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We propose a multivariate GARCH model for non‐stationary health time series by modifying the observation‐level variance of the standard state space model. The proposed model provides an intuitive and novel way of dealing with heteroskedastic data using the conditional nature of state‐space models. We follow the Bayesian paradigm to perform the inference procedure. In particular, we use Markov chain Monte Carlo methods to obtain samples from the resultant posterior distribution. We use the forward filtering backward sampling algorithm to efficiently obtain samples from the posterior distribution of the latent state. The proposed model also handles missing data in a fully Bayesian fashion. We validate our model on synthetic data and analyze a data set obtained from an intensive care unit in a Montreal hospital and the MIMIC dataset. We further show that our proposed models offer better performance, in terms of WAIC than standard state space models. The proposed model provides a new way to model multivariate heteroskedastic non‐stationary time series data. Model comparison can then be easily performed using the WAIC. [ABSTRACT FROM AUTHOR]

Quadratic Phase Coupling (QPC) serves as an essential statistical instrument for evaluating nonlinear synchronization within multivariate time series data, especially in signal processing and neuroscience fields. This study explores the precision of QPC detection using numerical estimates derived from cross-bicoherence and bivariate Granger causality within a straightforward, yet noisy, instantaneous multiplier model. It further assesses the impact of accidental statistically significant bifrequency interactions, introducing new metrics such as the ratio of bispectral quadratic phase coupling and the ratio of bivariate Granger causality quadratic phase coupling. Ratios nearing 1 signify a high degree of accuracy in detecting QPC. The coupling strength between interacting channels is identified as a key element that introduces nonlinearities, influencing the signal-to-noise ratio in the output channel. The model is tested across 59 experimental conditions of simulated recordings, with each condition evaluated against six coupling strength values, covering a wide range of carrier frequencies to examine a broad spectrum of scenarios. The findings demonstrate that the bispectral method outperforms bivariate Granger causality, particularly in identifying specific QPC under conditions of very weak couplings and in the presence of noise. The detection of specific QPC is crucial for neuroscience applications aimed at better understanding the temporal and spatial coordination between different brain regions. [ABSTRACT FROM AUTHOR]

Many industrial companies capture high volume of time series data from their industrial processes. However, to visualize it, regular visualization approaches require specialized hardware. Thus, downsampling algorithms are required to create a simplified view of the original data. Although industrial processes involve synchronized variables that should be visualized together for their analysis, existing downsampling algorithms tackle visualization of univariate data series. This paper proposes an adaptive extension of the M4 algorithm for multivariate datasets. The algorithm has been validated successfully with data from a conventional 3D turning operation and commodity hardware. For validation, a direct image comparison has been performed. [ABSTRACT FROM AUTHOR]

To guarantee the safety and reliability of equipment operation, such as liquid rocket engine (LRE), carrying out system-level anomaly detection (AD) is crucial. However, current methods ignore the prior knowledge of mechanical system itself, and seldom unite the observations with the inherent relation in data tightly. Meanwhile, they neglect the weakness and nonindependence of system-level anomaly which is different from component fault. To overcome above limitations, we propose a separate reconstruction framework using worsened tendency for system-level AD. To prevent anomalous feature being attenuated, we first propose to divide single sample into two equal-length parts along the temporal dimension. And we maximize the mean maximum discrepancy (MMD) between feature segments to force encoders to learn normal features with different distributions. Then, to fully explore the multivariate time series, we model temporal-spatial dependence by temporal convolution and graph attention. Besides, a joint graph learning strategy is proposed to handle prior knowledge and data characteristics simultaneously. Finally, the proposed method is evaluated on two real multi-sensor datasets from LRE and the results demonstrate the effectiveness and potential of the proposed method on system-level AD. • A novel neural network based on segmenting and reconstructing temporal-spatial feature for system anomaly detection. • Segmenting operation is designed to overcome the weakness of anomaly, which is simple and universal for time series data. • A joint graph learning strategy and a novel temporal-spatial feature extraction module are proposed for multi-source data. • Experiments on two different real-world datasets are conducted and demonstrated the superiority of proposed method. [ABSTRACT FROM AUTHOR]

Multivariate Time Series Anomaly Detection (MTS-AD) is crucial for the effective management and maintenance of devices in complex systems, such as server clusters, spacecrafts, and financial systems, and so on. However, upgrade or cross-platform deployment of these devices will introduce the issue of cross-domain distribution shift, which leads to the prototypical problem of domain adaptation for MTS-AD. Compared with general domain adaptation problems, MTS-AD domain adaptation presents two peculiar challenges: (1) the dimensions of data from the source domain and the target domain are usually different, so alignment without losing any information is necessary; and (2) the association between different variates plays a vital role in the MTS-AD task, which is overlooked by traditional domain adaptation approaches. Aiming at addressing the above issues, we propose a Variate Associated Domain Adaptation Method Combined with a Graph Deviation Network (VANDA) for MTS-AD, which includes two major contributions. First, we characterize the intra-domain variate associations of the source domain by a graph deviation network (GDN), which can share parameters across domains without dimension alignment. Second, we propose a sliding similarity to measure the inter-domain variate associations and perform joint training by minimizing the optimal transport distance between source and target data for transferring variate associations across domains. VANDA achieves domain adaptation by transferring both variate associations and GDN parameters from the source domain to the target domain. We construct two pairs of MTS-AD datasets from existing MTS-AD data and combine three domain adaptation strategies with six MTS-AD backbones as the benchmark methods for experimental evaluation and comparison. Extensive experiments demonstrate the effectiveness of our approach, which outperforms the benchmark methods, and significantly improves the AD performance of the target domain by effectively utilizing the source domain knowledge. [ABSTRACT FROM AUTHOR]

