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Register a new userDirect estimation of differential Granger causality between two high-dimensional time series
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Added by
Yue Wang
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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Differential Granger causality, that is understanding how Granger causal relations differ between two related time series, is of interest in many scientific applications. Modeling each time series by a vector autoregressive (VAR) model, we propose a new method to directly learn the difference between the corresponding transition matrices in high dimensions. Key to the new method is an estimating equation constructed based on the Yule-Walker equation that links the difference in transition matrices to the difference in the corresponding precision matrices. In contrast to separately estimating each transition matrix and then calculating the difference, the proposed direct estimation method only requires sparsity of the difference of the two VAR models, and hence allows hub nodes in each high-dimensional time series. The direct estimator is shown to be consistent in estimation and support recovery under mild assumptions. These results also lead to novel consistency results with potentially faster convergence rates for estimating differences between precision matrices of i.i.d observations under weaker assumptions than existing results. We evaluate the finite sample performance of the proposed method using simulation studies and an application to electroencephalogram (EEG) data.
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We study causality between bivariate curve time series using the Granger causality generalized measures of correlation. With this measure, we can investigate which curve time series Granger-causes the other; in turn, it helps determine the predictability of any two curve time series. Illustrated by a climatology example, we find that the sea surface temperature Granger-causes the sea-level atmospheric pressure. Motivated by a portfolio management application in finance, we single out those stocks that lead or lag behind Dow-Jones industrial averages. Given a close relationship between S&P 500 index and crude oil price, we determine the leading and lagging variables.
In the study of complex physical and biological systems represented by multivariate stochastic processes, an issue of great relevance is the description of the system dynamics spanning multiple temporal scales. While methods to assess the dynamic complexity of individual processes at different time scales are well-established, multiscale analysis of directed interactions has never been formalized theoretically, and empirical evaluations are complicated by practical issues such as filtering and downsampling. Here we extend the very popular measure of Granger causality (GC), a prominent tool for assessing directed lagged interactions between joint processes, to quantify information transfer across multiple time scales. We show that the multiscale processing of a vector autoregressive (AR) process introduces a moving average (MA) component, and describe how to represent the resulting ARMA process using state space (SS) models and to combine the SS model parameters for computing exact GC values at arbitrarily large time scales. We exploit the theoretical formulation to identify peculiar features of multiscale GC in basic AR processes, and demonstrate with numerical simulations the much larger estimation accuracy of the SS approach compared with pure AR modeling of filtered and downsampled data. The improved computational reliability is exploited to disclose meaningful multiscale patterns of information transfer between global temperature and carbon dioxide concentration time series, both in paleoclimate and in recent years.
Continuous, automated surveillance systems that incorporate machine learning models are becoming increasingly more common in healthcare environments. These models can capture temporally dependent changes across multiple patient variables and can enhance a clinicians situational awareness by providing an early warning alarm of an impending adverse event such as sepsis. However, most commonly used methods, e.g., XGBoost, fail to provide an interpretable mechanism for understanding why a model produced a sepsis alarm at a given time. The black-box nature of many models is a severe limitation as it prevents clinicians from independently corroborating those physiologic features that have contributed to the sepsis alarm. To overcome this limitation, we propose a generalized linear model (GLM) approach to fit a Granger causal graph based on the physiology of several major sepsis-associated derangements (SADs). We adopt a recently developed stochastic monotone variational inequality-based estimator coupled with forwarding feature selection to learn the graph structure from both continuous and discrete-valued as well as regularly and irregularly sampled time series. Most importantly, we develop a non-asymptotic upper bound on the estimation error for any monotone link function in the GLM. We conduct real-data experiments and demonstrate that our proposed method can achieve comparable performance to popular and powerful prediction methods such as XGBoost while simultaneously maintaining a high level of interpretability.
Granger causality has been employed to investigate causality relations between components of stationary multiple time series. We generalize this concept by developing statistical inference for local Granger causality for multivariate locally stationary processes. Our proposed local Granger causality approach captures time-evolving causality relationships in nonstationary processes. The proposed local Granger causality is well represented in the frequency domain and estimated based on the parametric time-varying spectral density matrix using the local Whittle likelihood. Under regularity conditions, we demonstrate that the estimators converge to multivariate normal in distribution. Additionally, the test statistic for the local Granger causality is shown to be asymptotically distributed as a quadratic form of a multivariate normal distribution. The finite sample performance is confirmed with several simulation studies for multivariate time-varying autoregressive models. For practical demonstration, the proposed local Granger causality method uncovered new functional connectivity relationships between channels in brain signals. Moreover, the method was able to identify structural changes in financial data.
Introduced more than a half century ago, Granger causality has become a popular tool for analyzing time series data in many application domains, from economics and finance to genomics and neuroscience. Despite this popularity, the validity of this notion for inferring causal relationships among time series has remained the topic of continuous debate. Moreover, while the original definition was general, limitations in computational tools have primarily limited the applications of Granger causality to simple bivariate vector auto-regressive processes or pairwise relationships among a set of variables. Starting with a review of early developments and debates, this paper discusses recent advances that address various shortcomings of the earlier approaches, from models for high-dimensional time series to more recent developments that account for nonlinear and non-Gaussian observations and allow for sub-sampled and mixed frequency time series.
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