Granger Causality on forward and Reversed Time Series.

In this study, the information flow time arrow is investigated for stochastic data defined by vector autoregressive models. The time series are analyzed forward and backward by different Granger causality detection methods. Besides the normal distribution, which is usually required for the validity of Granger causality analysis, several other distributions of predictive errors are considered. A clear effect of a change in the order of cause and effect on the time-reversed series of unidirectionally connected variables was detected with standard Granger causality test (GC), when the product of the connection strength and the ratio of the predictive errors of the driver and the recipient was below a certain level, otherwise bidirectional causal connection was detected. On the other hand, opposite causal link was detected unconditionally by the methods based on the time reversal testing, but they were not able to detect correct bidirectional connection. The usefulness of the backward analysis is manifested in cases where falsely detected unidirectional connections can be rejected by applying the result obtained after the time reversal, and in cases of uncorrelated causally independent variables, where the absence of a causal link detected by GC on the original series should be confirmed on the time-reversed series.

Implementation of two causal methods based on predictions in reconstructed state spaces.

If deterministic dynamics is dominant in the data, then methods based on predictions in reconstructed state spaces can serve to detect causal relationships between and within the systems. Here we introduce two algorithms for such causal analysis. They are designed to detect causality from two time series but are potentially also applicable in a multivariate context. The first method is based on cross-predictions, and the second one on the so-called mixed predictions. In terms of performance, the cross-prediction method is considerably faster and less prone to false negatives. The predictability improvement method is slower, but in addition to causal detection, in a multivariate scenario, it also reveals which specific observables can help the most if we want to improve prediction. The study also highlights cases where our methods and state-space approaches generally seem to lose reliability. We propose a new perspective on these situations, namely that the variables under investigation have weak observability due to the complex nonlinear information flow in the system. Thus, in such cases, the failure of causality detection cannot be attributed to the methods themselves but to the use of data that do not allow reliable reconstruction of the underlying dynamics.

Causality, dynamical systems and the arrow of time.

Using several methods for detection of causality in time series, we show in a numerical study that coupled chaotic dynamical systems violate the first principle of Granger causality that the cause precedes the effect. While such a violation can be observed in formal applications of time series analysis methods, it cannot occur in nature, due to the relation between entropy production and temporal irreversibility. The obtained knowledge, however, can help to understand the type of causal relations observed in experimental data, namely, it can help to distinguish linear transfer of time-delayed signals from nonlinear interactions. We illustrate these findings in causality detected in experimental time series from the climate system and mammalian cardio-respiratory interactions.

Comparison of six methods for the detection of causality in a bivariate time series.

In this comparative study, six causality detection methods were compared, namely, the Granger vector autoregressive test, the extended Granger test, the kernel version of the Granger test, the conditional mutual information (transfer entropy), the evaluation of cross mappings between state spaces, and an assessment of predictability improvement due to the use of mixed predictions. Seven test data sets were analyzed: linear coupling of autoregressive models, a unidirectional connection of two Hénon systems, a unidirectional connection of chaotic systems of Rössler and Lorenz type and of two different Rössler systems, an example of bidirectionally connected two-species systems, a fishery model as an example of two correlated observables without a causal relationship, and an example of mediated causality. We tested not only 20000 points long clean time series but also noisy and short variants of the data. The standard and the extended Granger tests worked only for the autoregressive models. The remaining methods were more successful with the more complex test examples, although they differed considerably in their capability to reveal the presence and the direction of coupling and to distinguish causality from mere correlation.

Detection of coupling delay: A problem not yet solved.

Nonparametric detection of coupling delay in unidirectionally and bidirectionally coupled nonlinear dynamical systems is examined. Both continuous and discrete-time systems are considered. Two methods of detection are assessed-the method based on conditional mutual information-the CMI method (also known as the transfer entropy method) and the method of convergent cross mapping-the CCM method. Computer simulations show that neither method is generally reliable in the detection of coupling delays. For continuous-time chaotic systems, the CMI method appears to be more sensitive and applicable in a broader range of coupling parameters than the CCM method. In the case of tested discrete-time dynamical systems, the CCM method has been found to be more sensitive, while the CMI method required much stronger coupling strength in order to bring correct results. However, when studied systems contain a strong oscillatory component in their dynamics, results of both methods become ambiguous. The presented study suggests that results of the tested algorithms should be interpreted with utmost care and the nonparametric detection of coupling delay, in general, is a problem not yet solved.

