# Information Transfer in Linear Multivariate Processes Assessed through Penalized Regression Techniques: Validation and Application to Physiological Networks

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Vector Autoregressive Model Identification

#### 2.2. Measures of Information Transfer

#### 2.3. Computation of the Measures of Information Transfer for Multivariate Gaussian Processes

#### Formulation of State–Space Models

**K**, the state matrix

**A**and the observation matrix

**C**, which can all be computed from the original VAR parameters in (1) as reported in ([54]). Starting from the parameters of an ISS model is possible to compute any partial variance ${\lambda}_{j|a}$, where the subscript a denotes any combination of indexes $\in (1,\dots ,M)$, by evaluating the innovation of a “submodel” obtained removing from the observation Equation (14) the variables not included in a. Furthermore, in this formulation the state Equation (15) remains unaltered and the observation equation of relevant submodel becomes:

#### 2.4. Testing the Significance of the Conditional Transfer Entropy

^{th}percentile of its distribution on the surrogates was determined for each directed link, and the link was detected as statistically significant when the original cTE was above the threshold. In the case of LASSO, the statistical significance of the estimated cTE values was determined exploiting the sparseness of the identification procedure. Since LASSO model identification always produces a sparse matrix with several VAR coefficients equal to zero, the cTE values result exactly zero when the coefficients along the investigated direction are zero at each time lag; on the contrary, cTE is positive, and was considered to be statistically significant in this study, when at least one coefficient is non-zero along the considered direction.

## 3. Simulation Experiments

#### 3.1. Simulation Study I

#### 3.1.1. Simulation Design and Realization

#### 3.1.2. Simulation Results

#### 3.2. Simulation Study II

#### 3.2.1. Simulation Design and Realization

#### 3.2.2. Performance Evaluation

#### 3.2.3. Statistical Analysis

#### 3.2.4. Results of the Simulation Study

## 4. Application to Physiological Time Series

#### 4.1. Data Acquisition and Pre-Processing

#### 4.2. Information Transfer Analysis

- First, a PID analysis was performed for OLS and LASSO through the computation of the joint information transfer ${T}_{ik\to j}$ and the terms of its decomposition ${U}_{i\to j}$, ${U}_{k\to j}$, ${R}_{ik\to j}$, ${S}_{ik\to j}$. The analysis was performed collecting in the first source (index i) the processes $[\eta ,\rho ,\pi ]$ forming the so-called “body” sub-network that accounts for cardiac, cardiovascular and respiratory dynamics, and in the second source (index k) the processes $[\delta ,\theta ,\alpha ,\beta ]$ forming the “brain” sub-network that accounts for the different brain wave amplitudes; the analysis was repeated considering each one of the seven processes as the target process $(j=[\eta ,\rho ,\pi ,\delta ,\theta ,\alpha ,\beta \left]\right)$ and excluding it from the set of sources.
- Second, the topological structure of the network of physiological interactions was detected computing the conditional transfer entropy ${T}_{i\to j|s}$ based on the two VAR identification methods combined with their method for assessing the statistical significance of cTE (i.e., using surrogate data for OLS and exploiting the intrinsic sparseness for LASSO). The analysis was performed between each pair of processes as driver and target $(i,j=[\eta ,\rho ,\pi ,\delta ,\theta ,\alpha ,\beta ],i\ne j)$ and collecting the remaining five processes in the conditioning vector with index s. As a quantitative descriptor of the network was used the in-strength, defined as the sum of all weighted inward links connected to one node [63]. Moreover, to describe the overall brain–body interactions the in-strength of the body sub-network due to brain sub-network (and vice-versa) was computed considering as link weights the percentage of subjects showing at least one statistically significant brain-to-body connection (and vice-versa). To study the involvement of each specific node in the network, the in-strength of each node was computed considering as link weights the cTE values of all network links pointing into the considered node.

