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.

Early time series classification is a variant of the time series classification task, in which a label must be assigned to the incoming time series as quickly as possible without necessarily screening through the whole sequence. It needs to be realized on the algorithmic level by fusing a decision-making method that detects the right moment to stop and a classifier that assigns a class label. The contribution addressed in this paper is twofold. Firstly, we present a new method for finding the best moment to perform an action (terminate/continue). Secondly, we propose a new learning scheme using classifier calibration to estimate classification accuracy. The new approach, called CALIMERA, is formalized as a cost minimization problem. Using two benchmark methodologies for early time series classification, we have shown that the proposed model achieves better results than the current state-of-the-art. Two most serious competitors of CALIMERA are ECONOMY and TEASER. The empirical comparison showed that the new method achieved a higher accuracy than TEASER for 35 out of 45 datasets and it outperformed ECONOMY in 20 out of 34 datasets.

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.

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.

A fuzzy cognitive map (FCM) is a graph-based knowledge representation model wherein the connections of the nodes (edges) represent casual relationships between the knowledge items associated with the nodes. This model has been applied to solve various modeling tasks including forecasting time series. In the original FCM-based forecasting model, causal relationships among concepts of the FCM remain unchanged. However, causal relationships may change in time. Therefore, we propose a new learning method for training an FCM resulting in an adaptive FCM which consists of several sub-FCMs. It can select different sub-FCMs at different moments. In an active processing scenario, in which we deal with a large-scale time series with new data being continuously generated, a forecasting model built on the old data should be updated when the new data arrive. Furthermore, retraining an FCM from scratch entails increasing computing overhead that will become a serious obstacle in many practical scenarios. To overcome the above-mentioned shortcomings, this study offers an original design setting in which the FCM is updated by knowledge-guidance learning mechanism for the first time. Compared with the existing classical forecasting models, the proposed model shows higher accuracy and efficiency. Its increased performance is demonstrated through a series of reported experimental studies.

Classification of sequences of temporal intervals is a part of time series analysis which concerns series of events. We propose a new method of transforming the problem to a task of multivariate series classification. We use one of the state-of-the-art algorithms from the latter domain on the new representation to obtain significantly better accuracy than the state-of-the-art methods from the former field. We discuss limitations of this workflow and address them by developing a novel method for classification termed COSTI (short for Classification of Sequences of Temporal Intervals) operating directly on sequences of temporal intervals. The proposed method remains at a high level of accuracy and obtains better performance while avoiding shortcomings connected to operating on transformed data. We propose a generalized version of the problem of classification of temporal intervals, where each event is supplemented with information about its intensity. We also provide two new data sets where this information is of substantial value.

The article analyses the similarity of business cycles among European Union countries. For this purpose, we design a new method – a fuzzy concept-based model, operating on time-series windows. We showcase the capability of the tool to detect similarities in the phase and the amplitude of the cycle, and we compare it with the commonly used measure of synchronization, i.e., with the correlation of business cycles. On the methodological level, the new method uses fuzzy clustering and various similarity measures. An application of the proposed method lets us revisit the co-movements of the cycles in the EU.

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.

One drawback of using the existing one-step forecasting models for long-term time series prediction is the cumulative errors caused by iterations. In order to overcome this shortcoming, this article proposes a trend-fuzzy-granulation-based adaptive fuzzy cognitive map (FCM) for long-term time series forecasting. Different from the original FCM-based forecasting models, a class of trend fuzzy information granules is built to represent the trend, fluctuation range, and trend persistence of various segments of time series, which are more instrumental and comprehensive than simple magnitude information. Thus, the proposed forecasting model is a granular model according to the form of its inputs and outputs. In an original FCM-based forecasting model, the causal relationships among concepts remain unchanged throughout the training of the whole dataset, however, in reality, the causal relationships may change with the state of concepts. Therefore, it is unreasonable to use the invariable causal relationships which often result in poor predictions. In view of this, we construct an adaptive FCM where different causal relationships are built to forecast concepts of different states. This is the first time to forecast trend fuzzy information granules using an adaptive FCM. Compared with the existing classical forecasting models, the proposed forecasting model achieves superior performance which is verified through a series of experimental studies.

