The causality measure of partial mutual information from mixed embedding (PMIME) revisited.

The measure of partial mutual information from mixed embedding (PMIME) is an information theory-based measure to accurately identify the direct and directional coupling, termed Granger causality or simply causality, between the observed variables or subsystems of a high-dimensional dynamical and complex system, without any a priori assumptions about the nature of the coupling relationship. In its core, it is a forward selection procedure that aims to iteratively identify the lag-dependence structure of a given observed variable (response) to all the other observed variables (candidate drivers). This model-free approach is capable of detecting nonlinear interactions, abundantly present in real-world complex systems, and it was shown to perform well on multivariate time series of moderately high dimension. However, the PMIME presents some inefficiencies in its performance mainly when applied on strongly stochastic (linear or nonlinear) systems as it may falsely detect non-existent relationships. Moreover, and by construction, the measure cannot extract purely synergetic relationships present in a system. In the current work, the issue of false detections is addressed by introducing an improved resampling significance test and a procedure of rechecking the identified drivers (backward revision). Regarding the inability to detect synergetic relationships, the PMIME is further enhanced by checking pairs as candidate drivers for the response variable after having considered all drivers individually. The effects of these modifications are investigated in a systematic simulation study on properly designed systems involving strong stochasticity, regressor terms with synergetic effects, and a system dimension ranging from 3 to 30. The overall results of the simulations indicate that these modifications indeed improve the performance of PMIME and alleviate to a significant degree the issues of the original algorithm. Guidelines for balancing between accuracy and computational efficiency are also given, particularly relevant for real-world applications. Finally, the measure performance is investigated in the study of futures of various government bonds and stock market indices in the period around COVID-19 pandemic.

Comparison of discretization strategies for the model-free information-theoretic assessment of short-term physiological interactions.

This work presents a comparison between different approaches for the model-free estimation of information-theoretic measures of the dynamic coupling between short realizations of random processes. The measures considered are the mutual information rate (MIR) between two random processes X and Y and the terms of its decomposition evidencing either the individual entropy rates of X and Y and their joint entropy rate, or the transfer entropies from X to Y and from Y to X and the instantaneous information shared by X and Y. All measures are estimated through discretization of the random variables forming the processes, performed either via uniform quantization (binning approach) or rank ordering (permutation approach). The binning and permutation approaches are compared on simulations of two coupled non-identical Hènon systems and on three datasets, including short realizations of cardiorespiratory (CR, heart period and respiration flow), cardiovascular (CV, heart period and systolic arterial pressure), and cerebrovascular (CB, mean arterial pressure and cerebral blood flow velocity) measured in different physiological conditions, i.e., spontaneous vs paced breathing or supine vs upright positions. Our results show that, with careful selection of the estimation parameters (i.e., the embedding dimension and the number of quantization levels for the binning approach), meaningful patterns of the MIR and of its components can be achieved in the analyzed systems. On physiological time series, we found that paced breathing at slow breathing rates induces less complex and more coupled CR dynamics, while postural stress leads to unbalancing of CV interactions with prevalent baroreflex coupling and to less complex pressure dynamics with preserved CB interactions. These results are better highlighted by the permutation approach, thanks to its more parsimonious representation of the discretized dynamic patterns, which allows one to explore interactions with longer memory while limiting the curse of dimensionality.

Detecting Nonlinear Interactions in Complex Systems: Application in Financial Markets.

Emerging or diminishing nonlinear interactions in the evolution of a complex system may signal a possible structural change in its underlying mechanism. This type of structural break may exist in many applications, such as in climate and finance, and standard methods for change-point detection may not be sensitive to it. In this article, we present a novel scheme for detecting structural breaks through the occurrence or vanishing of nonlinear causal relationships in a complex system. A significance resampling test was developed for the null hypothesis (H0) of no nonlinear causal relationships using (a) an appropriate Gaussian instantaneous transform and vector autoregressive (VAR) process to generate the resampled multivariate time series consistent with H0; (b) the modelfree Granger causality measure of partial mutual information from mixed embedding (PMIME) to estimate all causal relationships; and (c) a characteristic of the network formed by PMIME as test statistic. The significance test was applied to sliding windows on the observed multivariate time series, and the change from rejection to no-rejection of H0, or the opposite, signaled a non-trivial change of the underlying dynamics of the observed complex system. Different network indices that capture different characteristics of the PMIME networks were used as test statistics. The test was evaluated on multiple synthetic complex and chaotic systems, as well as on linear and nonlinear stochastic systems, demonstrating that the proposed methodology is capable of detecting nonlinear causality. Furthermore, the scheme was applied to different records of financial indices regarding the global financial crisis of 2008, the two commodity crises of 2014 and 2020, the Brexit referendum of 2016, and the outbreak of COVID-19, accurately identifying the structural breaks at the identified times.

