Current measures of causality and temporal precedence have limited frequency and time resolution and therefore may not be viable in the detection of short periods of causality in specific frequencies. In addition, the presence of nonstationarities hinders the causality estimation of current techniques as they are based on Fourier transforms or autoregressive model estimation. In this work we present a combination of techniques to measure causality and temporal precedence between stationary and nonstationary time series, that is sensitive to frequency-specific short episodes of causality. This methodology provides a highly informative time-frequency representation of causality with existing causality measures. This is done by decomposing each time series into intrinsic oscillatory modes with an empirical mode decomposition algorithm and, subsequently, calculating their complex Hilbert spectrum. At each time point the cross-spectrum is calculated between time series and used to measure coherency and compute the transfer function and error covariance matrices using the Wilson-Burg method for spectral factorization. The imaginary part of coherency can then be computed as well as several Granger causality measures in the previous matrices. This work covers the most important theoretical background of these techniques and tries to prove the usefulness of this new approach while pointing out some of its qualities and drawbacks.

Identifying interactions of different temporal scales is a recurrent topic in fields such as neuroscience and meteorological or financial research. The concept of functional connectivity, which is defined as the statistical dependence between different variables, is widely used. However, if activity from one variable directly or indirectly exerts influence on other variables, functional connectivity measures will not identify this dependence [

The spectral (frequency domain) decomposition of GC was formulated by Geweke [

Recently, nonparametric methods for spectral GC have been developed allowing this measure and its variants that depend on the autoregressive (AR) model estimation to be computed based on Fourier transform (FT) or wavelet transform (WT) [

The Hilbert-Huang transform (HHT) can also be used for the time-frequency representation of a time-series amplitude and provides greater time-frequency resolution than the aforementioned methods by calculating the instantaneous frequency (IF) and amplitude on a set of orthogonal functions in which the time series is decomposed, intrinsic mode functions (IMFs) [

Therefore, in this work we propose the necessary methodology for the estimation of instantaneous causality and test the computation of the instantaneous spectral GC (ISGC) and instantaneous imaginary coherence (IIC) with the HHT.

Measures of GC can be assessed in the time and frequency domain in a pairwise or conditional fashion. For multichannel datasets, conditional and pairwise analysis can be performed. In the former case, an AR model is fitted to each distinct pair of data channels whereas in the latter, a multivariate autoregressive (MVAR) model can be fitted to the whole dataset. Pairwise analysis is expected to suffer from the drawback that it is not possible to discern whether the influence between two channels is direct or mediated by other channels [

Considering two variables

Spectral GC has the advantage of providing the causality values for each frequency component making it possible to distinguish different processes within each interaction. Besides Geweke’s spectral GC [

A nonparametric alternative has been proposed by Dhamala et al. [

From the minimum phase spectral factor it is possible to obtain a noise covariance matrix and a transfer function (minimum phase):

The coherency between two time series is a measure of their linear relationship at a specific frequency, and it is given by (

The EMD is an iterative algorithm that removes the highest frequency oscillation from the analyzed data at each iteration. After each repetition, a lower frequency information residue remains that is further decomposed until only a trend remains. The resulting components of this adaptive decomposition are the IMFs and represent the intrinsic oscillations of the signal, so that when added, they should result in the original signal. These IMFs are defined as functions with equal number of extrema and zero crossings (they may not differ by more than one) with zero average between their upper and lower envelopes. As they represent a simple oscillatory mode, they can be seen as the equivalent to a spectral line in the Fourier estimated spectrum with the difference that IMFs may be frequency modulated. The EMD is a fully data-driven mechanism and does not require previous knowledge about the signal, contrary to filtering. It was developed [

Recently this method has known several improvements in the envelope computation and applicability to complex and multivariate time series. Among these improvements we find the ensemble empirical mode decomposition (EEMD) and the multivariate empirical mode decomposition (MEMD). The EEMD is a recent improvement by Wu and Huang [

The MEMD algorithm has been developed by Rehman and Mandic [

After all the IMFs are determined, the IF of each IMF at each time point can be calculated. For a given real valued time series

In this work IIC is computed in two different ways: in the first the computation is based on the noise-assisted instantaneous coherence in [

In the second method, IMFs must be aligned according to their time scale (mode aligned) and IIC is computed for pairs of IMFs according to (

ISGC computation follows a similar implementation to [

The null hypothesis of lack of causal interaction is tested. Hence, a null population has to be created using the original variables, for which the causal relationship between the new variables is removed, but their time and spectral distributions are preserved. This null hypothesis is tested with a one-tailed test, the connection being deemed significant if the measure, for a given frequency, falls above a chosen percentile of the null distribution for that frequency. For both IIC and ISGC measures, the method for creating surrogates consisted in phase randomization and correlation nullification between variables using Fourier transformed surrogates.