#### 4.3. Statistical Analysis

#### 4.4. Results of Real Data Application

#### 4.4.1. Partial Information Decomposition

#### 4.4.2. Conditional Information Transfer

## 5. Discussion

#### 5.1. Simulation Study I

#### 5.2. Simulation Study II

#### 5.3. Real Data Application

#### 5.3.1. Partial Information Decomposition Analysis

#### 5.3.2. Conditional Information Transfer Analysis

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Bashan, A.; Bartsch, R.P.; Kantelhardt, J.W.; Havlin, S.; Ivanov, P.C. Network physiology reveals relations between network topology and physiological function. Nat. Commun.
**2012**, 3, 1–9. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Faes, L.; Porta, A.; Nollo, G. Information decomposition in bivariate systems: Theory and application to cardiorespiratory dynamics. Entropy
**2015**, 17, 277–303. [Google Scholar] [CrossRef] - Zanetti, M.; Faes, L.; Nollo, G.; De Cecco, M.; Pernice, R.; Maule, L.; Pertile, M.; Fornaser, A. Information dynamics of the brain, cardiovascular and respiratory network during different levels of mental stress. Entropy
**2019**, 21, 275. [Google Scholar] [CrossRef] [Green Version] - Malik, M. Heart rate variability: Standards of measurement, physiological interpretation, and clinical use: Task force of the European Society of Cardiology and the North American Society for Pacing and Electrophysiology. Ann. Noninvasive Electrocardiol.
**1996**, 1, 151–181. [Google Scholar] [CrossRef] - Pereda, E.; Quiroga, R.Q.; Bhattacharya, J. Nonlinear multivariate analysis of neurophysiological signals. Prog. Neurobiol.
**2005**, 77, 1–37. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schulz, S.; Adochiei, F.C.; Edu, I.R.; Schroeder, R.; Costin, H.; Bär, K.J.; Voss, A. Cardiovascular and cardiorespiratory coupling analyses: A review. Philos. Trans. R. Soc. A
**2013**, 371, 20120191. [Google Scholar] [CrossRef] [Green Version] - Faes, L.; Nollo, G.; Jurysta, F.; Marinazzo, D. Information dynamics of brain–heart physiological networks during sleep. New J. Phys.
**2014**, 16, 105005. [Google Scholar] [CrossRef] [Green Version] - Bartsch, R.P.; Liu, K.K.; Bashan, A.; Ivanov, P.C. Network physiology: How organ systems dynamically interact. PLoS ONE
**2015**, 10, e0142143. [Google Scholar] [CrossRef] - Lizier, J.T.; Prokopenko, M.; Zomaya, A.Y. Local measures of information storage in complex distributed computation. Inf. Sci.
**2012**, 208, 39–54. [Google Scholar] [CrossRef] - Faes, L.; Javorka, M.; Nollo, G. Information-Theoretic Assessment of Cardiovascular Variability During Postural and Mental Stress. In Proceedings of the XIV Mediterranean Conference on Medical and Biological Engineering and Computing 2016, Paphos, Cyprus, 31 March–2 April 2016; pp. 67–70. [Google Scholar]
- Schreiber, T. Measuring information transfer. Phys. Rev. Lett.
**2000**, 85, 461. [Google Scholar] [CrossRef] [Green Version] - Lizier, J.T.; Prokopenko, M.; Zomaya, A.Y. Information modification and particle collisions in distributed computation. Chaos Interdiscip. J. Nonlinear Sci.
**2010**, 20, 037109. [Google Scholar] [CrossRef] [PubMed] - Schneidman, E.; Bialek, W.; Berry, M.J. Synergy, redundancy, and independence in population codes. J. Neurosci.
**2003**, 23, 11539–11553. [Google Scholar] [CrossRef] [PubMed] - Barnett, L.; Barrett, A.B.; Seth, A.K. Granger causality and transfer entropy are equivalent for Gaussian variables. Phys. Rev. Lett.
**2009**, 103, 238701. [Google Scholar] [CrossRef] [Green Version] - Barrett, A.B. Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems. Phys. Rev. E
**2015**, 91, 052802. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lombardi, F.; Wang, J.W.; Zhang, X.; Ivanov, P.C. Power-law correlations and coupling of active and quiet states underlie a class of complex systems with self-organization at criticality. EPJ Web Conf.
**2020**, 230, 00005. [Google Scholar] [CrossRef] - Lombardi, F.; Gómez-Extremera, M.; Bernaola-Galván, P.; Vetrivelan, R.; Saper, C.B.; Scammell, T.E.; Ivanov, P.C. Critical dynamics and coupling in bursts of cortical rhythms indicate non-homeostatic mechanism for sleep-stage transitions and dual role of VLPO neurons in both sleep and wake. J. Neurosci.
