# Normalized Multivariate Time Series Causality Analysis and Causal Graph Reconstruction

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## Abstract

**:**

## 1. Introduction

## 2. An Overview of the Theory of Information Flow-Based Causality Analysis

#### 2.1. Directed Graph, Uncertainty Propagation, and Causality

**Definition**

**1.**

**Definition**

**2.**

#### 2.2. A Brief Stroll through the Theory and Recent Advances

If the evolution of an event, say, ${X}_{1}$, is independent of another one, ${X}_{2}$, then the information flow from ${X}_{2}$ to ${X}_{1}$ is zero.

**Theorem**

**1.**

**Corollary**

**1.**

**Theorem**

**3.**

**Theorem**

**4.**

**Corollary**

**2.**

**Theorem**

**5.**

## 3. Information Flow among Time Series and Algorithm for Multivariate Causal Inference

**Theorem**

**6.**

**Proof.**

Algorithm 1: Quantitative causal inference |

Input : d time seriesOutput: a DG $\mathcal{G}=(V,E)$, and IFs along edges initialize $\mathcal{G}$ such that all vertexes are isolated;set a significance level $\alpha $; for each $(i,j)\in V\times V$ docompute ${\widehat{T}}_{i\to j}$ by (14); if ${\widehat{T}}_{i\to j}$ is significant at level $\alpha $ thenadd $i\to j$ to $\mathcal{G}$; record ${\widehat{T}}_{i\to j}$; endendreturn $\mathcal{G}$, together with the IFs ${\widehat{T}}_{i\to j}$ |

## 4. Normalization of the Causality among Multivariate Time Series

## 5. Application to Causal Graph Reconstruction

#### 5.1. A Noisy Causal Network from Autoregressive Processes

#### 5.2. A Network of Nearly Synchronized Chaotic Series

## 6. Conclusions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(

**a**) A schematic of the directed network generated with the vector autoregressive processes (21). (

**b**) The directed graph reconstructed from the six time series. Overlaid numbers are the respective significant information flows (in nats per time step); also overlaid are the inferred self loops or trivial cycles of length 1 (in light blue).

**Figure 2.**The information flows among the oscillators X, Y, and Z (in nats/unit time) versus the coupling strength $\u03f5$: (

**a**) $|{T}_{X\to Y}|$ (blue) and $|{T}_{Y\to X}|$ (red); (

**b**) $|{T}_{X\to Z}|$ (blue) and $|{T}_{Z\to X}|$ (red); (

**c**) $|{T}_{Y\to Z}|$ (blue) and $|{T}_{Y\to Z}|$ (red). (

**d**) The series of ${X}_{1}$, ${Y}_{1}$, and ${Z}_{1}$ on a time interval when the coupling parameter $\u03f5=0.25$. (Note, in solving for $(X,Y,Z)$, the initial conditions are randomly chosen, some of which may happen to make a highly singular matrix and hence cause large errors. In that case, simply re-run the program.)

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**MDPI and ACS Style**

Liang, X.S.
Normalized Multivariate Time Series Causality Analysis and Causal Graph Reconstruction. *Entropy* **2021**, *23*, 679.
https://doi.org/10.3390/e23060679

**AMA Style**

Liang XS.
Normalized Multivariate Time Series Causality Analysis and Causal Graph Reconstruction. *Entropy*. 2021; 23(6):679.
https://doi.org/10.3390/e23060679

**Chicago/Turabian Style**

Liang, X. San.
2021. "Normalized Multivariate Time Series Causality Analysis and Causal Graph Reconstruction" *Entropy* 23, no. 6: 679.
https://doi.org/10.3390/e23060679