize feature contributions, but their interpretability is limited to predictions made by black-box models. An open challenge is to develop models that have intrinsic interpretability and produce their own explanations, even for classes of models that are traditionally considered black boxes like (recurrent) neural networks. In this article, we propose a long-term cognitive network (LTCN) for interpretable pattern classification of structured data. Our method brings its own mechanism for providing explanations by quantifying the relevance of each feature in the decision process. For supporting the interpretability without affecting the performance, the model incorporates more flexibility through a quasi-nonlinear reasoning rule that allows controlling nonlinearity. Besides, we propose a recurrence-aware decision model that evades the issues posed by the unique fixed point while introducing a deterministic learning algorithm to compute the tunable parameters. The simulations show that our interpretable model obtains competitive results when compared to state-of-the-art white and black-box models.

Neural networks are considered a black-box model as their strength in modeling complex interactions makes its operation almost impossible to explain. Still, neural networks remain very interesting tools as they have shown promising performance in various classification tasks. Layer-wise relevance propagation is a technique that, based on a propagation approach, is able to explain the predictions obtained by a neural network. In this work, we propose four adaptations of this technique to operate on multi-label neural networks. The proposed methods provide new ways of distributing the relevance between the output layer and the preceding ones. The efficacy of these adaptations is demonstrated after an experimental study. The study is carried out based on existing evaluation criteria in the literature that measure the explanation's quality. These methods are applied to a case study in which a neural network is used to detect secondary coinfections in patients infected with SARS-CoV-2. Overall, the proposed methods provide a post-hoc interpretability stage of the results.

Forecasting windmill time series is often the basis of other processes such as anomaly detection, health monitoring, or maintenance scheduling. The amount of data generated by windmill farms makes online learning the most viable strategy to follow. Such settings require retraining the model each time a new batch of data is available. However, updating the model with new information is often very expensive when using traditional Recurrent Neural Networks (RNNs). In this paper, we use Long Short-term Cognitive Networks (LSTCNs) to forecast windmill time series in online settings. These recently introduced neural systems consist of chained Short-term Cognitive Network blocks, each processing a temporal data chunk. The learning algorithm of these blocks is based on a very fast, deterministic learning rule that makes LSTCNs suitable for online learning tasks. The numerical simulations using a case study involving four windmills showed that our approach reported the lowest forecasting errors with respect to a simple RNN, a Long Short-term Memory, a Gated Recurrent Unit, and a Hidden Markov Model. What is perhaps more important is that the LSTCN approach is significantly faster than these state-of-the-art models.

This paper presents a Fuzzy Cognitive Map model to quantify implicit bias in structured datasets where features can be numeric or discrete. In our proposal, problem features are mapped to neural concepts that are initially activated by experts when running what-if simulations, whereas weights connecting the neural concepts represent absolute correlation/association patterns between features. In addition, we introduce a new reasoning mechanism equipped with a normalization-like transfer function that prevents neurons from saturating. Another advantage of this new reasoning mechanism is that it can easily be controlled by regulating nonlinearity when updating neurons’ activation values in each iteration. Finally, we study the convergence of our model and derive analytical conditions concerning the existence and unicity of fixed-point attractors.

Abstract In this paper, we present a recurrent neural system named long short-term cognitive networks (LSTCNs) as a generalization of the short-term cognitive network (STCN) model. Such a generalization is motivated by the difficulty of forecasting very long time series efficiently. The LSTCN model can be defined as a collection of STCN blocks, each processing a specific time patch of the (multivariate) time series being modeled. In this neural ensemble, each block passes information to the subsequent one in the form of weight matrices representing the prior knowledge. As a second contribution, we propose a deterministic learning algorithm to compute the learnable weights while preserving the prior knowledge resulting from previous learning processes. As a third contribution, we introduce a feature influence score as a proxy to explain the forecasting process in multivariate time series. The simulations using three case studies show that our neural system reports small forecasting errors while being significantly faster than state-of-the-art recurrent models.