In the industrial Internet, industrial software plays a central role in enhancing the level of intelligent manufacturing. It enables the promotion of digital collaborative services. Effective anomaly detection of multivariate time series can ensure the quality of industrial software. Extensive research has been conducted on time series anomaly detection to identify abnormal data. However, detecting anomalies in multivariate time series, which consist of high-dimensional, high-noise, and random data, poses significant challenges. The states of different timestamps within a time series sample can influence the overall correlation of sensor features. Unfortunately, existing methods often overlook this impact, making it difficult to capture subtle variations in the delayed response of attacked sensors.Consequently, there are false alarms and abnormal omissions. To address these limitations, this paper proposes an anomaly detection method called DGINet. DGINet leverages a dynamic graph attention network and Informer to capture and integrate feature correlation across different time states. By combining GRU and Informer, DGINet effectively captures continuous correlations in long time series. Moreover, DGINet simultaneously optimizes the reconstruction and forecasting modules, enhancing its overall performance. Experimental results on four benchmark datasets demonstrate that DGINet outperforms state-of-the-art methods by achieving up to a 2 % improvement in accuracy. Further analysis reveals that DGINet excels in accurately detecting anomalies in long time series and locating candidate abnormal attack points. [ABSTRACT FROM AUTHOR]

Electronic and digital trading models have made stock trading more accessible and convenient, leading to exponential growth in trading data. With a wealth of trading data available, researchers have found opportunities to extract valuable insights by uncovering patterns in stock price movements and market dynamics. Deep learning models are increasingly being employed for stock price prediction. While neural networks offer superior computational capabilities compared with traditional statistical methods, their results often lack interpretability, limiting their utility in explaining stock price volatility and investment behavior. To address this challenge, we propose a causality‐based method that incorporates a multivariate approach, integrating news event attention sequences and sentiment index sequences. The goal is to capture the intricate and multifaceted relationships among news events, media sentiment, and stock prices. We illustrate the application of this proposed approach using a Global Database of Events, Language, and Tone global event database, demonstrating its benefits through the analysis of attention sequences and media sentiment index sequences for news events across various categories. This research not only identifies promising directions for further exploration but also offers insights with implications for informed investment decisions. [ABSTRACT FROM AUTHOR]

The Industrial Internet of Things has enabled the integration and analysis of vast volumes of data across various industries, with the maritime sector being no exception. Advances in cloud computing and deep learning (DL) are continuously reshaping the industry, particularly in optimizing maritime operations such as Predictive Maintenance (PdM). In this study, we propose a novel DL-based framework focusing on the fault detection task of PdM in marine operations, leveraging time-series data from sensors installed on shipboard machinery. The framework is designed as a scalable and cost-efficient software solution, encompassing all stages from data collection and pre-processing at the edge to the deployment and lifecycle management of DL models. The proposed DL architecture utilizes Graph Attention Networks (GATs) to extract spatio-temporal information from the time-series data and provides explainable predictions through a feature-wise scoring mechanism. Additionally, a custom evaluation metric with real-world applicability is employed, prioritizing both prediction accuracy and the timeliness of fault identification. To demonstrate the effectiveness of our framework, we conduct experiments on three types of open-source datasets relevant to PdM: electrical data, bearing datasets, and data from water circulation experiments. [ABSTRACT FROM AUTHOR]