Adaptation of Partial Mutual Information from Mixed Embedding to Discrete-Valued Time Series.

A causality analysis aims at estimating the interactions of the observed variables and subsequently the connectivity structure of the observed dynamical system or stochastic process. The partial mutual information from mixed embedding (PMIME) is found appropriate for the causality analysis of continuous-valued time series, even of high dimension, as it applies a dimension reduction by selecting the most relevant lag variables of all the observed variables to the response, using conditional mutual information (CMI). The presence of lag components of the driving variable in this vector implies a direct causal (driving-response) effect. In this study, the PMIME is appropriately adapted to discrete-valued multivariate time series, called the discrete PMIME (DPMIME). An appropriate estimation of the discrete probability distributions and CMI for discrete variables is implemented in the DPMIME. Further, the asymptotic distribution of the estimated CMI is derived, allowing for a parametric significance test for the CMI in the DPMIME, whereas for the PMIME, there is no parametric test for the CMI and the test is performed using resampling. Monte Carlo simulations are performed using different generating systems of discrete-valued time series. The simulation suggests that the parametric significance test for the CMI in the progressive algorithm of the DPMIME is compared favorably to the corresponding resampling significance test, and the accuracy of the DPMIME in the estimation of direct causality converges with the time-series length to the accuracy of the PMIME. Further, the DPMIME is used to investigate whether the global financial crisis has an effect on the causality network of the financial world market.

Phase-based causality analysis with partial mutual information from mixed embedding.

Instantaneous phases extracted from multivariate time series can retain information about the relationships between the underlying mechanisms that generate the series. Although phases have been widely used in the study of nondirectional coupling and connectivity, they have not found similar appeal in the study of causality. Herein, we present a new method for phase-based causality analysis, which combines ideas from the mixed embedding technique and the information-theoretic approach to causality in coupled oscillatory systems. We then use the introduced method to investigate causality in simulated datasets of bivariate, unidirectionally paired systems from combinations of Rössler, Lorenz, van der Pol, and Mackey-Glass equations. We observe that causality analysis using the phases can capture the true causal relation for coupling strength smaller than the analysis based on the amplitudes can capture. On the other hand, the causality estimation based on the phases tends to have larger variability, which is attributed more to the phase extraction process than the actual phase-based causality method. In addition, an application on real electroencephalographic data from an experiment on elicited human emotional states reinforces the usefulness of phases in causality identification.

Removing artifacts from TMS-evoked EEG: A methods review and a unifying theoretical framework.

Transcranial magnetic stimulation (TMS) combined with electroencephalography (EEG) is a technique for studying cortical excitability and connectivity in health and disease, allowing basic research and potential clinical applications. A major methodological issue, severely limiting the applicability of TMS-EEG, relates to the contamination of EEG signals by artifacts of biologic or non-biologic origin. To solve this problem, several methods, based on independent component analysis (ICA), principal component analysis (PCA), signal space projection (SSP), and other approaches, have been developed over the last decade. This article is divided into two parts. In the first part, we review the theoretical background of the currently available TMS-EEG artifact removal methods. In the second part, we formally introduce the mathematics underpinnings of the cleaning methods. We classify them into spatial and temporal filters based on their properties. Since the most frequently used TMS-EEG cleaning approach are spatial filter methods, we focus on them and introduce beamforming as a unified framework of the most popular spatial filtering techniques. This unifying approach enables the comparative assessment of these methods by highlighting their differences in terms of assumptions, challenges, and applicability for different types of artifacts and data. The different properties and challenges of the methods discussed are illustrated with both simulated and recorded data. This article targets non-mathematical and mathematical audiences. Accordingly, those readers interested in essential background information on these methods can focus on Section 2. Whereas theory-oriented readers may find Section 3 helpful for making informed decisions between existing methods and developing the methodology further.

TMS-induced brain connectivity modulation in Genetic Generalized Epilepsy.

