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An approach is proposed for inferring Granger causality between jointly stationary, Gaussian signals from quantized data. First, a necessary and sufficient rank criterion for the equality of two conditional Gaussian distributions is proved. Assuming a partial finite-order Markov property, conditions are then derived under which Granger causality between them can be reliably inferred from the second order moments of the quantized processes. A necessary and sufficient condition is proposed for Granger causality inference under binary quantization. Furthermore, sufficient conditions are introduced to infer Granger causality between jointly Gaussian signals through measurements quantized via non-uniform, uniform or high resolution quantizers. This approach does not require the statistics of the underlying Gaussian signals to be estimated, or a system model to be identified. No assumptions are made on the identifiability of the jointly Gaussian random processes through the quantized observations. The effectiveness of the proposed method is illustrated by simulation results.
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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.
Granger causality is a statistical notion of causal influence based on prediction via vector autoregression. For Gaussian variables it is equivalent to transfer entropy, an information-theoretic measure of time-directed information transfer between jointly dependent processes. We exploit such equivalence and calculate exactly the local Granger causality, i.e. the profile of the information transfer at each discrete time point in Gaussian processes; in this frame Granger causality is the average of its local version. Our approach offers a robust and computationally fast method to follow the information transfer along the time history of linear stochastic processes, as well as of nonlinear complex systems studied in the Gaussian approximation.
Networked dynamic systems are often abstracted as directed graphs, where the observed system processes form the vertex set and directed edges are used to represent non-zero transfer functions. Recovering the exact underlying graph structure of such a networked dynamic system, given only observational data, is a challenging task. Under relatively mild well-posedness assumptions on the network dynamics, there are state-of-the-art methods which can guarantee the absence of false positives. However, in this article we prove that under the same well-posedness assumptions, there are instances of networks for which any method is susceptible to inferring false negative edges or false positive edges. Borrowing a terminology from the theory of graphical models, we say those systems are unfaithful to their networks. We formalize a variant of faithfulness for dynamic systems, called Granger-faithfulness, and for a large class of dynamic networks, we show that Granger-unfaithful systems constitute a Lebesgue zero-measure set. For the same class of networks, under the Granger-faithfulness assumption, we provide an algorithm that reconstructs the network topology with guarantees for no false positive and no false negative edges in its output. We augment the topology reconstruction algorithm with orientation rules for some of the inferred edges, and we prove the rules are consistent under the Granger-faithfulness assumption.
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.
Climate system teleconnections, which are far-away climate responses to perturbations or oscillations, are difficult to quantify, yet understanding them is crucial for improving climate predictability. Here we leverage Granger causality in a novel method of identifying teleconnections. Because Granger causality is explicitly defined as a statistical test between two time series, our method allows for immediate interpretation of causal relationships between any two fields and provides an estimate of the timescale of the teleconnection response. We demonstrate the power of this new method by recovering known seasonal precipitation responses to the sea surface temperature pattern associated with the El Ni~{n}o Southern Oscillation, with accuracy comparable to previously used correlation-based methods. By adjusting the maximum lag window, Granger causality can evaluate the strength of the teleconnection (the seasonal precipitation response) on different timescales; the lagged correlation method does not show ability to differentiate signals at different lags. We also identify candidates for previously unexplored teleconnection responses, highlighting the improved sensitivity of this method over previously used ones.
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