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We suggest specification tests for the conditional mean function in vector multiplicative error models. The test statistics are easy to compute given a suitable estimator of the model parameters. Consistency of the test statistic is proved, the asymptotic distribution of the test under the null hypothesis is studied, while a bootstrap resampling is used in order to approximate critical points and actually carry out the test. Finite-sample results are presented as well as applications of the proposed procedures to real data from the financial markets. [ABSTRACT FROM AUTHOR]

Multivariate time series is ubiquitous in real-world applications, yet it often suffers from missing values that impede downstream analytical tasks. In this paper, we introduce the Long Short-Term Memory Network based Recurrent Autoencoder with Imputation Units and Temporal Attention Imputation Model (RATAI), tailored for multivariate time series. RATAI is designed to address certain limitations of traditional RNN-based imputation methods, which often focus on predictive modeling to estimate missing values, sometimes neglecting the contextual impact of observed data at and beyond the target time step. Drawing inspiration from Kalman smoothing, which effectively integrates past and future information to refine state estimations, RATAI aims to extract feature representations from time series data and use them to reconstruct a complete time series, thus overcoming the shortcomings of existing approaches. It employs a dual-stage imputation process: the encoder utilizes temporal information and attribute correlations to predict and impute missing values, and extract feature representation of imputed time series. Subsequently, the decoder reconstructs the series from the feature representation, and the reconstructed values are used as the final imputation values. Additionally, RATAI incorporates a temporal attention mechanism, allowing the decoder to focus on highly relevant inputs during reconstruction. This model can be trained directly using data that contains missing values, avoiding the misleading effects on model training that can arise from setting initial values for missing values. Our experiments demonstrate that RATAI outperforms benchmark models in multivariate time series imputation. [ABSTRACT FROM AUTHOR]

As a multivariate time series, the prediction of curling trajectories is crucial for athletes to devise game strategies. However, the wide prediction range and complex data correlations present significant challenges to this task. This paper puts forward an innovative deep learning approach, CasLSTM, by introducing integrated inter-layer memory, and establishes an encoder-predictor curling trajectory forecasting model accordingly. Additionally, tailored training techniques involving non-teacher-forcing, ExMSE loss and incremental multi-trajectory learning are devised to enhance model performance. Notably, the model demonstrates astounding accuracy, achieving sub-1cm average errors over 30m trajectories, outperforming vanilla LSTM by 41.8%. It also showcases robustness across various curling settings, with strict validation metrics on a static test set further verifying precision. Field test results reveal promising predictive capabilities for real-world scenarios as well, exhibiting applicability. The proposed technique liberates data-driven curling stone trajectory prediction from sole reliance on analytical models and tackles key challenges of long sequence forecasting. The presented technologies and insights could also generalize to prediction tasks in other remote trajectories and multivariate time series domains. [ABSTRACT FROM AUTHOR]

This study introduces a novel multivariate time series dataset of 228 outdoor sport activities recorded by individual non-competitive athlete in uncontrolled environments. The dataset includes three features: Heart Rate, Speed, and Altitude, and covers five sport categories: walking, running, skiing, roller-skiing, and biking. The data was collected using two types of Garmin sport watches. The original dataset was carefully pre-processed using typical data cleansing methods such as gaps filling, and value format transformations. Furthermore, activity filtering was implemented for missing sensor value data and using domain knowledge of sport categories. Full length sequences, varying from 10 min to several hours, were split into equal length segments, approximately 1 min. To address the small number of instances data was augmented using several consecutive segments from the same activity. However, only a small part of the whole original data was used as a computational cost–information gain tradeoff. Three-dimensional dataset is divided into three parts, each dimension to its own comma separated value (CSV) file. The dataset aims to provide a unique resource for researchers and practitioners in the field of sports science, human performance analysis, and activity recognition. It aims to complement the very limited or non-existent publicly available sport activity datasets. [ABSTRACT FROM AUTHOR]

