Partial Directed Coherence: Some Estimation Issues Baccalá LA 1, Sameshima K 2

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1 Partial Directed Coherence: Some Estimation Issues Baccalá LA 1, Sameshima K 2 1 Telecommunications and Control Eng. Dept., Escola Politécnica and 2 Discipline of Medical Informatics & Functional Neurosurgery Laboratory, School of Medicine University of São Paulo baccala@lcs.poli.usp.br ksameshi@usp.br Summary This paper addresses some estimation performance issues of the Partial Directed Coherence (PDC), a newly introduced quantity related to Granger Causality, that is applicable to the functional and structural inference in neurobiology by processing many simultaneously measured signals. Exhaustive Monte Carlo simulations of simple models of interest show the asymptotic validity of previously used frequency independent thresholds in rejecting the hypothesis of lack of unidirectional connection between structures. When only short stationary time-series are available, we show that frequency dependent thresholds for PDC are more appropriate. Finally our results suggest that PDC converges faster to the correct structural inferences compared to classical Granger Causality tests. Introduction Recently, we proposed a new multiple-time-series based method to investigate structural relationships between simultaneous measurements taken from multiple electrodes (Baccalá and Sameshima, 21). The new method was named Partial Directed Coherence because, being an evolution of the well known frequency domain concept of partial coherence, it also permits determining the direction of information flow between each member of a pair of neural structures by subtracting the interactions and possible common influences due to other remaining yet simultaneously observed time series. The ability of PDC to disclose the direction of information flow is intimately related to the fundamental notion of Granger causality (Granger, 1969) which, roughly, refers to how much the predictability of the present of a time-series is enhanced by the exclusive knowledge of the past of another time series (Baccalá and Sameshima, 21). As a matter of fact, zero PDC from structure A to structure B at all frequencies is synonimous with lack of Granger causality from A to B. In turn this allows one to infer whether some possibly existing physical link from structure A to B is inactive during a given span of signal observation. As such, the value of the functional connectivity information PDC can provide is self evident in that it offers a glimpse on active signal pathways within the brain and deducible from the simultaneous measurement of multiple sensors. While a number of alternatives exist to gauge Granger causality (Lütkepohl, 1993, Baccalá et al., 1998), a number of open questions concerning reliable PDC inference remain. So much so that, as a conservative measure, to test the null

2 hypothesis of zero PDC in our previously published illustrative examples (Baccalá and Sameshima, 21,21b), we used the arbitrary threshold of.1 beyond which we deemed PDC significantly non-zero. This value was adopted because PDC reduces to the so called Directed Coherence proposed by Saito and Harashima (1983) when just two time series (normalized to unit power) are simultaneously analyzed; the value.1 itself was derived from extensive simulations reported by Schnider et al. (1989) for pairs of simultaneously analyzed time series. (Note that Directed Coherence was later generalized to dealing with more than two simultaneously processed time series (Baccalá et al., 1998; Baccalá and Sameshima, 1998) and is closely related to the estimator proposed by Kaminski and Blinowska (1991) known as Directed Transfer Function (see Baccalá and Sameshima, 21, for further comparisons)). As such, by contrast with Granger Causality tests (GCT) whose large sample behaviour is well known (Lütkepohl, 1993), more research is necessary to assess PDC s statistical effectiveness and this paper represents a step towards that goal. Here, rather than pursue a detailed mathematical investigation of the asymptotic statistics, we chose to investigate and illustrate PDC s behaviour via extensive Monte Carlo simulations of some scenarios of interest from a very simple model comprising just a pair of time series to a more complicated scenario involving more structures and varying complexities in dynamical behaviour. With neuroscience applications in mind, performing Monte Carlo studies is justifiable because many functional connectivity issues among brain structures, as in behaviour related studies, take place in small sample contexts, where large sample statistics are just a rough guide. Background and Methods In this paper, three scenarios of interest were simulated; their data were obtained by from multiple time series derived from autoregressive models. The first example covered just two time series with unidirectional connection only from x 1 (k) to x 2 (k) described by: x 1 (k) x 2 (k) =.75.5 x 1 (k 1) x 2 (k 1) + w 1(k) w 2 (k) (Model A) where we allowed the dynamics of x 2 (k) to vary through the parameter. Note that lack of Granger causality from x 2 (k) to x 1 (k) came from the zero entry in the matrix. The w i (k) are uncorrelated zero mean gaussian innovations processes. To measure the effect of a more complex time structure in the series we repeated PDC estimation after pre-processing Model A signals through identical moving average filters: y i (k) = 1, (Model A ) 2 (x i(k) +x i (k 1)) i = 1, 2 Remark: Though, as one can easily show, the theoretical values of PDC remain unaffected in going from Model A to A, estimability changes severely. Finally, we consider a scenario, borrowed from Baccalá and Sameshima (21b) involving a signal source and a signal sink in which information travels via

