# Detecting Causality in Multivariate Time Series via Non-Uniform Embedding

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## Abstract

**:**

## 1. Introduction

## 2. Method

#### 2.1. PMIME Method

- An empty embedding vector $\mathbf{v}{}_{\mathbf{t}}^{\mathbf{0}}=\varnothing $ is initialized.
- At the first iteration k = 1, the embedding vector $w{}_{t}^{1}\in {W}_{t}$ is selected most related to ${y}_{t}$:$$w{}_{t}^{1}=\underset{w\in {W}_{t}}{\mathrm{arg}\mathrm{max}}I({y}_{t};w),$$
- At the iteration $k>1$, the mixed embedding vector is augmented by the component $w{}_{t}^{k}$ of ${W}_{t}$, giving most information about ${y}_{t}$ additionally to the information already contained in $\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}}=[w{}_{t}^{1},\dots ,w{}_{t}^{k-1}]$. $w{}_{t}^{k}$ will be selected by a standard through calculating the maximum value of the conditional mutual information, $w{}_{t}^{k}={\mathrm{arg}\mathrm{max}}_{w\in {W}_{t}}I({y}_{t};w|\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}})$, i.e., at the iteration $k=2$, $w{}_{t}^{2}={\mathrm{arg}\mathrm{max}}_{w\in {W}_{t}}I({y}_{t};w|\mathbf{v}{}_{\mathbf{t}}^{\mathbf{1}})$, where the CMI is calculated by the k-NNs estimator, and the mixed embedding vector is $\mathbf{v}{}_{\mathbf{t}}^{\mathbf{2}}=[w{}_{t}^{1},w{}_{t}^{2}]$. By using greedy forward method, each $w{}_{t}^{k}$ will be embedded in the already embedded vector $\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}}$ until the process stops. The termination criterion is quantified as:$$I({y}_{t};\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}})/I({y}_{t};\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}})>A,$$
- To calculate the causality strength of X on Y conditioned by the other variables in Z, the index is defined as$$R{}_{X\to Y|Z}=\frac{I({y}_{t};\mathbf{v}{}_{\mathbf{t}}^{\mathbf{x}}|\mathbf{v}{}_{\mathbf{t}}^{\mathbf{y}},\mathbf{v}{}_{\mathbf{t}}^{\mathbf{z}})}{I({y}_{t};\mathbf{v}{}_{\mathbf{t}})},$$

#### 2.2. The Proposed Method

#### 2.2.1. Low Dimensional Approximation of CMI

#### 2.2.2. Mixed Search Strategy

#### 2.2.3. LM-PMIME Method

- Initialize an empty embedding vector $\mathbf{v}{}_{\mathbf{t}}^{\mathbf{0}}=\varnothing $.
- At the first iteration k = 1, the embedding vector $w{}_{t}^{1}\in {W}_{t}$ is selected most related to ${y}_{t}$:$$w{}_{t}^{1}=\underset{w\in {W}_{t}}{\mathrm{arg}\mathrm{max}}I({y}_{t};w)$$Then we have $\mathbf{v}{}_{\mathbf{t}}^{\mathbf{1}}=\left[w{}_{t}^{1}\right]$.
- At the iteration $1<k\le m$, $w{}_{t}^{k}$ will be selected by a standard through calculating the maximum value of the low dimensional approximation of CMI.$$\begin{array}{c}\hfill w{}_{t}^{k}\phantom{\rule{-0.166667em}{0ex}}=\phantom{\rule{-0.166667em}{0ex}}\underset{w\in {W}_{t}}{\mathrm{arg}\mathrm{max}}I(w;{y}_{t})\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\beta \sum _{{w}_{i}\in \mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}}}I(w;{w}_{i})\phantom{\rule{-0.166667em}{0ex}}+\phantom{\rule{-0.166667em}{0ex}}\gamma \sum _{{w}_{i}\in \mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}}}I(w;{w}_{i}|{y}_{t})\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\delta \sum _{{w}_{i}^{}\in \mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}}}\sum _{{w}_{j}^{}\in \mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}};i\ne j}I(w;{w}_{j}|{w}_{i}),\end{array}$$
- At the iteration $k>m$, greedy strategy is used. Each $w{}_{t}^{k}$ will be embedded in the already embedded vector $\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}}$ until the process stops. The standard of low dimensional approximation is still used before stopping.
- The termination criterion is quantified as:$$I({y}_{t};\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}-\mathbf{1}})/I({y}_{t};\mathbf{v}{}_{\mathbf{t}}^{\mathbf{k}})>A,$$
- To calculate the causality strength of X on Y conditioned by the other variables in Z, the index is defined as:$$R{}_{X\to Y|Z}=\frac{I({y}_{t};\mathbf{v}{}_{\mathbf{t}}^{\mathbf{x}}|\mathbf{v}{}_{\mathbf{t}}^{\mathbf{y}},\mathbf{v}{}_{\mathbf{t}}^{\mathbf{z}})}{I({y}_{t};\mathbf{v}{}_{\mathbf{t}})},$$

