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Entropy
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Advance Methods for the Quantification of Correlations and Causal Relations...
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Advance Methods for the Quantification of Correlations and Causal Relations between Processes

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: closed (18 April 2021) | Viewed by 26414

Special Issue Editors

Dr. Andrea Murari

E-Mail Website
Guest Editor
Consorzio RFX (CNR, ENEA, INFN, Universita’ di Padova, Acciaierie Venete SpA), 35127 Padova, Italy
Interests: nuclear fusion; entropy; information theory; machine learning; evolutionary computation; tomography; image processing
Special Issues, Collections and Topics in MDPI journals
Dr. Teddy Craciunescu

E-Mail Website
Guest Editor
National Institute for Laser, Plasma and Radiation Physics, RO-077125 Magurele-Bucharest, Romania
Interests: computed tomography; imagine processing; time series analysis; complex networks; data mining; Monte Carlo simulations
Special Issues, Collections and Topics in MDPI journals
Dr. Michela Gelfusa

E-Mail Website
Guest Editor
Department of Industrial Engineering, University of Rome “Tor Vergata”, via del Politecnico 1, 00133 Roma, Italy
Interests: plasma diagnostics; inverse problems; data mining; time series analysis; genetic programming
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Two of the most relevant characteristics of modern societies are their complexity and the huge amounts of data that they produce. Unfortunately, in the investigation of complex systems, large datasets can become a liability, instead of an asset, if they are not analysed with adequate tools. One of the first steps in the formulation of scientific models and theories is certainly the assessment of the correlations between the quantities potentially involved. More advanced is the goal of determining their actual causal relations and relative strengths. In various domains, performing experiments and interventions to establish direct causal relationships could be unethical, extremely expensive, or even impossible. In the last few years, many efforts have been made to improve the techniques and methodologies for identifying and quantifying the correlations and the causal influences between processes based on time-series and cross-sectional data; they range from causal networks to phase space reconstructions and information-theoretic tools. For practical applications, the limited number of observations and the noise that inherently accompanies the measurements represent additional challenges.

This Special Issue aims to collect papers that describe new solutions for the above-mentioned problems. The contributions can be based on (but not limited to) the following fields:

  • Information Theory;
  • Network Theory;
  • Statistical Inference;
  • Machine Learning;
  • Neural Computation;
  • Genetic Programing.

Theoretical approaches as well as practical applications are welcome.

Best regards,

Prof. Dr. Andrea Murari
Dr. Teddy Craciunescu
Dr. Michela Gelfusa
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (7 papers)

