Grouped graphical Granger modeling for gene expression regulatory networks discovery

We consider the problem of discovering gene regulatory networks from time-series microarray data. Recently, graphical Granger modeling has gained considerable attention as a promising direction for addressing this problem. These methods apply graphical modeling methods on time-series data and invoke the notion of ‘Granger causality’ to make assertions on causality through inference on time-lagged effects. Existing algorithms, however, have neglected an important aspect of the problem—the group structure among the lagged temporal variables naturally imposed by the time series they belong to. Specifically, existing methods in computational biology share this shortcoming, as well as additional computational limitations, prohibiting their effective applications to the large datasets including a large number of genes and many data points. In the present article, we propose a novel methodology which we term ‘grouped graphical Granger modeling method’, which overcomes the limitations mentioned above by applying a regression method suited for high-dimensional and large data, and by leveraging the group structure among the lagged temporal variables according to the time series they belong to. We demonstrate the effectiveness of the proposed methodology on both simulated and actual gene expression data, specifically the human cancer cell (HeLa S3) cycle data. The simulation results show that the proposed methodology generally exhibits higher accuracy in recovering the underlying causal structure. Those on the gene expression data demonstrate that it leads to improved accuracy with respect to prediction of known links, and also uncovers additional causal relationships uncaptured by earlier works

Longitudinal LASSO: Jointly Learning Features and Temporal Contingency for Outcome Prediction

Longitudinal analysis is important in many disciplines, such as the study of behavioral transitions in social science. Only very recently, feature selection has drawn adequate attention in the context of longitudinal modeling. Standard techniques, such as generalized estimating equations, have been modified to select features by imposing sparsity-inducing regularizers. However, they do not explicitly model how a dependent variable relies on features measured at proximal time points. Recent graphical Granger modeling can select features in lagged time points but ignores the temporal correlations within an individual's repeated measurements. We propose an approach to automatically and simultaneously determine both the relevant features and the relevant temporal points that impact the current outcome of the dependent variable. Meanwhile, the proposed model takes into account the non-{\em i.i.d} nature of the data by estimating the within-individual correlations. This approach decomposes model parameters into a summation of two components and imposes separate block-wise LASSO penalties to each component when building a linear model in terms of the past $\tau$ measurements of features. One component is used to select features whereas the other is used to select temporal contingent points. An accelerated gradient descent algorithm is developed to efficiently solve the related optimization problem with detailed convergence analysis and asymptotic analysis. Computational results on both synthetic and real world problems demonstrate the superior performance of the proposed approach over existing techniques.Comment: Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining. ACM, 201

Discovering Graphical Granger Causality Using the Truncating Lasso Penalty

Components of biological systems interact with each other in order to carry out vital cell functions. Such information can be used to improve estimation and inference, and to obtain better insights into the underlying cellular mechanisms. Discovering regulatory interactions among genes is therefore an important problem in systems biology. Whole-genome expression data over time provides an opportunity to determine how the expression levels of genes are affected by changes in transcription levels of other genes, and can therefore be used to discover regulatory interactions among genes. In this paper, we propose a novel penalization method, called truncating lasso, for estimation of causal relationships from time-course gene expression data. The proposed penalty can correctly determine the order of the underlying time series, and improves the performance of the lasso-type estimators. Moreover, the resulting estimate provides information on the time lag between activation of transcription factors and their effects on regulated genes. We provide an efficient algorithm for estimation of model parameters, and show that the proposed method can consistently discover causal relationships in the large $p$, small $n$ setting. The performance of the proposed model is evaluated favorably in simulated, as well as real, data examples. The proposed truncating lasso method is implemented in the R-package grangerTlasso and is available at http://www.stat.lsa.umich.edu/~shojaie.Comment: 12 pages, 4 figures, 1 tabl

Forecasting and Granger Modelling with Non-linear Dynamical Dependencies

Traditional linear methods for forecasting multivariate time series are not able to satisfactorily model the non-linear dependencies that may exist in non-Gaussian series. We build on the theory of learning vector-valued functions in the reproducing kernel Hilbert space and develop a method for learning prediction functions that accommodate such non-linearities. The method not only learns the predictive function but also the matrix-valued kernel underlying the function search space directly from the data. Our approach is based on learning multiple matrix-valued kernels, each of those composed of a set of input kernels and a set of output kernels learned in the cone of positive semi-definite matrices. In addition to superior predictive performance in the presence of strong non-linearities, our method also recovers the hidden dynamic relationships between the series and thus is a new alternative to existing graphical Granger techniques.Comment: Accepted for ECML-PKDD 201

Sparse Vector Autoregressive Modeling

The vector autoregressive (VAR) model has been widely used for modeling temporal dependence in a multivariate time series. For large (and even moderate) dimensions, the number of AR coefficients can be prohibitively large, resulting in noisy estimates, unstable predictions and difficult-to-interpret temporal dependence. To overcome such drawbacks, we propose a 2-stage approach for fitting sparse VAR (sVAR) models in which many of the AR coefficients are zero. The first stage selects non-zero AR coefficients based on an estimate of the partial spectral coherence (PSC) together with the use of BIC. The PSC is useful for quantifying the conditional relationship between marginal series in a multivariate process. A refinement second stage is then applied to further reduce the number of parameters. The performance of this 2-stage approach is illustrated with simulation results. The 2-stage approach is also applied to two real data examples: the first is the Google Flu Trends data and the second is a time series of concentration levels of air pollutants.Comment: 39 pages, 7 figure