OBJECTIVE: In epilepsy patients, Transcranial Magnetic Stimulation (TMS) may result in the induction and modulation of epileptiform discharges (EDs). We hereby investigate the modulatory effects of TMS on brain connectivity in Genetic Generalized Epilepsy (GGE) and explore their potential as a diagnostic biomarker in GGE. METHODS: Patients with GGE (n=18) and healthy controls (n=11) were investigated with a paired-pulse TMS-EEG protocol. The brain network was studied at local and at global levels using Coherence as an EEG connectivity measure. Comparison of patients vs controls was performed in a time-resolved manner by analyzing comparatively pre- vs post-TMS brain networks. RESULTS: There was statistically significant TMS-induced modulation of connectivity at specific frequency bands within groups and difference in TMS-induced modulation between the two groups. The most significant difference between patients and controls related to connectivity modulation in the γ band at 1-3 sec post-TMS (p=0.004). CONCLUSIONS: TMS modulates the healthy and epileptic brain connectivity in different ways. Our results indicate that TMS-EEG connectivity analysis can be a basis for a diagnostic biomarker of GGE. SIGNIFICANCE: The analysis identifies specific time periods and frequency bands of interest of TMS-induced connectivity modulation and elucidates the effect of TMS on the healthy and epileptic brain connectivity.

The Effect of a Hidden Source on the Estimation of Connectivity Networks from Multivariate Time Series.

Many methods of Granger causality, or broadly termed connectivity, have been developed to assess the causal relationships between the system variables based only on the information extracted from the time series. The power of these methods to capture the true underlying connectivity structure has been assessed using simulated dynamical systems where the ground truth is known. Here, we consider the presence of an unobserved variable that acts as a hidden source for the observed high-dimensional dynamical system and study the effect of the hidden source on the estimation of the connectivity structure. In particular, the focus is on estimating the direct causality effects in high-dimensional time series (not including the hidden source) of relatively short length. We examine the performance of a linear and a nonlinear connectivity measure using dimension reduction and compare them to a linear measure designed for latent variables. For the simulations, four systems are considered, the coupled Hénon maps system, the coupled Mackey-Glass system, the neural mass model and the vector autoregressive (VAR) process, each comprising 25 subsystems (variables for VAR) at close chain coupling structure and another subsystem (variable for VAR) driving all others acting as the hidden source. The results show that the direct causality measures estimate, in general terms, correctly the existing connectivity in the absence of the source when its driving is zero or weak, yet fail to detect the actual relationships when the driving is strong, with the nonlinear measure of dimension reduction performing best. An example from finance including and excluding the USA index in the global market indices highlights the different performance of the connectivity measures in the presence of hidden source.

Comparison of Causality Network Estimation in the Sensor and Source Space: Simulation and Application on EEG.

The usage of methods for the estimation of the true underlying connectivity among the observed variables of a system is increasing, especially in the domain of neuroscience. Granger causality and similar concepts are employed for the estimation of the brain network from electroencephalogram (EEG) data. Also source localization techniques, such as the standardized low resolution electromagnetic tomography (sLORETA), are widely used for obtaining more reliable data in the source space. In this work, connectivity structures are estimated in the sensor and in the source space making use of the sLORETA transformation for simulated and for EEG data with episodes of spontaneous epileptiform discharges (ED). From the comparative simulation study on high-dimensional coupled stochastic and deterministic systems originating in the sensor space, we conclude that the structure of the estimated causality networks differs in the sensor space and in the source space. Moreover, different network types, such as random, small-world and scale-free, can be better discriminated on the basis of the data in the original sensor space than on the transformed data in the source space. Similarly, in EEG epochs containing epileptiform discharges, the discriminative ability of network topological indices was significantly better in the sensor compared to the source level. In conclusion, causality networks constructed at the sensor and source level, for both simulated and empirical data, exhibit significant structural differences. These observations indicate that further studies are warranted in order to clarify the exact relationship between data registered in the sensor and source space.

A Novel Connectome-Based Electrophysiological Study of Subjective Cognitive Decline Related to Alzheimer's Disease by Using Resting-State High-Density EEG EGI GES 300.

