Research Article Instantaneous Granger Causality with the...

Hindawi Publishing CorporationISRN Signal ProcessingVolume 2013 Article ID 374064 9 pageshttpdxdoiorg1011552013374064

Research ArticleInstantaneous Granger Causality withthe Hilbert-Huang Transform

Joatildeo Rodrigues and Alexandre Andrade

Institute of Biophysics and Biomedical Engineering Faculty of Sciences University of Lisbon CampoGrande 1749-016 Lisboa Portugal

Correspondence should be addressed to Joao Rodrigues pmrjoaogmailcom

Received 30 November 2012 Accepted 6 January 2013

Academic Editors G A Tsihrintzis and M Wicks

Copyright copy 2013 J Rodrigues and A AndradeThis is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

Current measures of causality and temporal precedence have limited frequency and time resolution and therefore may not beviable in the detection of short periods of causality in specific frequencies In addition the presence of nonstationarities hindersthe causality estimation of current techniques as they are based on Fourier transforms or autoregressive model estimation In thiswork we present a combination of techniques to measure causality and temporal precedence between stationary and nonstationarytime series that is sensitive to frequency-specific short episodes of causalityThis methodology provides a highly informative time-frequency representation of causality with existing causality measures This is done by decomposing each time series into intrinsicoscillatorymodes with an empiricalmode decomposition algorithm and subsequently calculating their complexHilbert spectrumAt each time point the cross-spectrum is calculated between time series and used to measure coherency and compute the transferfunction and error covariance matrices using the Wilson-Burg method for spectral factorization The imaginary part of coherencycan then be computed as well as several Granger causality measures in the previous matrices This work covers the most importanttheoretical background of these techniques and tries to prove the usefulness of this new approach while pointing out some of itsqualities and drawbacks

1 Introduction

Identifying interactions of different temporal scales is a recur-rent topic in fields such as neuroscience and meteorologicalor financial research The concept of functional connectiv-ity which is defined as the statistical dependence betweendifferent variables is widely used However if activity fromone variable directly or indirectly exerts influence on othervariables functional connectivity measures will not identifythis dependence [1] This influence is interpreted as theeffective connectivity or causal influence and one solution forthis problem in time series inference (TSI) is the concept ofcausality introduced by Wiener and formulated by Granger[2] the Granger causality (GC) measure According to theconcept of causality one stochastic process is causal to asecond if the autoregressive predictability of the secondprocess at a given time point is improved by includingmeasurements from the past of the first GC has shown

to be suitable for the study of directionality in neuronalinteractions by assessment of neurophysiologic data in boththe frequency and time domains [3] as well as of simulatedsimple systems where the direction and relative strength ofcausal influence could be simulated [4]

The spectral (frequency domain) decomposition of GCwas formulated by Geweke [5] and it is compatible with thetotal time-domain GC as the latter is equal to the sum ofspectral GC components over all frequencies from zero to theNyquist frequency Spectral GC is important in neurophys-iologic studies because causal influences between neuronalpopulations often depend on oscillatory synchrony [3] andspectral decomposition helps to identify the dependencebetween them

Recently nonparametric methods for spectral GC havebeen developed allowing this measure and its variants thatdepend on the autoregressive (AR) model estimation tobe computed based on Fourier transform (FT) or wavelet

2 ISRN Signal Processing

transform (WT) [6] This methodology has evident advan-tages for the GC computation as it does not rely on the ARmodel estimation and therefore does not depend on esti-mating the correct model order or having appropriate modelconsistency Furthermore usingWTenables the possibility ofa time-frequency representation of causality with improvedtime and frequency resolution

TheHilbert-Huang transform (HHT) can also be used forthe time-frequency representation of a time-series amplitudeand provides greater time-frequency resolution than theaforementioned methods by calculating the instantaneousfrequency (IF) and amplitude on a set of orthogonal functionsin which the time series is decomposed intrinsic modefunctions (IMFs) [7] Also it has the advantage of greatlyreducing spectral leakage replacing the effects of harmonicdistortion by intrawave modulation and it can be used fornonstationary time series However it is highly dependenton the quality of the empirical mode decomposition (EMD)algorithm used to compute the IMFs [7] Although Hilbertspectra and Hilbert marginal spectra have been widely usedfor the spectral characterization of data [8ndash10] this methodhas also been used to study phase locking [11 12] andcoherence [13 14] Until now neither GC nor any causalityindex has been computed with the HHT in spite of previouspropositions [14] We have previously demonstrated promis-ing results by computing GC between IMFs [15]

Therefore in this workwe propose the necessarymethod-ology for the estimation of instantaneous causality and testthe computation of the instantaneous spectralGC (ISGC) andinstantaneous imaginary coherence (IIC) with the HHT

2 Methods

21 Background

211 Granger Causality Measures of GC can be assessed inthe time and frequency domain in a pairwise or conditionalfashion For multichannel datasets conditional and pairwiseanalysis can be performed In the former case an AR modelis fitted to each distinct pair of data channels whereas in thelatter a multivariate autoregressive (MVAR) model can befitted to the whole dataset Pairwise analysis is expected tosuffer from the drawback that it is not possible to discernwhether the influence between two channels is direct ormediated by other channels [16]

Considering two variables 1198831and 119883

2we try to predict

1198831(119905 + 1) using only past terms of 119883

1(restricted model)

Then the same prediction is made using past terms of both1198831and 119883

2(unrestricted model) If the second prediction is

significantly more successful then the past of 1198832contains

useful information for predicting 1198831(119905 + 1) that is not

in the past of 1198831 If so 119883

2G-causes 119883

1 Geweke [17]

showed that a conditional GCmay be calculated by includinganother variable 119883

3that can be a scalar or vector time

series This variable contains the remaining channels thatmust be included in the model to discern between direct ormediated influences between 119883

1and 119883

2 Equation (1) shows

the calculation of conditional time GC using a MVARmodel

of order 119901 For pairwise analysis 1198833is never used

1198831

(119905) =

119901

sum

119895=1

1198601119895

1198831

(119905 minus 119895) +

119901

sum

119895=1

1198602119895

1198832

(119905 minus 119895)

+

119901

sum

119895=1

1198603119895

1198833

(119905 minus 119895) + 11986411015840 (119905)

(1)