The need to measure bias encoded in tabular data that are used to solve pattern recognition problems is widely recognized by academia, legislators and enterprises alike. In previous work, we proposed a bias quantification measure, called fuzzy-rough uncertainty, which relies on the fuzzy-rough set theory. The intuition dictates that protected features should not change the fuzzy-rough boundary regions of a decision class significantly. The extent to which this happens is a proxy for bias expressed as uncertainty in a decision-making context. Our measure’s main advantage is that it does not depend on any machine learning prediction model but a distance function. In this paper, we extend our study by exploring the existence of bias encoded implicitly in non-protected features as defined by the correlation between protected and unprotected attributes. This analysis leads to four scenarios that domain experts should evaluate before deciding how to tackle bias. In addition, we conduct a sensitivity analysis to determine the fuzzy operators and distance function that best capture change in the boundary regions.

Fuzzy-rough cognitive networks (FRCNs) are recurrent neural networks (RNNs) intended for structured classification purposes in which the problem is described by an explicit set of features. The advantage of this granular neural system relies on its transparency and simplicity while being competitive to state-of-the-art classifiers. Despite their relative empirical success in terms of prediction rates, there are limited studies on FRCNs' dynamic properties and how their building blocks contribute to the algorithm's performance. In this article, we theoretically study these issues and conclude that boundary and negative neurons always converge to a unique fixed-point attractor. Moreover, we demonstrate that negative neurons have no impact on the algorithm's performance and that the ranking of positive neurons is invariant. Moved by our theoretical findings, we propose two simpler fuzzy-rough classifiers that overcome the detected issues and maintain the competitive prediction rates of this classifier. Toward the end, we present a case study concerned with image classification, in which a convolutional neural network is coupled with one of the simpler models derived from the theoretical analysis of the FRCN model. The numerical simulations suggest that once the features have been extracted, our granular neural system performs as well as other RNNs.

This paper elaborates on the modeling and simulation of complex systems involving uncertainty. More explicitly, we are interested in situations in which experts hesitate about the exact values of variables when designing the model. Such situations can be modeled using Interval-valued Long-term Cognitive Networks (IVLTCNs). In this model, the activation values and the weights between neural concepts are expressed as interval grey numbers. Unlike other grey cognitive networks, our model neither imposes restrictions on the weights nor performs a whitenization process. The second contribution of this paper is a nonsynaptic grey backpropagation algorithm, which allows adjusting the learnable parameters of IVLTCNs under uncertainty conditions. Moreover, this learning algorithm does not alter the linear knowledge representations provided by domain experts during the modeling phase.

This paper presents an interpretable neural system—termed Evolving Long-term Cognitive Network—for pattern classification. The proposed model was inspired by Fuzzy Cognitive Maps, which are interpretable recurrent neural networks for modeling and simulation. The network architecture is comprised of two neural blocks: a recurrent input layer and an output layer. The input layer is a Long-term Cognitive Network that gets unfolded in the same way as other recurrent neural networks, thus producing a sort of abstract hidden layers. In our model, we can attach meaningful linguistic labels to each neuron since the input neurons correspond to features in a given classification problem and the output neurons correspond to class labels. Moreover, we propose a variant of the backpropagation learning algorithm to compute the required parameters. This algorithm includes two new regularization components that are aimed at obtaining more interpretable knowledge representations. The numerical simulations using 58 datasets show that our model achieves higher prediction rates when compared with traditional white boxes while remaining competitive with the black boxes. Finally, we elaborate on the interpretability of our neural system using a proof of concept.

This paper presents a neural system to deal with multi-label classification problems that might involve sparse features. The architecture of this model involves three sequential blocks with well-defined functions. The first block consists of a multilayered feed-forward structure that extracts hidden features, thus reducing the problem dimensionality. This block is useful when dealing with sparse problems. The second block consists of a Long-term Cognitive Network-based model that operates on features extracted by the first block. The activation rule of this recurrent neural network is modified to prevent the vanishing of the input signal during the recurrent inference process. The modified activation rule combines the neurons' state in the previous abstract layer (iteration) with the initial state. Moreover, we add a bias component to shift the transfer functions as needed to obtain good approximations. Finally, the third block consists of an output layer that adapts the second block's outputs to the label space. We propose a backpropagation learning algorithm that uses a squared hinge loss function to maximize the margins between labels to train this network. The results show that our model outperforms the state-of-the-art algorithms in most datasets.