**2020**, 40, 171–190. [Google Scholar] [CrossRef] - Porta, A.; Bari, V.; De Maria, B.; Baumert, M. A network physiology approach to the assessment of the link between sinoatrial and ventricular cardiac controls. Physiol. Meas.
**2017**, 38, 1472. [Google Scholar] [CrossRef] - Krohova, J.; Faes, L.; Czippelova, B.; Turianikova, Z.; Mazgutova, N.; Pernice, R.; Busacca, A.; Marinazzo, D.; Stramaglia, S.; Javorka, M. Multiscale information decomposition dissects control mechanisms of heart rate variability at rest and during physiological stress. Entropy
**2019**, 21, 526. [Google Scholar] [CrossRef] [Green Version] - Widjaja, D.; Montalto, A.; Vlemincx, E.; Marinazzo, D.; Van Huffel, S.; Faes, L. Cardiorespiratory information dynamics during mental arithmetic and sustained attention. PLoS ONE
**2015**, 10, e0129112. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zanetti, M.; Mizumoto, T.; Faes, L.; Fornaser, A.; De Cecco, M.; Maule, L.; Valente, M.; Nollo, G. Multilevel assessment of mental stress via network physiology paradigm using consumer wearable devices. J. Ambient Intell. Hum. Comput.
**2019**. [Google Scholar] [CrossRef] - Valenza, G.; Greco, A.; Gentili, C.; Lanata, A.; Sebastiani, L.; Menicucci, D.; Gemignani, A.; Scilingo, E. Combining electroencephalographic activity and instantaneous heart rate for assessing brain–heart dynamics during visual emotional elicitation in healthy subjects. Philos. Trans. R. Soc. A
**2016**, 374, 20150176. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Greco, A.; Faes, L.; Catrambone, V.; Barbieri, R.; Scilingo, E.P.; Valenza, G. Lateralization of directional brain-heart information transfer during visual emotional elicitation. Am. J. Physiol.-Regul. Integr. and Comp. Physiol.
**2019**, 317, R25–R38. [Google Scholar] [CrossRef] [PubMed] - Wibral, M.; Lizier, J.; Vögler, S.; Priesemann, V.; Galuske, R. Local active information storage as a tool to understand distributed neural information processing. Front. Neuroinf.
**2014**, 8, 1. [Google Scholar] [CrossRef] [Green Version] - Barnett, L.; Lizier, J.T.; Harré, M.; Seth, A.K.; Bossomaier, T. Information flow in a kinetic Ising model peaks in the disordered phase. Phys. Rev. Lett.
**2013**, 111, 177203. [Google Scholar] [CrossRef] [Green Version] - Dimpfl, T.; Peter, F.J. Using transfer entropy to measure information flows between financial markets. Stud. Nonlinear Dyn. Econom.
**2013**, 17, 85–102. [Google Scholar] [CrossRef] [Green Version] - Stramaglia, S.; Cortes, J.M.; Marinazzo, D. Synergy and redundancy in the Granger causal analysis of dynamical networks. New J. Phys.
**2014**, 16, 105003. [Google Scholar] [CrossRef] [Green Version] - Barnett, L.; Seth, A.K. Granger causality for state-space models. Phys. Rev. E
**2015**, 91, 040101. [Google Scholar] [CrossRef] [Green Version] - Solo, V. State-space analysis of Granger-Geweke causality measures with application to fMRI. Neural Comput.
**2016**, 28, 914–949. [Google Scholar] [CrossRef] [Green Version] - Faes, L.; Marinazzo, D.; Stramaglia, S. Multiscale information decomposition: Exact computation for multivariate Gaussian processes. Entropy
**2017**, 19, 408. [Google Scholar] [CrossRef] [Green Version] - Schlögl, A. A comparison of multivariate autoregressive estimators. Signal Process.
**2006**, 86, 2426–2429. [Google Scholar] [CrossRef] - Hoerl, A.E.; Kennard, R.W. Ridge regression: Biased estimation for nonorthogonal problems. Technometrics
**1970**, 12, 55–67. [Google Scholar] [CrossRef] - Antonacci, Y.; Toppi, J.; Caschera, S.; Anzolin, A.; Mattia, D.; Astolfi, L. Estimating brain connectivity when few data points are available: Perspectives and limitations. In Proceedings of the 2017 39th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Seogwipo, Korea, 11–15 July 2017; pp. 4351–4354. [Google Scholar]
- Antonacci, Y.; Toppi, J.; Mattia, D.; Pietrabissa, A.; Astolfi, L. Single-trial Connectivity Estimation through the Least Absolute Shrinkage and Selection Operator. In Proceedings of the 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Berlin, Germany, 23–27 July 2019; pp. 6422–6425. [Google Scholar]