## 3. Simulation Study

#### 3.1. Linear Multivariate Stochastic Process

#### 3.2. Nonlinear Multivariate Stochastic Process

#### 3.3. Coupled Henon Maps

#### 3.4. Coupled Lorenz System

## 4. Application to Epilespy ECoG Signals

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The flowchart of the low-dimensional approximation of CMI and the mixed search strategy(LM)-partial conditional mutual information from mixed embedding (PMIME) method.

**Figure 2.**Matrix representation of causality for the linear vector autoregressive (VAR) process. Retrieved by (

**a**) traditional PMIME method, (

**b**) mixed search strategy (M)-PMIME method, (

**c**) and LM-PMIME method with k-nearest neighbors (k-NNs) estimator. The length of the time series is 512. $m=2$ is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same ($L=6,A=0.97$). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the linear VAR process. The direction of causal influence is from row to column in the matrix. The true causal connections in this linear VAR process are at the matrix elements (1, 2), (1, 4), (2, 4), (4, 5), (5, 1), (5, 2) and (5, 3).

**Figure 3.**Matrix representation of causality for ${\mathrm{NLVAR}}_{3}\left(1\right)$. Retrieved by (

**a**) traditional PMIME method, (

**b**) M-PMIME method, (

**c**) and LM-PMIME method with k-NNs estimator. The time series length is 512. $m=3$ is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same ($L=6$, $A=0.97$). Color maps for the mean values of coupling measurements are obtained from 100 realizations of ${\mathrm{NLVAR}}_{3}\left(1\right)$. The true causal connections in ${\mathrm{NLVAR}}_{3}\left(1\right)$ are at the matrix elements (1,2), (1,3), (2,3).

**Figure 4.**Matrix representation of causality for $K=6$ variables of the coupled Henon maps ($C=0.1$). Retrieved by (

**a**) traditional PMIME method, (

**b**) M-PMIME method, (

**c**) and LM-PMIME method with k-NNs estimator. The time series length is 1024. $m=2$ is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same ($L=5$, $A=0.95$). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the coupled Henon maps. The true causal connections in the coupled Henon maps are at the matrix elements ($i-1,i$), where $i=2,\cdots \phantom{\rule{4pt}{0ex}},6$.

**Figure 5.**Matrix representation of causality for $K=6$ variables of the coupled Henon maps ($C=0.3$). Retrieved by (

**a**) traditional PMIME method, (

**b**) M-PMIME method, (

**c**) and LM-PMIME method with k-NNs estimator. The time series length is 1024. $m=2$ is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same ($L=5$, $A=0.95$). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the coupled Henon maps. The true causal connections in the coupled Henon maps are at the matrix elements ($i-1,i$), where $i=2,\cdots \phantom{\rule{4pt}{0ex}},6$.

**Figure 6.**Matrix representation of causality for the three coupled Lorenz oscillators. Retrieved by (

**a**) traditional PMIME method, (

**b**) M-PMIME method, (

**c**) and LM-PMIME method with k-NNs estimator. The length of the time series is 512 with coupling strength $C=3$. $m=3$ is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same ($L=5$, $A=0.95$). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the three coupled Lorenz oscillators. The true causal connections in the three coupled Lorenz oscillators are at the matrix elements ($i-1,i$), where $i=2,3$.

**Figure 7.**Matrix representation of causality for the three coupled Lorenz oscillators. Retrieved by (

**a**) traditional PMIME method, (

**b**) M-PMIME method, (

**c**) and LM-PMIME method with k-NNs estimator. The length of the time series is 512 with coupling strength $C=5$. $m=3$ is used for the M-PMIME method and the LM-PMIME method. The remaining parameters of the three methods are the same ($L=5$, $A=0.95$). Color maps for the mean values of coupling measurements are obtained from 100 realizations of the three coupled Lorenz oscillators. The direction of causal influence is from row to column in the matrix. The true causal connections in the three coupled Lorenz oscillators are at the matrix elements ($i-1,i$), where $i=2,3$.