Modeling a real-world system by means of a neural model involves numerous challenges that range from formulating transparent knowledge representations to obtaining reliable simulation errors. However, that knowledge is often difficult to formalize in a precise way using crisp numbers. In this paper, we present the long-term grey cognitive networks which expands the recently proposed long-term cognitive networks (LTCNs) with grey numbers. One advantage of our neural system is that it allows embedding knowledge into the network using weights and constricted neurons. In addition, we propose two procedures to construct the network in situations where only historical data are available, and a regularization method that is coupled with a nonsynaptic backpropagation algorithm. The results have shown that our proposal outperforms the LTCN model and other state-of-the-art methods in terms of accuracy.

Fuzzy cognitive maps (FCMs) are recurrent neural networks comprised of well-defined concepts and causal relations. While the literature about real-world FCM applications is prolific, the studies devoted to understanding the foundations behind these neural networks are rather scant. In this article, we introduce several definitions and theorems that unveil the dynamic behavior of FCM-based models equipped with transfer F-functions. These analytical expressions allow estimating bounds for the activation value of each neuron and analyzing the covering and proximity of feasible activation spaces. The main theoretical findings suggest that the state space of any FCM model equipped with transfer F-functions shrinks infinitely with no guarantee for the FCM to converge to a fixed point but to its limit state space. This result in conjunction with the covering and proximity values of FCM-based models helps understand their poor performance when solving complex simulation problems.

Fuzzy-Rough Cognitive Networks (FRCNs) are neural networks that use rough information granules with soft boundaries to perform the classification process. Unlike other neural systems, FRCNs are lazy learners in the sense that we can build the whole model when classifying a new instance. This is possible because the weight matrix connecting the neurons is prescriptively programmed. Similar to other lazy learners, the processing time of FRCNs notably increases with the number of instances in the training set, while their performance deteriorates in noisy environments. Aiming at coping with these issues, this paper presents a new FRCN-based algorithm termed Fast k-Fuzzy-Rough Cognitive Network. This variant employs a multi-thread approach for building the information granules as computed by k-fuzzy-rough sets. Numerical simulations on 35 classification datasets show a notable reduction on FRCNs' processing time, while also delivering competitive results when compared to other lazy learners in noisy environments.

In this paper, we build a recommender system based on Long-term Cognitive Networks (LTCNs), which are a type of recurrent neural network that allows reasoning with prior knowledge structures. Given that our approach is context-free and that we did not involve human experts in our study, the prior knowledge is replaced with Pearson’s correlation coefficients. The proposed architecture expands the LTCN model by adding Gaussian kernel neurons that compute estimates for the missing ratings. These neurons feed the recurrent structure that corrects the estimates and makes the predictions. Moreover, we present an extension of the non-synaptic backpropagation algorithm to compute the proper non-linearity of each neuron together with its activation boundaries. Numerical results using several case studies have shown that our proposal outperforms most state-of-the-art methods. Towards the end, we explain how can we inject expert knowledge to the proposed neural system.

We introduce a neural cognitive mapping technique named long-term cognitive network (LTCN) that is able to memorize long-term dependencies between a sequence of input and output vectors, especially in those scenarios that require predicting the values of multiple dependent variables at the same time. The proposed technique is an extension of a recently proposed method named short-term cognitive network that aims at preserving the expert knowledge encoded in the weight matrix while optimizing the nonlinear mappings provided by the transfer function of each neuron. A nonsynaptic, backpropagation-based learning algorithm powered by stochastic gradient descent is put forward to iteratively optimize four parameters of the generalized sigmoid transfer function associated with each neuron. Numerical simulations over 35 multivariate regression and pattern completion data sets confirm that the proposed LTCN algorithm attains statistically significant performance differences with respect to other well-known state-of-the-art methods.

