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Kernel Granger causality and the analysis of dynamical networks
Daniele Marinazzo, Mario Pellicoro, Sebastiano Stramaglia
TL;DR
The paper addresses how to infer topology and directed interactions in dynamical networks from time series, including nonlinear and cyclic systems. It generalizes kernel Granger causality to multivariate data, controlling model complexity and filtering additional features. The approach reconstructs network structure when sufficient samples are available and identifies causal relationships in simulated and HeLa gene-expression data.
Problem
The paper addresses the need for effective inference of dynamical-network structure and causal influences from time-series data.
Method
The method generalizes kernel Granger causality to multivariate time series, with kernel choice controlling nonlinearity and eigenvector selection addressing false causalities.
Results
The approach perfectly reconstructs a chaotic-map network with sufficient samples, outperforms DBN in a simulated genetic network, and identifies 19 causal relationships in HeLa expression data.
Takeaways & Limitations
Kernel Granger analysis provides a statistically robust tool for assessing drive-response relationships across scientific fields, including networks with cycles.
Takeaways & Limitations
The approach requires many samples for reliable answers, and its performance depends on sampling time and the suitability of the chosen kernel.
Abstract
from arXiv · showhide
We propose a method of analysis of dynamical networks based on a recent measure of Granger causality between time series, based on kernel methods. The generalization of kernel Granger causality to the multivariate case, here presented, shares the following features with the bivariate measures: (i) the nonlinearity of the regression model can be controlled by choosing the kernel function and (ii) the problem of false-causalities, arising as the complexity of the model increases, is addressed by a selection strategy of the eigenvectors of a reduced Gram matrix whose range represents the additional features due to the second time series. Moreover, there is no {\it a priori} assumption that the network must be a directed acyclic graph. We apply the proposed approach to a network of chaotic maps and to a simulated genetic regulatory network: it is shown that the underlying topology of the network can be reconstructed from time series of node's dynamics, provided that a sufficient number of samples is available. Considering a linear dynamical network, built by preferential attachment scheme, we show that for limited data use of bivariate Granger causality is a better choice w.r.t methods using $L1$ minimization. Finally we consider real expression data from HeLa cells, 94 genes and 48 time points. The analysis of static correlations between genes reveals two modules corresponding to well known transcription factors; Granger analysis puts in evidence nineteen causal relationships, all involving genes related to tumor development.
I. INTRODUCTION
The paper develops kernel Granger causality for inferring dynamical-network topology and drive-response relationships from multivariate time series. It extends prior nonlinear and over-fitting-controlled Granger methods to multivariate data.
- Dynamical networks span physical and biological systems, motivating methods that infer network structure from time-series data.
- Granger causality assesses whether adding one time series reduces prediction error for another in a regression model.
- Kernel methods embed data into a Hilbert space, enabling nonlinear regression through an implicitly specified inner product.
- The paper generalizes kernel Granger causality from bivariate to multivariate time series for estimating network topology and drive-response relationships.
- The paper analyzes bivariate kernel Granger causality before extending the method to multivariate data and dynamical-network applications.
A. Linear Granger causality
Linear Granger causality compares autoregressive predictions with and without a second time series, using projections onto nested feature spaces. Statistical filtering selects additional features associated with the candidate cause.
- Linear Granger causality models a stationary target series with an autoregressive model of order m.
- Including a second series produces a bivariate autoregressive model whose coefficients are estimated by standard least squares.
- η Granger-causes ξ when the augmented model has significantly smaller residual variance than the target-only model.
- The target and augmented predictions are projections onto H and H′, with H′ decomposed into the original space H and additional features H⊥.
- The filtered index sums contributions from reduced-Gram-matrix eigenvectors whose correlations with the residual pass Bonferroni correction, using q = 0.05.
B. Kernel Granger causality
Kernel Granger causality replaces linear regression features with kernel-induced Hilbert-space features, allowing nonlinear models while filtering statistically unsupported additional components. Polynomial and Gaussian kernels provide distinct complexity controls.
- Kernel Granger causality projects the target onto the range of a centered Gram matrix, where regression is nonlinear in the original variables.
- The inhomogeneous polynomial kernel represents monomials up to degree p, with p = 1 corresponding to linear regression.
- The additional feature space is obtained from a reduced Gram matrix formed by removing the projection associated with the target-only inputs.
- Eigenvectors of the reduced Gram matrix are retained only when their correlations pass a Bonferroni test, limiting false causalities as model complexity increases.
- For the Gaussian kernel, width σ controls model complexity because the Gram-matrix range dimension decreases as σ increases.
- The Gaussian-kernel procedure retains eigenvectors of the augmented Gram matrix whose eigenvalues are at least µ times its largest eigenvalue.
III. MULTIVARIATE KERNEL CAUSALITY
The multivariate formulation evaluates whether removing one source series changes prediction of a target while retaining all other recorded inputs. Repeating this across source-target pairs yields a network-wide causality pattern.
- The method processes M simultaneously recorded time series jointly rather than evaluating only isolated pairs.
- For testing ξ^(a) → ξ^(b), one input representation contains all variables, while the reduced representation excludes variables related to ξ^(a).
- The target vector consists of lagged samples of ξ^(b), and the resulting index measures the strength of a → b while accounting for all available variables.
- Repeating the calculation for every ordered source-target pair reconstructs the causality pattern in the dataset.
- The multivariate Bonferroni threshold is lowered according to the number of tested pairs, M(M − 1)/2.
