Effective multivariate time series forecasting often benefits from accurately modeling complex inter-variable dependencies. However, existing attention- or graph-based methods face three key issues: (a) strong temporal self-dependencies are often disrupted by irrelevant variables; (b) softmax normalization ignores and reverses negative correlations; (c) variables struggle to perceive their temporal positions. To address these, we propose **SEED**, a Spectral Entropy-guided evaluation framework for spatial-temporal dependency modeling. SEED introduces a Dependency Evaluator, a key innovation that leverages spectral entropy to dynamically provide a preliminary evaluation of the spatial and temporal dependencies of each variable, enabling the model to adaptively balance Channel Independence (CI) and Channel Dependence (CD) strategies. To account for temporal regularities originating from the influence of other variables rather than intrinsic dynamics, we propose Spectral Entropy-based Fuser to further refine the evaluated dependency weights, effectively separating this part. Moreover, to preserve negative correlations, we introduce a Signed Graph Constructor that enables signed edge weights, overcoming the limitations of softmax. Finally, to help variables perceive their temporal positions and thereby construct more comprehensive spatial features, we introduce the Context Spatial Extractor, which leverages local contextual windows to extract spatial features. Extensive experiments on 12 real-world datasets from various application domains demonstrate that SEED achieves state-of-the-art performance, validating its effectiveness and generality.

Class incremental learning (CIL) offers a promising framework for continuous fault diagnosis (CFD), allowing networks to accumulate knowledge from streaming industrial data and recognize new fault classes. However, current CIL methods assume a balanced data stream, which does not align with the long-tail distribution of fault classes in real industrial scenarios. To fill this gap, this article investigates the impact of long-tail bias in the data stream on the CIL training process through the experimental analysis. Observations show that long-tail bias in the data stream has a cascading effect, affecting the retention of old task knowledge and learning new tasks. Concurrently, the incremental model encounters challenges in identifying samples that conflict with its biases. Accordingly, we propose a CFD method called long-tail CIL via bias calibration (LTCIL-BC), which aims to improve the learning of bias-conflicting samples through bias exploration and debiasing. Specifically, LTCIL-BC simultaneously trains a primary debiased network and an auxiliary biased network. Then, a bias-indicating score is developed to provide insight into model bias and data bias based on the prediction error of the primary and auxiliary models, respectively. LTCIL-BC subsequently adjusts the logits of the debiased network using the bias-indicating score to guide optimization, thereby better utilizing the role of old class exemplars and reducing catastrophic forgetting. Experiments on power system (PS) and secure water treatment (SWaT) datasets demonstrate the superior performance of LTCIL-BC in CFD, achieving up to 9% improvement over state-of-the-art baselines in multiple long-tailed CIL setting. Comprehensive results demonstrate the effectiveness of LTCIL-BC in jointly addressing data and model bias during calibration and prioritizing bias-conflicting samples.

Deep learning has significantly advanced time series forecasting through its powerful capacity to capture sequence relationships. However, training these models with the Mean Square Error (MSE) loss often results in over-smooth predictions, making it challenging to handle the complexity and learn high-entropy features from time series data with high variability and unpredictability. In this work, we introduce a novel approach by tokenizing time series values to train forecasting models via cross-entropy loss, while considering the continuous nature of time series data. Specifically, we propose a Hierarchical Classification Auxiliary Network, HCAN, a general model-agnostic component that can be integrated with any forecasting model. HCAN is based on a Hierarchy-Aware Attention module that integrates multi-granularity high-entropy features at different hierarchy levels. At each level, we assign a class label for timesteps to train an Uncertainty-Aware Classifier. This classifier mitigates the over-confidence in softmax loss via evidence theory. We also implement a Hierarchical Consistency Loss to maintain prediction consistency across hierarchy levels. Extensive experiments integrating HCAN with state-of-the-art forecasting models demonstrate substantial improvements over baselines on several real-world datasets.