Computing technologies have already established their place in various areas of public transport control in smart cities. While the analysis of signals coming from various sensors is executed at a very high level of sophistication, information expressed by humans in natural language is still not being used in a way that takes advantage of its full potential. Existing research on text mining in public transport monitoring is focused mainly on event detection. In this paper, we present a novel approach to vehicle delay prediction based on text data. The proposed method fuses information coming from standard sources (sensors) with text messages, to construct a regression model, that predicts delays for previously unseen messages describing road conditions. The method has been implemented based on an existing public transport monitoring system in Warsaw, Poland. In the paper, we discuss it briefly. Delay prediction based on information expressed in natural language will not replace standard methods for delay prediction that involve the use of vehicle sensors. However, it offers an attractive alternative to mine for knowledge from sources such as social media.

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.

In this study, we discuss the problem of interpretability of scale versus polarity in multicriteria decision-making problem. Decision making requires aggregation of premises of different characters and types. The influence of premises on a decision to be made may have a different characteristics as well. Some premises may have a positive character, i.e. they vote/agitate towards making a decision, others may fight against a decision. On the other hand, premises can be tempered by priorities, which may affect their character. Therefore, there is a need to discuss different configurations of premises and their priorities. This is the first aspect of our discussion on multicriteria decision making. The second one under discussion is the interpretability of all aspects mentioned so far. In this case, we discuss the representation problems of both premises and priorities. They are usually exhibited as numbers taken from some scale as, for instance, the unipolar unit interval [0,1] or the bipolar unit interval [-1,1]. On the other hand, there is a question raised about a character of premises/priorities, that is, whether they vote pro or contra a decision to be taken and what relations between scales and polarities are.

In the plethora of conceptual and algorithmic developments supporting system modeling, we encounter growing challenges associated with the complexity of systems, diversity of available data and a variety of requests imposed on the quality of the models. The accuracy of models is important. At the same time, the interpretability and explainability of models are equally important and of high practical relevance. We advocate that the level of abstraction at which models are constructed (and which could be flexibly adjusted), is conveniently realized through Granular Computing. Granular Computing is concerned with the development and processing information granules - formal entities that facilitate a way of organizing and representing knowledge about the available data and relationships existing there. This study identifies the principles of Granular Computing, shows how information granules are constructed and subsequently used in the realization of models.

This paper presents a time-series classification method based on fuzzy cognitive maps. We advocate that fuzzy cognitive maps provide a sound representation of time series, and we can construct a classification mechanism based on them. The classifier has to distinguish maps constructed for time series belonging to different classes. The proposed classification procedure evaluates similarity of fuzzy cognitive maps, and it is done by comparing weight matrices based on the same set of concepts. A weight matrix describes relationships between concepts in a map. Concepts represent the underlying data, because they are extracted via a data-driven clustering procedure. Each data point of a time series is related to each concept, and we evaluate the strength of relationships with a membership function. This paper investigates performance of the proposed approach on a suite of real-world datasets. We compare classification accuracy of our method with 37 state-of-the-art time-series classification methods. Experiments show that the proposed method is performing well. In many cases, it is better than its competitors.

This study elaborates on a comprehensive design methodology of fuzzy cognitive maps (FCMs). Here, the maps are regarded as a modeling vehicle of time series. It is apparent that whereas time series are predominantly numeric, FCMs are abstract constructs operating at the level of abstract entities referred to as concepts and represented by the individual nodes of the map. We introduce a mechanism to represent a numeric time series in terms of information granules constructed in the space of amplitude and change of amplitude of the time series, which, in turn, gives rise to a collection of concepts forming the corresponding nodes of the FCMs. Each information granule is mapped onto a node (concept) of the map. We identify two fundamental design phases of FCMs, namely 1) formation of information granules mapping numeric data (time series) into activation levels of information granules (viz., the nodes of the map), and 2) optimization of information granules at the parametric level, viz., learning (estimating) the weights between the nodes of the map. The learning is typically realized in a supervised mode on a basis of some experimental data. A construction of information granules is realized with the aid of fuzzy clustering, namely fuzzy C-means. The optimization is realized with the use of particle swarm optimization. The proposed approach is illustrated in detail by a series of experiments using a collection of publicly available data.