- Marinazzo, D.; Pellicoro, M.; Stramaglia, S. Causal information approach to partial conditioning in multivariate data sets. Comput. Math. Methods Med.
**2012**, 2012, 303601. [Google Scholar] [CrossRef] - Siggiridou, E.; Kugiumtzis, D. Granger causality in multivariate time series using a time-ordered restricted vector autoregressive model. IEEE Trans. Signal Process.
**2015**, 64, 1759–1773. [Google Scholar] [CrossRef] - Hastie, T.; Tibshirani, R.; Wainwright, M. Statistical Learning with Sparsity: The Lasso and Generalizations; CRC Press: Boca Raton, FL, USA, 2015. [Google Scholar]
- Tibshirani, R. Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B (Methodol.)
**1996**, 58, 267–288. [Google Scholar] [CrossRef] - Haufe, S.; Müller, K.R.; Nolte, G.; Krämer, N. Sparse causal discovery in multivariate time series. arXiv
**2009**, arXiv:0901.2234. [Google Scholar] - Antonacci, Y.; Toppi, J.; Mattia, D.; Pietrabissa, A.; Astolfi, L. Estimation of brain connectivity through Artificial Neural Networks. In Proceedings of the 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Berlin, Germany, 23–27 July 2019; pp. 636–639. [Google Scholar]
- Billinger, M.; Brunner, C.; Müller-Putz, G.R. Single-trial connectivity estimation for classification of motor imagery data. J. Neural Eng.
**2013**, 10, 046006. [Google Scholar] [CrossRef] [PubMed] - Valdés-Sosa, P.A.; Sánchez-Bornot, J.M.; Lage-Castellanos, A.; Vega-Hernández, M.; Bosch-Bayard, J.; Melie-García, L.; Canales-Rodríguez, E. Estimating brain functional connectivity with sparse multivariate autoregression. Philos. Trans. R. Soc. B Biol. Sci.
**2005**, 360, 969–981. [Google Scholar] [CrossRef] [Green Version] - Smeekes, S.; Wijler, E. Macroeconomic forecasting using penalized regression methods. Int. J. Forecasting
**2018**, 34, 408–430. [Google Scholar] [CrossRef] [Green Version] - Pernice, R.; Zanetti, M.; Nollo, G.; De Cecco, M.; Busacca, A.; Faes, L. Mutual Information Analysis of Brain-Body Interactions during different Levels of Mental stress. In Proceedings of the 2019 41st Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Berlin, Germany, 23–27 July 2019; pp. 6176–6179. [Google Scholar]
- Lütkepohl, H. Introduction to Multiple Time Series Analysis; Springer Science & Business Media: Berlin, Germany, 2013. [Google Scholar]
- Zou, H.; Hastie, T.; Tibshirani, R. On the “degrees of freedom” of the lasso. Ann. Stat.
**2007**, 35, 2173–2192. [Google Scholar] [CrossRef] - Sun, X. The Lasso and Its Implementation for Neural Networks. Ph.D. Thesis, National Library of Canada = Bibliothèque nationale du Canada, University of Toronto, Toronto, ON, Canada, February 1999. [Google Scholar]
- Tibshirani, R.J.; Taylor, J. Degrees of freedom in lasso problems. Ann. Stat.
**2012**, 40, 1198–1232. [Google Scholar] [CrossRef] - Faes, L.; Porta, A.; Nollo, G.; Javorka, M. Information decomposition in multivariate systems: Definitions, implementation and application to cardiovascular networks. Entropy
**2017**, 19, 5. [Google Scholar] [CrossRef] - Bossomaier, T.; Barnett, L.; Harré, M.; Lizier, J. An Introduction to Transfer Entropy: Information Flow in Complex Systems; Springer International Publishing: Berlin, Germany; Cham, Switzerland, 2016. [Google Scholar]
- Williams, P.L.; Beer, R.D. Nonnegative decomposition of multivariate information. arXiv
**2010**, arXiv:1004.2515. [Google Scholar] - Barrett, A.B.; Barnett, L.; Seth, A.K. Multivariate Granger causality and generalized variance. Phys. Rev. E
**2010**, 81, 041907. [Google Scholar] [CrossRef] [Green Version] - Faes, L.; Nollo, G.; Porta, A. Information decomposition: A tool to dissect cardiovascular and cardiorespiratory complexity. In Complexity Nonlinearity Cardiovasc. Signals; Springer: Cham, Switzerland, 2017; pp. 87–113. [Google Scholar]
- Faes, L.; Nollo, G.; Stramaglia, S.; Marinazzo, D. Multiscale granger causality. Phys. Rev. E