In recent years, social media popularity forecasting has become a research hotspot. There are two mainstream methods: cascade-based and sequence-based ones. Cascade-based methods are inefficient for large-scale data because of the cascade-graph representation and learning, while sequence-based methods face challenges in capturing global temporal features due to the inputs of short time series. Actually, there exists strong correlated history data with the new short input. What’s more, the complete trend information in similar historical data contain rich temporal feature for the popularity prediction, which often neglects in the existing methods. Thus, this paper proposes a Frequency-domain Popularity Forecasting model based on Shape Retrieval (FPF-SR). A shape similarity retrieval is used in FPF-SR to select strong correlated history data efficiently with only the new short input. Furthermore, from the retrieved history data and the new post, FPF-SR extracts the inter-sequence and temporal features in the frequency domain. Therefore, FPF-SR uses not only the complete history trend information but also the frequency features for popularity forecasting. Detailedly FPF-SR first retrieves the top K relevant sequences from historical data based on shape correlation, then applies a Fourier transform to the retrieved target sequences, and finally extracts and fuses similar information through linear operations in the frequency domain. Additionally, multi-point trend forecasting is introduced to improve the accuracy of single-point predictions. Experimental results demonstrate that FPF-SR achieves excellent forecasting performance on four real-world datasets. Code is available at: https://anonymous.4open.science/r/FPF-SR-AE10.

Time-series forecasting has gained significant attention in machine learning due to its crucial role in various domains. However, most existing forecasting models rely heavily on point-wise loss functions like Mean Squared Error, which treat each time step independently and neglect the structural dependencies inherent in time series data, making it challenging to capture complex temporal patterns accurately. To address these challenges, we propose a novel Patch-wise Structural (PS) loss, designed to enhance structural alignment by comparing time series at the patch level. Through leveraging local statistical properties, such as correlation, variance, and mean, PS loss captures nuanced structural discrepancies overlooked by traditional point-wise losses. Furthermore, it integrates seamlessly with point-wise loss, simultaneously addressing local structural inconsistencies and individual time-step errors. PS loss establishes a novel benchmark for accurately modeling complex time series data and provides a new perspective on time series loss function design. Extensive experiments demonstrate that PS loss significantly improves the performance of state-of-the-art models across diverse real-world datasets. The data and code are publicly available at: https://github.com/Dilfiraa/PS_Loss.

Recent research has shown an increasing interest in utilizing pre-trained large language models (LLMs) for a variety of time series applications. However, there are three main challenges when using LLMs as foundational models for time series forecasting: (1) Cross-domain generalization. (2) Cross-modality alignment. (3) Error accumulation in autoregressive frameworks. To address these challenges, we proposed LangTime, a language-guided unified model for time series forecasting that incorporates cross-domain pre-training with reinforcement learning-based fine-tuning. Specifically, LangTime constructs Temporal Comprehension Prompts (TCPs), which include dataset-wise and channel-wise instructions, to facilitate domain adaptation and condense time series into a single token, enabling LLMs to understand better and align temporal data. To improve autoregressive forecasting, we introduce TimePPO, a reinforcement learning-based fine-tuning algorithm. TimePPO mitigates error accumulation by leveraging a multidimensional rewards function tailored for time series and a repeat-based value estimation strategy. Extensive experiments demonstrate that LangTime achieves state-of-the-art cross-domain forecasting performance, while TimePPO fine-tuning effectively enhances the stability and accuracy of autoregressive forecasting.

Vision-and-Language Navigation in Continuous Environments (VLN-CE) requires agents to navigate with lowlevel actions following natural language instructions in 3D environments. Most existing approaches utilize observation features from the current step to represent the viewpoint. However, these representations often conflate redundant and essential information for navigation, introducing ambiguity into the agent's action prediction. To address the problem of inadequate representation, we propose a Knowledge-andHistory Aware Visual Representation for Continuous Vision-and-Language Navigation (VLN-KHVR). The proposed approach constructs enriched visual representations tailored to navigation instructions, enhancing agents' navigation performance. Specifically, VLN-KHVR extracts image features from the current observation, retrieves relevant knowledge in the knowledge base, and obtains the history of the navigation episode. Subsequently, the knowledge and history features are filtered to eliminate the information irrelevant to navigation instruction. These refined features are integrated with the instruction for further interaction. Finally, the aggregated features are used to guide navigation. Our model outperforms previous methods on the VLN-CE benchmark, demonstrating the effectiveness of the proposed method.