The objective of this paper is to present methodology for similarity evaluation of structured spaces of sets inspired by human cognitive processes. In contrast to classical similarity relations, which can operate only within the same space, our method can be applied to separate spaces. Proposed formulas are designed to compare two families of sets belonging to separate spaces. Unlike in set-theoretic approach to similarity present in literature, fundamental knowledge, which we use is sets and subsets cardinalities and division of spaces into subsets combined with appropriate minimum and maximum as aggregation operators. Theoretical discussion is supported with a case study, where we apply designed formulas to calculate similarities of four cities. Introduced method has been constructed after an analysis how humans perform similarity evaluation for hard to compare concepts and phenomena.

Fuzzy Cognitive Maps are recognized knowledge modeling tool. FCMs are visualized with directed graphs. Nodes represent information, edges represent relations within information. The core element of each Fuzzy Cognitive Map is weights matrix, which contains evaluations of connections between map's nodes. Typically, weights matrix is constructed by experts. Fuzzy Cognitive Map can be also reconstructed in an unmanned mode. In this article authors present their own, new approach to time series modeling with Fuzzy Cognitive Maps. Developed methodology joins Fuzzy Cognitive Map reconstruction procedure with moving window approach to time series prediction. Authors train Fuzzy Cognitive Maps to model and forecast time series. The size of the map corresponds to the moving window size and it informs about the length of historical data, which produces time series model. Developed procedure is illustrated with a series of experiments on three real-life time series. Obtained results are compared with other approaches to time series modeling. The most important contribution of this paper is description of the methodology for time series modeling with Fuzzy Cognitive Maps and moving windows.

Cognitive Maps are abstract knowledge representation framework, suitable to model complex systems. Cognitive Maps are visualized with directed graphs, where nodes represent phenomena and edges represent relationships. Granular Cognitive Maps are augmented Cognitive Maps, which use knowledge granules as information representation model. Conceptually, GCMs originated as an extension of Fuzzy Cognitive Maps. The contribution presented in this paper is a methodology for Granular Cognitive Map reconstruction. The goal of the procedure is to construct a weights matrix - and thereby the GCM, which outputs best describe the phenomena of interest. The article addresses the conflict between generality and specificity of various Granular Cognitive Maps. Balance between generality and specificity is the most important architectural aspect of a model built with knowledge granules. A series of experiments illustrates, how various optimization techniques allow improvement in map's quality without a loss in map's precision.

The paper is focused on fuzzy cognitive maps - abstract soft computing models, which can be applied to model complex systems with uncertainty. The authors present two distinct methodologies for fuzzy cognitive map reconstruction. Both theoretical and practical issues involved in the process of a map reconstruction are discussed. Among researched and described aspects are: map sizes, data dimensionality, distortions, optimization procedure, etc. Theoretical results are supported by a series of experiments, that allow to evaluate the quality of the developed approach. The authors compare both procedures characteristics and discuss practical issues, that are entailed in the developed methodology. The goal of this study is to investigate theoretical and practical problems, that are relevant in the fuzzy cognitive map reconstruction process. Proposed two methodologies for FCM reconstruction are based on gradient learning. A series of experiments allows to illustrate important characteristics of the fuzzy cognitive map reconstruction procedure.

This paper is focused on modeling consumer's behavior related to decision making with some models of imperfect information. Our approach is based on the Lewin's field theory with our own expansion stemming from it. We use a few imperfect information models, namely: fuzzy sets, triangular norms and balanced norms. The former two models are applied in aggregating positive premisses to indicate a decision. The latter one involves aggregation of both negative and positive premisses in the decision making process. The goal of the research is to investigate how, using imprecise information representation models, neoclassical understanding of the consumer's choice theory can be reformed. We believe that applying named tools might be beneficial for better description of human behavior on the market.