Abstract— Fuzzy Cognitive Maps (FCMs) are a convenient tool for modeling of dynamic systems by means of concepts connected by cause-effect relationships. The FCM models can be developed either manually (by the experts) or using an automated learning method (from data). Some of the methods from the latter group, including recently proposed Nonlinear Hebbian Learning (NHL) algorithm, use Hebbian law and a set of conditions imposed on output concepts. In this paper, we propose a novel approach named data-driven NHL (DD-NHL) that extends NHL method by using historical data of the input concepts to provide improved quality of the learned FCMs. DD-NHL is tested on both synthetic and real-life data, and the experiments show that if historical data are available, then the proposed method produces better FCM models when compared with those formed by the generic NHL method. I. INTRODUCTION uzzy Cognitive Maps (FCMs), introduced by Kosko [1] in 1986, are a convienient conceptual and computing machinery for modeling and simulation of dynamic systems. They represent knowledge in a symbolic manner and relate states, variables, events, outputs and inputs using a cause and effect approach. When compared to other techniques, FCMs exhibit a number of highly appealing properties. In particular, knowledge representation becomes easy and intuitive. One can easily model feedback relationships and capture hidden dependencies between the concepts. [2]. Applications of FCMs are in various areas including engineering [3], [4], medicine [5], political science [6], economics [7], earth and environmental sciences [8], etc. There are two main groups of approaches to develop Fuzzy Cognitive Maps: (1) manual methods carried out by expert(s) who have knowledge of both FCMs and the domain of application, and (2) automated or semi-automated methods, which use learning algorithms to establish models from historical data (simulations of concept values). The methods from the latter group exhibit numerous advantages over the manual methods, such as independence of the domain of application which may lead to the development of unbiased models [9]. One of the paradigms used to automate development of FCMs stems from the Hebbian law. The first attempt to learn FCMs using this approach was This work was supported in part by the Alberta Ingenuity, and by the Natural Sciences & Engineering Research Council of Canada (NSERC) W. Stach, L. Kurgan, and W. Pedrycz are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Canada (e-mail: {wstach, lkurgan, pedrycz}@ece.ualberta.ca) proposed by Dickerson and Kosko in 1994, and was referred to as Differential Hebbian Learning (DHL) [10]. This method was further extended into Nonlinear Hebbian Learning (NHL) [11]. The NHL algorithm learns FCMs from initial expert-derived FCM model and a set of conditions imposed on output concepts. The algorithm does not use historical data and requires an expert to develop an initial map. To this end, we propose a novel extension to NHL method, called data-driven NHL (DD-NHL) which uses historical data to improve the quality of learned FCM models when compared with generic NHL method. It is also worth emphasizing that the proposed method does not rely on some initial, expert-derived FCM model. The study is organized as follows. Section II presents background information on Fuzzy Cognitive Maps and motivation of this research. In Section III our algorithm, DD-NHL, is introduced. Section IV and Section V describe experiments that have been performed and elaborate on the results, whereas Section VI summarizes this paper. II. BACKGROUND AND MOTIVATION A. Fuzzy Cognitive Maps FCMs define a given dynamic system by means of concepts associated by mutual cause-effect relations. Each relation is described by a number from interval [-1, 1], which corresponds to its strength. Positive values reflect promoting effect, whereas negative values correspond to inhibiting effect. The value of –1 represents full negative, +1 full positive and 0 denotes neutral relation. Other values correspond to different intermediate levels of causal effect. FCMs are conveniently expressed in the form of graphs. In a graph, the nodes correspond to states, and arrows associated with numbers correspond to relations. The graph representation is equivalent to a square matrix, called connection matrix, which stores all weight values of edges between corresponding concepts. Figure 1 shows an example a process control problem [11], which is modeled by the FCM shown in Figure 2. Data-Driven Nonlinear Hebbian Learning Method for Fuzzy Cognitive Maps Wojciech Stach, Lukasz Kurgan, and Witold Pedrycz F