Time series forecasting, which aims to predict future values based on historical data, has garnered significant attention due to its broad range of applications. However, real-world time series often exhibit heterogeneous pattern evolution across segments, such as seasonal variations, regime changes, or contextual shifts, making accurate forecasting challenging. Existing approaches, which typically train a single model to capture all these diverse patterns, often struggle with the pattern drifts between patches and may lead to poor generalization. To address these challenges, we propose TFPS, a novel architecture that leverages pattern-specific experts for more accurate and adaptable time series forecasting. TFPS employs a dual-domain encoder to capture both time-domain and frequency-domain features, enabling a more comprehensive understanding of temporal dynamics. It then performs subspace clustering to dynamically identify distinct patterns across data segments. Finally, these patterns are modeled by specialized experts, allowing the model to learn multiple predictive functions. Extensive experiments on real-world datasets demonstrate that TFPS outperforms state-of-the-art methods, particularly on datasets exhibiting significant distribution shifts. The data and code are available: https://github.com/syrGitHub/TFPS.

Time series forecasting (TSF) is an essential branch of machine learning with various applications. Most methods for TSF focus on constructing different networks to extract better information and improve performance. However, practical application data contain different internal mechanisms, resulting in a mixture of multiple patterns. That is, the model's ability to fit different patterns is different and generates different errors. In order to solve this problem, we propose an end-to-end framework, namely probability pattern-guided time series forecasting (PPGF). PPGF reformulates the TSF problem as a forecasting task guided by probabilistic pattern classification. First, we propose the grouping strategy to approach forecasting problems as classification and alleviate the impact of data imbalance on classification. Second, we predict the corresponding class interval to guarantee the consistency of classification and forecasting. In addition, true class probability (TCP) is introduced to pay more attention to the difficult samples to improve the classification accuracy. Detailedly, PPGF classifies the different patterns to determine which one the target value may belong to and estimates it accurately in the corresponding interval. To demonstrate the effectiveness of the proposed framework, we conduct extensive experiments on real-world datasets, and PPGF achieves significant performance improvements over several baseline methods. Furthermore, the effectiveness of TCP and the necessity of consistency between classification and forecasting are proved in the experiments. All data and codes are available online: https://github.com/syrGitHub/PPGF.

Deep learning (DL) methods have been widely applied to intelligent fault diagnosis of industrial processes and achieved state-of-the-art performance. However, fault diagnosis with point estimate may provide untrustworthy decisions. Recently, Bayesian inference shows to be a promising approach to trustworthy fault diagnosis by quantifying the uncertainty of the decisions with a DL model. The uncertainty information is not involved in the training process, which does not help the learning of highly uncertain samples and has little effect on improving the fault diagnosis performance. To address this challenge, we propose a Bayesian hierarchical graph neural network (BHGNN) with an uncertainty feedback mechanism, which formulates a trustworthy fault diagnosis on the Bayesian DL (BDL) framework. Specifically, BHGNN captures the epistemic uncertainty and aleatoric uncertainty via a variational dropout approach and utilizes the uncertainty information of each sample to adjust the strength of the temporal consistency (TC) constraint for robust feature learning. Meanwhile, the BHGNN method models the process data as a hierarchical graph (HG) by leveraging the interaction-aware module and physical topology knowledge of the industrial process, which integrates data with domain knowledge to learn fault representation. Moreover, the experiments on a three-phase flow facility (TFF) and secure water treatment (SWaT) show superior and competitive performance in fault diagnosis and verify the trustworthiness of the proposed method.

