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.

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 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.

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.

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.

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.

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.

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.