Background: Describing the transmission dynamics of infectious diseases across different regions is crucial for effective disease surveillance. The multivariate time series (MTS) model has been widely adopted for constructing cross-regional infectious disease transmission networks due to its strengths in interpretability and predictive performance. Nevertheless, the assumption of constant parameters frequently disregards the dynamic shifts in disease transmission rates, thereby compromising the accuracy of early warnings. This study investigated the applicability of time-varying MTS models in multi-regional infectious disease monitoring and explored strategies for model selection. Methods: This study focused on two prominent time-varying MTS models: the time-varying parameter-stochastic volatility-vector autoregression (TVP-SV-VAR) model and the time-varying VAR model using the generalized additive framework (tvvarGAM), and intended to explore and verify their applicable conditions for the surveillance of infectious diseases. For the first time, this study proposed the time delay coefficient and spatial sparsity indicators for model selection. These indicators quantify the temporal lags and spatial distribution of infectious disease data, respectively. Simulation study adopted from real-world infectious disease surveillance was carried out to compare model performances under various scenarios of spatio-temporal variation as well as random volatility. Meanwhile, we illustrated how the modelling process could help the surveillance of infectious diseases with an application to the influenza-like case in Sichuan Province, China. Results: When the spatio-temporal variation was small (time delay coefficient: 0.1–0.2, spatial sparsity:0.1–0.3), the TVP-SV-VAR model was superior with smaller fitting residuals and standard errors of parameter estimation than those of the tvvarGAM model. In contrast, the tvvarGAM model was preferable when the spatio-temporal variation increased (time delay coefficient: 0.2–0.3, spatial sparsity: 0.6–0.9). Conclusion: This study emphasized the importance of considering spatio-temporal variations when selecting appropriate models for infectious disease surveillance. By incorporating our novel indicators—the time delay coefficient and spatial sparsity—into the model selection process, the study could enhance the accuracy and effectiveness of infectious disease monitoring efforts. This approach was not only valuable in the context of this study, but also has broader implications for improving time-varying MTS analyses in various applications. [ABSTRACT FROM AUTHOR]

Summary: This paper briefly reviews the recent research in matrix‐variate time series analysis, discusses some new developments, especially for seasonal time series, and demonstrates some applications. A general matrix autoregressive moving‐average model is introduced. The paper narrates a simple approach for understanding the model, identifiability issues, and estimation. Real examples are used to illustrate the theory. [ABSTRACT FROM AUTHOR]

Air quality analysis helps analysts understand the state of atmospheric pollution and its changing trends, providing robust data and theoretical support for developing and implementing environmental policies. Air quality data are typically represented as multivariate time series, which poses challenges due to the large amount of data, high dimensionality, and lack of labeled information. Analysts often struggle to discover internal relationships and patterns within the data. There is still significant room for improvement in related data mining and exploration methods, as issues such as perceptual burden and low efficiency must be addressed. To assist analysts in atmospheric pollution analysis, we propose an air quality visualization scheme based on feature extraction of multivariate time series data. We utilize the automated data modeling capability of deep learning and intuitive data visualization to help analysts explore and analyze complex air quality datasets. To extract features of air quality data effectively, we transform the multivariate time series feature extraction task into an automated deep learning self-supervised task and propose a feature extraction method called CTDCN for multivariate time series. Finally, we design and implement a visualization and analysis system for air quality multivariate time series. This system helps analysts discover potential information and patterns in air quality data, providing support and a foundation for informed decision-making. The system offers rich visualization views, allows users to change data modeling parameters, and interactively analyze and extract insights from the data through multiple views. Extensive experiments on UEA public datasets confirm CTDCN's superior feature extraction capabilities, while case studies and user studies validate the effectiveness and practicality of our visualization approach. [ABSTRACT FROM AUTHOR]

Anomaly detection for multivariate time series is an essential task in the modern industrial field. Although several methods have been developed for anomaly detection, they usually fail to effectively exploit the metrical-temporal correlation and the other dependencies among multiple variables. To address this problem, we propose a stacked attention autoencoder for anomaly detection in multivariate time series (SA2E-AD); it focuses on fully utilizing the metrical and temporal relationships among multivariate time series. We design a multiattention block, alternately containing the temporal attention and metrical attention components in a hierarchical structure to better reconstruct normal time series, which is helpful in distinguishing the anomalies from the normal time series. Meanwhile, a two-stage training strategy is designed to further separate the anomalies from the normal data. Experiments on three publicly available datasets show that SA2E-AD outperforms the advanced baseline methods in detection performance and demonstrate the effectiveness of each part of the process in our method. [ABSTRACT FROM AUTHOR]

In the Industrial Internet of Things (IIoT), sensors generate time series data to reflect the working state. When the systems are attacked, timely identification of outliers in time series is critical to ensure security. Although many anomaly detection methods have been proposed, the temporal correlation of the time series over the same sensor and the state (spatial) correlation between different sensors are rarely considered simultaneously in these methods. Owing to the superior capability of Transformer in learning time series features. This paper proposes a time series anomaly detection method based on a spatial-temporal network and an improved Transformer. Additionally, the methods based on graph neural networks typically include a graph structure learning module and an anomaly detection module, which are interdependent. However, in the initial phase of training, since neither of the modules has reached an optimal state, their performance may influence each other. This scenario makes the end-to-end training approach hard to effectively direct the learning trajectory of each module. This interdependence between the modules, coupled with the initial instability, may cause the model to find it hard to find the optimal solution during the training process, resulting in unsatisfactory results. We introduce an adaptive graph structure learning method to obtain the optimal model parameters and graph structure. Experiments on two publicly available datasets demonstrate that the proposed method attains higher anomaly detection results than other methods. [ABSTRACT FROM AUTHOR]