We tested the proposed methods and both of their variants in two simple situations: stationary time series and a nonstationary time series. Due to the novelty of these methods and the absence of any previous applications, the time series were designed to have few oscillatory modes, and no noise was added. If more oscillatory modes or noise had been added, it would have been impossible to tell whether poor results were due to this methodology or due to a poor EMD performance. We need to infer the viability of the methodology as EMD problems can be mitigated with the improvements explained in Section

A network comprised of two nodes was used. Two time series were constructed presenting a causal influence from the first time series to the second. Each time series had two oscillatory modes for 10 Hz and 30 Hz each, and the causal influence was only present in the 10 Hz frequency. Time series length was 1 second and assume a sampling frequency of 200 Hz. Both time series can be seen in Figure

Stationary time series showing causal influence in the 10 Hz from node 1 to node 2.

We chose to use the EEMD algorithm for IMF computation. After having these IMFs significance tested against the null hypothesis of being IMFs of Gaussian noise, they were aligned according to their temporal scale. In this case no alignment was needed. The cross-spectrum is calculated in both ways as explained in Section

IIC based on the standard cross-spectrum is computed for each time point. Spectral resolution was set to 1 Hz for visualization effects. Positive values of IIC from time series 1 to 2 at 10 Hz can be observed. End effects are present due to the EEMD.

IIC based on the IMF pairs cross-spectrum is computed for each time point. Spectral resolution was set to 1 Hz for visualization effects. Positive values of IIC from time series 1 to 2 at 10 Hz can be observed. End effects are present due to the EEMD.

Significant positive values of IIC_{12} (10 Hz) point to a temporal precedence of time series 1 around the 10 Hz frequency for both IIC methods as expected, however IIC based on the standard cross-spectrum has higher frequency precision. We opted for a 1 Hz spectral resolution for visualization effects, although a much higher resolution could be used.

After adding Gaussian noise of infinitesimal magnitude to the cross-spectral matrix, the Wilson-Burg method for spectral factorization was applied at each time point, and the transfer matrix and noise covariance matrix in (

ISGC based on the standard cross-spectrum is computed for each time point. Spectral resolution was set to 1 Hz for visualization effects. Positive values of ISGC from time series 1 to 2 at 10 Hz can be observed.

ISGC based on the IMF pairs cross-spectrum is computed for each time point. Spectral resolution was set to 1 Hz for visualization effects. Positive values of ISGC from time series 1 to 2 at 10 Hz can be observed.

Significant positive values of ISGC_{12} (10 Hz) point to a causal relation of time series 1 in time series 2 around the 10 Hz frequency for both ISGC methods. While IIC is consistent for the temporal delay, ISGC sometimes presents false positives of causality from time series 2 to time series 1. Both alternative implementations show similar results, but the ISGC based on the standard cross-spectrum produces fewer false positives.

A network comprised of two nodes is used. Two nonstationary time series are constructed presenting a causal influence from the first time series to the second. In this case three oscillatory modes are used. Two oscillatory modes are stationary and have frequencies of 3 and 8 Hz. The third oscillatory mode is nonstationary: from 0 to 0.5 seconds it oscillates at 20 Hz, and from 0.5 to 1 seconds it oscillates at 36 Hz. Also, from 0.5 to 1 seconds, this mode’s amplitude increases linearly. The causal influence is also only present in this mode. In Figure

Nonstationary time series at nodes 1 and 2 showing causal influence in the 20 Hz from 0 to 0.5 seconds and in 36 Hz from 0.5 to 1 Hz.