Pattern classification is a popular research field within the Machine Learning discipline. Black-box models have proven to be potent classifiers in this particular field. However, their inability to provide a transparent decision mechanism is often regarded as an undesirable feature. Fuzzy-Rough Cognitive Networks are granular classifiers that have proven competitive and effective in such tasks. In this paper, we examine the contribution of the FRCN's main building blocks, being the causal weight matrix and the activation values of the neurons, to the model's average performance. Noise injection is employed to this end. Our findings suggest that optimising the weight matrix might not be as beneficial to the model's performance as suggested in previous research. Furthermore, we found that a powerful activation of the neurons included in the model topology is crucial to performance, as expected. Further research should as such focus on finding more powerful ways to activate these neurons, rather than focus on optimising the causal weight matrix.

In contrast with the extense variety of machine learning algorithms, to fully automate the reasoning process, only a few can take advantage of the expert knowledge. Fuzzy Cognitive Maps (FCMs) are neural networks that can naturally integrate this kind of knowledge in the inference process. Nevertheless, FCMs have serious drawbacks difficult to overcome from the absence of an intrinsically learning algorithm or limited prediction horizon of the activation space of the neurons. Recently, some variants of the FCMs like Short-Term Cognitive Networks (STCN) and Long Term Cognitive Networks (LTCN) have been proposed to solve these problems. In this paper, we propose a new neural network model as a variant of LTCNs called Long-Term Cognitive Networks with Inputs (LTCNIs). A new kind of input neuron which is not present in the traditional FCMs approach or the derived algorithms STCNs and LTCNs is introduced, in order to model inputs like energy or mass in physical systems. The performance of the method is discussed through the modeling of a passive circuit problem. As a second contribution, a new flexible reasoning strategy, which preserves the expert knowledge through synaptic learning is presented. A synaptic learning based on a gradient descent method is implemented limited by a set of restrictions that preserves the model semantics.

Recently, a learning procedure to improve the overall convergence of sigmoid fuzzy cognitive maps used in pattern classification was proposed. The algorithm estimates the slope of each sigmoid neuron while preserving the causal weights. This paper proposes a more realistic error function for this algorithm, which is based on 1) the dissimilarity between two consecutive responses, and 2) the dissimilarity between the current output and the expected one. As a second contribution, we introduce sufficient conditions to arrive at stability features. These conditions allow assessing the accuracy-convergence tradeoff attached to the proposed learning procedure.

After 30 years of research, challenges and solutions, Fuzzy Cognitive Maps (FCMs) have become a suitable knowledgebased methodology for modeling and simulation. This technique is especially attractive when modeling systems that are characterized by ambiguity, complexity and non-trivial causality. FCMs are well-known due to the transparency achieved during modeling tasks. The literature reports successful studies related to the modeling of complex systems using FCMs. However, the situation is not the same when it comes to software implementations where domain experts can design FCM-based systems, run simulations or perform more advanced experiments. The existing implementations are not proficient in providing many options to adjust essential parameters during the modeling steps. The gap between the theoretical advances and the development of accurate, transparent and sound FCM-based systems advocates for the creation of more complete and flexible software products. Therefore, the goal of this paper is to introduce FCM Expert, a software tool for fuzzy cognitive modeling oriented to scenario analysis and pattern classification. The main features of FCM Expert rely on Machine Learning algorithms to compute the parameters defining the model, optimize the network topology and improve the system convergence without losing information. On the other hand, FCM Expert allows performing WHAT-IF simulations and studying the system behavior through a friendly, intuitive and easy-to-use graphical user interface.

Many papers describe the use of Fuzzy Cognitive Maps as a modeling/representation technique for real-life scenarios’ simulation or prediction. However, not many real software implementations are described neither found. In this proposal the authors describe a modeling and experimentation framework where realistic problems can be recreated using Fuzzy Cognitive Maps as a knowledge representation form. Design elements, and descriptions of the algorithms that have been incorporated into the software, and hybridized with Fuzzy Cognitive Maps, are presented in this paper. Case studies were conducted and are illustrated with the intention of demonstrating the success and practical value of the general approach together with the implementation tool.