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Research

22 pages, 1814 KiB  
Article
Granger-Causality Inference of the Existence of Unobserved Important Components in Network Analysis
by Heba Elsegai
Entropy 2021, 23(8), 994; https://doi.org/10.3390/e23080994 - 30 Jul 2021
Cited by 1 | Viewed by 2177
Abstract
Detecting causal interrelationships in multivariate systems, in terms of the Granger-causality concept, is of major interest for applications in many fields. Analyzing all the relevant components of a system is almost impossible, which contrasts with the concept of Granger causality. Not observing some [...] Read more.
Detecting causal interrelationships in multivariate systems, in terms of the Granger-causality concept, is of major interest for applications in many fields. Analyzing all the relevant components of a system is almost impossible, which contrasts with the concept of Granger causality. Not observing some components might, in turn, lead to misleading results, particularly if the missing components are the most influential and important in the system under investigation. In networks, the importance of a node depends on the number of nodes connected to this node. The degree of centrality is the most commonly used measure to identify important nodes in networks. There are two kinds of degree centrality, which are in-degree and out-degree. This manuscrpt is concerned with finding the highest out-degree among nodes to identify the most influential nodes. Inferring the existence of unobserved important components is critical in many multivariate interacting systems. The implications of such a situation are discussed in the Granger-causality framework. To this end, two of the most recent Granger-causality techniques, renormalized partial directed coherence and directed partial correlation, were employed. They were then compared in terms of their performance according to the extent to which they can infer the existence of unobserved important components. Sub-network analysis was conducted to aid these two techniques in inferring the existence of unobserved important components, which is evidenced in the results. By comparing the results of the two conducted techniques, it can be asserted that renormalized partial coherence outperforms directed partial correlation in the inference of existing unobserved important components that have not been included in the analysis. This measure of Granger causality and sub-network analysis emphasizes their ubiquitous successful applicability in such cases of the existence of hidden unobserved important components. Full article
(This article belongs to the Special Issue Advance Methods for the Quantification of Correlations and Causal Relations between Processes)
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21 pages, 691 KiB  
Article
Estimating Conditional Transfer Entropy in Time Series Using Mutual Information and Nonlinear Prediction
by Payam Shahsavari Baboukani, Carina Graversen, Emina Alickovic and Jan Østergaard
Entropy 2020, 22(10), 1124; https://doi.org/10.3390/e22101124 - 3 Oct 2020
Cited by 13 | Viewed by 3702
Abstract
We propose a new estimator to measure directed dependencies in time series. The dimensionality of data is first reduced using a new non-uniform embedding technique, where the variables are ranked according to a weighted sum of the amount of new information and improvement [...] Read more.
We propose a new estimator to measure directed dependencies in time series. The dimensionality of data is first reduced using a new non-uniform embedding technique, where the variables are ranked according to a weighted sum of the amount of new information and improvement of the prediction accuracy provided by the variables. Then, using a greedy approach, the most informative subsets are selected in an iterative way. The algorithm terminates, when the highest ranked variable is not able to significantly improve the accuracy of the prediction as compared to that obtained using the existing selected subsets. In a simulation study, we compare our estimator to existing state-of-the-art methods at different data lengths and directed dependencies strengths. It is demonstrated that the proposed estimator has a significantly higher accuracy than that of existing methods, especially for the difficult case, where the data are highly correlated and coupled. Moreover, we show its false detection of directed dependencies due to instantaneous couplings effect is lower than that of existing measures. We also show applicability of the proposed estimator on real intracranial electroencephalography data. Full article
(This article belongs to the Special Issue Advance Methods for the Quantification of Correlations and Causal Relations between Processes)
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23 pages, 5842 KiB  
Article
A Refinement of Recurrence Analysis to Determine the Time Delay of Causality in Presence of External Perturbations
by Emmanuele Peluso, Teddy Craciunescu and Andrea Murari
Entropy 2020, 22(8), 865; https://doi.org/10.3390/e22080865 - 6 Aug 2020
Cited by 8 | Viewed by 3055
Abstract
This article describes a refinement of recurrence analysis to determine the delay in the causal influence between a driver and a target, in the presence of additional perturbations affecting the time series of the response observable. The methodology is based on the definition [...] Read more.
This article describes a refinement of recurrence analysis to determine the delay in the causal influence between a driver and a target, in the presence of additional perturbations affecting the time series of the response observable. The methodology is based on the definition of a new type of recurrence plots, the Conditional Joint Recurrence plot. The potential of the proposed approach resides in the great flexibility of recurrence plots themselves, which allows extending the technique to more than three quantities. Autoregressive time series, both linear and nonlinear, with different couplings and percentage of additive Gaussian noise have been investigated in detail, with and without outliers. The approach has also been applied to the case of synthetic periodic signals, representing realistic situations of synchronization experiments in thermonuclear fusion. The results obtained have been very positive; the proposed Conditional Joint Recurrence plots have always managed to identify the right interval of the causal influences and are very competitive with alternative techniques such as the Conditional Transfer Entropy. Full article
(This article belongs to the Special Issue Advance Methods for the Quantification of Correlations and Causal Relations between Processes)
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16 pages, 3521 KiB  
Article
Image-Based Methods to Investigate Synchronization between Time Series Relevant for Plasma Fusion Diagnostics
by Teddy Craciunescu, Andrea Murari, Ernesto Lerche, Michela Gelfusa and JET Contributors
Entropy 2020, 22(7), 775; https://doi.org/10.3390/e22070775 - 16 Jul 2020
Cited by 4 | Viewed by 2717
Abstract
Advanced time series analysis and causality detection techniques have been successfully applied to the assessment of synchronization experiments in tokamaks, such as Edge Localized Modes (ELMs) and sawtooth pacing. Lag synchronization is a typical strategy for fusion plasma instability control by pace-making techniques. [...] Read more.
