Multi-output time series forecasting with randomized multivariate Fuzzy Cognitive Maps

Fuzzy Cognitive Maps (FCMs) have become a relevant technique for modeling and forecasting time series due to their advantages in dealing with uncertainty and simulating the dynamics of complex systems. Although numerous univariate and multivariate FCM-based forecasting models have been presented in the literature, one of the still open questions is how to enable FCMs to forecast multivariate time series for multiple-input, multiple-output (MIMO) systems with an efficient learning mechanism. from a computational point of view. This paper suggests a randomized MIMO FCM-based forecasting approach called MO-RHFCM to predict low-dimensional multivariate time series. More specifically, MO-RHFCM is a hybrid model merging the concepts of multivariate fuzzy time series, high order FCM (HFCM), and Echo State Networks (ESN). The structure of MO-RHFCM consists of three layers: input layer, reservoir (internal) layer, and output layer. Only the output layer is trainable using the Least Squares minimization algorithm; hence training the proposed MO-RHFCM method is fast and simple. The weights inside each sub-reservoir are selected randomly and remain fixed during the training process. The obtained results indicate the efficacy and validity of the proposed MO-RHFCM technique compared with some machine learning and deep learning baseline models.