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.

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.

Structured regression on graphs aims to predict response variables from multiple nodes by discovering and exploiting the dependency structure among response variables. This problem is challenging since dependencies among response variables are always unknown, and the associated prior knowledge is non-symmetric. In previous studies, various promising solutions were proposed to improve structured regression by utilizing symmetric prior knowledge, learning sparse dependency structure among response variables, or learning representations of attributes of multiple nodes. However, none of them are capable of efficiently learning dependency structure while incorporating non-symmetric prior knowledge. To achieve these objectives, we proposed Continuous Conditional Dependency Network (CCDN) for structured regression. The intuitive idea behind this model is that each response variable is not only dependent on attributes from the same node, but also on response variables from all other nodes. This results in a joint modeling of local conditional probabilities. The parameter learning is formulated as a convex optimization problem and an effective sampling algorithm is proposed for inference. CCDN is flexible in absorbing non-symmetric prior knowledge. The performance of CCDN on multiple datasets provides evidence of its structure recovery ability and superior effectiveness and efficiency as compared to the state-of-the-art alternatives.

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.

Gaussian Conditional Random Fields (GCRF) are atype of structured regression model that incorporatesmultiple predictors and multiple graphs. This isachieved by defining quadratic term feature functions inGaussian canonical form which makes the conditionallog-likelihood function convex and hence allows findingthe optimal parameters by learning from data. In thiswork, the parameter space for the GCRF model is extendedto facilitate joint modelling of positive and negativeinfluences. This is achieved by restricting the modelto a single graph and formulating linear bounds on convexitywith respect to the models parameters. In addition,our formulation for the model using one networkallows calculating gradients much faster than alternativeimplementations. Lastly, we extend the model onestep farther and incorporate a bias term into our linkweight. This bias is solved as part of the convex optimization.Benefits of the proposed model in terms ofimproved accuracy and speed are characterized on severalsynthetic graphs with 2 million links as well as on ahospital admissions prediction task represented as a humandisease-symptom similarity network correspondingto more than 35 million hospitalization records inCalifornia over 9 years.

When used for structured regression, powerful Conditional Random Fields (CRFs) are typically restricted to modeling effects of interactions among examples in local neighborhoods. Using more expressive representation would result in dense graphs, making these methods impractical for large-scale applications. To address this issue, we propose an effective CRF model with linear scale-up properties regarding approximate learning and inference for structured regression on large, fully connected graphs. The proposed method is validated on real-world large-scale problems of image de-noising and remote sensing. In conducted experiments, we demonstrated that dense connectivity provides an improvement in prediction accuracy. Inference time of less than ten seconds on graphs with millions of nodes and trillions of edges makes the proposed model an attractive tool for large-scale, structured regression problems.

This study considers the problem of feature selection in incomplete data. The intuitive approach is to first impute the missing values, and then apply a standard feature selection method to select relevant features. In this study, we show how to perform feature selection directly, without imputing missing values. We define the objective function of the uncertainty margin-based feature selection method to maximize each instance’s uncertainty margin in its own relevant subspace. In optimization, we take into account the uncertainty of each instance due to the missing values. The experimental results on synthetic and 6 benchmark data sets with few missing values (less than 25%) provide evidence that our method can select the same accurate features as the alternative methods which apply an imputation method first. However, when there is a large fraction of missing values (more than 25%) in data, our feature selection method outperforms the alternatives, which impute missing values first.

Analog neurons with limited precision essentially compute k-ary weighted multilinear threshold functions.

Analog neurons with limited precision essentially compute k-ary weighted multilinear threshold functions.