If 1198832helps prediction of 119883

1 then consider 119864

1to be the error

of a similar AR model without the past of 1198832

1198652rarr1|3

= lnvar (119864

1)

var (11986411015840)

(2)

where 1198652rarr1|3

is positiveSpectral GC has the advantage of providing the causality

values for each frequency component making it possibleto distinguish different processes within each interactionBesides Gewekersquos spectral GC [5] DTF [18] and PDC [19]can also be used to measure causal influence in the frequencydomain All the aforementioned methods depend on theFourier components of the MVAR model estimated in thetime domain they differ in how to calculate the influencebetween time series In Gewekersquos spectral GC the MVARspectrum is decomposed into intrinsic and causal parts andthe GC measure is calculated as the natural logarithm of theratio of the total spectral power to the intrinsic power Inthe bivariate analysis spectral GC is calculated by fitting anMVAR model into the dataset

1198831

(119905) =

119901

sum

119895=1

11986011119895

1198831

(119905 minus 119895) +

119901

sum

119895=1

11986012119895

1198832

(119905 minus 119895) + 1198641

(119905)

1198832

(119905) =

119901

sum

119895=1

11986021119895

1198831

(119905 minus 119895) +

119901

sum

119895=1

11986022119895

1198832

(119905 minus 119895) + 1198642

(119905)

(3)

The Fourier components of this MVARmodel are calculated

119860119897119898

(119891) = 120575119897119898

minus

119901

sum

119895=1

119860119897119898

(119895) 119890minus1198942120587119891119895

120575119897119898

1 997888rarr 119897 = 119898

0 997888rarr 119897 = 119898

(4)

From the Fourier components of the MVAR model it ispossible to calculate its transfer matrix H and obtain thefollowing representation for the MVAR model

(1198831

(119891)

1198832

(119891)) = (

11986711

(119891) 11986712

(119891)

11986721

(119891) 11986722

(119891)) (

1198641

(119891)

E2

(119891))H = Aminus1

(5)

The spectral matrix 119878 can then be calculated

S (119891) = ⟨X (119891)Xlowast (119891)⟩ = ⟨H (119891) ΣHlowast (119891)⟩ (6)

ISRN Signal Processing 3

where Σ is the model error covariance matrix The influenceof variable 119895 on variable 119894 in the frequency119891 can be calculatedas follows

119868119895997888rarr119894

= ln119878119894119894

(119891)

119878119894119894

(119891) minus (Σ119895119895

minus (Σ2

119894119895Σ119894119894))

10038161003816100381610038161003816119867119894119895

(119891)10038161003816100381610038161003816

2119895 119894 isin 1 2

(7)

Subtracting the causal part from the total spectral power inthe denominator results in the intrinsic spectral power of thevariable 119894Thismethod can only be used for pairwise analysisIf there are more than two variables in theMVARmodel thisequation will not be true because the denominator would notcontain only the intrinsic power For the conditional analysisGeweke [5] proposed a different method that will not bediscussed here For improvements of this method refer to[20]

A nonparametric alternative has been proposed byDhamala et al [6] and consists in computing the spectraldecomposition matrices used by the previous spectral GCformulations with the Wilson-Burg method for spectralfactorization [21] The basic principle of this method is thefactorization of the spectral density matrix into a unique setofminimumphase functionsΨThe following steps representthe analytic implementation of the solution proposed in [21]

119878 (119891) = ΨΨlowast

Ψ =

infin

sum

119896=0

1198601198961198901198941198962120587119891

(8)

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

Σ = 1198600119860119879

0 119867 = Ψ119860

minus1

0

such that

119878 = ΨΨlowast

= 119867Σ119867lowast

(9)

With these values GC can be calculated as in the parametricformulation

212 Imaginary Part of Coherency The coherency betweentwo time series is a measure of their linear relationship at aspecific frequency and it is given by (10)

119862119894119895

(119891) =

119878119894119895

(119891)

radic119878119894119894

(119891) 119878119895119895

(119891)

(10)

where 119878119894119895(119891) is the cross-spectrum between time series 119894 and

119895 and 119878119894119894(119891) and 119878

119895119895(119891) are their autospectra respectively

Because the coherencyrsquos phase 120601(119891) is proportional to thetime delay between these time series [22] its imaginarypart can be seen as the time relation weighted by thecoherence (absolute value of coherency) |119862

119894119895(119891)| sin(120601(119891))

and is insensitive to noninteracting sources [23]

213 Empirical Mode Decomposition The EMD is an itera-tive algorithm that removes the highest frequency oscillationfrom the analyzed data at each iteration After each repeti-tion a lower frequency information residue remains that isfurther decomposed until only a trend remainsThe resultingcomponents of this adaptive decomposition are the IMFs andrepresent the intrinsic oscillations of the signal so that whenadded they should result in the original signal These IMFsare defined as functions with equal number of extrema andzero crossings (they may not differ by more than one) withzero average between their upper and lower envelopes Asthey represent a simple oscillatory mode they can be seenas the equivalent to a spectral line in the Fourier estimatedspectrum with the difference that IMFs may be frequencymodulated The EMD is a fully data-driven mechanismand does not require previous knowledge about the signalcontrary to filtering It was developed [7] for the analysisof nonlinear nonstationary geophysical time series whichmight have been the motivation for the data-driven characterof this method and has spread to different applications frombiomedical to financial fields For nonstationary signals withits frequency content quickly changing across time the HHThas advantages when compared to a classical spectral analysislike Fourier or wavelet transform to yield higher frequencyresolution [10]

Recently thismethod has known several improvements inthe envelope computation and applicability to complex andmultivariate time series Among these improvements we findthe ensemble empirical mode decomposition (EEMD) andthe multivariate empirical mode decomposition (MEMD)The EEMD is a recent improvement by Wu and Huang [24]to the EMD algorithm The major drawback resolved by thisnew improvement is the frequent occurrence ofmodemixingin each IMF one IMF having signals of widely separate scalesor the same signal scale residing in different IMFs Whenmode mixing occurs the physical meaning of each IMF canbe difficult to interpret as several oscillatory modes can bepresent in one IMF as long as they do not intersect in time orthe same oscillatory mode can be present in more than oneIMF Due to previous studies [25] where the EMD showeda dyadic filter bank behavior when applied to white noiseWu andHuang decided to test the possibility of modemixingreduction by a noise-assisted data analysis (NADA) methodTherefore the EEMD algorithm consists in creating a largenumber of new signals that are the sum of the original dataand a set of different realizations of white noise Each one ofthese new signals is subjected to EMD resulting in a largenumber of IMF sets The resulting IMF set is an averageof all the previous IMF sets providing a cancelation of thenoise in each IMF Because of the dyadic filter behavior whendecomposing white noise each IMF is forced to have itsown scale and mode mixing is avoided Wu and Huang [24]proved that having the capability of isolating and extractingphysically meaningful signals facilitates the examination ofhow a system interacts with the correlation between IMFs ofdifferent data