3 multiple pathways. The network comprises N = 7 structures and has two disconnected portions that are the object of our investigation here through x 1 (k)=.95 2 x 1 (k 1).925x 1 (k 2)+w 1 (k) x 2 (k)=.5x 1 (k 1)+w 2 (k) x 3 (k)=+.4x 1 (k 4).4x 2 (k 2)+w 3 (k) x 4 (k)=.5x 3 (k 1) x 4 (k 1) x 5 (k 1)+w 4 (k) (Model B) x 5 (k)=.25 2 x 4 (k 1) x 5 (k 1)+w 5 (k) x 6 (k)=.95 2 x 6 (k 1).925x 6 (k 2)+w 6 (k) x 7 (k)=.1x 6 (k 2)+w 7 (k) (see also Baccalá and Sameshima (21b) for a directed graph connectivity structure and a figure of the overall theoretical PDC). Here we concentrated on the interaction behaviour between x 1 (k) and x 6 (k) that represent identical but unlinked oscillators. Each model was simulated using random white zero mean unit variance gaussian uncorrelated innovations w i (k) and various signal lengths n. Simulations were repeated N r times to obtain the sample behaviour of PDC (from each x j (k) to each x i (k)) as defined in Baccalá and Sameshima (21) via ij (f) = A ij (f) N A & kj (f)a kj (f) i=1 with p 1 aij (r)e j2 fr, if i = j A ij (f) = r=1 p aij (r)e j2 fr, otherwise r=1 which was computed through the Nutall-Strand algorithm (Marple Jr.,1985) fit of the model: x 1 (k) x N (k) p = r=1 a 11 (r) a 12 (r) a 1N (r) a ij (r) a N1 (r) a NN (r) x 1 (k r) x N (k r) + w 1 (k) w N (k). Model orders p were estimated using Akaike s Information criterion (AIC), except for Model A where other criteria (Schwartz s (SC) and Hannan-Quinn s (HQ)) were also investigated (Lütkepohl, 1993). Model order was limited to p = 2 for practical computational reasons. In all cases, the corresponding Granger Causality tests (GCT) (see Baccalá et al (1998) for details) were performed at 5% confidence levels as comparison benchmarks. We also used the conservative PDC based criterion: ij (f) 2 >.1

4 (called the Spectral Causality Criterion (SCC)) to look for the consistency of its previous use as evidence of existing connectivity. Results The average behaviour of N r = 256 PDC estimates from x 2 to x 1 as a function of sample size n is depicted in Fig. 1 for Model A ( =.75). For smaller sample sizes the graph reveals the frequency dependence of PDC under the null hypothesis of no connection. In comparing GCT to SCC, the rough independence of GCT on sample size becomes evident (Fig. 2) and contrasts with SCC whose number of false positives decreases as n grows. Obviously this failure of SCC is a by-product of the inadequacy of a frequency independent threshold for small samples sizes. Note also that very few instances of false positive SCC inferences were observed for sufficiently large sample sizes. 1 π 12 (f) n=32 n=128 n= f Fig. 1 - Average behaviour of the estimated squared PDC, which is theoretically zero, as a function of the number of samples for the non-existing connection in Model A ( =.75). Note the explicit dependence on frequency (represented in normalized scale from to.5 of the Nyquist sampling frequency), Percentage of False Positives SCC Samples GCT Fig. 2 - Comparison as a function of signal length n between a canonical Granger Causality test (GCT) and SCC, i. e. assuming PDC connectivity detection based on a frequency independent decision threshold for Model A with. =.75