**2017**, 96, 042150. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schreiber, T.; Schmitz, A. Improved surrogate data for nonlinearity tests. Phys. Rev. Lett.
**1996**, 77, 635. [Google Scholar] [CrossRef] [Green Version] - Toppi, J.; Mattia, D.; Risetti, M.; Formisano, R.; Babiloni, F.; Astolfi, L. Testing the significance of connectivity networks: Comparison of different assessing procedures. IEEE Trans. Biomed. Eng.
**2016**, 63, 2461–2473. [Google Scholar] [PubMed] - Porta, A.; Faes, L.; Nollo, G.; Bari, V.; Marchi, A.; De Maria, B.; Takahashi, A.C.; Catai, A.M. Conditional self-entropy and conditional joint transfer entropy in heart period variability during graded postural challenge. PLoS ONE
**2015**, 10, e0132851. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anzolin, A.; Astolfi, L. Statistical Causality in the EEG for the Study of Cognitive Functions in Healthy and Pathological Brains; Sapienza University of Rome: Rome, Italy, 2018; Available online: https://iris.uniroma1.it (accessed on 19 February 2018).
- Kim, S.; Kim, H. A new metric of absolute percentage error for intermittent demand forecasts. Int. J. Forecast.
**2016**, 32, 669–679. [Google Scholar] [CrossRef] - Toppi, J.; Sciaraffa, N.; Antonacci, Y.; Anzolin, A.; Caschera, S.; Petti, M.; Mattia, D.; Astolfi, L. Measuring the agreement between brain connectivity networks. In Proceedings of the 2016 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Orlando, FL, USA, 16–20 August 2016; pp. 68–71. [Google Scholar]
- Porta, A.; D’addio, G.; Guzzetti, S.; Lucini, D.; Pagani, M. Testing the presence of non stationarities in short heart rate variability series. In Proceedings of the Computers in Cardiology, Chicago, IL, USA, 19–22 September 2004; pp. 645–648. [Google Scholar]
- Schwarz, G. Estimating the dimension of a model. Annu. Stat.
**1978**, 6, 461–464. [Google Scholar] [CrossRef] - Rubinov, M.; Sporns, O. Complex network measures of brain connectivity: Uses and interpretations. Neuroimage
**2010**, 52, 1059–1069. [Google Scholar] [CrossRef] - Lizier, J.T.; Bertschinger, N.; Jost, J.; Wibral, M. Information decomposition of target effects from multi-source interactions: Perspectives on previous, current and future work. Entropy
**2018**, 220, 307. [Google Scholar] [CrossRef] [Green Version] - Silvey, S. Multicollinearity and imprecise estimation. J. R. Stat. Soc. Ser. B (Methodol.)
**1969**, 31, 539–552. [Google Scholar] [CrossRef] - Rish, I.; Grabarnik, G. Sparse Modeling: Theory, Algorithms, and Applications; CRC Press: Boca Raton, FL, USA, 2014. [Google Scholar]
- Zou, H. The adaptive lasso and its oracle properties. J. Am. Stat. Assoc.
**2006**, 101, 1418–1429. [Google Scholar] [CrossRef] [Green Version] - Fan, J.; Li, R. Variable selection via nonconcave penalized likelihood and its oracle properties. J. Am. Stat. Assoc.
**2001**, 96, 1348–1360. [Google Scholar] [CrossRef] - Irfan, M.; Javed, M.; Raza, M.A. Comparison of shrinkage regression methods for remedy of multicollinearity problem. Middle-East J. Sci. Res.
**2013**, 14, 570–579. [Google Scholar] - Abd Elrahman, S.M.; Abraham, A. A review of class imbalance problem. J. Netw. Innov. Comput.
**2013**, 1, 332–340. [Google Scholar] - Chetverikov, D.; Liao, Z.; Chernozhukov, V. On Cross-Validated LASSO in High Dimensions; Technical Report, Working Paper; UCLA: Los Angeles, CA, USA, February 2020. [Google Scholar]
- Schulz, S.; Haueisen, J.; Bär, K.J.; Voss, A. Multivariate assessment of the central-cardiorespiratory network structure in neuropathological disease. Physiol. Meas.