The MEMD algorithm has been developed by Rehmanand Mandic [26] in order to find common oscillatorymodes in multivariate time series This technique calculates


the envelopes and the local mean of multivariate time seriesby projecting them along multiple directions on hyper-spheres A NADA extension (NA-MEMD) similar to EEMDhas also been developed which improves mode alignment[27]

214 Hilbert-Huang Spectrum After all the IMFs are deter-mined the IF of each IMF at each time point can be calcu-lated For a given real valued time series 119909(119905) this calculationuses its analytical signal 119911(119905) defined as 119911(119905) = 119909(119905) +

119894119867[119909(119905)] = 119886(119905) exp[119894120579(119905)] where 119886(119905) and 120579(119905) are the instan-taneous amplitude and phase respectively of the analyticalsignal and 119867[119909(119905)] is a signal orthogonal to 119909(119905) its Hilberttransform Now the IF can be obtained as the temporalderivative of the instantaneous phase IF(119905) = (12120587)119889120579(119905)119889119905This way it is possible to obtain a time-frequency distributionof each IMFrsquos amplitude and by combining these the Hilbertspectrum 119867119878(119891 119905) = 119886(119891 119905) is obtained In this workwe need to determine the complex Hilbert spectrum bycomputing 119888119867119878(119891 119905) = 119886(119891 119905) exp[119894120579(119891 119905)] to calculate thecross-spectrum necessary for the causality measures

22 Instantaneous Imaginary Coherency In this work IIC iscomputed in two different ways in the first the computationis based on the noise-assisted instantaneous coherence in[14] except that we calculate the imaginary part and avoidthe noise-assisted process In alternatives it is computedsimilarly to the magnitude squared coherence from [13]except that we use the imaginary part instead of the absolutevalue In the first method the cross-spectrum 119878

119894119895(119891 119905) is

calculated as the product of the complex Hilbert spectrumfrom variable 119894 119888119867119878

119894(119891 119905) and the complex conjugate of

119888119867119878119894(119891 119905) from variable 119895 The IIC can then be calculated by

(11)

II119862119894119895

(119891 119905) =

⟨119878119894119895

(119891 119905)⟩

radic⟨119878119894119894

(119891 119905)⟩ ⟨119878119895119895

(119891 119905)⟩

(11)

where the ⟨ ⟩ represents averaging multiple time measure-ments In case the denominator of (11) is zero IIC is assumedto be also zero as the cross-correlation is also zero

In the secondmethod IMFsmust be aligned according totheir time scale (mode aligned) and IIC is computed for pairsof IMFs according to (12) for time instant 119905 and pair 119899

II119862119894119895

(119899 119905) =

⟨119886119894119899

(119905) 119886119895119899

(119905) 119890[120579119894119899(119905)minus120579119895119899(119905)]

⟩

radic⟨119886119894119899

(119905)⟩ ⟨119886119895119899

(119905)⟩

(12)

The IIC for each IMF pair at each time instant is thencomplemented by the instantaneous frequency of each IMFin the pair Again when the denominator of (12) is zero sowill the numerator and the indetermination is resolved byassuming that IIC is also zero

23 Instantaneous Spectral Granger Causality ISGC com-putation follows a similar implementation to [6] exceptthat here 119878

119894119895(119891 119905) is calculated in the same way as the two

implementations of IIC in the previous section Thereforewe also have two alternatives for ISGC However there isone additional crucial step before using the nonparametricalternative proposed by Dhamala et al [6] Gaussian randomnoise of infinitesimal magnitude needs to be added to thecross-spectrum 119878

119894119895(119891 119905) at each time instant in order to

guarantee the correct solution of the spectral decompositionusing the analytic implementation proposed in [21] In [14]an amount of noise of four orders of magnitude less thanthe clean time series results in satisfactory root meanssquare error (RMSE) between estimated coherence and itstheoretical value thus we opted to use a noise magnitudefive times less than the original time series When noise isadded these methods can be computed several times andthe result averaged in order to average out the noise-inducedcausalities Causality can also be averaged across time pointsusing an averaging window

24 Significance Testing The null hypothesis of lack of causalinteraction is tested Hence a null population has to becreated using the original variables for which the causal rela-tionship between the new variables is removed but their timeand spectral distributions are preservedThis null hypothesisis tested with a one-tailed test the connection being deemedsignificant if the measure for a given frequency falls above achosen percentile of the null distribution for that frequencyFor both IIC and ISGC measures the method for creatingsurrogates consisted in phase randomization and correlationnullification between variables using Fourier transformedsurrogates

3 Results

We tested the proposed methods and both of their variantsin two simple situations stationary time series and a nonsta-tionary time series Due to the novelty of these methods andthe absence of any previous applications the time series weredesigned to have few oscillatory modes and no noise wasadded If more oscillatory modes or noise had been added itwould have been impossible to tell whether poor results weredue to this methodology or due to a poor EMD performanceWe need to infer the viability of the methodology as EMDproblems can be mitigated with the improvements explainedin Section 213

31 Stationary Time Series A network comprised of twonodes was used Two time series were constructed presentinga causal influence from the first time series to the secondEach time series had two oscillatory modes for 10Hz and30Hz each and the causal influence was only present in the10Hz frequency Time series length was 1 second and assumea sampling frequency of 200Hz Both time series can be seenin Figure 1

We chose to use the EEMD algorithm for IMF compu-tation After having these IMFs significance tested againstthe null hypothesis of being IMFs of Gaussian noise theywere aligned according to their temporal scale In this caseno alignment was needed The cross-spectrum is calculated


0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

minus3

minus2

minus1

0

1

2

3

Data points (119889119905 = 5ms)