Aim: To investigate for the first time the brain network in the Alzheimer's disease (AD) spectrum by implementing a high-density electroencephalography (HD-EEG - EGI GES 300) study with 256 channels in order to seek if the brain connectome can be effectively used to distinguish cognitive impairment in preclinical stages. Methods: Twenty participants with AD, 30 with mild cognitive impairment (MCI), 20 with subjective cognitive decline (SCD) and 22 healthy controls (HC) were examined with a detailed neuropsychological battery and 10 min resting state HD-EEG. We extracted correlation matrices by using Pearson correlation coefficients for each subject and constructed weighted undirected networks for calculating clustering coefficient (CC), strength (S) and betweenness centrality (BC) at global (256 electrodes) and local levels (29 parietal electrodes). Results: One-way ANOVA presented a statistically significant difference among the four groups at local level in CC [F (3, 88) = 4.76, p = 0.004] and S [F (3, 88) = 4.69, p = 0.004]. However, no statistically significant difference was found at a global level. According to the independent sample t-test, local CC was higher for HC [M (SD) = 0.79 (0.07)] compared with SCD [M (SD) = 0.72 (0.09)]; t (40) = 2.39, p = 0.02, MCI [M (SD) = 0.71 (0.09)]; t (50) = 0.41, p = 0.004 and AD [M (SD) = 0.68 (0.11)]; t (40) = 3.62, p = 0.001 as well, while BC showed an increase at a local level but a decrease at a global level as the disease progresses. These findings provide evidence that disruptions in brain networks in parietal organization may potentially represent a key factor in the ability to distinguish people at early stages of the AD continuum. Conclusions: The above findings reveal a dynamically disrupted network organization of preclinical stages, showing that SCD exhibits network disorganization withintermediate values between MCI and HC. Additionally, these pieces of evidence provide information on the usefulness of the 256 HD-EEG in network construction.

Influence of Preparation Depth and Design on Stress Distribution in Maxillary Central Incisors Restored with Ceramic Veneers: A 3D Finite Element Analysis.

PURPOSE: To evaluate the influence of different preparation designs and depths on the stress field developed in maxillary central incisors restored with veneers made with different ceramic materials using finite element analysis (FEA). MATERIALS AND METHODS: A linear static three-dimensional finite element analysis model was used with the aid of reverse engineering to develop digital models of maxillary central incisors restored with ceramic veneers, according to two different preparation depths (thin vs deep) and two different preparation designs (feather edge vs butt joint). Three ceramic systems were tested: (i) feldspathic porcelain, (ii) heat pressed glass ceramic IPS Empress 2 (Ivoclar Vivadent AG), and (iii) heat pressed glass ceramic IPS e.max-Press (Ivoclar Vivadent AG). Each model was subjected to a compressive force of 200N applied to the palatal surface 2 mm below the incisal edge. The longitudinal axis of the restored tooth formed an angle of 130o with the direction of the force. The biomechanical behavior of the different models was examined according to the von Mises stress criterion. Statistical analysis was performed using nonparametric confidence interval estimation using bootstrapping. RESULTS: The maximum observed stress values were calculated and found to be similar between prepared and intact teeth. The cervical margin of the veneers displayed the highest von Mises stress values. Irrespectively of the depth and preparation design, the biggest von Mises stress values were observed at the veneer structures with the following order: (i) IPS Empress 2, (ii) IPS e.max-Press, (iii) feldspathic (p = 0.001). Preparation depth resulted in statistically significant differences (p = 0.001) in the stress distribution in the majority of tested structures. As the preparation depth was increased, the stresses within the veneer structure and the tooth structures were decreased. No statistically significant differences were detected in the stresses among the different restored models, when the preparation design was considered. CONCLUSIONS: This FEA study suggests that ceramic veneers could restore the biomechanical behavior of prepared central incisors and made it similar of that of an intact tooth. Regardless of the preparation depth and design and the ceramic system used, the cervical margin of ceramic veneers presents the highest von Mises stress values. When feldspathic porcelain was compared with lithium disilicate (IPS e.max Press), the latter displayed the lowest transfer of stresses to dental tissues. An increase in preparation depth resulted in a statistically significant stress decrease in both the veneer and the tooth.

Evaluation of algorithms for correction of transcranial magnetic stimulation-induced artifacts in electroencephalograms.