We also chose to use the EEMD algorithm for IMFs computation, and after having these IMFs’ significance tested against the null hypothesis of being IMFs of Gaussian noise, their alignment was checked. No alignment was required, and no mode mixing was present. The cross-spectrum was calculated in both ways as for the stationary time series, and IIC was computed for each time point. IIC based on the standard cross-spectrum can be seen in Figure

IIC based on the standard cross-spectrum is computed for each time point. Spectral resolution was set to 0.5 Hz for visualization effects. Positive values of IIC from time series 1 to 2 at 20 Hz can be observed until 0.5 seconds and after this time at 36 Hz. End effects are present due to the EEMD.

IIC based on the IMF pairs cross-spectrum is computed for each time point. Spectral resolution was set to 0.5 Hz for visualization effects. Positive values of IIC from time series 1 to 2 at 20 Hz can be observed until 0.5 seconds and after this time at 36 Hz. Spurious values of causality are present. End effects are present due to the EEMD.

IIC based on the standard cross-spectrum presents consistent values of time precedence; however the instantaneous frequencies tend to oscillate around the correct value due to intrawave modulation. The same effect is also present in the IIC based on the IMF pairs cross-spectrum. End effects are always present in both alternatives suggesting that the first and last time points of the EEMD algorithm need to be discarded. This can be avoided by starting the EEMD some points earlier and finishing it a few points later in case the number of time points is not limited. In the IIC based on the IMF pairs, cross-spectrum spurious significant temporal precedence is found at 20 Hz although only one time series presents oscillations in this frequency range.

After adding Gaussian noise of infinitesimal magnitude (five times less than the original time series) to the cross-spectral matrix, the Wilson-Burg method for spectral factorization was applied at each time point, and the transfer matrix and noise covariance matrix in (

ISGC based on the standard cross-spectrum is computed for each time point. Spectral resolution was set to 0.5 Hz for visualization effects. Positive values of ISGC from time series 1 to 2 at 20 Hz can be observed until 0.5 seconds and at 36 Hz from 0.5 to 1 seconds. Spurious causality 2→1 is also present in lower frequencies.

ISGC based on the IMF pairs cross-spectrum is computed for each time point. Spectral resolution was set to 0.5 Hz for visualization effects. Positive values of ISGC from time series 1 to 2 at 20 Hz and 8 Hz can be observed until 0.5 seconds and at 20 Hz and 36 Hz from 0.5 to 1 seconds. Spurious causality 2→1 is also present.

Significant positive values of ISGC_{12} (20 Hz) from 0 to 0.5 seconds and ISGC_{12} (36 Hz) from 0.5 to 1 seconds are present in both alternatives. However, in ISGC based on the IMF pairs cross-spectrum additional causality can be observed: ISGC_{12} (8 Hz) from 0 to 0.5 seconds and ISGC_{12} (20 Hz) from 0.5 to 1 seconds which is not consistent with the simulated data. Both methods present end effects and from 200 to 400 milliseconds spurious ISGC_{21} (8 Hz).

This work is the first attempt (known to the authors) to formulate a spectral causality methodology that makes use of the highly informative (in both time and frequency) Hilbert-Huang spectrum. This is a highly data-driven method. The only assumption made is that the IMFs are represented as sinusoids with their amplitude and phase modulated through time. Therefore, the nonsinusoidal features will be represented by intrawave or amplitude modulations [

The decision of using simple time series and a causal network with only two nodes is made to avoid limitations specific to the EMD algorithm used. There are several extensions to the EMD algorithm, and it is reasonable to think that they will keep being developed. We decided to use the EEMD to avoid mode mixing and then check for any necessary mode alignment. Nonetheless, we could have used the bivariate EMD [

We did not present the application of this methodology to real data as we wanted to focus on simple and controlled data. Its use with electroencephalography (EEG) data seems promising especially in the case where nonstationaries are present (e.g., epilepsy or sleep [

Finally, with this work we were able to enumerate the necessary steps for causality estimation with the highest possible frequency and time resolution and demonstrate its initial promising results and limitations to overcome. Furthermore, an additional instantaneous measure of time precedence with arbitrary frequency resolution, based on the imaginary part of coherency, is also presented. It can be used individually or as a complement to GC. Other measures of phase were considered like the phase slope index [

This work was possible thanks to Project FCT, PTDC/SAU-ENB/112294/2009 financed by Fundação para a Ciência e Tecnologia.