Integrated Fuzzy-Gated Long Short-Term Memory Networks for Enhanced Multivariate Time Series Uncertainty Quantification
DOI:
https://doi.org/10.68050/JAMS.2026.253Abstract
Multivariate time series (MTS) modeling in critical domains—such as financial forecasting, smart grid management, and anomaly detection—demands robust quantification of both aleatoric and epistemic uncertainties. While Long Short-Term Memory (LSTM) networks excel at learning temporal dependencies, standard architectures rely on deterministic, real-valued operations that cannot intrinsically represent linguistic vagueness, hesitation, or cyclical phase information. To overcome these limitations, this paper introduces the mathematical foundations for three integrated Fuzzy-Gated LSTM (F-LSTM) architectures: the Hesitant Fuzzy LSTM (HFS-LSTM), the Spherical Fuzzy LSTM (SFS-LSTM), and the Complex Fuzzy LSTM (CFS-LSTM). The HFS-LSTM employs Hesitant Fuzzy Aggregation Operators within the gating mechanisms to capture epistemic uncertainty and upstream multi-fuzzification ambiguity. The SFS-LSTM integrates membership, non-membership, and explicit indeterminacy dimensions under spherical geometric constraints to quantify aleatoric uncertainty and facilitate reliable three-way diagnostic decisions. The CFS-LSTM leverages the complex plane to track cyclical and directional dynamics through phase coherence while mitigating vanishing gradients via norm-preserving state transitions. Probabilistic predictive distributions are constructed via defuzzification to enable comprehensive uncertainty evaluation using the Continuous Ranked Probability Score (CRPS). Together, these architectures establish an expressive theoretical framework for intrinsic uncertainty quantification in deep recurrent networks.
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