119868 (10 Hz)(1 rarr 2)

Figure 1 Stationary time series showing causal influence in the10Hz from node 1 to node 2

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)

Imag (coherency)

minus05Freq

uenc

y (H

z)

0

50

100

0

052 rarr 1

0 50 100 150Data points (119889119905 = 5ms)

(b)

Figure 2 IIC based on the standard cross-spectrum is computed foreach time point Spectral resolution was set to 1Hz for visualizationeffects Positive values of IIC from time series 1 to 2 at 10Hz can beobserved End effects are present due to the EEMD

in both ways as explained in Section 22 so that IIC canbe readily computed for each time point IIC based on thestandard cross-spectrum can be seen in Figure 2 and IICbased on the IMF pairs cross-spectrum is shown in Figure 3

Significant positive values of IIC12(10Hz) point to a tem-

poral precedence of time series 1 around the 10Hz frequencyfor both IIC methods as expected however IIC based on thestandard cross-spectrum has higher frequency precision Weopted for a 1Hz spectral resolution for visualization effectsalthough a much higher resolution could be used

After adding Gaussian noise of infinitesimal magnitudeto the cross-spectral matrix the Wilson-Burg method forspectral factorization was applied at each time point andthe transfer matrix and noise covariance matrix in (9) werecomputed and were used as in (7)This way a time-frequency

20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

1 rarr 2

(a)

20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

2 rarr 1

(b)

Figure 3 IIC based on the IMF pairs cross-spectrum is computedfor each time point Spectral resolution was set to 1Hz for visualiza-tion effects Positive values of IIC from time series 1 to 2 at 10Hz canbe observed End effects are present due to the EEMD

distribution of causality was obtained In case 3 or more timeseries were present methods formultivariate Granger causal-ity [20] could be used These computations were repeated 20times and their results averaged in order to average out noise-induced causalities A time averaging window of 5 sampleswas passed through the ISGC matrices ISGC based on thestandard cross-spectrum can be seen in Figure 4 and ISGCbased on the IMF pairs cross-spectrum is shown in Figure 5In both figures only significant ISGC values with positivedifference of influence (DOI) are presented

Significant positive values of ISGC12

(10Hz) point toa causal relation of time series 1 in time series 2 aroundthe 10Hz frequency for both ISGC methods While IIC isconsistent for the temporal delay ISGC sometimes presentsfalse positives of causality from time series 2 to time series1 Both alternative implementations show similar results butthe ISGC based on the standard cross-spectrum producesfewer false positives

32 Nonstationary Time Series A network comprised of twonodes is used Two nonstationary time series are constructedpresenting a causal influence from the first time series to thesecond In this case three oscillatory modes are used Twooscillatorymodes are stationary andhave frequencies of 3 and8Hz The third oscillatory mode is nonstationary from 0 to05 seconds it oscillates at 20Hz and from 05 to 1 secondsit oscillates at 36Hz Also from 05 to 1 seconds this modersquosamplitude increases linearly The causal influence is also onlypresent in this mode In Figure 6 the resulting time series canbe consulted


0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)

Figure 4 ISGC based on the standard cross-spectrum is computedfor each time point Spectral resolution was set to 1Hz for visualiza-tion effects Positive values of ISGC from time series 1 to 2 at 10Hzcan be observed

We also chose to use the EEMD algorithm for IMFscomputation and after having these IMFsrsquo significance testedagainst the null hypothesis of being IMFs of Gaussian noisetheir alignment was checked No alignment was requiredand no mode mixing was present The cross-spectrum wascalculated in both ways as for the stationary time series andIIC was computed for each time point IIC based on thestandard cross-spectrum can be seen in Figure 7 and IICbased on the IMF pairs cross-spectrum is shown in Figure 8

IIC based on the standard cross-spectrum presents con-sistent values of time precedence however the instantaneousfrequencies tend to oscillate around the correct value dueto intrawave modulation The same effect is also present inthe IIC based on the IMF pairs cross-spectrum End effectsare always present in both alternatives suggesting that thefirst and last time points of the EEMD algorithm need to bediscarded This can be avoided by starting the EEMD somepoints earlier and finishing it a few points later in case thenumber of time points is not limited In the IIC based onthe IMF pairs cross-spectrum spurious significant temporalprecedence is found at 20Hz although only one time seriespresents 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 fac-torization was applied at each time point and the transfermatrix and noise covariance matrix in (9) were computedand used as in (7) leading to a time-frequency distribution ofcausality Computationswere repeated for 20 times to averageout noise-induced causalities A time averaging window of 5samples was also passed through the ISGC matrices ISGCbased on the standard cross-spectrumcan be seen in Figure 9

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 1 rarr 2

(a)

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 2 rarr 1

(b)

Figure 5 ISGC based on the IMF pairs cross-spectrum is computedfor each time point Spectral resolution was set to 1Hz for visualiza-tion effects Positive values of ISGC from time series 1 to 2 at 10Hzcan be observed

0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

Data points (119889119905 = 5ms)

minus60

minus40

minus20

0

20

40

60

119868 (20 Hz)(1 rarr 2)

119868 (36 Hz)(1 rarr 2)

Figure 6 Nonstationary time series at nodes 1 and 2 showing causalinfluence in the 20Hz from 0 to 05 seconds and in 36Hz from 05to 1Hz

and ISGC based on the IMF pairs cross-spectrum is shown inFigure 10 In both figures only significant ISGC values withpositive difference of influence (DOI) are presented

Significant positive values of ISGC12(20Hz) from 0 to 05

seconds and ISGC12(36Hz) from 05 to 1 seconds are present

in both alternativesHowever in ISGCbased on the IMFpairscross-spectrum additional causality can be observed ISGC

12

(8Hz) from 0 to 05 seconds and ISGC12(20Hz) from 05 to 1

seconds which is not consistent with the simulated data Bothmethods present end effects and from200 to 400millisecondsspurious ISGC

21(8Hz)


0 50 100 150

Imag (coherency)Fr

eque

ncy

(Hz)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)Imag (coherency) 2 rarr 1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

(b)

Figure 7 IIC based on the standard cross-spectrum is computedfor each time point Spectral resolution was set to 05Hz forvisualization effects Positive values of IIC from time series 1 to 2 at20Hz can be observed until 05 seconds and after this time at 36HzEnd effects are present due to the EEMD

