High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes (1910.03002v2)

Abstract: Predicting the dependencies between observations from multiple time series is critical for applications such as anomaly detection, financial risk management, causal analysis, or demand forecasting. However, the computational and numerical difficulties of estimating time-varying and high-dimensional covariance matrices often limits existing methods to handling at most a few hundred dimensions or requires making strong assumptions on the dependence between series. We propose to combine an RNN-based time series model with a Gaussian copula process output model with a low-rank covariance structure to reduce the computational complexity and handle non-Gaussian marginal distributions. This permits to drastically reduce the number of parameters and consequently allows the modeling of time-varying correlations of thousands of time series. We show on several real-world datasets that our method provides significant accuracy improvements over state-of-the-art baselines and perform an ablation study analyzing the contributions of the different components of our model.

• The paper proposes a novel method for high-dimensional multivariate time series forecasting combining RNNs with a low-rank Gaussian copula process.
• The method uses a low-rank covariance structure and Gaussian copulas to model thousands of time series with non-Gaussian marginals efficiently.
• Empirical validation shows the model outperforms baselines on real-world datasets and scales effectively to dimensions up to 2,000.

High-Dimensional Multivariate Forecasting with Low-Rank Gaussian Copula Processes

The paper under review tackles the significant challenge of high-dimensional multivariate time series forecasting. This complex problem is pervasive across several essential application areas such as financial risk management, anomaly detection, and demand forecasting. Traditional methods often fall short due to computational burdens associated with estimating high-dimensional covariance matrices and assumptions of conditional independence.

The authors propose an innovative solution by integrating an RNN-based time series model with a Gaussian copula process that utilizes a low-rank covariance structure. This model addresses the computational complexities and allows for the modeling of thousands of time series with non-Gaussian marginals. The core contribution is the effective reduction in the number of parameters needed, which facilitates significant accuracy improvements over existing state-of-the-art methods, as validated against multiple real-world datasets.

Technical Approach

The proposed method employs a combination of a Recurrent Neural Network (RNN) for temporal dynamics modeling and a Gaussian copula for dependency modeling across time series. The low-rank plus diagonal parametrization of the covariance matrix enables tractability for high-dimensional data by reducing both the number of parameters and computational load, thus allowing scalability to previously unattainable dimensions.

The framework operates by transforming each time series' marginal distribution separately using a non-parametric CDF estimate, feeding into a Gaussian copula for capturing dependencies. This decouples marginal and joint distribution estimations, offering robustness to varying scales and non-Gaussian distributions typical in real-world data.

Empirical Validation

The effectiveness of the proposed framework is evidenced through comprehensive experiments across synthetic and real-world datasets, comprising exchange rates, solar power production, electricity consumption, and traffic metrics. The results consistently demonstrate superior performance over baseline models such as Vector Autoregression (VAR) and multivariate GARCH models. The use of the continuous ranked probability score (CRPS) as a metric underscores the predictive accuracy improvements.

Particularly notable is the scalability of the model, with empirical tests confirming its applicability to dimensions as large as 2,000, surpassing prior approaches constrained to hundreds. The ablation studies further underscore the impact of the Gaussian copula approach in handling scale variations and improving model robustness.

Implications and Future Directions

Practically, this research holds promise for various domains where Reliable time series forecasting is critical. Financial institutions can leverage the model to enhance risk portfolio strategies with better covariance forecasting, while e-commerce platforms might improve demand predictions, incorporating complex interdependencies among product sales. Anomaly detection systems, too, gain from the model's ability to account for correlated deviations across multiple metrics.

Theoretical implications suggest a broader applicability of copula processes in AI-driven multivariate analysis, beyond just time series. Future research could extend this work by exploring adaptive low-rank structures or integrating additional non-linear transformations for marginals to further enhance model robustness.

This paper presents a compelling case for the integration of deep learning architectures with copula-based statistical approaches, paving the way for novel solutions in high-dimensional data forecasting. Such interdisciplinary approaches are essential as AI continues to tackle ever-more complex real-world challenges.