Trends in terrestrial temperature variability are perhaps more relevant for species viability than trends in mean temperature. In this paper, we develop methodology for estimating such trends using multi-resolution climate data from polar orbiting weather satellites. We derive two novel algorithms for computation that are tailored for dense, gridded observations over both space and time. We evaluate our methods with a simulation that mimics these data's features and on a large, publicly available, global temperature dataset with the eventual goal of tracking trends in cloud reflectance temperature variability.
Our methodology is similar in flavor to @cite_7 or related work in , but with several fundamental differences. These papers aim to discover the time-varying structure of a network. To achieve this goal, they use Gaussian likelihood with unknown precision matrix and introduce penalty terms which (1) encourage sparsity among the off-diagonal elements and (2) discourage changes in the estimated inverse covariance matrix from one time-step to the next. Our goal in the present work is to detect the temporal trend in the variance of each point in the network, but the network is known (corresponding to the grid over the earth) and fixed in time. To apply these methods in our context (e.g., HallacPark2017 , Eq. 2), we would enforce complete sparsity on the off-diagonal elements (since they are not estimated) and add a new penalty to enforce spatial behavior across the diagonal elements. Thus, is not simply a special case of these existing methods. Finally, these papers examine networks with hundreds of nodes and dozens to hundreds of time points. As discussed next, our data are significantly larger than these networks and attempting to estimate a full covariance would be prohibitive, were it necessary.
@cite_7
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Many important problems can be modeled as a system of interconnected entities, where each entity is recording time-dependent observations or measurements. In order to spot trends, detect anomalies, and interpret the temporal dynamics of such data, it is essential to understand the relationships between the different entities and how these relationships evolve over time. In this paper, we introduce the time-varying graphical lasso (TVGL), a method of inferring time-varying networks from raw time series data. We cast the problem in terms of estimating a sparse time-varying inverse covariance matrix, which reveals a dynamic network of interdependencies between the entities. Since dynamic network inference is a computationally expensive task, we derive a scalable message-passing algorithm based on the Alternating Direction Method of Multipliers (ADMM) to solve this problem in an efficient way. We also discuss several extensions, including a streaming algorithm to update the model and incorporate new observations in real time. Finally, we evaluate our TVGL algorithm on both real and synthetic datasets, obtaining interpretable results and outperforming state-of-the-art baselines in terms of both accuracy and scalability.