A. Three coupled maps
The paper evaluates multivariate and bivariate Granger causality on three coupled logistic maps with known directed relationships. Using time-series segments of length N = 1000 and an IP kernel, it compares results across polynomial degrees.
- The system contains three logistic maps with implemented causal relationships 2 →1 and 1 →3.
- Segments of length N = 1000 are analyzed using multivariate and bivariate causality for all map pairs.
- The analysis uses the IP kernel with various values of p, with results displayed in figure (1).
IV. ANALYSIS OF DYNAMICAL NETWORKS
The paper applies multivariate Granger analysis to simulated dynamical networks to estimate their topology from node time series. The simulations cover interacting chaotic maps and a genetic regulatory network.
- Multivariate Granger analysis is applied to simulated dynamical networks to estimate topology from time series data.
- The simulated systems include interacting chaotic maps and a model of gene regulatory networks.
A. Network of chaotic maps
A 34-node directed network of coupled chaotic maps is reconstructed from node measurements using multivariate kernel Granger analysis. Reconstruction is perfect with sufficient samples, but causality strengths vary by link and should not be equated with coupling magnitudes.
- The 34-node network assigns directions randomly to Zachary-network links, with coupling c_ij = 0.05 on directed links and zero otherwise.
- Perfect recovery of the underlying network is obtained with the IP kernel at p = 2, m = 1, and N = 10000.
- At a threshold plateau around r = 0.1, 428 projections are selected, matching Bonferroni’s test, and the original network is perfectly reconstructed.
- The reconstructed network typically becomes perfect when N ≥5000, while errors appear as the sample size decreases.
- Causality strengths depend on the link even when all couplings have equal magnitude, because Granger causality measures information transfer rather than coupling itself.
B. Genetic regulatory network
The method is tested on a ten-gene regulatory network with nonlinear bounded expression dynamics and two independent pathways. Reconstruction is strong under the reported sampling setup, but performance depends on sampling rate.
- Expression dynamics are nonlinear because gene values are restricted to [0, 100] by floor and ceiling functions.
- The simulated genetic network contains ten genes arranged in two independent regulatory pathways, including one large feedback structure.
- With N = 2000, ts = 5, and nc = 3, linear multivariate Granger causality perfectly reconstructs the network in 90 of 100 simulations.
- IP kernels with p > 1 and Gaussian kernels achieve performances similar to the linear kernel on the reported examples.
- Reconstruction performance degrades as the sampling interval ts moves away from 5 for both linear and Gaussian kernels.
- At large ts, Granger causality becomes ineffective, whereas static correlations still reveal groups of genes within the same regulatory pathway.
V. SPARSELY CONNECTED DYNAMICAL NETWORKS AND LIMITED DATA
The multivariate Granger approach needs many samples, making it unsuitable when variables outnumber observations. Under sparse connectivity and limited data, bivariate Granger causality can outperform L1 minimization, although detected links may be mediated.
- The multivariate Granger approach requires a large number of samples, and is unfeasible when the number of samples is smaller than the number of genes.This setting is frequent in Bioinformatics.
- Bivariate Granger causality can retain statistically robust pairwise information-flow estimates with limited data, but some detected links may be indirect and mediated.This trade-off applies under the assumption of sparse connectivity.
- A 100-node, 100-link directed network was generated by preferential attachment and used to evolve a linear dynamical system.Link directions were assigned randomly with probability 1/2.
- The bivariate Granger approach found more true-positive connections than L1 minimization as the number of samples increased.The comparison is reported for the preferential-attachment linear network.
A. HeLa gene expression regulatory network
HeLa gene-expression correlations reveal two related modules associated with NFkB and p53/STAT3, while bivariate Granger analysis identifies a smaller set of directed relationships on the sampling-time scale. The detected causalities involve genes associated with tumor development.
- 800 pairs of HeLa genes were significantly correlated, forming two related modules associated with NFkB and p53/STAT3.The first module contains 23 genes, the second 62, and c-myc is central between them with 12 significant correlations.
- Bivariate Granger causality was applied to all gene pairs using the linear version with p = 1 and m = 1.Surrogate permutations were used instead of a t-test because the dataset contained few samples.
- 19 causal relationships were selected from 8742 tested relationships.The detected links include IkappaBa → NFkB, NFkB relationships with IAP and B99, and relationships involving apoptosis and tumor-related genes.
- A prior sparse multivariate L1 analysis recovered only A20 → Bcl-XL among the 19 relationships listed in this study.That prior analysis selected its regularization parameter λ by cross-validation.
VI. DISCUSSION
The kernel Granger approach reconstructs dynamical-network structure and identifies causal relationships, while its reliability depends on sample size, sampling rate, and kernel choice.
- Perfect reconstruction of a directed chaotic-map network is achieved when a sufficient number of samples is available.The typical case requires N ≥5000 samples; reducing N introduces reconstruction errors.
- The method outperforms DBN on the simulated genetic regulatory network, but performance depends strongly on sampling time.Performance degrades as the sampling interval moves away from ts = 5.
- For sparse linear networks with limited data, bivariate Granger analysis can outperform multivariate L1 minimization when the multivariate approach is unfeasible.The comparison was made on a preferential-attachment network with known connectivity.
- HeLa expression analysis identified two static-correlation modules and 19 causal relationships involving genes related to tumor development.The causal relationships acted on the time scale of one hour and showed little overlap with multivariate L1-minimization results.
- The approach is intended as a statistically robust tool for assessing drive-response relationships across scientific fields.The paper emphasizes cause-effect detection between components of complex systems.