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The prediction of time series in multi-steps is of significance in reality. However, considering the uncertainty and high noise existing in time series, the long-term forecasting is still an open problem. By means of granular computing, in this article, a novel spatial–temporal fuzzy information granule (STFIG) model is proposed to achieve the multi-step forecasting of time series. From the perspective of time dimension, by using unequal division method, time series is converted into generalized time-varying fuzzy information granules, where the trend information and dispersion degree of sequence data can be quantitatively described. Moreover, in terms of spatial dimension, the fluctuation information of time series is also calculated and involved into information granules, which can further enhance the semantic representation of sequential data. In order to improve the ability of dealing with uncertainties and fuzziness in time series, the interval type-2 fuzzy set is applied in the granules model. By using synthetic data and real-life time series, experiments are carried out to verify the effectiveness of the proposed scheme, where abundant semantic information and better long-term predictive performance can be obtained. [ABSTRACT FROM AUTHOR]

Time series exist widely in either nature or society such that the research on analysis of time series has great significance. However, considering the nonlinearity and uncertainty, the prediction of time series is still an open problem. In this paper, by means of the intuitionistic fuzzy set theory, we proposed a novel time series prediction scheme based on intuitionistic fuzzy cognitive map. In the previous research, intuitionistic fuzzy cognitive map, as a kind of knowledge-based modeling tool, is mainly used in decision-making field, where concept structure and weight matrix are usually obtained from experience of experts. To tackle with the diversity of time series, the proposed algorithm constructs the conceptual structure of cognitive map and weight matrix directly from raw sequential data, which effectively enlarges the application range by reducing human participation. Moreover, in order to appropriately calculate the hesitation degree, which is the key role for the application of intuitionistic fuzzy sets, we propose a real-time adjustable hesitation degree calculation scheme. By using this proposed method, hesitation degree can be adaptively adjusted by combining Femi formula with dynamic membership degree. A number of experiments are implemented to reveal feasibility and effectiveness of the proposed schemes. [ABSTRACT FROM AUTHOR]

In reality, some kinds of time series have the characteristics of high volatility, where fluctuation patterns of sequential data contain rich semantic knowledge and represent the spatial features from two-dimensional perspective. Especially, some non-trivial fluctuations may provide key information for the forecasting of time series. However, how to appropriately represent the fluctuation patterns and achieve the causal reasoning among them remains open. In this work, by learning the causal knowledge of fluctuation patterns, a novel spatial attention fuzzy cognitive map with high-order structure is proposed for the interpretable prediction of time series with high volatility. Firstly, a kind of extended polar fuzzy information granules is utilized to convert time series into granule sequences with interpretable fluctuation features, based on which fuzzy cognitive maps can be constructed by using full data-driven way. Secondly, in order to capture the key fluctuation patterns, the attention mechanism is first introduced into fuzzy cognitive map, where the spatial features of the focused patterns can be taken fully utilized. Thirdly, the high-order structure is involved into the proposed model for the learning of the temporal knowledge existing in the pattern sequences. Finally, real-world financial time series with strong noises and high volatility are empirically utilized to verify the promising performance of the proposed method. • The attention mechanism is first introduced into FCM. • A novel method to construct FCM based on fan-shape FIGs. • A kind of improved fan-shape FIGs in polar coordinates is proposed. • Two-stage training for the sparse learning of the large-scale FCM. [ABSTRACT FROM AUTHOR]

As to the long-term time series forecasting, it is more challenging and practical to obtain the trend information and fluctuation range of sequence data than single-step prediction values. In this article, by means of fuzzy information granules (FIGs) and recurrent fuzzy neural networks, a novel long-term prediction model for time series is proposed. Based on a variable-length division method, generalized zonary time-variant fuzzy information granule (GZT-FIG) is constructed, which can express the variation trend, fluctuation range and dispersion degree of sequence data. Furthermore, in order to improve the anti-noise ability and memory ability, type-2 fuzzy sets and long short-term memory mechanism are introduced into the prediction scheme, based on which a self-evolving interval type-2 LSTM fuzzy neural network (eIT2FNN-LSTM) is provided. Compared with the existing works related to fuzzy-neural models, the involvement of long short-term memory mechanism effectively improves the memory ability to achieve the long-term prediction. In order to verify the validation and effectiveness of the proposed scheme, several groups of experiments, including synthetic sequences and real-life time series, are carried out. The experimental results reveal the better predictive performance and rich semantic information. • A method of unequal dividing the time series based on stepwise linear division is proposed. • A generalized time-variant fuzzy information granule can reflect the trend and the range of data changing. • A distance algorithm to measure the similarity and characterize the correlation degree between two GZT-FIGs. [ABSTRACT FROM AUTHOR]