D-Tracker: Modeling Interest Diffusion in Social Activity Tensor Data Streams

TL;DR

D-Tracker addresses the challenge of forecasting high-dimensional social activity tensor streams that exhibit time-varying trends, seasonality, and diffusion across locations. It combines a tensor decomposition with a reaction-diffusion system to model latent dynamics, where $oldsymbol{X}^c \approx \hat{\boldsymbol{X}}_d + \hat{\boldsymbol{X}}_s + \hat{\boldsymbol{X}}_o$ and $\hat{\boldsymbol{X}}_d$ is generated by a low-rank latent system evolving according to reaction-diffusion equations with parameters $\mathbf{A}$ and $\mathcal{D}$. Model estimation uses alternating least squares to fit trend and seasonal components, while MDL-based criteria drive automatic model selection and rank adaptation in a streaming setting. The framework yields interpretable diffusion patterns across locations and keywords and demonstrates superior forecasting accuracy and computational efficiency on Google Trends and COVID-19 data compared with state-of-the-art baselines. The MDL-based automatic compression and model-switching mechanism enable scalable, parameter-free operation suitable for real-time monitoring of evolving social activity patterns.

Abstract

Large quantities of social activity data, such as weekly web search volumes and the number of new infections with infectious diseases, reflect peoples' interests and activities. It is important to discover temporal patterns from such data and to forecast future activities accurately. However, modeling and forecasting social activity data streams is difficult because they are high-dimensional and composed of multiple time-varying dynamics such as trends, seasonality, and interest diffusion. In this paper, we propose D-Tracker, a method for continuously capturing time-varying temporal patterns within social activity tensor data streams and forecasting future activities. Our proposed method has the following properties: (a) Interpretable: it incorporates the partial differential equation into a tensor decomposition framework and captures time-varying temporal patterns such as trends, seasonality, and interest diffusion between locations in an interpretable manner; (b) Automatic: it has no hyperparameters and continuously models tensor data streams fully automatically; (c) Scalable: the computation time of D-Tracker is independent of the time series length. Experiments using web search volume data obtained from GoogleTrends, and COVID-19 infection data obtained from COVID-19 Open Data Repository show that our method can achieve higher forecasting accuracy in less computation time than existing methods while extracting the interest diffusion between locations. Our source code and datasets are available at {https://github.com/Higashiguchi-Shingo/D-Tracker.

Figures (6)

• Figure 1: Modeling power of D-Tracker for a tensor stream related to devices: (a) It fits the original tensor stream very well owing to model switching (at time points shown by vertical dotted lines) for continuous modeling and future forecasting. (b) It decomposes the most recent tensor into the latent dynamics $\mathcal{W}^{\rm{(core)}}$ (top) and the location/keyword factors $\mathbf{W}^{\rm{(loc)}}$/$\mathbf{W}^{\rm{(key)}}$ (middle). More importantly, we can easily interpret patterns of interest diffusion over locations from the obtained model to generate $\mathcal{W}^{\rm{(core)}}$ (bottom).
• Figure 2: An overview of the D-Tracker model: The current tensor $\mathcal{X}^{c}$ is approximated by the sum of the trend tensor $\hat{\mathcal{X}}_{d}$, the seasonal tensor $\hat{\mathcal{X}}_{s}$, and the outlier tensor $\hat{\mathcal{X}}_{o}$.
• Figure 3: Ablation study results. The numbers under the bars indicate the improvement rate for each. Each mechanism in our model provides an improvement in accuracy.
• Figure 5: Modeling results for the Pythonlib data over two periods ((a) Jun 2013 - May 2015; (b) Mar 2020 - Feb 2022). Our model decomposes the input tensor into the latent dynamics $\mathcal{W}^{\rm{(core)}}$ (top), the location factor $\mathbf{W}^{\rm{(loc)}}$, and the keyword factor $\mathbf{W}^{\rm{(key)}}$ (middle). Also, our reaction-diffusion system captures the interest diffusion of "pandas" (bottom).
• Figure 6: Forecasting power of D-Tracker : Our model accurately captures trends and seasonal patterns that vary by location and keyword, and performs stream forecasting.

Theorems & Definitions (6)

• Definition 1: Reaction-diffusion system: $\theta$
• Definition 2: Parameters for trend tensor: $\theta_d$
• Definition 3: Parameter set for seasonal tensor: $\theta_s$
• Definition 4: Single model parameter set : $\Theta$
• Definition 5: Full parameter set: $\mathcal{F}$
• Lemma 1