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)

Imag (coherency) 2 rarr 1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

(b)

Figure 8 IIC based on the IMF pairs cross-spectrum is computedfor each time point Spectral resolution was set to 05Hz forvisualization effects Positive values of IIC from time series 1 to 2 at20Hz can be observed until 05 seconds and after this time at 36HzSpurious values of causality are present End effects are present dueto the EEMD

4 Discussion

This work is the first attempt (known to the authors) toformulate a spectral causality methodology that makes use ofthe highly informative (in both time and frequency) Hilbert-Huang spectrum This is a highly data-driven method The

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)

Figure 9 ISGC based on the standard cross-spectrum is computedfor each time point Spectral resolution was set to 05Hz forvisualization effects Positive values of ISGC from time series 1 to2 at 20Hz can be observed until 05 seconds and at 36Hz from05 to 1 seconds Spurious causality 2rarr 1 is also present in lowerfrequencies

only assumption made is that the IMFs are representedas sinusoids with their amplitude and phase modulatedthrough time Therefore the nonsinusoidal features will berepresented by intrawave or amplitude modulations [7] Thepossibility of being able to identify causal relations at eachtime step or a small average of contiguous time steps offersmany advantages and we assume that the most relevant arethe detection of causality between nonstationary processesand the ability to unfold in time bilateral causal relationsConcerning the first advantage all previous methods thatdepend on AR models or cross-spectral estimation based onthe Fourier transform can only be used in stationary dataAt most if data is wide-sense stationary these measures canbe calculated in several intervals similarly to the short timeFourier transform Nevertheless each interval needs to havesufficient points and this process can degrade frequencyresolution Also temporal resolution is typically coarse Theonly exception is the nonparametric GC computation withWT proposed by Dhamala et al [6] Indeed if it werenot for these authorsrsquo innovative idea of using the Wilson-Burg spectral factorization algorithm to obtain the transfermatrix 119867 and the error covariance matrix Σ (see (9)) thisimplementation would has not be possible Still the timeand frequency resolution of the HHT is far superior tothe WT Concerning the latter advantage since previouscausalitymethods need a considerable number of points to becomputed bilateral causal relations can be assumed to occursimultaneously Additionally it is common practice to use thedifference of influence (DOI) between two variables henceonly unilateral causal relations can be identified Using our


0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)

Figure 10 ISGC based on the IMF pairs cross-spectrum is com-puted for each time point Spectral resolution was set to 05Hz forvisualization effects Positive values of ISGC from time series 1 to 2 at20Hz and 8Hz can be observed until 05 seconds and at 20Hz and36Hz from 05 to 1 seconds Spurious causality 2rarr 1 is also present

proposed implementation it may be possible to identify atwhich intervals each causal relation of the bilateral causalityoccurs

The decision of using simple time series and a causalnetwork with only two nodes is made to avoid limitationsspecific to the EMD algorithm used There are severalextensions to the EMD algorithm and it is reasonable tothink that they will keep being developed We decided touse the EEMD to avoid mode mixing and then check forany necessary mode alignment Nonetheless we could haveused the bivariate EMD [28] for example In case more timeseries were present the NA-MEMD [27] seems to be thecorrect choice as it aligns oscillatory modes and avoids modemixing The ideal spectral GC measure could be the onedeveloped by Chen et al based on a partition matrix method[20] although other simpler measures can be used [18 19]Anyhow it is extremely important that the EMD algorithmsdo not alter the time relations between IMFs and do notintroduce spurious information In case they introduce spu-rious information it must be subsequently removed like inthe EEMDorNA-MEMDAlthough the results of the presentstudymay seemnaıve due to the simplicity of the simulationsthey add something new to the existing literature namelythat spectral GC can be calculated from IMFs Also theresponsibility to deal with more complex data relies on theEMD algorithm and the study of the best EMD algorithmfor causality estimation is beyond the scope of this paper

We did not present the application of this methodologyto real data as we wanted to focus on simple and controlleddata Its use with electroencephalography (EEG) data seemspromising especially in the case where nonstationaries are

present (eg epilepsy or sleep [13 29]) With our results bothIIC and ISGC were able to detect time lagged and causaleffects in stationary and nonstationary data with great timeand frequency resolutionWhen thesemethods use the cross-spectrum based on the IMF pairs results with nonstationarydata show an elevated number of causal relations in thewrongfrequenciesWhen thesemethods are computed based on thestandard cross-spectrum of complex HHTs these effects arenot noticeable except in the extremities due to EMD endeffects and both stationary and nonstationary influences arecorrectly detected Therefore for now we must recommendthe cross-spectrum calculation procedure

Finally with this work we were able to enumerate thenecessary steps for causality estimation with the highestpossible frequency and time resolution and demonstrateits initial promising results and limitations to overcomeFurthermore an additional instantaneous measure of timeprecedence with arbitrary frequency resolution based onthe imaginary part of coherency is also presented It can beused individually or as a complement to GC Other measuresof phase were considered like the phase slope index [22]though it depends on the slope of the coherency phase inthe frequency domain Because HHT presents instantaneousfrequencies this measure would not be applicable

Acknowledgment

This work was possible thanks to Project FCT PTDCSAU-ENB1122942009 financed by Fundacao para a Ciencia eTecnologia

References

[1] K J Friston ldquoFunctional and effective connectivity in neu-roimaging a synthesisrdquo Human Brain Mapping vol 2 no 1-2pp 56ndash78 1994

[2] C W J Granger ldquoInvestigating causal relations by econometricmodels and cross-spectral methodsrdquo Econometrica vol 37 no3 p 424 1969

[3] S L Bressler and A K Seth ldquoWiener-Granger Causality a wellestablished methodologyrdquo NeuroImage vol 58 no 2 pp 323ndash329 2011

[4] G Deshpande K Sathian and X Hu ldquoEffect of hemodynamicvariability on Granger causality analysis of fMRIrdquo NeuroImagevol 52 no 3 pp 884ndash896 2010

[5] J Geweke ldquoMeasurement of linear dependence and feedbackbetweenmultiple time seriesrdquo Journal of the American StatisticalAssociation vol 77 pp 304ndash313 1982