The article will focus on vector autoregressive‐moving average (VARMA) models with time‐dependent coefficients (td) to represent general nonstationary time series, not necessarily Gaussian. The coefficients depend on time, possibly on the length of the series n, hence the name tdVARMA (n) for the models, but not necessarily on the rescaled time t/n. As a consequence of the dependency on n of the model, we need to consider array processes instead of stochastic processes. Under appropriate assumptions, it is shown that a Gaussian quasi‐maximum likelihood estimator is consistent in probability and asymptotically normal. The theoretical results are illustrated using three examples of bivariate processes, the first two with marginal heteroscedasticity. The first example is a tdVAR (n)(1) process while the second example is a tdVMA (n)(1) process. In these two cases, the finite‐sample behavior is checked via a Monte Carlo simulation study. The results are compatible with the asymptotic properties even for small n. A third example shows the application of the tdVARMA (n) models for a real time series. [ABSTRACT FROM AUTHOR]

Accurately estimating the Remaining Useful Life (RUL) of lithium-ion batteries is crucial for maintaining the safe and stable operation of rechargeable battery management systems. However, this task is often challenging due to the complex temporal dynamics. Recently, attention-based networks, such as Transformers and Informer, have been the popular architecture in time series forecasting. Despite their effectiveness, these models with abundant parameters necessitate substantial training time to unravel temporal patterns. To tackle these challenges, we propose a straightforward MLP-Mixer-based architecture named "Intra–Inter Patch Mixer" (IIP-Mixer), which leverages the strengths of multilayer perceptron (MLP) models to capture both local and global temporal patterns in time series data. Specifically, it extracts information using an MLP and performs mixing operations along both intra-patch and inter-patch dimensions for battery RUL prediction. The proposed IIP-Mixer comprises parallel dual-head mixer layers: the intra-patch mixing MLP, capturing local temporal patterns in the short-term period, and the inter-patch mixing MLP, capturing global temporal patterns in the long-term period. Notably, to address the varying importance of features in RUL prediction, we introduce a weighted loss function in the MLP-Mixer-based architecture, marking the first time such an approach has been employed. Our experiments demonstrate that IIP-Mixer achieves competitive performance in battery RUL prediction, outperforming other popular time series frameworks, such as Informer and DLinear, with relative reductions in mean absolute error (MAE) of 24% and 10%, respectively. [ABSTRACT FROM AUTHOR]

In multivariate and multistep time series prediction research, we often face the problems of insufficient spatial feature extraction and insufficient time-dependent mining of historical series data, which also brings great challenges to multivariate time series analysis and prediction. Inspired by the attention mechanism and residual module, this study proposes a multivariate time series prediction method based on a convolutional-residual gated recurrent hybrid model (CNN-DA-RGRU) with a two-layer attention mechanism to solve the multivariate time series prediction problem in these two stages. Specifically, the convolution module of the proposed model is used to extract the relational features among the sequences, and the two-layer attention mechanism can pay more attention to the relevant variables and give them higher weights to eliminate the irrelevant features, while the residual gated loop module is used to extract the time-varying features of the sequences, in which the residual block is used to achieve the direct connectivity to enhance the expressive power of the model, to solve the gradient explosion and vanishing scenarios, and to facilitate gradient propagation. Experiments were conducted on two public datasets using the proposed model to determine the model hyperparameters, and ablation experiments were conducted to verify the effectiveness of the model; by comparing it with several models, the proposed model was found to achieve good results in multivariate time series-forecasting tasks. [ABSTRACT FROM AUTHOR]

In the field of photovoltaic (PV) power prediction, long-term forecasting, which is more challenging than short-term forecasting, can provide more comprehensive and forward-looking guidance. Currently, significant achievements have been made in the field of short-term forecasting for PV power, but inadequate attention has been paid to long-term forecasting. Additionally, multivariate global forecasting across multiple sites and the limited historical time series data available further increase the difficulty of prediction. To address these challenges, we propose a variable–adaptive channel-independent architecture (VACI) and design a deep tree-structured multi-scale gated component named DTM block for this architecture. Subsequently, we construct a specific forecasting model called DTMGNet. Unlike channel-independent modeling and channel-dependent modeling, the VACI integrates the advantages of both and emphasizes the diversity of training data and the model's adaptability to different variables across channels. Finally, the effectiveness of the DTM block is empirically validated using the real-world solar energy benchmark dataset. And on this dataset, the multivariate long-term forecasting performance of DTMGNet achieved state-of-the-art (SOTA) levels, particularly making significant breakthroughs in the 720-step ultra-long forecasting window, where it reduced the MSE metric below 0.2 for the first time (from 0.215 to 0.199), representing a reduction of 7.44%. [ABSTRACT FROM AUTHOR]