Multivariate time series classification is a key problem in many fields, but the current research on multivariate time series classification is faced with some problems, such as high dimensionality of original data, low accuracy, and lack of interpretability, which limits the performance improvement of models and makes it difficult to meet the actual requirements. Aiming at above problem, a multivariate time series classification method based on Shapelets was proposed. Firstly, unsupervised Shapelet learning of adaptive neighbors was used to automatically learn significant multivariate Shapelets by combining Shapelets transform and adaptive weights. Then, the method was combined with Shapelet similarity and class label constraint to enhance the interpretability and classification accuracy of the model. Finally, the optimization strategy of the model was proposed to obtain the best Shapelets to further improve the classification accuracy of the model. Three different types of 11 algorithms were compared on 11 public data sets, and the experimental results show that the proposed algorithm has high classification accuracy. [ABSTRACT FROM AUTHOR]

Although data-driven machine learning methods have been successfully applied to predict complex nonlinear dynamics, forecasting future evolution based on incomplete past information remains a significant challenge. This paper proposes a novel data-driven approach that leverages the dynamical relationships among variables. By integrating Non-Stationary Transformers with LightGBM, we construct a robust model where LightGBM builds a fitting function to capture and simulate the complex coupling relationships among variables in dynamically evolving chaotic systems. This approach enables the reconstruction of missing data, restoring sequence completeness and overcoming the limitations of existing chaotic time series prediction methods in handling missing data. We validate the proposed method by predicting the future evolution of variables with missing data in both dissipative and conservative chaotic systems. Experimental results demonstrate that the model maintains stability and effectiveness even with increasing missing rates, particularly in the range of 30% to 50%, where prediction errors remain relatively low. Furthermore, the feature importance extracted by the model aligns closely with the underlying dynamic characteristics of the chaotic system, enhancing the method's interpretability and reliability. This research offers a practical and theoretically sound solution to the challenges of predicting chaotic systems with incomplete datasets. [ABSTRACT FROM AUTHOR]

Multivariate time series anomaly detection (MTSAD) can effectively identify and analyze anomalous behavior in complex systems, which is particularly important in fields such as financial monitoring, industrial equipment fault detection, and cybersecurity. MTSAD requires simultaneously analyze temporal dependencies and inter-variable relationships have prompted researchers to develop specialized deep learning models to detect anomalous patterns. In this paper, we conducted a structured and comprehensive overview of the latest techniques in deep learning for multivariate time series anomaly detection methods. Firstly, we proposed a taxonomy for the anomaly detection strategies from the perspectives of learning paradigms and deep learning models, and then provide a systematic review that emphasizes their advantages and drawbacks. We also organized the public datasets for time series anomaly detection along with their respective application domains. Finally, open issues for future research on MTSAD were identified. [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]

In the Internet of Things (IoT) system, sensors generate a vast amount of multivariate time series data and transmit it to the data center for aggregation and analysis. However, due to equipment failure or attacks, the collected data may contain anomalies, which in turn affect the overall performance and reliability of IoT services. Therefore, an effective multivariate time series anomaly detection (MTSAD) method is a crucial issue to ensure the quality of service. Graph structure learning (GSL)-based methods become a promising technology in MTSAD, which learns an optimal graph structure joint with the anomaly detection task. However, most existing methods disregard the causal and dynamic relationships between sensors during the processing of IoT and assume that the data is devoid of any missing values. Therefore, we propose a uni-direction graph structure learning-based multivariate time-series anomaly detection with dynamic prior knowledge (DPGLAD), which learns the uni-directional relationships between sensors under the constraint of the dynamic prior graph and utilizes diffusion convolutional recurrent neural networks (DCRNN) based on timestamp mask to extract temporal and spatial features. Extensive experiments show that our method has better detection performance and shorter training times than state-of-the-art techniques on four real-world datasets. Compared with the best GSL-based method GTA, DPGLAD achieves 4.16–7.29% more F1-score. [ABSTRACT FROM AUTHOR]

Multivariate time series (MTS) clustering has been an essential research topic in various domains over the past decades. However, inherent properties of MTS data—namely, temporal dynamics and inter-variable correlations—make MTS clustering challenging. These challenges can be addressed in Grassmann manifold learning combined with state-space dynamical modeling, which allows existing clustering techniques to be applicable using similarity measures defined on MTS data. In this paper, we present a systematic overview of Grassmann MTS clustering from a geometrical perspective, categorizing the methods into three approaches: (i) extrinsic, (ii) intrinsic, and (iii) semi-intrinsic. Consequently, we outline 11 methods for Grassmann clustering and demonstrate their effectiveness through a comparative experimental study using human motion gesture-derived MTS data. [ABSTRACT FROM AUTHOR]

