Detecting coupling directions with transcript mutual information: A comparative study

José M. Amigó; Beata Graff; Grzegorz Graff; Roberto Monetti; Katarzyna Tessmer

doi:10.3934/dcdsb.2019051

Article Contents

2019, Volume 24, Issue 8: 4079-4097. Doi: 10.3934/dcdsb.2019051

This issue Previous Article Stabilisation by noise on the boundary for a Chafee-Infante equation with dynamical boundary conditions Next Article Stabilisation by delay feedback control for highly nonlinear hybrid stochastic differential equations

Detecting coupling directions with transcript mutual information: A comparative study

1.
Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain
2.
Department of Hypertension and Diabetology, Medical University of Gdańsk, 80-952 Gdańsk, Poland
3.
Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, 80-233 Gdańsk, Poland
4.
IngSoft GmbH. Irrerstrasse 17. 90403 Nuernberg, Germany
5.
Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, 80-233 Gdańsk, Poland

^* Corresponding author: jm.amigo@umh.es
^* Corresponding author: jm.amigo@umh.es

Dedicated to Peter E. Kloeden on the occasion of his 70th birthday

Received: February 28, 2018

Revised: June 30, 2018

Early access: February 2019

Published: July 2019

J.M.A. was financially supported by the Spanish Ministry of Economy, Industry and Competitivity, grant MTM2016-74921-P (AEI/FEDER, EU). B.G. was financially supported by the National Science Centre, Poland, grant MAESTRO UMO-2011/02/A/NZ5/00329. G.G. was financially supported by the National Science Centre, Poland, grant UMO-2014/15/B/ST1/01710

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Abstract

Causal relationships are important to understand the dynamics of coupled processes and, moreover, to influence or control the effects by acting on the causes. Among the different approaches to determine cause-effect relationships and, in particular, coupling directions in interacting random or deterministic processes, we focus in this paper on information-theoretic measures. So, we study in the theoretical part the difference between directionality indicators based on transfer entropy as well as on its dimensional reduction via transcripts in algebraic time series representations. In the applications we consider specifically the lowest dimensional case, i.e., 3-dimensional transfer entropy, which is currently one of the most popular causality indicators, and the (2-dimensional) mutual information of transcripts. Needless to say, the lower dimensionality of the transcript-based indicator can make a difference in practice, where datasets are usually small. To compare numerically the performance of both directionality indicators, synthetic data (obtained with random processes) and real world data (in the form of biomedical recordings) are used. As happened in previous related work, we found again that the transcript mutual information performs as good as, and in some cases even better than, the lowest dimensional binned and symbolic transfer entropy, the symbols being ordinal patterns.

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Mathematics Subject Classification: Primary: 37M10, 94A17; Secondary: 60G10.

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Figure 1. Plots of $ \Delta AT_{\mathbf{\hat{Z}}\rightarrow \mathbf{\hat{Y}}}(\Lambda ) $ (solid line) and $ \Delta TI_{\mathbf{\hat{Z}}\rightarrow \mathbf{\hat{Y}}}(\Lambda ) $ (dash-dotted line) vs $ \Lambda $, $ 1\leq \Lambda \leq T-1 $, for $ T = 3,4,...,10 $

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Figure 2. Plots of $ \Delta AT_{\mathbf{\hat{Y}}\rightarrow \mathbf{\hat{X}}}(\Lambda ) $ (solid line) and $ \Delta TI_{\mathbf{\hat{Y}}\rightarrow \mathbf{\hat{X}}}(\Lambda ) $ (dash-dotted line) vs $ \Lambda $, $ 1\leq \Lambda \leq T-1 $, for $ T = 3,4,...,10 $

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Figure 3. Plots of $ \Delta AT_{\mathbf{\hat{Z}}\rightarrow \mathbf{\hat{X}}}(\Lambda ) $ (solid line) and $ \Delta TI_{\mathbf{\hat{Z}}\rightarrow \mathbf{\hat{X}}}(\Lambda ) $ (dash-dotted line) vs $ \Lambda $, $ 1\leq \Lambda \leq T-1 $, for $ T = 3,4,...,10 $

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Figure 4. Plots of $ \Delta AT_{\mathbf{\hat{D}}\rightarrow \mathbf{\hat{R}}}(\Lambda ) $ (continuous lines) and $ \Delta TI_{\mathbf{\hat{D}}\rightarrow \mathbf{\hat{R}}}(\Lambda ) $ (dash-dotted lines) for $ T = 8 $, $ 1\leq \Lambda \leq 7 $, and $ k_{zy} = 0.0 $ (first column), $ k_{zy} = 0.1 $ (second column), $ k_{zy} = 0.5 $ (third column), and $ k_{zy} = 0.9 $ (fourth column). Top row corresponds to the coupling direction $ \mathbf{\hat{Z}}\rightarrow \mathbf{\hat{Y}} $, middle row to $ \mathbf{\hat{Z}}\rightarrow \mathbf{\hat{X}} $, and bottom row to $ \mathbf{\hat{Y}}\rightarrow \mathbf{\hat{X}} $

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Figure 5. The directionality indicators $ \Delta TE_{RR\rightarrow BP}(\Lambda ) $ (top row), $ \Delta STE_{RR\rightarrow BP}(\Lambda ) $ (middle row), and $ \Delta TMI_{RR\rightarrow BP}(\Lambda ) $ (bottom row) for the patient Groups Ⅰ (left column), Ⅱ (middle column), and ⅡB (right column) with $ T = 6 $ and $ 1\leq \Lambda \leq 5 $. For convenience, here $ TE $ stands for binned transfer entropy, $ STE $ for symbolic transfer entropy, and $ TMI $ for transcript mutual information. See the text for more details, the description of the patient Groups, and the data $ (RR_{n}) $ and $ (BP_{n}) $

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