**Figure 8.**Results for multivariate electrocorticographic (ECoG) data. Matrices of causalities reflect the pre-seizure state (

**top**) and the seizure state (

**bottom**)) estimated by the PMIME method and the LM-PMIME method. The causal strengths are averaged (the mean values of the coupling measurements over all epochs in the same physiological state). Contacts 1 to 64 belong to an eight-by-eight electrode grid, and contacts 65 to 76 belong to two depth electrodes. The direction of causal influence is from row to column in the matrices. The brighter colors indicate more significant values. The key contact is marked by a rectangular box. The parameter $A=0.95$ and $m=2$ are set for the different methods.

**Figure 9.**Results for multivariate ECoG data. Matrices reflect the difference of total numbers of significant connections between the seizure state and the pre-seizure state (seizure minus pre-seizure). The numbers are respectively summed from 8 seizure epochs and eight pre-seizure epochs. Contacts 1 to 64 belong to an eight-by-eight electrode grid, and contacts 65 to 76 belong to two depth electrodes. The brighter colors indicate more significant values. The key contact is marked by a rectangular box. The parameter $A=0.95$ and $m=2$ are set for the different methods.

**Table 1.**Evaluation indicators are obtained from 100 realizations of linear VAR process with varying time series length for the three different methods. $A=0.97$ and $L=6$ are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter $m=2$.

Sensitivity | Specificity | F1 Score | |
---|---|---|---|

$n=256$ | |||

PMIME | $0.988$ | $0.492$ | $0.600$ |

M-PMIME | $0.989$ | $0.481$ | $0.596$ |

LM-PMIME | $0.797$ | $0.741$ | $0.647$ |

$n=512$ | |||

PMIME | $1.000$ | $0.567$ | $0.643$ |

M-PMIME | $0.994$ | $0.727$ | $0.645$ |

LM-PMIME | $0.855$ | $0.763$ | $0.693$ |

$n=1024$ | |||

PMIME | $1.000$ | $0.697$ | $0.719$ |

M-PMIME | $0.940$ | $0.729$ | $0.713$ |

LM-PMIME | $0.877$ | $0.807$ | $0.739$ |

**Table 2.**Evaluation indicators are obtained from 100 realizations of ${\mathrm{NLVAR}}_{3}\left(1\right)$ with varying time series length for the three different methods. $A=0.97$ and $L=6$ are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter $m=3$.

Sensitivity | Specificity | F1 Score | |
---|---|---|---|

$n=256$ | |||

PMIME | $0.973$ | $0.650$ | $0.737$ |

M-PMIME | $0.976$ | $0.615$ | $0.712$ |

LM-PMIME | $0.860$ | $0.844$ | $0.792$ |

$n=512$ | |||

PMIME | $1.000$ | $0.681$ | $0.758$ |

M-PMIME | $1.000$ | $0.662$ | $0.748$ |

LM-PMIME | $0.950$ | $0.887$ | $0.873$ |

$n=1024$ | |||

PMIME | $1.000$ | $0.860$ | $0.877$ |

M-PMIME | $1.000$ | $0.790$ | $0.827$ |

LM-PMIME | $0.989$ | $0.892$ | $0.896$ |

**Table 3.**Evaluation indicators are obtained from 100 realizations of K variables of the Henon maps ($C=0.1$) for the three different methods. $A=0.95$ and $L=5$ are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter $m=2$.

Sensitivity | Specificity | F1 Score | |
---|---|---|---|

$K=3$ | |||

PMIME | $0.175$ | $1.000$ | $0.297$ |

M-PMIME | $0.715$ | $1.000$ | $0.834$ |

LM-PMIME | $0.945$ | $1.000$ | $0.972$ |

$K=6$ | |||

PMIME | $0.217$ | $1.000$ | $0.357$ |

M-PMIME | $0.674$ | $1.000$ | $0.806$ |

LM-PMIME | $0.926$ | $0.998$ | $0.950$ |

$K=9$ | |||

PMIME | $0.204$ | $1.000$ | $0.338$ |

M-PMIME | $0.700$ | $1.000$ | $0.824$ |

LM-PMIME | $0.895$ | $0.998$ | $0.904$ |

**Table 4.**Evaluation indicators are obtained from 100 realizations of K variables of the Henon maps ($C=0.3$) for the three different methods. $A=0.95$ and $L=5$ are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter $m=2$.