Government supervision is the highest level of construction quality management system. Due to a large number of constructions in progress, timely and accurate risk early warning is imperative for improving the efficiency of supervision. Aiming at the small-scale, ordinal, and unequal length multivariate time series of government supervision data, this paper proposes a construction quality risk early warning method based on ordinal multivariate grey incidence analysis. Firstly, to measure the dynamic similarity between risk indicators of projects, the proximity grey incidence model based on ordinal dynamic time warping (DTW) and the similarity grey incidence model based on ordinal L1 norm DTW are constructed respectively. Then, the two models are integrated to construct a comprehensive similarity model for construction quality risk warning. Combining the comprehensive similarity and k-nearest neighbour (k-NN) algorithm, a method of construction quality risk level classification and early warning is constructed. Finally, the method is applied to the quality supervision of water conservancy and hydropower projects in Zhejiang Province, and the results show that the proposed method can effectively solve the problem of construction quality risk early warning based on small-scale and ordinal data. [ABSTRACT FROM AUTHOR]

The objective of basic oxygen furnace (BOF) steelmaking is to achieve molten steel with final carbon content, temperature, and phosphorus content meeting the requirements. Accurate prediction of the above properties is crucial for end-point control in BOF steelmaking. Traditional prediction models typically use multi-variable input and single-variable output approaches, neglecting the coupling relationships between different property indicators, making it difficult to predict multiple outputs simultaneously. Consequently, a multi-output prediction model based on the fusion of deep convolution and attention mechanism networks (FDCAN) is proposed. The model inputs include scalar data, such as the properties of raw materials and target molten steel, and time series data, such as lance height, oxygen supply intensity, and bottom air supply intensity during the blowing process. The FDCAN model utilizes a fully connected module to extract nonlinear features from scalar data and a deep convolution module to process time series data, capturing high-dimensional feature representations. The attention mechanism then assigns greater weight to significant features. Finally, multiple multi-layer perceptron modules predict the outputs—final carbon content, temperature, and phosphorus content. This structure allows FDCAN to learn complex relationships within the input data and between input and output variables. The effectiveness of the FDCAN model is validated using actual BOF steelmaking data, achieving hit rates of 95.14% for final carbon content within ±0.015 wt%, 84.72% for final temperature within ±15 °C, and 88.89% for final phosphorus content within ±0.005 wt%. [ABSTRACT FROM AUTHOR]

Without imposing prior distributional knowledge underlying multivariate time series of interest, we propose a nonparametric change-point detection approach to estimate the number of change points and their locations along the temporal axis. We develop a structural subsampling procedure such that the observations are encoded into multiple sequences of Bernoulli variables. A maximum likelihood approach in conjunction with a newly developed searching algorithm is implemented to detect change points on each Bernoulli process separately. Then, aggregation statistics are proposed to collectively synthesize change-point results from all individual univariate time series into consistent and stable location estimations. We also study a weighting strategy to measure the degree of relevance for different subsampled groups. Simulation studies are conducted and shown that the proposed change-point methodology for multivariate time series has favorable performance comparing with currently available state-of-the-art nonparametric methods under various settings with different degrees of complexity. Real data analyses are finally performed on categorical, ordinal, and continuous time series taken from fields of genetics, climate, and finance. [ABSTRACT FROM AUTHOR]

Multivariate time series anomaly detection is a crucial data mining technique with a wide range of applications in areas such as IT applications. Currently, the majority of anomaly detection methods for time series data rely on unsupervised approaches due to the rarity of anomaly labels. However, in real-world scenarios, obtaining a limited number of anomaly labels is feasible and affordable. Effective usage of these labels can offer valuable insights into the temporal characteristics of anomalies and play a pivotal role in guiding anomaly detection efforts. To improve the performance of multivariate time series anomaly detection, we proposed a novel deep learning model named EDD (Encoder-Decoder-Discriminator) that leverages limited anomaly samples. The EDD model innovatively integrates a graph attention network with long short term memory (LSTM) to extract spatial and temporal features from multivariate time series data. This integrated approach enables the model to capture complex patterns and dependencies within the data. Additionally, the model skillfully maps series data into a latent space, utilizing a carefully crafted loss function to cluster normal data tightly in the latent space while dispersing abnormal data randomly. This innovative design results in distinct probability distributions for normal and abnormal data in the latent space, enabling precise identification of anomalous data. To evaluate the performance of our EDD model, we conducted extensive experimental validation across three diverse datasets. The results demonstrate the significant superiority of our model in multivariate time series anomaly detection. Specifically, the average F1-Score of our model outperformed the second-best method by 2.7% and 73.4% in both evaluation approaches, respectively, highlighting its superior detection capabilities. These findings validate the effectiveness of our proposed EDD model in leveraging limited anomaly samples for accurate and robust anomaly detection in multivariate time series data. [ABSTRACT FROM AUTHOR]

