Gaussian Conditional Random Fields (GCRF), as a structured regression model, is designed to achieve higher regression accuracy than unstructured predictors at the expense of execution time, taking into account the objects similarities and the outputs of unstructured predictors simultaneously. As most structural models, the GCRF model does not scale well with large networks. One of the approaches consists of performing calculations on factor graphs (if it is possible) rather than on the full graph, which is more computationally efficient. The Kronecker product of the graphs appears to be a natural choice for a graph decomposition. However, this idea is not straightforwardly applicable for GCRF, since characterizing a Laplacian spectrum of the Kronecker product of graphs, which GCRF is based on, from spectra of its factor graphs has remained an open problem. In this paper we apply new estimations for the Laplacian eigenvalues and eigenvectors, and achieve high prediction accuracy of the proposed models, while the computational complexity of the models, compared to the original GCRF model, is improved from $O(n_{1}^{3}n_{2}^{3})$ to $O(n_{1}^{3} + n_{2}^{3})$. Furthermore, we study the GCRF model with a non-Kronecker graph, where the model consists of finding the nearest Kronecker product of graph for an initial graph. Although the proposed models are more complex, they achieve high prediction accuracy too, while the execution time is still much better compare to the original GCRF model. The effectiveness of the proposed models is characterized on three types of random networks where the proposed models were consistently away more accurate than the previously presented GCRF model for multiscale networks [Jesse Glass and Zoran Obradovic. Structured regression on multiscale networks. IEEE Intelligent Systems, 32(2):23-30, 2017.].

For highly sensitive real-world predictive analytic applications such as healthcare and medicine, having good prediction accuracy alone is often not enough. These kinds of applications require a decision making process which uses uncertainty estimation as input whenever possible. Quality of uncertainty estimation is a subject of over or under confident prediction, which is often not addressed in many models. In this paper we show several extensions to the Gaussian Conditional Random Fields model, which aim to provide higher quality uncertainty estimation. These extensions are applied to the temporal disease graph built from the State Inpatient Database (SID) of California, acquired from the HCUP. Our experiments demonstrate benefits of using graph information in modeling temporal disease properties as well as improvements in uncertainty estimation provided by given extensions of the Gaussian Conditional Random Fields method.

It is of high interest for a company to identify customers expected to bring the largest profit in the upcoming period. Knowing as much as possible about each customer is crucial for such predictions. However, their demographic data, preferences, and other information that might be useful for building loyalty programs is often missing. Additionally, modeling relations among different customers as a network can be beneficial for predictions at an individual level, as similar customers tend to have similar purchasing patterns. We address this problem by proposing a robust framework for structured regression on deficient data in evolving networks with a supervised representation learning based on neural features embedding. The new method is compared to several unstructured and structured alternatives for predicting customer behavior (e.g. purchasing frequency and customer ticket) on user networks generated from customer databases of two companies from different industries. The obtained results show $4\%$ to $130\%$ improvement in accuracy over alternatives when all customer information is known. Additionally, the robustness of our method is demonstrated when up to $80\%$ of demographic information was missing where it was up to several folds more accurate as compared to alternatives that are either ignoring cases with missing values or learn their feature representation in an unsupervised manner.

Conditional probabilistic graphical models provide a powerful framework for structured regression in spatio-temporal datasets with complex correlation patterns. However, in real-life applications a large fraction of observations is often missing, which can severely limit the representational power of these models. In this paper we propose a Marginalized Gaussian Conditional Random Fields (m-GCRF) structured regression model for dealing with missing labels in partially observed temporal attributed graphs. This method is aimed at learning with both labeled and unlabeled parts and effectively predicting future values in a graph. The method is even capable of learning from nodes for which the response variable is never observed in history, which poses problems for many state-of-the-art models that can handle missing data. The proposed model is characterized for various missingness mechanisms on 500 synthetic graphs. The benefits of the new method are also demonstrated on a challenging application for predicting precipitation based on partial observations of climate variables in a temporal graph that spans the entire continental US. We also show that the method can be useful for optimizing the costs of data collection in climate applications via active reduction of the number of weather stations to consider. In experiments on these real-world and synthetic datasets we show that the proposed model is consistently more accurate than alternative semi-supervised structured models, as well as models that either use imputation to deal with missing values or simply ignore them altogether.

Conditional probabilistic graphical models provide a powerful Thus, a particular interest of this paper is long-term forecasting framework for structured regression in spatiotemporal on non-static networks with continuous target variables datasets with complex correlation patterns. It has been (structured regression) and proper uncertainty propagation shown that models utilizing underlying correlation patterns estimate in such evolving networks. This is motivated (structured models) can significantly improve predictive accuracy by climate modeling of long-term precipitation prediction in as compared to models not utilizing such information spatiotemporal weather station networks, as well as prediction (Radosavljevic, Vucetic, and Obradovic 2010; 2014; of different disease trends in temporal disease-disease Ristovski et al. 2013; Wytock and Kolter 2013; Stojanovic networks.