Advanced time series analysis and causality detection techniques have been successfully applied to the assessment of synchronization experiments in tokamaks, such as Edge Localized Modes (ELMs) and sawtooth pacing. Lag synchronization is a typical strategy for fusion plasma instability control by pace-making techniques. The major difficulty, in evaluating the efficiency of the pacing methods, is the coexistence of the causal effects with the periodic or quasi-periodic nature of the plasma instabilities. In the present work, a set of methods based on the image representation of time series, are investigated as tools for evaluating the efficiency of the pace-making techniques. The main options rely on the Gramian Angular Field (GAF), the Markov Transition Field (MTF), previously used for time series classification, and the Chaos Game Representation (CGR), employed for the visualization of large collections of long time series. The paper proposes an original variation of the Markov Transition Matrix, defined for a couple of time series. Additionally, a recently proposed method, based on the mapping of time series as cross-visibility networks and their representation as images, is included in this study. The performances of the method are evaluated on synthetic data and applied to JET measurements. Full article
(This article belongs to the Special Issue Advance Methods for the Quantification of Correlations and Causal Relations between Processes)
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12 pages, 1804 KiB  
Article
On the Potential of Time Delay Neural Networks to Detect Indirect Coupling between Time Series
by Riccardo Rossi, Andrea Murari and Pasquale Gaudio
Entropy 2020, 22(5), 584; https://doi.org/10.3390/e22050584 - 21 May 2020
Cited by 7 | Viewed by 3082
Abstract
Determining the coupling between systems remains a topic of active research in the field of complex science. Identifying the proper causal influences in time series can already be very challenging in the trivariate case, particularly when the interactions are non-linear. In this paper, [...] Read more.
Determining the coupling between systems remains a topic of active research in the field of complex science. Identifying the proper causal influences in time series can already be very challenging in the trivariate case, particularly when the interactions are non-linear. In this paper, the coupling between three Lorenz systems is investigated with the help of specifically designed artificial neural networks, called time delay neural networks (TDNNs). TDNNs can learn from their previous inputs and are therefore well suited to extract the causal relationship between time series. The performances of the TDNNs tested have always been very positive, showing an excellent capability to identify the correct causal relationships in absence of significant noise. The first tests on the time localization of the mutual influences and the effects of Gaussian noise have also provided very encouraging results. Even if further assessments are necessary, the networks of the proposed architecture have the potential to be a good complement to the other techniques available in the market for the investigation of mutual influences between time series. Full article
(This article belongs to the Special Issue Advance Methods for the Quantification of Correlations and Causal Relations between Processes)
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13 pages, 1875 KiB  
Article
Upgrading Model Selection Criteria with Goodness of Fit Tests for Practical Applications
by Riccardo Rossi, Andrea Murari, Pasquale Gaudio and Michela Gelfusa
Entropy 2020, 22(4), 447; https://doi.org/10.3390/e22040447 - 15 Apr 2020
Cited by 24 | Viewed by 3090
Abstract
The Bayesian information criterion (BIC), the Akaike information criterion (AIC), and some other indicators derived from them are widely used for model selection. In their original form, they contain the likelihood of the data given the models. Unfortunately, in many applications, it is [...] Read more.
The Bayesian information criterion (BIC), the Akaike information criterion (AIC), and some other indicators derived from them are widely used for model selection. In their original form, they contain the likelihood of the data given the models. Unfortunately, in many applications, it is practically impossible to calculate the likelihood, and, therefore, the criteria have been reformulated in terms of descriptive statistics of the residual distribution: the variance and the mean-squared error of the residuals. These alternative versions are strictly valid only in the presence of additive noise of Gaussian distribution, not a completely satisfactory assumption in many applications in science and engineering. Moreover, the variance and the mean-squared error are quite crude statistics of the residual distributions. More sophisticated statistical indicators, capable of better quantifying how close the residual distribution is to the noise, can be profitably used. In particular, specific goodness of fit tests have been included in the expressions of the traditional criteria and have proved to be very effective in improving their discriminating capability. These improved performances have been demonstrated with a systematic series of simulations using synthetic data for various classes of functions and different noise statistics. Full article
(This article belongs to the Special Issue Advance Methods for the Quantification of Correlations and Causal Relations between Processes)
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19 pages, 705 KiB  
Article
Association Factor for Identifying Linear and Nonlinear Correlations in Noisy Conditions
by Nezamoddin N. Kachouie and Wejdan Deebani
Entropy 2020, 22(4), 440; https://doi.org/10.3390/e22040440 - 13 Apr 2020
Cited by 5 | Viewed by 7622
Abstract
Background: In data analysis and machine learning, we often need to identify and quantify the correlation between variables. Although Pearson’s correlation coefficient has been widely used, its value is reliable only for linear relationships and Distance correlation was introduced to address this shortcoming. [...] Read more.
Background: In data analysis and machine learning, we often need to identify and quantify the correlation between variables. Although Pearson’s correlation coefficient has been widely used, its value is reliable only for linear relationships and Distance correlation was introduced to address this shortcoming. Methods: Distance correlation can identify linear and nonlinear correlations. However, its performance drops in noisy conditions. In this paper, we introduce the Association Factor (AF) as a robust method for identification and quantification of linear and nonlinear associations in noisy conditions. Results: To test the performance of the proposed Association Factor, we modeled several simulations of linear and nonlinear relationships in different noise conditions and computed Pearson’s correlation, Distance correlation, and the proposed Association Factor. Conclusion: Our results show that the proposed method is robust in two ways. First, it can identify both linear and nonlinear associations. Second, the proposed Association Factor is reliable in both noiseless and noisy conditions. Full article
(This article belongs to the Special Issue Advance Methods for the Quantification of Correlations and Causal Relations between Processes)
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