In this paper, we present a new approach for modelling matrix-variate time series data that accounts for smooth changes in the dynamics of matrices. Although stylized facts in several fields suggest the existence of smooth nonlinearities, the existing matrix-variate models do not account for regime switches that are not abrupt. To address this gap, we introduce the matrix smooth transition autoregressive model, a flexible regime-switching model capable of capturing abrupt, smooth and no regime changes in matrix-valued data. We provide a thorough examination of the estimation process and evaluate the finite-sample performance of the matrix-variate smooth transition autoregressive model estimators with simulated data. Finally, the model is applied to real-world data. [ABSTRACT FROM AUTHOR]

Purpose: The proposed model aims to tackle the data quality issues in multivariate time series caused by missing values. It preserves data set integrity by accurately imputing missing data, ensuring reliable analysis outcomes. Design/methodology/approach: The Conv-DMSA model employs a combination of self-attention mechanisms and convolutional networks to handle the complexities of multivariate time series data. The convolutional network is adept at learning features across uneven time intervals through an imputation feature map, while the Diagonal Mask Self-Attention (DMSA) block is specifically designed to capture time dependencies and feature correlations. This dual approach allows the model to effectively address the temporal imbalance, feature correlation and time dependency challenges that are often overlooked in traditional imputation models. Findings: Extensive experiments conducted on two public data sets and a real project data set have demonstrated the adaptability and effectiveness of the Conv-DMSA model for imputing missing data. The model outperforms baseline methods by significantly reducing the Root Mean Square Error (RMSE) metric, showcasing its superior performance. Specifically, Conv-DMSA has been found to reduce RMSE by 37.2% to 63.87% compared to other models, indicating its enhanced accuracy and efficiency in handling missing data in multivariate time series. Originality/value: The Conv-DMSA model introduces a unique combination of convolutional networks and self-attention mechanisms to the field of missing data imputation. Its innovative use of a diagonal mask within the self-attention block allows for a more nuanced understanding of the data's temporal and relational aspects. This novel approach not only addresses the existing shortcomings of conventional imputation methods but also sets a new standard for handling missing data in complex, multivariate time series data sets. The model's superior performance and its capacity to adapt to varying levels of missing data make it a significant contribution to the field. [ABSTRACT FROM AUTHOR]

Multivariate time series anomaly detection is a crucial technology to prevent unexpected errors from causing critical impacts. Effective anomaly detection in such data requires accurately capturing temporal patterns and ensuring the availability of adequate data. This study proposes a patch-wise framework for anomaly detection. The proposed approach comprises four key components: (i) maintaining continuous features through patching, (ii) incorporating various temporal information by learning channel dependencies and adding relative positional bias, (iii) achieving feature representation learning through self-supervised learning, and (iv) supervised learning based on anomaly augmentation for downstream tasks. The proposed method demonstrates strong anomaly detection performance by leveraging patching to maintain temporal continuity while effectively learning data representations and handling downstream tasks. Additionally, it mitigates the issue of insufficient anomaly data by supporting the learning of diverse types of anomalies. The experimental results show that our model achieved a 23% to 205% improvement in the F1 score compared to existing methods on datasets such as MSL, which has a relatively small amount of training data. Furthermore, the model also delivered a competitive performance on the SMAP dataset. By systematically learning both local and global dependencies, the proposed method strikes an effective balance between feature representation and anomaly detection accuracy, making it a valuable tool for real-world multivariate time series applications. [ABSTRACT FROM AUTHOR]

The advancement of the computer and information industry has led to the emergence of new demands for multivariate time series anomaly detection (MTSAD) models, namely, the necessity for unsupervised anomaly detection that is both efficient and accurate. However, long-term time series data typically encompass a multitude of intricate temporal pattern variations and noise. Consequently, accurately capturing anomalous patterns within such data and establishing precise and rapid anomaly detection models pose challenging problems. In this paper, we propose a decomposition GAN-based transformer for anomaly detection (DGTAD) in multivariate time series data. Specifically, DGTAD integrates a time series decomposition structure into the original transformer model, further decomposing the extracted global features into deep trend information and seasonal information. On this basis, we improve the attention mechanism, which uses decomposed time-dependent features to change the traditional focus of the transformer, enabling the model to reconstruct anomalies of different types in a targeted manner. This makes it difficult for anomalous data to adapt to these changes, thereby amplifying the anomalous features. Finally, by combining the GAN structure and using multiple generators from different perspectives, we alleviate the mode collapse issue, thereby enhancing the model's generalizability. DGTAD has been validated on nine benchmark datasets, demonstrating significant performance improvements and thus proving its effectiveness in unsupervised anomaly detection. [ABSTRACT FROM AUTHOR]

