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PopescuAs we can see in both Figure 2 and Table 1, all methods are almost equally robustto unmixed colored additive noise (except PDC). However, while addition of mixedcolored noise induces a mild gap in maximum accuracy, it creates a large gap in termsof TP/FP rates. Note the dramatic drop-off of the TP rate of VAR/SVAR based methodsPDC and DTF. Figure 3 shows this most clearly, by a wide scatter of STAT outputsfor DTF around β = 0 that is to say with no actual causality in the time series and acorresponding fall-off of TP vs. FP rates. Note also that PSI methods still allow a fairlyreasonable TP rate determination at low FP rates of 10% even at 100 points per timeseries,while the CSI method was also robust to the addition of colored mixed noise,not showing any significant difference with respect to PSI except a higher FP rate forlonger time series (N=5000). The advantage of PSIcardinal was near PSI in overallaccuracy. In conclusion, DTF (or weak Granger causality) and PDC are not robust withrespect to additive mixed colored noise, although they perform similarly to PSI and CSIfor independent colored noise. 19. Conditional causality assignmentIn multivariate time series analysis we are often concerned with inference of causalrelationship among more than 2 variables, in which the role of a potential commoncause must be accounted for, analogously to vanishing partial correlation in the staticdata case. For this reason the PAIRS data set was extended into a set called TRIPLESin which the degree of common driver influence versus direct coupling was controlled.In effect, the TRIPLES DGP is similar to PAIRS, in that additive noise is mixedcolored noise (in 3 dimensions) but in this case another variable x 3 may drive the pairx 1 , x 2 independently of each other, also with random coefficients (but either one set to1/10 of the other randomly). That is to say, the signal is itself a mixture of one wherethere is a direct one sided causal link among x 1 , x 2 as in PAIRS and one where they areactually independent but commonly driven, according to a parameter χ which at 0 iscommonly driven and at 1 is causal.β < 0 y C,i =⎡K∑︁k=1⎢⎣a 11 a 12 00 a 22 0 y⎤⎥⎦ C,i−k + w C,i0 0 a 33 C,k1. Correlation and rank correlation analysis was performed (for N=5000) to shed light on the reasonfor the discrepancy between PSI and CSI. The linear correlation between rawSTAT and STATwas .87 and .89 for PSI and CSI. No influence of model order K of the simulated system wasseen in the error of either PSI or CSI, where error is estimated as the difference in rank ofrankerr(S T AT) = |rank(β) − rank(S T AT)|. There were however significant correlations betweenrank(|β|) and rankerr(S T AT), -.13 for PSI and -.27 for CSI. Note that as expected, standard Grangercausality (GC) performed the same as DTF (TP=0.116 for FP
Robust Statistics for CausalityFigure 4: Diagram of TRIPLES dataset with common driver61
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Causality in Time SeriesVolume 5:Ca
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PrefaceThe following is a print ver
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JMLR: Workshop and Conference Proce
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Linking Granger Causality and the P
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Page 42 and 43: Popescu1. IntroductionCausality is
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Page 46 and 47: PopescuType III error prob. γ = P
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Page 56 and 57: PopescuFigure 1: SVAR causality and
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Page 62 and 63: Popescu8.1. The cardinal transform
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Page 66 and 67: Popescu(a) Unmixed colored noise(b)
Page 70 and 71: Popescuβ > 0 y C,i =x N,i =⎡K∑
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