**2018**, 39, 074004. [Google Scholar] [CrossRef] - Lin, A.; Liu, K.K.; Bartsch, R.P.; Ivanov, P.C. Dynamic network interactions among distinct brain rhythms as a hallmark of physiologic state and function. Commun. Biol.
**2020**, 3, 1–11. [Google Scholar] - Kuipers, N.T.; Sauder, C.L.; Carter, J.R.; Ray, C.A. Neurovascular responses to mental stress in the supine and upright postures. J. Appl. Physiol.
**2008**, 104, 1129–1136. [Google Scholar] [CrossRef] - Berntson, G.G.; Cacioppo, J.T.; Quigley, K.S. Respiratory sinus arrhythmia: Autonomic origins, physiological mechanisms, and psychophysiological implications. Psychophysiology
**1993**, 30, 183–196. [Google Scholar] [CrossRef] - Schäfer, C.; Rosenblum, M.G.; Kurths, J.; Abel, H.H. Heartbeat synchronized with ventilation. Nature
**1998**, 392, 239–240. [Google Scholar] [CrossRef] - Drinnan, M.J.; Allen, J.; Murray, A. Relation between heart rate and pulse transit time during paced respiration. Physiol. Meas.
**2001**, 22, 425. [Google Scholar] [CrossRef] - Vrieze, S.I. Model selection and psychological theory: A discussion of the differences between the Akaike information criterion (AIC) and the Bayesian information criterion (BIC). Psychol. Methods
**2012**, 17, 228. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Faes, L.; Stramaglia, S.; Marinazzo, D. On the interpretability and computational reliability of frequency-domain Granger causality. F1000Research
**2017**, 6, 1710. [Google Scholar] [CrossRef] - Kuo, T.B.; Chen, C.Y.; Hsu, Y.C.; Yang, C.C. EEG beta power and heart rate variability describe the association between cortical and autonomic arousals across sleep. Auton. Neurosci.
**2016**, 194, 32–37. [Google Scholar] [CrossRef] [PubMed] - Kubota, Y.; Sato, W.; Toichi, M.; Murai, T.; Okada, T.; Hayashi, A.; Sengoku, A. Frontal midline theta rhythm is correlated with cardiac autonomic activities during the performance of an attention demanding meditation procedure. Cognit. Brain Res.
**2001**, 11, 281–287. [Google Scholar] [CrossRef] - Behzadnia, A.; Ghoshuni, M.; Chermahini, S. EEG Activities and the Sustained Attention Performance. Neurophysiology
**2017**, 49, 226–233. [Google Scholar] [CrossRef] - Tort, A.B.; Ponsel, S.; Jessberger, J.; Yanovsky, Y.; Brankačk, J.; Draguhn, A. Parallel detection of theta and respiration-coupled oscillations throughout the mouse brain. Sci. Rep.
**2018**, 8, 1–14. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pernice, R.; Javorka, M.; Krohova, J.; Czippelova, B.; Turianikova, Z.; Busacca, A.; Faes, L. Comparison of short-term heart rate variability indexes evaluated through electrocardiographic and continuous blood pressure monitoring. Med. Biol. Eng. Comput.
**2019**, 57, 1247–1263. [Google Scholar] [CrossRef] - Silvani, A.; Calandra-Buonaura, G.; Dampney, R.A.; Cortelli, P. Brain–heart interactions: Physiology and clinical implications. Philos. Trans. R. Soc. A
**2016**, 374, 20150181. [Google Scholar] [CrossRef] - Jurysta, F.; Lanquart, J.P.; Sputaels, V.; Dumont, M.; Migeotte, P.F.; Leistedt, S.; Linkowski, P.; Van De Borne, P. The impact of chronic primary insomnia on the heart rate–EEG variability link. Clin. Neurophysiol.
**2009**, 120, 1054–1060. [Google Scholar] [CrossRef] [PubMed] - Jurysta, F.; Lanquart, J.P.; Van De Borne, P.; Migeotte, P.F.; Dumont, M.; Degaute, J.P.; Linkowski, P. The link between cardiac autonomic activity and sleep delta power is altered in men with sleep apnea-hypopnea syndrome. Am. J. Physiol.-Regul. Integr. Comp. Physiol.
**2006**, 291, R1165–R1171. [Google Scholar] [CrossRef] [Green Version] - Zou, H.; Hastie, T. Regularization and variable selection via the elastic net. J. R. Stat. Soc. Seri. B (Stat. Method.)
**2005**, 67, 301–320. [Google Scholar] [CrossRef] [Green Version] - Milde, T.; Leistritz, L.; Astolfi, L.; Miltner, W.H.; Weiss, T.; Babiloni, F.; Witte, H. A new Kalman filter approach for the estimation of high-dimensional time-variant multivariate AR models and its application in analysis of laser-evoked brain potentials. Neuroimage
**2010**, 50, 960–969. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Graphical representation of the four-variate VAR (Vector Autoregressive) process realized in the first simulation according to Equation (17). Network nodes represent the four simulated processes, and arrows represent the imposed causal interactions (self-loops depict influences from the past to the present sample of a process).