Abstract Background Describing the transmission dynamics of infectious diseases across different regions is crucial for effective disease surveillance. The multivariate time series (MTS) model has been widely adopted for constructing cross-regional infectious disease transmission networks due to its strengths in interpretability and predictive performance. Nevertheless, the assumption of constant parameters frequently disregards the dynamic shifts in disease transmission rates, thereby compromising the accuracy of early warnings. This study investigated the applicability of time-varying MTS models in multi-regional infectious disease monitoring and explored strategies for model selection. Methods This study focused on two prominent time-varying MTS models: the time-varying parameter-stochastic volatility-vector autoregression (TVP-SV-VAR) model and the time-varying VAR model using the generalized additive framework (tvvarGAM), and intended to explore and verify their applicable conditions for the surveillance of infectious diseases. For the first time, this study proposed the time delay coefficient and spatial sparsity indicators for model selection. These indicators quantify the temporal lags and spatial distribution of infectious disease data, respectively. Simulation study adopted from real-world infectious disease surveillance was carried out to compare model performances under various scenarios of spatio-temporal variation as well as random volatility. Meanwhile, we illustrated how the modelling process could help the surveillance of infectious diseases with an application to the influenza-like case in Sichuan Province, China. Results When the spatio-temporal variation was small (time delay coefficient: 0.1–0.2, spatial sparsity:0.1–0.3), the TVP-SV-VAR model was superior with smaller fitting residuals and standard errors of parameter estimation than those of the tvvarGAM model. In contrast, the tvvarGAM model was preferable when the spatio-temporal variation increased (time delay coefficient: 0.2–0.3, spatial sparsity: 0.6–0.9). Conclusion This study emphasized the importance of considering spatio-temporal variations when selecting appropriate models for infectious disease surveillance. By incorporating our novel indicators—the time delay coefficient and spatial sparsity—into the model selection process, the study could enhance the accuracy and effectiveness of infectious disease monitoring efforts. This approach was not only valuable in the context of this study, but also has broader implications for improving time-varying MTS analyses in various applications.

Modelling multivariate time series of counts in a parsimonious way is a popular topic. In this paper, we consider an integer‐valued network autoregressive model with a non‐random neighbourhood structure, which uses negative binomial distribution as the conditional marginal distribution and the softplus function as the link function. The new model generalizes existing ones in the literature and has a great flexibility in modelling. Stationary conditions in cases of fixed dimension and increasing dimension are given. Parameters are estimated by maximizing the quasi‐likelihood function, and related asymptotic properties of the estimators are established. A simulation study is conducted to assess performances of the estimators, and a real data example is analysed to show superior performances of the proposed model compared with existing ones. [ABSTRACT FROM AUTHOR]

Modeling multivariate integer-valued time series with appropriate methods is currently a popular research topic. In this paper, we propose a multivariate integer-valued autoregressive time series model based on a fixed network community structure. We use the negative binomial distribution as the conditional marginal distribution and a copula to construct the conditional joint distribution. The newly proposed model introduces the heterogeneity of nodes. Stability conditions are provided for both fixed and increasing dimensions. We estimate the parameters of the proposed model by maximizing the quasi-likelihood function with known and unknown community membership matrices, respectively. Corresponding asymptotic properties of parameter estimates are also provided. A simulation study is conducted to demonstrate the asymptotic behavior of the proposed model, and two real datasets are employed to compare the proposed model with other competitive models. • Model multivariate integer-valued time series with suitable methods is hot. • A model with network community structure is proposed, which has node heterogeneity. • Stability conditions for both fixed and increasing dimensions are provided. • Parameters are estimated with known and unknown community membership matrices. • Two real datasets are used for illustrating flexibility. [ABSTRACT FROM AUTHOR]

In intelligent machining, tool condition monitoring (TCM) is crucial to improving tool efficiency and machining accuracy, which requires the real-time analysis and feature extraction of multivariate time series signals collected by multiple sensors. However, multivariate time series are ultra-high-dimensional and difficult to perform representation learning directly, requiring sampling and typical feature extraction. The existing deep feature extractors based on Sequential sampling, Random sampling, or Window sampling, are poor at capturing the critical information from the huge amount of time series data, and ignore the temporal associations, so the actual results are not satisfactory in terms of prediction accuracy and efficiency. Therefore, we propose an appending-inspired multivariate time series association fusion method for TCM tasks: after the necessary denoising, we capture typical time-domain, frequency-domain, and time-frequency-domain features of multivariate time series based on the proposed appending-inspired feature capturer to fully consider the temporal associations, and employ the ACNNs (Attention-based Convolutional Neural Networks) to extract and fuse the multivariate time series features for real-time TCM tasks. The experimental results on NASA and PHM2010 datasets show that our method can real-time and effectively monitor the tool condition and accurately predict the tool wear state. [ABSTRACT FROM AUTHOR]

Abstract Quadratic Phase Coupling (QPC) serves as an essential statistical instrument for evaluating nonlinear synchronization within multivariate time series data, especially in signal processing and neuroscience fields. This study explores the precision of QPC detection using numerical estimates derived from cross-bicoherence and bivariate Granger causality within a straightforward, yet noisy, instantaneous multiplier model. It further assesses the impact of accidental statistically significant bifrequency interactions, introducing new metrics such as the ratio of bispectral quadratic phase coupling and the ratio of bivariate Granger causality quadratic phase coupling. Ratios nearing 1 signify a high degree of accuracy in detecting QPC. The coupling strength between interacting channels is identified as a key element that introduces nonlinearities, influencing the signal-to-noise ratio in the output channel. The model is tested across 59 experimental conditions of simulated recordings, with each condition evaluated against six coupling strength values, covering a wide range of carrier frequencies to examine a broad spectrum of scenarios. The findings demonstrate that the bispectral method outperforms bivariate Granger causality, particularly in identifying specific QPC under conditions of very weak couplings and in the presence of noise. The detection of specific QPC is crucial for neuroscience applications aimed at better understanding the temporal and spatial coordination between different brain regions.