Social media popularity forecasting has become a hot research topic in recent years. It is of great significance in assisting public opinion monitoring and advertising placement. Time series prediction is one of the simple and commonly used methods for popularity forecasting, which takes the popularity of the first few time steps in the observed data as inputs. However, the complete popularity trend of each social media is known in the training dataset, while the historical time series information except for the first few time steps is neglected in the existing models. In order to utilize the complete historical information from the observed data, a retrieval method is introduced in this paper. Therefore, how to retrieve similar social media based on the first few steps time series and how to integrate the similar historical information have become two challenges. A two-stage prediction method named Angle Feature Retrieval based Forecasting (AFRF) is proposed in this paper to solve the upper two problems. In the first stage, based on the angle features of series, we retrieve K similar series from the historical posts and concatenate them with the target series as the model's input. In the second stage, an attention mechanism is used to learn the temporal relationships among the series and generate future popularity forecasts. We evaluated the multi-step and single-point forecasting performance of AFRF on three real-world datasets and compared it with state-of-the-art popularity forecasting methods, such as temporal feature-based and cascade-based methods, verifying the effectiveness of AFRF.

Time series prediction is applied in many fields as a fundamental task. Although existing methods have achieved satisfactory accuracy, making their prediction more credible is still a significant challenge. In the existing research, the uncertainty estimate method is relatively mature. However, the uncertainty is generally calibrated with post-processing so that the prediction probability of the model matches the actual likelihood. In this paper, we present a novel approach to predict and calibrate the uncertainty of existing state-of-the-art time series models. We regularize the model through multi-expert gating mechanisms while taking advantage of the differences between experts to build new uncertainty estimation methods and propose the AU-Loss function, which combines accuracy and uncertainty terms. Experiments across different real-world datasets with various state-of-the-art time series models show that our method can effectively improve the calibration effect and credibility for regression problems.

Prediction interval (PI) as a method of probabilistic prediction can output the prediction range with a certain degree of confidence. It can give the users more information than point prediction. The noise of data in PI is usually assumed as a Gaussian, Laplace or other single distribution. However, these assumptions are not suitable for all the applications. In order to solve this problem, a mixed approach based on Long Short Term Memory Network with bootstrap (LSTM-bootstrapping) and mixed Gaussian distribution (MGD) with Expectation-Maximization (EM) algorithm is proposed to fore-cast intervals for time series. LSTM is chosen here because of its extremely effectiveness for time series prediction. Firstly, LSTM-bootstrapping is employed to calculate the model uncertainties and the point prediction. Afterwards, we assume that the noise satisfies a mixed Gaussian distribution and the EM algorithm is applied to estimate the noise uncertainty. Then PI can be acquired by the variances of model and noise uncertainty. The proposed predictive approach is evaluated on wind speed, heteroscedastic wind power and reg capacity price datasets. The results show that our method can solve the uncertainty problem of arbitrary distribution and obtain better performance.

Some errors in our original paper in defining relative reduct with information measures are pointed out in this paper. It is shown that in our original work, Theorems 10 and 19 hold just under the condition that decision tables are consistent. We also show that an attribute reduction algorithm based on the entropies can be used to select features in practical applications although the two theorems do not hold in inconsistent cases.

Rough set theory has proven to be an efficient tool for modeling and reasoning with uncertainty information. By introducing probability into fuzzy approximation space, a theory about fuzzy probabilistic approximation spaces is proposed in this paper, which combines three types of uncertainty: probability, fuzziness, and roughness into a rough set model. We introduce Shannon's entropy to measure information quantity implied in a Pawlak's approximation space, and then present a novel representation of Shannon's entropy with a relation matrix. Based on the modified formulas, some generalizations of the entropy are proposed to calculate the information in a fuzzy approximation space and a fuzzy probabilistic approximation space, respectively. As a result, uniform representations of approximation spaces and their information measures are formed with this work.

Data usually exists with hybrid formats in real-world applications, and a unified data reduction for hybrid data is desirable. In this paper a unified information measure is proposed to computing discernibility power of a crisp equivalence relation and a fuzzy one, which is the key concept in classical rough set model and fuzzy rough set model. Based on the information measure, a general definition of significance of nominal, numeric and fuzzy attributes is presented. We redefine the independence of hybrid attribute subset, reduct, and relative reduct. Then two greedy reduction algorithms for unsupervised and supervised data dimensionality reduction based on the proposed information measure are constructed. Experiments show the reducts found by the proposed algorithms get a better performance compared with traditional rough set approaches.