[6] M Dhamala G Rangarajan and M Ding ldquoAnalyzing infor-mation flow in brain networks with nonparametric Grangercausalityrdquo NeuroImage vol 41 no 2 pp 354ndash362 2008

[7] N E Huang Z Shen S R Long et al ldquoThe empirical modedecomposition and the Hubert spectrum for nonlinear andnon-stationary time series analysisrdquo Proceedings of the RoyalSociety A vol 454 no 1971 pp 903ndash995 1998

[8] T Tanaka and D P Mandic ldquoComplex empirical mode decom-positionrdquo IEEE Signal Processing Letters vol 14 no 2 pp 101ndash104 2007

[9] C-L Yeh H-C Chang C-H Wu and P-L Lee ldquoExtractionof single-trial cortical beta oscillatory activities in EEG signals


using empirical mode decompositionrdquo BioMedical EngineeringOnline vol 9 article 25 2010

[10] H Liang S L Bressler E A Buffalo R Desimone and PFries ldquoEmpirical mode decomposition of field potentials frommacaque V4 in visual spatial attentionrdquo Biological Cyberneticsvol 92 no 6 pp 380ndash392 2005

[11] C M Sweeney-Reed and S J Nasuto ldquoA novel approach to thedetection of synchronisation in EEG based on empirical modedecompositionrdquo Journal of Computational Neuroscience vol 23no 1 pp 79ndash111 2007

[12] J K Kroger and J Lakey ldquoPhase locking and empirical modedecompositions in EEGrdquo Citeseer no 1978 2008

[13] D Liu X Yang G Wang et al ldquoHHT based cardiopulmonarycoupling analysis for sleep apnea detectionrdquo SleepMedicine vol13 no 5 pp 503ndash509 2012

[14] M Hu and H Liang ldquoNoise-assisted instantaneous coherenceanalysis of brain connectivityrdquo Computational Intelligence andNeuroscience vol 2012 Article ID 275073 12 pages 2012

[15] J Rodrigues and A Andrade ldquoA New Approach for GrangerCausality between Neuronal Signals using the Empirical ModeDecomposition Algorithmrdquo Biomedical Engineering 765Telehealth 766 Assistive Technologies 2012

[16] M Kaminski M Ding W A Truccolo and S L BresslerldquoEvaluating causal relations in neural systems granger causalitydirected transfer function and statistical assessment of signifi-cancerdquo Biological Cybernetics vol 85 no 2 pp 145ndash157 2001

[17] J F Geweke ldquoMeasures of conditional linear dependenceand feedback between time seriesrdquo Journal of the AmericanStatistical Association vol 79 no 388 pp 907ndash915 1984

[18] M Eichler ldquoOn the evaluation of information flow in mul-tivariate systems by the directed transfer functionrdquo BiologicalCybernetics vol 94 no 6 pp 469ndash482 2006

[19] B Schelter M Winterhalder M Eichler et al ldquoTesting fordirected influences among neural signals using partial directedcoherencerdquo Journal of NeuroscienceMethods vol 152 no 1-2 pp210ndash219 2006

[20] Y Chen S L Bressler andM Ding ldquoFrequency decompositionof conditional Granger causality and application tomultivariateneural field potential datardquo Journal of Neuroscience Methodsvol 150 no 2 pp 228ndash237 2006

[21] G T Wilson ldquoFactorization of matricial spectral densitiesrdquoSIAM Journal on Applied Mathematics vol 23 no 4 pp 420ndash426 1972

[22] G Nolte A Ziehe V V Nikulin et al ldquoRobustly estimatingthe flow direction of information in complex physical systemsrdquoPhysical Review Letters vol 100 no 23 pp 1ndash4 2008

[23] G Nolte O Bai L Wheaton Z Mari S Vorbach and MHallett ldquoIdentifying true brain interaction fromEEGdata usingthe imaginary part of coherencyrdquo Clinical Neurophysiology vol115 no 10 pp 2292ndash2307 2004

[24] Z Wu and N E Huang ldquoEnsemble empirical mode decom-position a noise-assisted data analysis methodrdquo Advances inAdaptive Data Analysis vol 1 no 1 pp 1ndash41 2009

[25] P Flandrin G Rilling and P Goncalves ldquoEmpirical modedecomposition as a filter bankrdquo IEEE Signal Processing Lettersvol 11 no 2 pp 112ndash114 2004

[26] N Rehman and D P Mandic ldquoMultivariate empirical modedecompositionrdquo Proceedings of the Royal Society A vol 466 no2117 pp 1291ndash1302 2010

[27] N Ur Rehman and D P Mandic ldquoFilter bank property of mul-tivariate empirical mode decompositionrdquo IEEE Transactions onSignal Processing vol 59 no 5 pp 2421ndash2426 2011

[28] C Park D Looney P Kidmose M Ungstrup andD PMandicldquoTime-frequency analysis of EEG asymmetry using bivariateempirical mode decompositionrdquo IEEE Transactions on NeuralSystems and Rehabilitation Engineering vol 19 no 4 pp 366ndash373 2011

[29] M Chavez M Le Van Quyen V Navarro M Baulac andJ Martinerie ldquoSpatio-temporal dynamics prior to neocorticalseizures amplitude versus phase couplingsrdquo IEEE Transactionson Bio-Medical Engineering vol 50 no 5 pp 571ndash583 2003

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



transform (WT) [6] This methodology has evident advan-tages for the GC computation as it does not rely on the ARmodel estimation and therefore does not depend on esti-mating the correct model order or having appropriate modelconsistency Furthermore usingWTenables the possibility ofa time-frequency representation of causality with improvedtime and frequency resolution

TheHilbert-Huang transform (HHT) can also be used forthe time-frequency representation of a time-series amplitudeand provides greater time-frequency resolution than theaforementioned methods by calculating the instantaneousfrequency (IF) and amplitude on a set of orthogonal functionsin which the time series is decomposed intrinsic modefunctions (IMFs) [7] Also it has the advantage of greatlyreducing spectral leakage replacing the effects of harmonicdistortion by intrawave modulation and it can be used fornonstationary time series However it is highly dependenton the quality of the empirical mode decomposition (EMD)algorithm used to compute the IMFs [7] Although Hilbertspectra and Hilbert marginal spectra have been widely usedfor the spectral characterization of data [8ndash10] this methodhas also been used to study phase locking [11 12] andcoherence [13 14] Until now neither GC nor any causalityindex has been computed with the HHT in spite of previouspropositions [14] We have previously demonstrated promis-ing results by computing GC between IMFs [15]

