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Register a new userMultiscale Granger causality analysis by `a trous wavelet transform
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Sebastiano Stramaglia
Publication date
2017
fields
Biology
and research's language is
English
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Since interactions in neural systems occur across multiple temporal scales, it is likely that information flow will exhibit a multiscale structure, thus requiring a multiscale generalization of classical temporal precedence causality analysis like Grangers approach. However, the computation of multiscale measures of information dynamics is complicated by theoretical and practical issues such as filtering and undersampling: to overcome these problems, we propose a wavelet-based approach for multiscale Granger causality (GC) analysis, which is characterized by the following properties: (i) only the candidate driver variable is wavelet transformed (ii) the decomposition is performed using the `a trous wavelet transform with cubic B-spline filter. We measure GC, at a given scale, by including the wavelet coefficients of the driver times series, at that scale, in the regression model of the target. To validate our method, we apply it to publicly available scalp EEG signals, and we find that the condition of closed eyes, at rest, is characterized by an enhanced GC among channels at slow scales w.r.t. eye open condition, whilst the standard Granger causality is not significantly different in the two conditions.
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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.
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.
Identifying directed interactions between species from time series of their population densities has many uses in ecology. This key statistical task is equivalent to causal time series inference, which connects to the Granger causality (GC) concept: $x$ causes $y$ if $x$ improves the prediction of $y$ in a dynamic model. However, the entangled nature of nonlinear ecological systems has led to question the appropriateness of Granger causality, especially in its classical linear Multivariate AutoRegressive (MAR) model form. Convergent-cross mapping (CCM), a nonparametric method developed for deterministic dynamical systems, has been suggested as an alternative. Here, we show that linear GC and CCM are able to uncover interactions with surprisingly similar performance, for predator-prey cycles, 2-species deterministic (chaotic) or stochastic competition, as well as 10- and 20-species interaction networks. There is no correspondence between the degree of nonlinearity of the dynamics and which method performs best. Our results therefore imply that Granger causality, even in its linear MAR($p$) formulation, is a valid method for inferring interactions in nonlinear ecological networks; using GC or CCM (or both) can instead be decided based on the aims and specifics of the analysis.
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.
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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