Three-way clustering has been an active research topic in the field of cluster analysis in recent years. Some efforts are focused on the technique due to its feasibility and rationality. We observe, however, that the existing three-way clustering algorithms struggle to obtain more information and limit the fault tolerance excessively. Moreover, although the one-step three-way allocation based on a pair of fixed, global thresholds is the most straightforward way to generate the three-way cluster representations, the clusters derived from a pair of global thresholds cannot exactly reveal the inherent clustering structure of the dataset, and the threshold values are often difficult to determine beforehand. Inspired by sequential three-way decisions, we propose an algorithm, called multistep three-way clustering (M3W), to address these issues. Specifically, we first use a progressive erosion strategy to construct a multilevel structure of data, so that lower levels (or external layers) can gather more available information from higher levels (or internal layers). Then, we further propose a multistep three-way allocation strategy, which sufficiently considers the neighborhood information of every eroded instance. We use the allocation strategy in combination with the multilevel structure to ensure that more information is gradually obtained to increase the probability of being assigned correctly, capturing adaptively the inherent clustering structure of the dataset. The proposed algorithm is compared with eight competitors using 18 benchmark datasets. Experimental results show that M3W achieves superior performance, verifying its advantages and effectiveness.

A probabilistic load forecast that is accurate and reliable is crucial to not only the efficient operation of power systems but also to the efficient use of energy resources. In order to estimate the uncertainties in forecasting models and nonstationary electric load data, this study proposes a probabilistic load forecasting model, namely BFEEMD-LSTM-TWSVRSOA. This model consists of a data filtering method named fast ensemble empirical model decomposition (FEEMD) method, a twin support vector regression (TWSVR) whose features are extracted by deep learning-based long short-term memory (LSTM) networks, and parameters optimized by seeker optimization algorithms (SOAs). We compared the probabilistic forecasting performance of the BFEEMD-LSTM-TWSVRSOA and its point forecasting version with different machine learning and deep learning algorithms on Global Energy Forecasting Competition 2014 (GEFCom2014). The most representative month data of each season, totally four monthly data, collected from the one-year data in GEFCom2014, forming four datasets. Several bootstrap methods are compared in order to determine the best prediction intervals (PIs) for the proposed model. Various forecasting step sizes are also taken into consideration in order to obtain the best satisfactory point forecasting results. Experimental results on these four datasets indicate that the wild bootstrap method and 24-h step size are the best bootstrap method and forecasting step size for the proposed model. The proposed model achieves averaged 46%, 11%, 36%, and 44% better than suboptimal model on these four datasets with respect to point forecasting, and achieves averaged 53%, 48%, 46%, and 51% better than suboptimal model on these four datasets with respect to probabilistic forecasting.

A bunch of stream clustering algorithms have been proposed recently to mine data streams generated at high speeds from hardware platforms and software applications. Density-based methods are widely used because they can handle outliers and capture clusters of arbitrary shapes. However, it is still hard to effectively identify multi-density clusters with ambiguous boundaries in a data stream. To address these limitations, this paper introduces a data stream clustering algorithm called TWStream, based on the three-way decision theory. It is a two-stage clustering algorithm based on density. In the online stage, an augmented $k$ nn graph is maintained incrementally to accelerate the update of the $k$ nn graph. In the offline stage, TWStream introduces the concept of boundary confidence to detect cluster boundaries efficiently and reveal potential cores of clusters. It integrates the skewness and sparsity of the data distribution, as well as the evolving trend of the stream.In the next step, a micro-cluster-based three-way clustering strategy is applied to reconstruct latent clusters. It improves the clustering quality of boundary-ambiguous clusters in a stream using a mutual reachability-based clustering approach and a three-way assignment approach. The proposed algorithm is compared with 9 competitors on 15 data streams. Experimental results show TWStream achieves competitive performance, verifying its effectiveness. The source code of the proposed TWStream can be available at https://github.com/Du-Team/TWStream .

Time series clustering is a crucial unsupervised technique for analyzing data, commonly used in various fields, including medicine and stock analysis. However, in real-world scenarios, time series data inevitably contain missing values, consequently reducing the efficiency of traditional clustering methods. In incomplete time series, existing clustering methods typically adopt a two-stage strategy, i.e., initially imputing missing values followed by clustering. However, this approach of separating imputation from clustering may lead to inconsistencies in the optimization objectives and increase the complexity of parameter tuning, potentially resulting in unsatisfactory clustering results. This paper proposes an end-to-end deep fuzzy clustering model for incomplete time series (EEDFC), which jointly optimizes imputation and clustering within a unified framework by integrating multiple losses. In the imputation part, an attention mechanism is integrated to tackle challenges associated with dependencies in extended sequences. In addition, an adversarial strategy is introduced to enhance the encoder's imputation and feature representation learning capability, thus reducing the error propagation from imputation to clustering. In the clustering part, EEDFC combines a feature weighting-based fuzzy clustering (FWFC), which considers intra-cluster compactness and inter-cluster separateness. Furthermore, exponential distance is adopted, and feature and cluster weighting are also integrated into the Kullback-Leibler (KL) divergence loss to improve clustering performance. We conduct extensive experiments comparing our proposed model with eleven other methods across ten benchmark datasets. The experimental results demonstrate that our proposed model performs better than eleven comparative methods.