5 To appreciate the effect of variation in the temporal dynamics, we changed in Model A and computed the corresponding percentage of false positives for n = 256 and N r = 256. GCT results were fairly uniform whereas SCC depended more strongly on the underlying signal dynamics (Fig. 3) having its effectiveness decreased for larger values. Percentage of False Positives SCC GCT θ Fig. 3 - The effect on GCT and SSC of variations in Model A s dynamics through changes in (n=256 and Nr=256). Only SCC was affected by not taking account of PDC s small sample dependence on frequency. For Model A (Fig. 4) comparable PDC performance was only achieved for larger sample sizes. Also, GCT computation produced unacceptably high ratios of false positives that increased with sample size. Of the model order criteria, HQ provided the best PDC performance. Note the intriguing reversal in model order criterion performance in Fig. 4 depending on whether PDC or GCT was used for causalilty detection. ( N r = 64). Percentage of False Positives HQ SC HQ SC AIC AIC Samples Fig. 4 - Impact of different model order estimators on GCT (dashed) and SCC (solid) as connectivity estimators as a function of signal length in samples for Model A. For the next scenario, our focus was the behaviour of possible causality inference errors between the structurally disconnected oscillators represented by x 1 (k) and x 6 (k) in Model B. It is easy to see (Fig. 5) that on average PDC estimates

6 are larger around that common frequency. As for the previous models, SCC s percentages of false positives decrease monotonically and eventually fall bellow GCT s percentages for a sample size of n = 512 points. By contrast GCT s percentages, as for Model A are fairly constant throughout the sample length range (Fig. 5), yet remain well above 5% ( N r = 256)..9 Percentage of False Positives GCT SCC Samples Fig. 5 - Comparison between GCT and SCC in falsely detecting connections from x 6 (k) to x 1 (k) as a function of signal duration in samples for Model B based on 256 independent simulations 5 π 16 (f) Increasing sample size f x 6 (k) x 1 (k) Fig. 6 - Average squared PDC from to shows that most of the partial directed coherence inference errors reflected in the SCC test of Fig. 5 come from the frequency common to both oscillators (N r = 256). The frequency scale is normalized from to.5 of the Nyquist sampling frequency. Discussion The first emerging pattern is that PDC and its frequency independent threshold based criterion (SCC) are good estimators of the unidirectional connectivity for a sufficiently long signal sample. For small samples sizes, PDC dependends on frequency; for Model A, under the assumption of an estimated correct p = 1 model order, it is easy to explicitly quantify this dependence by computing

7 12 (f) 2 = 2, Re{ } cos(2 f) where { results from imprecise estimates of a 12 (1) due to the finite sample duration. This expression also predicts the dependence of PDC errors due to changes in the inherent time series dynamics (Fig. 3) as the denominator approaches zero for large. In fact if is negative, PDC increases for high frequencies (as in Fig. 1); conversely PDC estimates are increased for small frequencies if is positive. From this, one also concludes that a sufficiently long sample for correct SCC-based connectivity detection is a concept that depends on the inherent temporal dynamics of the series under study via parameters like. When sufficiently long signals were used, negligible error rates in directional connectivity inference resulted from the appplication of SCC. The same level of performance was not witnessed for GCT. In the computations for Model A, at least 2 to 4 times as many samples (Fig. 4) were necessary to achieve the same rate of correct conclusions as for Model A which has the same underlying directional connectivity structure as Model A. The common moving average filter applied to both series complicated the dynamics so that it no longer conformed to a low order autoregressive dynamic structure. In this case, the best model order estimator to produce the most correct connectivity inferences based on SCC with fewer time samples was Hannan-Quinn s. The exactly reverse (and as whole generally unsatisfactory) effect was observed for GCT and merits further investigation. The intent behind Model B s illustration was to show that though many more N = 7 time series were analysed simultaneously, the effect on causality inference was less pronounced than for Model A. This leads to the conjecture that reliable PDC-based connectivity inference from short time samples through multivariate autoregressions is possible only in so far as the signals are parsimoniously represented by MAR models, being less affected by issues of how many series are simultaneously processed. Though beyond the scope of this paper, our investigations show that bootstraping can be used to introduce frequency dependent thresholds to PDC connectivity estimation that further decrease the number of time samples necessary for correct inference. Conclusion Partial Directed Coherence affords better means of correct direct unidirectional connectivity inference when compared to a more traditional Granger Causality Test. PDC alse achieves negligibly few incorrect connectivity decisions if the sample size is sufficiently long even if a frequency independent threshold is used. What constitutes a sufficiently long sample depends on the temporal dynamics of the signals analysed. Supported by Grants from FAPESP (99/ to LAB, 96/ to KS) and CNPq (31273/96- to LAB).

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