Therefore in this workwe propose the necessarymethod-ology for the estimation of instantaneous causality and testthe computation of the instantaneous spectralGC (ISGC) andinstantaneous imaginary coherence (IIC) with the HHT

2 Methods

21 Background

211 Granger Causality Measures of GC can be assessed inthe time and frequency domain in a pairwise or conditionalfashion For multichannel datasets conditional and pairwiseanalysis can be performed In the former case an AR modelis fitted to each distinct pair of data channels whereas in thelatter a multivariate autoregressive (MVAR) model can befitted to the whole dataset Pairwise analysis is expected tosuffer from the drawback that it is not possible to discernwhether the influence between two channels is direct ormediated by other channels [16]

Considering two variables 1198831and 119883

2we try to predict

1198831(119905 + 1) using only past terms of 119883

1(restricted model)

Then the same prediction is made using past terms of both1198831and 119883

2(unrestricted model) If the second prediction is

significantly more successful then the past of 1198832contains

useful information for predicting 1198831(119905 + 1) that is not

in the past of 1198831 If so 119883

2G-causes 119883

1 Geweke [17]

showed that a conditional GCmay be calculated by includinganother variable 119883

3that can be a scalar or vector time

series This variable contains the remaining channels thatmust be included in the model to discern between direct ormediated influences between 119883

1and 119883

2 Equation (1) shows

the calculation of conditional time GC using a MVARmodel

of order 119901 For pairwise analysis 1198833is never used

1198831

(119905) =

119901

sum

119895=1

1198601119895

1198831

(119905 minus 119895) +

119901

sum

119895=1

1198602119895

1198832

(119905 minus 119895)

+

119901

sum

119895=1

1198603119895

1198833

(119905 minus 119895) + 11986411015840 (119905)

(1)

If 1198832helps prediction of 119883

1 then consider 119864

1to be the error

of a similar AR model without the past of 1198832

1198652rarr1|3

= lnvar (119864

1)

var (11986411015840)

(2)

where 1198652rarr1|3

is positiveSpectral GC has the advantage of providing the causality

values for each frequency component making it possibleto distinguish different processes within each interactionBesides Gewekersquos spectral GC [5] DTF [18] and PDC [19]can also be used to measure causal influence in the frequencydomain All the aforementioned methods depend on theFourier components of the MVAR model estimated in thetime domain they differ in how to calculate the influencebetween time series In Gewekersquos spectral GC the MVARspectrum is decomposed into intrinsic and causal parts andthe GC measure is calculated as the natural logarithm of theratio of the total spectral power to the intrinsic power Inthe bivariate analysis spectral GC is calculated by fitting anMVAR model into the dataset

1198831

(119905) =

119901

sum

119895=1

11986011119895

1198831

(119905 minus 119895) +

119901

sum

119895=1

11986012119895

1198832

(119905 minus 119895) + 1198641

(119905)

1198832

(119905) =

119901

sum

119895=1

11986021119895

1198831

(119905 minus 119895) +

119901

sum

119895=1

11986022119895

1198832

(119905 minus 119895) + 1198642

(119905)

(3)

The Fourier components of this MVARmodel are calculated

119860119897119898

(119891) = 120575119897119898

minus

119901

sum

119895=1

119860119897119898

(119895) 119890minus1198942120587119891119895

120575119897119898

1 997888rarr 119897 = 119898

0 997888rarr 119897 = 119898

(4)

From the Fourier components of the MVAR model it ispossible to calculate its transfer matrix H and obtain thefollowing representation for the MVAR model

(1198831

(119891)

1198832

(119891)) = (

11986711

(119891) 11986712

(119891)

11986721

(119891) 11986722

(119891)) (

1198641

(119891)

E2

(119891))H = Aminus1

(5)

The spectral matrix 119878 can then be calculated

S (119891) = ⟨X (119891)Xlowast (119891)⟩ = ⟨H (119891) ΣHlowast (119891)⟩ (6)



119868119895997888rarr119894

= ln119878119894119894

(119891)

119878119894119894

(119891) minus (Σ119895119895

minus (Σ2

119894119895Σ119894119894))

10038161003816100381610038161003816119867119894119895

(119891)10038161003816100381610038161003816

2119895 119894 isin 1 2

(7)



119878 (119891) = ΨΨlowast

Ψ =

infin

sum

119896=0

1198601198961198901198941198962120587119891

(8)


Σ = 1198600119860119879

0 119867 = Ψ119860

minus1

0

such that

119878 = ΨΨlowast

= 119867Σ119867lowast

(9)



119862119894119895

(119891) =

119878119894119895

(119891)

radic119878119894119894

(119891) 119878119895119895

(119891)

(10)


119895 and 119878119894119894(119891) and 119878



119894119895(119891)| sin(120601(119891))










119894119895(119891 119905) is




(11)

II119862119894119895

(119891 119905) =

⟨119878119894119895

(119891 119905)⟩

radic⟨119878119894119894

(119891 119905)⟩ ⟨119878119895119895

(119891 119905)⟩

(11)



II119862119894119895

(119899 119905) =

⟨119886119894119899

(119905) 119886119895119899

(119905) 119890[120579119894119899(119905)minus120579119895119899(119905)]

⟩

radic⟨119886119894119899

(119905)⟩ ⟨119886119895119899

(119905)⟩

(12)








3 Results





0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

minus3

minus2

minus1

0

1

2

3

Data points (119889119905 = 5ms)

119868 (10 Hz)(1 rarr 2)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)

Imag (coherency)

minus05Freq

uenc

y (H

z)

0

50

100

0

052 rarr 1

0 50 100 150Data points (119889119905 = 5ms)

(b)






20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

1 rarr 2

(a)

20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

2 rarr 1

(b)







0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)





0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 1 rarr 2

(a)

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 2 rarr 1

(b)


0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

Data points (119889119905 = 5ms)

minus60

minus40

minus20

0

20

40

60

119868 (20 Hz)(1 rarr 2)

119868 (36 Hz)(1 rarr 2)






12



21(8Hz)