Testing for causality in reconstructed state spaces by an optimized mixed prediction method.

In this study, a method of causality detection was designed to reveal coupling between dynamical systems represented by time series. The method is based on the predictions in reconstructed state spaces. The results of the proposed method were compared with outcomes of two other methods, the Granger VAR test of causality and the convergent cross-mapping. We used two types of test data. The first test example is a unidirectional connection of chaotic systems of Rössler and Lorenz type. The second one, the fishery model, is an example of two correlated observables without a causal relationship. The results showed that the proposed method of optimized mixed prediction was able to reveal the presence and the direction of coupling and distinguish causality from mere correlation as well.

Automatic sleep scoring: a search for an optimal combination of measures.

OBJECTIVE: The objective of this study is to find the best set of characteristics of polysomnographic signals for the automatic classification of sleep stages. METHODS: A selection was made from 74 measures, including linear spectral measures, interdependency measures, and nonlinear measures of complexity that were computed for the all-night polysomnographic recordings of 20 healthy subjects. The adopted multidimensional analysis involved quadratic discriminant analysis, forward selection procedure, and selection by the best subset procedure. Two situations were considered: the use of four polysomnographic signals (EEG, EMG, EOG, and ECG) and the use of the EEG alone. RESULTS: For the given database, the best automatic sleep classifier achieved approximately an 81% agreement with the hypnograms of experts. The classifier was based on the next 14 features of polysomnographic signals: the ratio of powers in the beta and delta frequency range (EEG, channel C3), the fractal exponent (EMG), the variance (EOG), the absolute power in the sigma 1 band (EEG, C3), the relative power in the delta 2 band (EEG, O2), theta/gamma (EEG, C3), theta/alpha (EEG, O1), sigma/gamma (EEG, C4), the coherence in the delta 1 band (EEG, O1-O2), the entropy (EMG), the absolute theta 2 (EEG, Fp1), theta/alpha (EEG, Fp1), the sigma 2 coherence (EEG, O1-C3), and the zero-crossing rate (ECG); however, even with only four features, we could perform sleep scoring with a 74% accuracy, which is comparable to the inter-rater agreement between two independent specialists. CONCLUSIONS: We have shown that 4-14 carefully selected polysomnographic features were sufficient for successful sleep scoring. The efficiency of the corresponding automatic classifiers was verified and conclusively demonstrated on all-night recordings from healthy adults.

Discrimination ability of individual measures used in sleep stages classification.

OBJECTIVE: The paper goes through the basic knowledge about classification of sleep stages from polysomnographic recordings. The next goal was to review and compare a large number of measures to find the suitable candidates for the study of sleep onset and sleep evolution. METHODS AND MATERIAL: A huge number of characteristics, including relevant simple measures in time domain, characteristics of distribution, linear spectral measures, measures of complexity and interdependency measures were computed for polysomnographic recordings of 20 healthy subjects. Summarily, all-night evolutions of 818 measures (73 characteristics for various channels and channel combinations) were analysed and compared with visual scorings of experts (hypnograms). Our tests involved classification of the data into five classes (waking and four sleep stages) and 10 classification tasks to distinguish between two specific sleep stages. To discover measures of the best decision-making ability, discriminant analysis was done by Fisher quadratic classifier for one-dimensional case. RESULTS AND CONCLUSIONS: The most difficult decision problem, between S1 and REM sleep, were best managed by measures computed from electromyogram led by fractal exponent (classification error 23%). In the simplest task, distinction between wake and deep sleep, the power ratio between delta and beta band of electroencephalogram was the most successful measure (classification error 1%). Delta/beta ratio with mean classification error 42.6% was the best single-performing measure also in discrimination between all five stages. However, the error level shows impossibility to satisfactorily separate the five sleep stages by a single measure. Use of a few additional characteristics is necessary. Some novel measures, especially fractal exponent and fractal dimension turned up equally successful or even superior to the conventional scoring methods in discrimination between particular states of sleep. They seem to provide a very promising basis for automatic sleep analysis particularly in conjunction with some of the successful spectral standards.