Transcranial magnetic stimulation combined with electroencephalography (TMS-EEG) is widely used to study the reactivity and connectivity of brain regions for clinical or research purposes. The electromagnetic pulse of the TMS device generates at the instant of administration an artifact of large amplitude and a duration up to tens of milliseconds that overlaps with brain activity. Methods for TMS artifact correction have been developed to remove the artifact and recover the underlying, immediate response of the cerebral cortex to the magnetic stimulus. In this study, four such algorithms are evaluated. Since there is no ground truth for the masked brain activity, pilot data formed from the superposition of the isolated TMS artifact on EEG brain activity are used to evaluate the performance of the algorithms. Different scenarios of TMS-EEG experiments are considered for the evaluation: TMS at resting state, TMS inducing epileptiform discharges, and TMS administered during epileptiform discharges. We show that a proposed gap filling method is able to reproduce qualitative characteristics and, in many cases, closely resemble the hidden EEG signal. Finally, shortcomings of the TMS correction algorithms as well as the pilot data approach are discussed. Graphical abstract The transcranial magnetic stimulation (TMS) artifact on the electroencephalogram (EEG) and its correction.

Clinical utility and prospective of TMS-EEG.

Concurrent transcranial magnetic stimulation and electroencephalography (TMS-EEG) has emerged as a powerful tool to non-invasively probe brain circuits in humans, allowing for the assessment of several cortical properties such as excitability and connectivity. Over the past decade, this technique has been applied to various clinical populations, enabling the characterization and development of potential TMS-EEG predictors and markers of treatments and of the pathophysiology of brain disorders. The objective of this article is to present a comprehensive review of studies that have used TMS-EEG in clinical populations and to discuss potential clinical applications. To provide a technical and theoretical framework, we first give an overview of TMS-EEG methodology and discuss the current state of knowledge regarding the use of TMS-EEG to assess excitability, inhibition, plasticity and connectivity following neuromodulatory techniques in the healthy brain. We then review the insights afforded by TMS-EEG into the pathophysiology and predictors of treatment response in psychiatric and neurological conditions, before presenting recommendations for how to address some of the salient challenges faced in clinical TMS-EEG research. Finally, we conclude by presenting future directions in line with the tremendous potential of TMS-EEG as a clinical tool.

Classification methods can identify external constrains in swimming.

The purpose of the present study is to examine whether the use of fins is identifiable based on swimmers' technique and to find out technique-related features that depict fins' influence. First, a number of features were extracted from kinematic data given by movement sensors attached to swimmers' bodies during butterfly swimming technique. Then, dimensionality reduction, feature selection and classification methods were applied to the extracted features. Two classification tasks were defined, one for the three classes of long, short and no fins, attaining accuracy up to 70, 62 and 70%, respectively, and the two-class simplified version (long fins, no fins) with accuracy up to 78%. These high accuracy levels were also found statistically significant and suggest that the use of fins influences swimming technique in a recognizable way and that the selected features that depict those differences are swimming type depended.

Identification of Hidden Sources by Estimating Instantaneous Causality in High-Dimensional Biomedical Time Series.

The study of connectivity patterns of a system's variables, such as multi-channel electroencephalograms (EEG), is of utmost importance towards a better understanding of its internal evolutionary mechanisms. Here, the problem of estimating the connectivity network from multivariate time series in the presence of prominent unobserved variables is addressed. The causality measure of partial mutual information from mixed embedding (PMIME), designed to estimate direct lag-causal effects in the presence of many observed variables, is adapted to estimate also zero-lag effects, the so-called instantaneous causality. We term the proposed advanced method, PMIME0. The estimation of instantaneous causality by PMIME0 is a signature of the presence of hidden source in the observed system, as demonstrated analytically in a toy model. It is further demonstrated that the PMIME0 identifies the true instantaneous with great accuracy in a variety of high-dimensional dynamical systems. The method is applied to EEG data with epileptiform discharges (EDs), and the results imply a strong impact of unobserved confounders during the EDs. This finding comes as a possible explanation for the increased levels of causality during epileptic seizures estimated by some measures affected by the presence of a common source.

Decoding Motor Imagery through Common Spatial Pattern Filters at the EEG Source Space.

Brain-Computer Interface (BCI) is a rapidly developing technology that aims to support individuals suffering from various disabilities and, ultimately, improve everyday quality of life. Sensorimotor rhythm-based BCIs have demonstrated remarkable results in controlling virtual or physical external devices but they still face a number of challenges and limitations. Main challenges include multiple degrees-of-freedom control, accuracy, and robustness. In this work, we develop a multiclass BCI decoding algorithm that uses electroencephalography (EEG) source imaging, a technique that maps scalp potentials to cortical activations, to compensate for low spatial resolution of EEG. Spatial features were extracted using Common Spatial Pattern (CSP) filters in the cortical source space from a number of selected Regions of Interest (ROIs). Classification was performed through an ensemble model, based on individual ROI classification models. The evaluation was performed on the BCI Competition IV dataset 2a, which features 4 motor imagery classes from 9 participants. Our results revealed a mean accuracy increase of 5.6% with respect to the conventional application method of CSP on sensors. Neuroanatomical constraints and prior neurophysiological knowledge play an important role in developing source space-based BCI algorithms. Feature selection and classifier characteristics of our implementation will be explored to raise performance to current state-of-the-art.