0 50 100 150

Imag (coherency)Fr

eque

ncy

(Hz)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

1 rarr 2


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

(b)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

(b)


4 Discussion


0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)




0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)







Acknowledgment


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119868119895997888rarr119894

= ln119878119894119894

(119891)

119878119894119894

(119891) minus (Σ119895119895

minus (Σ2

119894119895Σ119894119894))

10038161003816100381610038161003816119867119894119895

(119891)10038161003816100381610038161003816

2119895 119894 isin 1 2

(7)



119878 (119891) = ΨΨlowast

Ψ =

infin

sum

119896=0

1198601198961198901198941198962120587119891

(8)


Σ = 1198600119860119879

0 119867 = Ψ119860

minus1

0

such that

119878 = ΨΨlowast

= 119867Σ119867lowast

(9)



119862119894119895

(119891) =

119878119894119895

(119891)

radic119878119894119894

(119891) 119878119895119895

(119891)

(10)


119895 and 119878119894119894(119891) and 119878



119894119895(119891)| sin(120601(119891))










119894119895(119891 119905) is




(11)

II119862119894119895

(119891 119905) =

⟨119878119894119895

(119891 119905)⟩

radic⟨119878119894119894

(119891 119905)⟩ ⟨119878119895119895

(119891 119905)⟩

(11)



II119862119894119895

(119899 119905) =

⟨119886119894119899

(119905) 119886119895119899

(119905) 119890[120579119894119899(119905)minus120579119895119899(119905)]

⟩

radic⟨119886119894119899

(119905)⟩ ⟨119886119895119899

(119905)⟩

(12)








3 Results





0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

minus3

minus2

minus1

0

1

2

3

Data points (119889119905 = 5ms)

119868 (10 Hz)(1 rarr 2)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)

Imag (coherency)

minus05Freq

uenc

y (H

z)

0

50

100

0

052 rarr 1

0 50 100 150Data points (119889119905 = 5ms)

(b)






20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

1 rarr 2

(a)

20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

2 rarr 1

(b)







0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)





0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 1 rarr 2

(a)

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 2 rarr 1

(b)


0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

Data points (119889119905 = 5ms)

minus60

minus40

minus20

0

20

40

60

119868 (20 Hz)(1 rarr 2)

119868 (36 Hz)(1 rarr 2)






12



21(8Hz)


0 50 100 150

Imag (coherency)Fr

eque

ncy

(Hz)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

1 rarr 2


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

(b)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

(b)


4 Discussion


0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)




0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)







Acknowledgment


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














119894119895(119891 119905) is




(11)

II119862119894119895

(119891 119905) =

⟨119878119894119895

(119891 119905)⟩

radic⟨119878119894119894

(119891 119905)⟩ ⟨119878119895119895

(119891 119905)⟩

(11)



II119862119894119895

(119899 119905) =

⟨119886119894119899

(119905) 119886119895119899

(119905) 119890[120579119894119899(119905)minus120579119895119899(119905)]

⟩

radic⟨119886119894119899

(119905)⟩ ⟨119886119895119899

(119905)⟩

(12)








3 Results





0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

minus3

minus2

minus1

0

1

2

3

Data points (119889119905 = 5ms)

119868 (10 Hz)(1 rarr 2)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)

Imag (coherency)

minus05Freq

uenc

y (H

z)

0

50

100

0

052 rarr 1

0 50 100 150Data points (119889119905 = 5ms)

(b)






20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

1 rarr 2

(a)

20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

2 rarr 1

(b)







0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)





0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 1 rarr 2

(a)

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 2 rarr 1

(b)


0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

Data points (119889119905 = 5ms)

minus60

minus40

minus20

0

20

40

60

119868 (20 Hz)(1 rarr 2)

119868 (36 Hz)(1 rarr 2)






12



21(8Hz)


0 50 100 150

Imag (coherency)Fr

eque

ncy

(Hz)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

1 rarr 2


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

(b)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

(b)


4 Discussion


0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)




0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)







Acknowledgment


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

minus3

minus2

minus1

0

1

2

3

Data points (119889119905 = 5ms)

119868 (10 Hz)(1 rarr 2)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)

Imag (coherency)

minus05Freq

uenc

y (H

z)

0

50

100

0

052 rarr 1

0 50 100 150Data points (119889119905 = 5ms)

(b)






20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

1 rarr 2

(a)

20406080

0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

100

minus2

0

2

Data points (119889119905 = 5ms)

2 rarr 1

(b)







0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)





0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 1 rarr 2

(a)

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 2 rarr 1

(b)


0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

Data points (119889119905 = 5ms)

minus60

minus40

minus20

0

20

40

60

119868 (20 Hz)(1 rarr 2)

119868 (36 Hz)(1 rarr 2)






12



21(8Hz)


0 50 100 150

Imag (coherency)Fr

eque

ncy

(Hz)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

1 rarr 2


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

(b)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

(b)


4 Discussion


0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)




0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)







Acknowledgment


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

05

1

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)





0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 1 rarr 2

(a)

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

02040608112

GC 2 rarr 1

(b)


0 20 40 60 80 100 120 140 160 180 200

Node 1Node 2

Data points (119889119905 = 5ms)

minus60

minus40

minus20

0

20

40

60

119868 (20 Hz)(1 rarr 2)

119868 (36 Hz)(1 rarr 2)






12



21(8Hz)


0 50 100 150

Imag (coherency)Fr

eque

ncy

(Hz)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

1 rarr 2


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

(b)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

(b)


4 Discussion


0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)




0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)







Acknowledgment


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0 50 100 150

Imag (coherency)Fr

eque

ncy

(Hz)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

1 rarr 2


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus050

05

Data points (119889119905 = 5ms)

(b)


0 50 100 150

Imag (coherency)

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

1 rarr 2

(a)


0 50 100 150

Freq

uenc

y (H

z)

0

50

100

minus05

0

05

Data points (119889119905 = 5ms)

(b)


4 Discussion


0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)




0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)







Acknowledgment


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 1 rarr 2

(a)

0

1

2

0 50 100 150

Freq

uenc

y (H

z)

0

50

100

Data points (119889119905 = 5ms)

GC 2 rarr 1

(b)







Acknowledgment


References


































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Instantaneous Granger Causality with the...

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Transcript of Research Article Instantaneous Granger Causality with the...