**Figure 2.**Accuracy of PID (Partial Information Decomposition) measures computed for the VAR processes of Simulation I when ${Y}_{4}$ is taken as the target process. Panels report the bias (

**a**–

**c**) and the variance (

**d**–

**f**) relevant the computation of the TE (Transfer Entropy) from ${Y}_{2}$ to ${Y}_{4}$ and from ${Y}_{3}$ to ${Y}_{4}$ (

**a**,

**d**), the unique TE from ${Y}_{2}$ to ${Y}_{4}$ and from ${Y}_{3}$ to ${Y}_{4}$ (

**b**,

**e**) and the redundant and synergistic TE from ${Y}_{2}$ and ${Y}_{3}$ to ${Y}_{4}$ (

**c**,

**f**).

**Figure 3.**Accuracy of PID measures computed for the VAR processes of Simulation I when ${Y}_{1}$ is taken as the target process. Panels report the bias (

**a**–

**c**) and the variance (

**d**–

**f**) relevant the computation of the TE from ${Y}_{2}$ to ${Y}_{1}$ and from ${Y}_{3}$ to ${Y}_{1}$ (

**a**,

**d**), the unique TE from ${Y}_{2}$ to ${Y}_{1}$ and from ${Y}_{3}$ to ${Y}_{1}$ (

**b**,

**e**) and the redundant and synergistic TE from ${Y}_{2}$ and ${Y}_{3}$ to ${Y}_{1}$ (

**c**,

**f**).

**Figure 4.**Graphical representation for one of the ground-truth networks of Simulation II. Arrows represent the existence of a link, randomly assigned, between two nodes in the network. The thickness of the arrows is proportional to the strength of the connection, with a maximum value for the cTE equal to 0.15. The number of connections for each network is set to 45 out of 90.

**Figure 5.**Distribution of the bias parameters computed for the null links ($BIAS$,

**a**) and for the non-null links ($BIA{S}_{N}$,

**b**) considering the interaction factor K × TYPE, expressed as mean value and 95% confidence interval of the parameter computed across 100 realizations of simulation II for OLS (blue line) and LASSO (red line) for different values of K.

**Figure 6.**Distributions of $FNR$ (

**a**), $FPR$ (

**b**) and $ACC$ (

**c**) parameters considering the interaction factor K x TYPE, expressed as mean value and 95% confidence interval of the parameter computed across 100 realizations of simulation II for OLS (blue line) and LASSO (red line) for different values of K.

**Figure 7.**Partial Information Decomposition of brain–body interactions directed to the body nodes of the physiological network, assessed using OLS VAR identification. Box plots report the distributions across subjects (median: red lines; interquartile range: box; 10

^{th}–90

^{th}percentiles: blue lines) as well as the individual values (circles or triangles) of the PID measures (

**a**,

**d**,

**g**: joint information transfer;

**b**,

**e**,

**h**: unique information transfer;

**c**,

**f**,

**i**: synergistic and redundant transfer) computed at rest (R), during mental stress (M) and during serious game (G) considering the RR interval ($\eta $), the respiratory amplitude ($\rho $), or the pulse arrival time ($\pi $) as the target process j, and the body and brain sub-networks as source processes i and k. Statistically significant differences between pairs of distributions are marked with * (R vs. M), with # (R vs. G), with § (R vs. R), with ∼ (M vs. M) and with ∘ (G vs. G).

**Figure 8.**Partial Information Decomposition of brain–body interactions directed to the body nodes of the physiological network, assessed using LASSO-VAR identification. Box plots report the distributions across subjects (median: red lines; interquartile range: box; 10

^{th}–90

^{th}percentiles: blue lines) as well as the individual values (circles or triangles) of the PID measures (

**a**,

**d**,

**g**: joint information transfer;

**b**,

**e**,

**h**: unique information transfer;

**c**,

**f**,

**i**: synergistic and redundant transfer) computed at rest (R), during mental stress (M) and during serious game (G) considering the RR interval ($\eta $), the respiratory amplitude ($\rho $), or the pulse arrival time ($\pi $) as the target process j, and the body and brain sub-networks as source processes i and k. Statistically significant differences between pairs of distributions are marked with * (R vs. M), with # (R vs. G), with § (R vs. R), with ∼ (M vs. M) and with ∘ (G vs. G).