This research paper explores the implementation of machine learning (ML) techniques in weather and climate forecasting, with a specific focus on predicting monthly precipitation. The study analyzes the efficacy of six multivariate machine learning models: Decision Tree, Random Forest, K-Nearest Neighbors (KNN), AdaBoost, XGBoost, and Long Short-Term Memory (LSTM). Multivariate time series models incorporating lagged meteorological variables were employed to capture the dynamics of monthly rainfall in Rabat, Morocco, from 1993 to 2018. The models were evaluated based on various metrics, including root mean square error (RMSE), mean absolute error (MAE), and coefficient of determination (R2). XGBoost showed the highest performance among the six individual models, with an RMSE of 40.8 (mm). In contrast, Decision Tree, AdaBoost, Random Forest, LSTM, and KNN showed relatively lower performances, with specific RMSEs ranging from 47.5 (mm) to 51 (mm). A novel multi-view stacking learning approach is introduced, offering a new perspective on various ML strategies. This integrated algorithm is designed to leverage the strengths of each individual model, aiming to substantially improve the precision of precipitation forecasts. The best results were achieved by combining Decision Tree, KNN, and LSTM to build the meta-base while using XGBoost as the second-level learner. This approach yielded a RMSE of 17.5 millimeters. The results show the potential of the proposed multi-view stacking learning algorithm to refine predictive results and improve the accuracy of monthly precipitation forecasts, setting a benchmark for future research in this field. [ABSTRACT FROM AUTHOR]

Solar flares are characterized by sudden bursts of electromagnetic radiation from the Sun's surface, and are caused by the changes in magnetic field states in active solar regions. Earth and its surrounding space environment can suffer from various negative impacts caused by solar flares, ranging from electronic communication disruption to radiation exposure-based health risks to astronauts. In this paper, we address the solar flare prediction problem from magnetic field parameter-based multivariate time series (MVTS) data using multiple state-of-the-art machine learning classifiers that include MINImally RandOm Convolutional KErnel Transform (MiniRocket), Support Vector Machine (SVM), Canonical Interval Forest (CIF), Multiple Representations Sequence Learner (Mr-SEQL), and a Long Short-Term Memory (LSTM)-based deep learning model. Our experiment is conducted on the Space Weather Analytics for Solar Flares (SWAN-SF) benchmark data set, which is a partitioned collection of MVTS data of active region magnetic field parameters spanning over nine years of operation of the Solar Dynamics Observatory (SDO). The MVTS instances of the SWAN-SF dataset are labeled by GOES X-ray flux-based flare class labels, and attributed to extreme class imbalance because of the rarity of the major flaring events (e.g., X and M). As a performance validation metric in this class-imbalanced dataset, we used the True Skill Statistic ( T S S ) score. Finally, we demonstrate the advantages of the MVTS learning algorithm MiniRocket, which outperformed the aforementioned classifiers without the need for essential data preprocessing steps such as normalization, statistical summarization, and class imbalance handling heuristics. [ABSTRACT FROM AUTHOR]

The accurate estimation of prediction errors in time series is an important problem, which has immediate implications for the accuracy of prediction intervals as well as the quality of a number of widely used time series model selection criteria such as the Akaike information criterion. Except for simple cases, however, it is difficult or even impossible to obtain exact analytical expressions for one-step and multi-step predictions. This may be one of the reasons that, unlike in the independent case (see Efron, 2004), up to now there has been no fully established methodology for time series prediction error estimation. Starting from an approximation to the bias-variance decomposition of the squared prediction error, a method for accurate estimation of prediction errors in both univariate and multivariate stationary time series is developed in this article. In particular, several estimates are derived for a general class of predictors that includes most of the popular linear, nonlinear, parametric and nonparametric time series models used in practice, with causal invertible autoregressive moving average and nonparametric autoregressive processes discussed as lead examples. Simulations demonstrate that the proposed estimators perform quite well in finite samples. The estimates may also be used for model selection when the purpose of modelling is prediction. [ABSTRACT FROM AUTHOR]