Sensitivity | Specificity | F1 Score | |
---|---|---|---|

$K=3$ | |||

PMIME | $1.000$ | $1.000$ | $1.000$ |

M-PMIME | $1.000$ | $1.000$ | $1.000$ |

LM-PMIME | $1.000$ | $1.000$ | $1.000$ |

$K=6$ | |||

PMIME | $1.000$ | $1.000$ | $1.000$ |

M-PMIME | $1.000$ | $1.000$ | $1.000$ |

LM-PMIME | $1.000$ | $1.000$ | $1.000$ |

$K=9$ | |||

PMIME | $1.000$ | $1.000$ | $1.000$ |

M-PMIME | $1.000$ | $1.000$ | $1.000$ |

LM-PMIME | $1.000$ | $1.000$ | $1.000$ |

**Table 5.**Evaluation indicators are obtained from 100 realizations of the three coupled Lorenz oscillators ($C=3$) with varying time series length for the three different methods. $A=0.95$ and $L=5$ are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter $m=3$.

Sensitivity | Specificity | F1 Score | |
---|---|---|---|

$n=256$ | |||

PMIME | $0.225$ | $0.997$ | $0.364$ |

M-PMIME | $0.660$ | $0.863$ | $0.617$ |

LM-PMIME | $0.805$ | $0.665$ | $0.541$ |

$n=512$ | |||

PMIME | $0.185$ | $1.000$ | $0.312$ |

M-PMIME | $0.640$ | $0.913$ | $0.658$ |

LM-PMIME | $0.875$ | $0.743$ | $0.631$ |

$n=1024$ | |||

PMIME | $0.175$ | $1.000$ | $0.297$ |

M-PMIME | $0.670$ | $0.909$ | $0.674$ |

LM-PMIME | $0.970$ | $0.756$ | $0.687$ |

**Table 6.**Evaluation indicators are obtained from 100 realizations of the three coupled Lorenz oscillators ($n=512$) with coupling strength C from 1 to 5 for the three different methods. $A=0.95$ and $L=5$ are the parameters common to the three methods. In addition, the LM-PMIME method and the M-PMIME method use the parameter $m=3$.

Sensitivity | Specificity | F1 Score | |
---|---|---|---|

$C=1$ | |||

PMIME | $0.000$ | $1.000$ | $0.000$ |

M-PMIME | $0.155$ | $0.926$ | $0.221$ |

LM-PMIME | $0.375$ | $0.830$ | $0.381$ |

$C=2$ | |||

PMIME | $0.075$ | $1.000$ | $0.141$ |

M-PMIME | $0.565$ | $0.893$ | $0.583$ |

LM-PMIME | $0.825$ | $0.765$ | $0.623$ |

$C=3$ | |||

PMIME | $0.185$ | $1.000$ | $0.312$ |

M-PMIME | $0.640$ | $0.913$ | $0.658$ |

LM-PMIME | $0.875$ | $0.743$ | $0.631$ |

$C=4$ | |||

PMIME | $0.260$ | $1.000$ | $0.413$ |

M-PMIME | $0.740$ | $0.892$ | $0.698$ |

LM-PMIME | $0.920$ | $0.710$ | $0.627$ |

$C=5$ | |||

PMIME | $0.320$ | $0.997$ | $0.481$ |

M-PMIME | $0.725$ | $0.873$ | $0.660$ |

LM-PMIME | $0.960$ | $0.731$ | $0.661$ |

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## Share and Cite

**MDPI and ACS Style**

Jia, Z.; Lin, Y.; Jiao, Z.; Ma, Y.; Wang, J. Detecting Causality in Multivariate Time Series via Non-Uniform Embedding. *Entropy* **2019**, *21*, 1233.
https://doi.org/10.3390/e21121233

**AMA Style**

Jia Z, Lin Y, Jiao Z, Ma Y, Wang J. Detecting Causality in Multivariate Time Series via Non-Uniform Embedding. *Entropy*. 2019; 21(12):1233.
https://doi.org/10.3390/e21121233

**Chicago/Turabian Style**

Jia, Ziyu, Youfang Lin, Zehui Jiao, Yan Ma, and Jing Wang. 2019. "Detecting Causality in Multivariate Time Series via Non-Uniform Embedding" *Entropy* 21, no. 12: 1233.
https://doi.org/10.3390/e21121233