Multivariate time series data classification has a wide range of applications in reality. With rapid development of deep learning, convolutional networks are widely used in this task and have achieved the current best performance. However, due to high difficulty and cost of collecting this type of data, labeled data is still scarce. In some tasks, the model shows overfitting, resulting in relatively poor classification performance. In order to improve the classification performance under such situation, we proposed a novel classification method based on transfer learning—similar domain knowledge transfer (call SDKT for short). Firstly, we designed a multivariate time series domain distance calculation method (call MTSDDC for short), which helped selecting the source domain that is most similar to target domain; Secondly, we used ResNet as a pre‐trained classifier, transferred the parameters of the similar domain network to the target domain network and continue to fine‐tune the parameters. To verify our method, we conducted experiments on several public datasets. Our study has also shown that the transfer effect from the source domain to the target domain is highly negatively correlated with the distance between them, with an average Pearson coefficient of −0.78. For the transfer of most similar source domain, compared to the ResNet model without transfer and the current best model, the average accuracy improvements on the datasets we used are 4.01% and 1.46% respectively. [ABSTRACT FROM AUTHOR]

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]

This paper develops a nonparametric method for estimating the spectral density of multivariate stationary time series using basis expansion. A likelihood-based approach is used to fit the model through the minimization of a penalized Whittle negative log-likelihood. Then, a Newton-type algorithm is developed for the computation. In this method, we smooth the Cholesky factors of the multivariate spectral density matrix in a way that the reconstructed estimate based on the smoothed Cholesky components is consistent and positive-definite. In a simulation study, we have illustrated and compared our proposed method with other competitive approaches. Finally, we apply our approach to two real-world problems, Electroencephalogram signals analysis, E l N i n ~ o Cycle. [ABSTRACT FROM AUTHOR]

Abstract This study introduces a novel multivariate time series dataset of 228 outdoor sport activities recorded by individual non-competitive athlete in uncontrolled environments. The dataset includes three features: Heart Rate, Speed, and Altitude, and covers five sport categories: walking, running, skiing, roller-skiing, and biking. The data was collected using two types of Garmin sport watches. The original dataset was carefully pre-processed using typical data cleansing methods such as gaps filling, and value format transformations. Furthermore, activity filtering was implemented for missing sensor value data and using domain knowledge of sport categories. Full length sequences, varying from 10 min to several hours, were split into equal length segments, approximately 1 min. To address the small number of instances data was augmented using several consecutive segments from the same activity. However, only a small part of the whole original data was used as a computational cost–information gain tradeoff. Three-dimensional dataset is divided into three parts, each dimension to its own comma separated value (CSV) file. The dataset aims to provide a unique resource for researchers and practitioners in the field of sports science, human performance analysis, and activity recognition. It aims to complement the very limited or non-existent publicly available sport activity datasets.

The state of health (SOH) of a Li-ion battery is determined by complex interactions among its internal components and external factors. Approaches leveraging deep learning architectures have been proposed to predict the SOH using convolutional networks, recurrent networks, and transformers. Recently, Mamba selective state space models have emerged as a new sequence model that combines fast parallel training with data efficiency and fast sampling. In this paper, we propose SambaMixer, a Mamba-based model for predicting the SOH of Li-ion batteries using multivariate time signals measured during the battery’s discharge cycle. Our model is designed to handle analog signals with irregular sampling rates and recuperation effects of Li-ion batteries. We introduce a novel anchor-based resampling method as an augmentation technique. Additionally, we improve performance and learn recuperation effects by conditioning the prediction on the sample time and cycle time difference using positional encodings. We evaluate our model on the NASA battery discharge dataset, reporting MAE, RMSE, and MAPE. Our model outperforms previous methods based on CNNs and recurrent networks, reducing MAE by 52%, RMSE by 43%, and MAPE by 7%.