Accurate prediction of traffic flow is essential for optimizing transportation resource allocation and enhancing urban mobility efficiency. However, traffic data generated daily are vast and complex, involving dynamic and intricate changes in the traffic road network and traffic flow. Therefore, real-time and accurate prediction of traffic flow is a challenging task that requires modeling the intricate spatial–temporal dynamics of traffic data. In this paper, we propose a novel approach for traffic flow prediction, based on a Multi-Scale Residual Graph Convolution Network with hierarchical attention. First, we design a novel encoder–decoder with multi-independent channels to capture traffic flow information from different time scales and diverse temporal dependencies. Second, we employ a coupled graph convolution network with residual graph attention to dynamically learn the varying spatial features among and within traffic stations. Third, we utilize channel attention to fuse the multi-scale spatial–temporal dependencies and accurately predict traffic flow. We evaluate the proposed approach on multiple benchmark datasets, and the experimental results demonstrate its superior performance compared to state-of-the-art approaches in terms of various metrics. [ABSTRACT FROM AUTHOR]

AbstractMultivariate Singular Spectrum Analysis (MSSA) is a powerful and widely used nonparametric method for multivariate time series, which allows the analysis of complex temporal data from diverse fields such as finance, healthcare, ecology, and engineering. However, MSSA lacks robustness against outliers because it relies on the singular value decomposition, which is very sensitive to the presence of anomalous values. MSSA can then give biased results and lead to erroneous conclusions. In this article a new MSSA method is proposed, named RObust Diagonalwise Estimation of SSA (RODESSA), which is robust against the presence of cellwise and casewise outliers. In particular, the decomposition step of MSSA is replaced by a new robust low-rank approximation of the trajectory matrix that takes its special structure into account. A fast algorithm is constructed, and it is proved that each iteration step decreases the objective function. In order to visualize different types of outliers, a new graphical display is introduced, called an enhanced time series plot. An extensive Monte Carlo simulation study is performed to compare RODESSA with competing approaches in the literature. A real data example about temperature analysis in passenger railway vehicles demonstrates the practical utility of the proposed approach. [ABSTRACT FROM AUTHOR]

The devices deployed across diverse industrial scenarios have generated significant network traffic related to time. The system's irregular operation could result in substantial bad influence. Anomaly detection technologies utilized for identifying possible non-standard behaviours are paramount; furthermore, multivariate time series exhibit complex dependencies besides temporal correlation. However, most previous methods merely consider the temporal and variable correlation of time series data, neglecting the distance metrics among the sequences, leading to a deficiency in the model's anomaly detection ability. We propose a multivariate time series anomaly detection model based on the encoder–decoder architecture (CCER-ED). The model considers the similarity measure between temporal subsequences and designs a multi-scale feature embedding module for leveraging more interrelated properties. Moreover, the interrelations among sensors are explicitly learned using a manifold regularization graph structure. On this basis, an improved data fusion approach based on a multi-head self-attention mechanism is designed for capturing global feature representation, effectively integrating various aspects of information. Evaluations using the real-world datasets SWAT and WADI and performance analysis show that the proposed approach achieves improvement over the baselines in the recall and F1-score of anomaly detection performance at 9.3% and 8.5% (maximum), respectively, outperforming other existing methods. [ABSTRACT FROM AUTHOR]

Transition networks have extended the existing concept of nonlinear time series analysis and offered new insights into the dynamical system analysis based on time series data. Existing methods mainly focus on one-dimensional time series. However, due to the complexity of real-world systems, the dynamic interrelation within multichannel data sequences of synchronous observation has attracted increasingly attentions. Thus, further study is needed to extend the transition network method from univariate time series to multivariate time series. In this paper, we propose two multivariate time series analysis methods. The first is multivariate hard c-means transition network which maps different time series into transition networks with the same size so that the time series characteristics can be captured more effectively by network measures. The second is combined hard c-means transition network whose size is different for various time series. The noise resistance, time cost and time series characterization ability of the proposed methods are illustrated by studying multivariate fractal series and coupled Rössler system. Finally, the application to multivariate electrocardiogram (ECG) data demonstrates the effectiveness of the proposed methods for distinguishing different heart conditions. [ABSTRACT FROM AUTHOR]

Drought forecasting is crucial for minimizing the effects of drought, alerting people to its dangers, and assisting decision-makers in taking preventative action. This article suggests an encoder–decoder framework for multivariate times series (EDFMTS) forecasting. EDFMTS is composed of three layers: a temporal attention context layer, a gated recurrent unit (GRU)-based decoder component, and a bidirectional gated recurrent unit (Bi-GRU)-based encoder component. The proposed framework was evaluated using multivariate gathered from various sources in China (remote-sensing sensors, climate sensors, biophysical sensors, and so on). According to experimental results, the proposed framework outperformed the baseline methods in univariate and multivariate times series (TS) forecasting. The correlation coefficient of determination ( R 2 ), root-mean-squared error (RMSE), and the mean absolute error (MAE) were used for the evaluation of the framework performance. The R 2 , RMSE, and MAE are 0.94, 0.20, and 0.13, respectively, for EDFMTS. In contrast, the RMSE provided by autoregressive integrated moving average (ARIMA), PROPHET, long short-term memory (LSTM), GRU and convolutional neural network (CNN)-LSTM are 0.72, 0.92, 0.36, 0.40, and 0.27, respectively. [ABSTRACT FROM AUTHOR]