In a multilabel classification problem, each object gets associated with multiple target labels. Graded multilabel classification (GMLC) problems go a step further in that they provide a degree of association between an object and each possible label. The goal of a GMLC model is to learn this mapping while minimizing a certain loss function. In this paper, we tackle GMLC problems from a Granular Computing perspective for the first time. The proposed schemes, termed as partitive granular cognitive maps (PGCMs), lean on Fuzzy Cognitive Maps (FCMs) whose input concepts represent cluster prototypes elicited via Fuzzy C-Means whereas the output concepts denote the set of existing labels. We consider three different linkages between the FCM's input and output concepts and learn the causal connections (weight matrix) through a Particle Swarm Optimizer (PSO). During the exploitation phase, the membership grades of a test object to each fuzzy cluster prototype in the PGCM are taken as the initial activation values of the recurrent network. Empirical results on 16 synthetically generated datasets show that the PGCM architecture is capable of accurately solving GMLC instances.

During the last decade Fuzzy Cognitive Maps (FCM) have become a useful tool for solving unstructured problems. In a few words they could be defined as Recurrent Neural Networks for simulating complex systems, where neurons denote concepts, objects or entities of the investigated system. Normally FCM are entirely designed using the best knowledge of a group of experts in a given domain, so frequently learning algorithms for tuning the model parameters are required. Despite the theoretical advances in such fields, the lack of a suitable computational framework for handling FCM-based systems is still an open problem. This paper introduces a novel tool for designing and simulating FCM which gathers several learning algorithms for adjusting the introduced parameters. More specifically, the framework includes supervised and unsupervised learning algorithms for computing the causal weights, algorithms for optimizing the network topology in large FCM (without losing significant information) and also methods for improving the global convergence on continuous FCM. It should be stated that these algorithms are oriented to prediction tasks, but they could be easily extended to other fields.

Fuzzy Cognitive Maps (FCM) may be defined as Recurrent Neural Networks that allow causal reasoning. According to the transformation function used for updating the activation value of concepts they can be characterized as discrete or continuous. It is remarkable that FCM having discrete neurons never exhibit chaotic states, but this premise cannot be guaranteed for FCM having continuous concepts. On the other hand, complex Sigmoid FCM resulting from experts or learning algorithms often show chaotic or cyclic patterns, therefore leading to confusing interpretation of the investigated system. The first contribution of this paper is focused on explaining why most studies on FCM stability are not applicable to FCM used on classification or decision-making tasks. Next we describe a non-direct learning methodology based on Swarm Intelligence for improving the system stability once the causal weight estimation is done. The objective here is to find a specific threshold function for each map neuron simulating an external stimulus, instead of using the same transformation function for all concepts. At the end, we can compute more stable maps, so better consistency in hidden patterns is achieved.

During the last couple of decades, evolutionary and swarm intelligence algorithms have significantly advanced the state of the art for both discrete and numerical optimization. Without niching strategies, they usually converge to a single optimum, even in multimodal search spaces where numerous global or local solutions exist. In the literature, several niching approaches have been proposed for simultaneously computing multiple optima, though most of them require some user-specified parameters that should be calculated a priori, i.e. additional knowledge about the problem domain is required. Recently, it was demonstrated that particle swarm optimization (PSO) using a ring topology for neighborhood definition can give rise to robust and parameterless niching methods. Nevertheless, their performance dramatically worsens when the dimensionality of the solution space hikes, thus increasing the number of local optima. This paper aims at enhancing the performance of these types of PSO-based algorithms by introducing two procedures: (1) a differential operator for improving the search ability and (2) a heuristic clearing operator for controlling the swarm diversity. Such operators are probabilistically activated through a novel self-adaptive learning strategy. Empirical results confirm the superiority of our proposed scheme with respect to six other competitive niching techniques.