Dynamics of Epileptiform Discharges Induced by Transcranial Magnetic Stimulation in Genetic Generalized Epilepsy.

OBJECTIVE: In patients with Genetic Generalized Epilepsy (GGE), transcranial magnetic stimulation (TMS) can induce epileptiform discharges (EDs) of varying duration. We hypothesized that (a) the ED duration is determined by the dynamic states of critical network nodes (brain areas) at the early post-TMS period, and (b) brain connectivity changes before, during and after the ED, as well as within the ED. METHODS: EEG recordings from two GGE patients were analyzed. For hypothesis (a), the characteristics of the brain dynamics at the early ED stage are measured with univariate and multivariate EEG measures and the dependence of the ED duration on these measures is evaluated. For hypothesis (b), effective connectivity measures are combined with network indices so as to quantify the brain network characteristics and identify changes in brain connectivity. RESULTS: A number of measures combined with specific channels computed on the first EEG segment post-TMS correlate with the ED duration. In addition, brain connectivity is altered from pre-ED to ED and post-ED and statistically significant changes were also detected across stages within the ED. CONCLUSION: ED duration is not purely stochastic, but depends on the dynamics of the post-TMS brain state. The brain network dynamics is significantly altered in the course of EDs.

Dimension reduction of frequency-based direct Granger causality measures on short time series.

BACKGROUND: The mainstream in the estimation of effective brain connectivity relies on Granger causality measures in the frequency domain. If the measure is meant to capture direct causal effects accounting for the presence of other observed variables, as in multi-channel electroencephalograms (EEG), typically the fit of a vector autoregressive (VAR) model on the multivariate time series is required. For short time series of many variables, the estimation of VAR may not be stable requiring dimension reduction resulting in restricted or sparse VAR models. NEW METHOD: The restricted VAR obtained by the modified backward-in-time selection method (mBTS) is adapted to the generalized partial directed coherence (GPDC), termed restricted GPDC (RGPDC). Dimension reduction on other frequency based measures, such the direct directed transfer function (dDTF), is straightforward. RESULTS: First, a simulation study using linear stochastic multivariate systems is conducted and RGPDC is favorably compared to GPDC on short time series in terms of sensitivity and specificity. Then the two measures are tested for their ability to detect changes in brain connectivity during an epileptiform discharge (ED) from multi-channel scalp EEG. COMPARISON WITH EXISTING METHOD(S): It is shown that RGPDC identifies better than GPDC the connectivity structure of the simulated systems, as well as changes in the brain connectivity, and is less dependent on the free parameter of VAR order. CONCLUSIONS: The proposed dimension reduction in frequency measures based on VAR constitutes an appropriate strategy to estimate reliably brain networks within short-time windows.

Assessment of resampling methods for causality testing: A note on the US inflation behavior.

Different resampling methods for the null hypothesis of no Granger causality are assessed in the setting of multivariate time series, taking into account that the driving-response coupling is conditioned on the other observed variables. As appropriate test statistic for this setting, the partial transfer entropy (PTE), an information and model-free measure, is used. Two resampling techniques, time-shifted surrogates and the stationary bootstrap, are combined with three independence settings (giving a total of six resampling methods), all approximating the null hypothesis of no Granger causality. In these three settings, the level of dependence is changed, while the conditioning variables remain intact. The empirical null distribution of the PTE, as the surrogate and bootstrapped time series become more independent, is examined along with the size and power of the respective tests. Additionally, we consider a seventh resampling method by contemporaneously resampling the driving and the response time series using the stationary bootstrap. Although this case does not comply with the no causality hypothesis, one can obtain an accurate sampling distribution for the mean of the test statistic since its value is zero under H0. Results indicate that as the resampling setting gets more independent, the test becomes more conservative. Finally, we conclude with a real application. More specifically, we investigate the causal links among the growth rates for the US CPI, money supply and crude oil. Based on the PTE and the seven resampling methods, we consistently find that changes in crude oil cause inflation conditioning on money supply in the post-1986 period. However this relationship cannot be explained on the basis of traditional cost-push mechanisms.