**Figure 9.**Partial Information Decomposition of brain–body interactions directed to the brain nodes of the physiological network, assessed using OLS VAR identification. Box plots report the distributions across subjects (median: red lines; interquartile range: box; 10

^{th}–90

^{th}percentiles: blue lines) as well as the individual values (circles or triangles) of the PID measures (

**a**,

**d**,

**g**,

**j**: joint information transfer;

**b**,

**e**,

**h**,

**k**: unique information transfer;

**c**,

**f**,

**i**,

**l**: synergistic and redundant transfer) computed at rest (R), during mental stress (M) and during serious game (G) considering the $\delta $, $\theta $, $\alpha $, or $\beta $ brain wave amplitude as the target process j, and the body and brain sub-networks as source processes i and k. Statistically significant differences between pairs of distributions are marked with * (R vs. M), with # (R vs. G), with § (R vs. R), with ∼ (M vs. M) and with ∘ (G vs. G).

**Figure 10.**Partial Information Decomposition of brain–body interactions directed to the brain nodes of the physiological network, assessed using LASSO-VAR identification. Box plots report the distributions across subjects (median: red lines; interquartile range: box; 10

^{th}–90

^{th}percentiles: blue lines) as well as the individual values (circles or triangles) of the PID measures (

**a**,

**d**,

**g**,

**j**: joint information transfer;

**b**,

**e**,

**h**,

**k**: unique information transfer;

**c**,

**f**,

**i**,

**l**: synergistic and redundant transfer) computed at rest (R), during mental stress (M) and during serious game (G) considering the $\delta $, $\theta $, $\alpha $, or $\beta $ brain wave amplitude as the target process j, and the body and brain sub-networks as source processes i and k. Statistically significant differences between pairs of distributions are marked with * (R vs. M), with # (R vs. G), with § (R vs. R), with ∼ (M vs. M) and with ∘ (G vs. G).

**Figure 11.**Topological structure for the networks of physiological interactions reconstructed during the three analyzes physiological states. Graphs depict significant directed interactions within the brain (yellow arrows) and body (red arrows) sub-networks as well as interactions between brain and body (blue arrows). Directed interactions were assessed counting the number of subjects for which the conditional transfer entropy (${T}_{i\to j|s}$) was detected as statistically significant using OLS (

**a**–

**c**) or LASSO (

**d**–

**f**) to perform VAR model identification. The arrow thickness is proportional to the number of subjects (n) for which the link is detected as statistically significant.

**Figure 12.**Bar plots reporting the in-strength index extracted from the cTE networks of Figure 11 by considering as link weights the percentage of subjects showing a brain-to-body connection (

**a**) or a body-to-brain connection (

**b**), computed at rest (R), during mental stress (M) and during serious game (G) for the two VAR identification methods. Please note that the in-strength computed along the direction from body to brain using LASSO is null in all conditions.

**Figure 13.**In-strength index computed for each node of the physiological network. Box plots report the distributions across subjects (median: red lines; interquartile range: box; 10

^{th}–90

^{th}percentiles: blue bars) as well as the individual values (circles) of the in-strength index (a-g) OLS, h-p LASSO) computed at rest (R), during mental stress (M) and during serious game (G) for each node ($\eta $,$\rho $,$\pi $,$\delta $,$\theta $,$\alpha $,$\beta $). Statistically significant differences between pairs of distributions are marked with # (R vs. G).

Factor | $\mathbf{BIAS}$ | ${\mathbf{BIAS}}_{\mathit{N}}$ | $\mathbf{FNR}$ | $\mathbf{FPR}$ | $\mathbf{ACC}$ |
---|---|---|---|---|---|

K | 8582 ** | 1694 ** | 2204 ** | 197.2 ** | 2492 ** |

TYPE | 1640 ** | 377 ** | 3538 ** | 223.4 ** | 1575 ** |

K × TYPE | 8633 ** | 848 ** | 1055 ** | 114.5 ** | 339 ** |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Antonacci, Y.; Astolfi, L.; Nollo, G.; Faes, L.
Information Transfer in Linear Multivariate Processes Assessed through Penalized Regression Techniques: Validation and Application to Physiological Networks. *Entropy* **2020**, *22*, 732.
https://doi.org/10.3390/e22070732

**AMA Style**

Antonacci Y, Astolfi L, Nollo G, Faes L.
Information Transfer in Linear Multivariate Processes Assessed through Penalized Regression Techniques: Validation and Application to Physiological Networks. *Entropy*. 2020; 22(7):732.
https://doi.org/10.3390/e22070732

**Chicago/Turabian Style**

Antonacci, Yuri, Laura Astolfi, Giandomenico Nollo, and Luca Faes.
2020. "Information Transfer in Linear Multivariate Processes Assessed through Penalized Regression Techniques: Validation and Application to Physiological Networks" *Entropy* 22, no. 7: 732.
https://doi.org/10.3390/e22070732