Causality in Time Series - ClopiNet

Robust Statistics for CausalityAll methods were statistically evaluated for robustness and generality by performinga 5-fold jackknife, which gave both a mean and standard deviation estimate for eachmethod and each simulation. All statistics reported below are mean normalized by standarddeviation (from jackknife). For all methods the final output could be -1, 0, or 1,corresponding to causality assignment 1→2, no assignment, and causality 2 → 1. A truepositive (TP) was the rate of correct causality assignment, while a false positive (FP)was the rate of incorrect causality assignment (type III error), such that TP+FP+NA=1,where NA stands for rate of no assignment (or neutral assignment). The TP and FPrates are be co-modulated by increasing/decreasing the threshold of the absolute valueof the mean/std statistic, under which no causality assignment is made:S T AT = rawS T AT/std(rawS T AT), rawSTAT=PSI, DTF, PDC ..c = sign(S T AT) if S T AT > TRES H, 0 otherwiseIn Table 1 we see results of overall accuracy and controlled True Positive rate forthe non-mixed colored noise case (meaning the matrix B above is diagonal). In Table 1and Table 2 methods are ordered according to the mean TP rate over time series length(highest on top).Table 1: Unmixed colored noise PAIRSMax. Accuracy TP , FP < 0.10100 500 1000 5000 100 500 1000 5000Ψ 0.62 0.73 0.83 0.88 0.25 0.56 0.75 0.85DTF 0.58 0.79 0.82 0.88 0.18 0.58 0.72 0.86CSI 0.62 0.72 0.79 0.89 0.23 0.53 0.66 0.88Ψ C 0.57 0.68 0.81 0.88 0.19 0.29 0.70 0.87PDC 0.64 0.67 0.75 0.78 0.23 0.33 0.48 0.57In Table 2 we can see results for a PAIRS, in which the noise mixing matrix B isnot strictly diagonal.Table 2: Mixed colored noise PAIRSMax. Accuracy TP , FP< 0.10N=100 500 1000 5000 N=100 500 1000 5000Ψ C 0.64 0.74 0.81 0.83 0.31 0.49 0.64 0.73Ψ 0.66 0.76 0.78 0.81 0.25 0.59 0.61 0.71CSI 0.63 0.77 0.79 0.80 0.27 0.62 0.59 0.66PDC 0.64 0.71 0.69 0.66 0.24 0.30 0.29 0.24DTF 0.55 0.61 0.66 0.66 0.11 0.10 0.09 0.1257