With the rapid development of web services, large amounts of time series data are generated and accumulated across various domains such as finance, healthcare, and online platforms. As such data often co-evolves with multiple variables interacting with each other, estimating the time-varying dependencies between variables (i.e., the dynamic network structure) has become crucial for accurate modeling. However, real-world data is often represented as tensor time series with multiple modes, resulting in large, entangled networks that are hard to interpret and computationally intensive to estimate. In this paper, we propose Kronecker Time-Varying Graphical Lasso (KTVGL), a method designed for modeling tensor time series. Our approach estimates mode-specific dynamic networks in a Kronecker product form, thereby avoiding overly complex entangled structures and producing interpretable modeling results. Moreover, the partitioned network structure prevents the exponential growth of computational time with data dimension. In addition, our method can be extended to stream algorithms, making the computational time independent of the sequence length. Experiments on synthetic data show that the proposed method achieves higher edge estimation accuracy than existing methods while requiring less computation time. To further demonstrate its practical value, we also present a case study using real-world data. Our source code and datasets are available at https://github.com/Higashiguchi-Shingo/KTVGL.

Many scientific data occur as sequences of multidimensional arrays called tensors. How can hidden, evolving trends in such data be extracted while preserving the tensor structure? The model that is traditionally used is the linear dynamical system (LDS), which treats the observation at each time slice as a vector. In this paper, we propose the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation-maximization (EM) algorithm to estimate the parameters. Compared to the LDS with an equal number of parameters, the MLDS achieves higher prediction accuracy and marginal likelihood for both simulated and real datasets.

Data in the sciences frequently occur as sequences of multidimensional arrays called tensors. How can hidden, evolving trends in such data be extracted while preserving the tensor structure? The model that is traditionally used is the linear dynamical system (LDS) with Gaussian noise, which treats the latent state and observation at each time slice as a vector. We present the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation-maximization (EM) algorithm to estimate the parameters.

Subsequence clustering of time series is an essential task in data mining, and interpreting the resulting clusters is also crucial since we generally do not have prior knowledge of the data. Thus, given a large collection of tensor time series consisting of multiple modes, including timestamps, how can we achieve subsequence clustering for tensor time series and provide interpretable insights? In this paper, we propose a new method, Dynamic Multi-network Mining (DMM), that converts a tensor time series into a set of segment groups of various lengths (i.e., clusters) characterized by a dependency network constrained with l1-norm. Our method has the following properties. (a) Interpretable: it characterizes the cluster with multiple networks, each of which is a sparse dependency network of a corresponding non-temporal mode, and thus provides visible and interpretable insights into the key relationships. (b) Accurate: it discovers the clusters with distinct networks from tensor time series according to the minimum description length (MDL). (c) Scalable: it scales linearly in terms of the input data size when solving a non-convex problem to optimize the number of segments and clusters, and thus it is applicable to long-range and high-dimensional tensors. Extensive experiments with synthetic datasets confirm that our method outperforms the state-of-the-art methods in terms of clustering accuracy. We then use real datasets to demonstrate that DMM is useful for providing interpretable insights from tensor time series.

Many scientific data occur as sequences of multidimensional arrays called tensors. How can hidden, evolving trends in such data be extracted while preserving the tensor structure? The model that is traditionally used is the linear dynamical system (LDS), which treats the observation at each time slice as a vector. In this paper, we propose the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation-maximization (EM) algorithm to estimate the parameters. Compared to the LDS with an equal number of parameters, the MLDS achieves higher prediction accuracy and marginal likelihood for both simulated and real datasets.

Co-evolving time series appears in a multitude of applications such as environmental monitoring, financial analysis, and smart transportation. This paper aims to address the following challenges, including (C1) how to incorporate explicit relationship networks of the time series; (C2) how to model the implicit relationship of the temporal dynamics. We propose a novel model called Network of Tensor Time Series, which is comprised of two modules, including Tensor Graph Convolutional Network (TGCN) and Tensor Recurrent Neural Network (TRNN). TGCN tackles the first challenge by generalizing Graph Convolutional Network (GCN) for flat graphs to tensor graphs, which captures the synergy between multiple graphs associated with the tensors. TRNN leverages tensor decomposition to model the implicit relationships among co-evolving time series. The experimental results on five real-world datasets demonstrate the efficacy of the proposed method.

Many scientific data occur as sequences of multidimensional arrays called tensors. How can hidden, evolving trends in such data be extracted while preserving the tensor structure? The model that is traditionally used is the linear dynamical system (LDS), which treats the observation at each time slice as a vector. In this paper, we propose the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation-maximization (EM) algorithm to estimate the parameters. Compared to the LDS with an equal number of parameters, the MLDS achieves higher prediction accuracy and marginal likelihood for both simulated and real datasets.

Many scientific data occur as sequences of multidimensional arrays called tensors. How can hidden, evolving trends in such data be extracted while preserving the tensor structure? The model that is traditionally used is the linear dynamical system (LDS), which treats the observation at each time slice as a vector. In this paper, we propose the multilinear dynamical system (MLDS) for modeling tensor time series and an expectation-maximization (EM) algorithm to estimate the parameters. The MLDS models each time slice of the tensor time series as the multilinear projection of a corresponding member of a sequence of latent, low-dimensional tensors. Compared to the LDS with an equal number of parameters, the MLDS achieves higher prediction accuracy and marginal likelihood for both simulated and real datasets.