Normalized Multivariate Time Series Causality Analysis and Causal Graph Reconstruction

Abstract

Causality analysis is an important problem lying at the heart of science, and is of particular importance in data science and machine learning. An endeavor during the past 16 years viewing causality as a real physical notion so as to formulate it from first principles, however, seems to have gone unnoticed. This study introduces to the community this line of work, with a long-due generalization of the information flow-based bivariate time series causal inference to multivariate series, based on the recent advance in theoretical development. The resulting formula is transparent, and can be implemented as a computationally very efficient algorithm for application. It can be normalized and tested for statistical significance. Different from the previous work along this line where only information flows are estimated, here an algorithm is also implemented to quantify the influence of a unit to itself. While this forms a challenge in some causal inferences, here it comes naturally, and hence the identification of self-loops in a causal graph is fulfilled automatically as the causalities along edges are inferred. To demonstrate the power of the approach, presented here are two applications in extreme situations. The first is a network of multivariate processes buried in heavy noises (with the noise-to-signal ratio exceeding 100), and the second a network with nearly synchronized chaotic oscillators. In both graphs, confounding processes exist. While it seems to be a challenge to reconstruct from given series these causal graphs, an easy application of the algorithm immediately reveals the desideratum. Particularly, the confounding processes have been accurately differentiated. Considering the surge of interest in the community, this study is very timely.

The Causal Interaction between Complex Subsystems

“…We only check the formula for T B→A . In (20), the deterministic part By the compactness of ρ, the whole deterministic part hence vanishes. Likewise, it can be proved that the stochastic part also vanishes.…”

“…Initialized with random numbers, we iterate the process for 20,000 steps and discard the first 10,000 steps to form six time series with a length of 10,000 steps. Using the algorithm by Liang (e.g., [ 16 , 18 , 19 , 20 ]), the information flows between and can be rather accurately obtained. As shown in Figure 2 a, the information flow/causality from X to Y increases with , and there is no causality the other way around, just as expected.…”

“…One may expect that the PCA method should yield more reasonable causal patterns. We have computed the first PCs for and , respectively, and estimated the information flows using the algorithm by Liang [ 20 ]. The resulting distributions, however, are no better than those with the averaged series ( Figure 6 ).…”

“…When two processes interact, IF provides not only the direction but also the strength of the interaction. Thus far, the formalism of the IF between two components has been well established (see [ 16 , 17 , 18 , 19 , 20 ], among others). It has been shown promising to extend the formalism to subspaces with many components involved.…”

The Role of Instrumental Variables in Causal Inference Based on Independence of Cause and Mechanism

“…The most studied causal inference case is probably the case of time series [27], where the Granger causality can be applied. We would like to underline that we consider the case of observational non-temporal data in the current contribution, and the results on the time series are beyond the scope of our paper.…”