SlowGuess/NeurIPS_2023_OCR huggingface dataset:
For discrete-valued time series, a longer time length is required. To evaluate performance, we conduct a series of experiments following the same methodology as described in section 4.1. Fig.4(b) illustrates the variation in comprehensive performance with respect to  $\omega_{\mathrm{max}}$ .  $\mathrm{PCMCI}_{\Omega}$  demonstrates a similar performance to PCMCI in terms of the  $F1$  score, indicating a well-balanced trade-off between precision and recall. This outcome is expected since discrete-valued time series demand larger sample sizes, and the increases in  $\omega_{\mathrm{max}}$  negatively impact the power of MCI tests. This observation is further supported by Fig.5(a), where an increase in time length  $T$  from 4000 to 12000 does not lead to a significant improvement in the accuracy rate of  $\hat{\omega}$ , while the accuracy decreases rapidly with higher values of  $\omega_{\mathrm{max}}$ .
Comparing these results to the experiments conducted on continuous-valued time series, it becomes evident that the demand for efficient samples is even more substantial for binary time series, and the influence of increasing  $\omega_{\mathrm{max}}$  on performance becomes more pronounced.
Fig.5(b) shows how the performance of the algorithm varies across  $\tau_{\mathrm{max}}$  and the same trade-off between recall and precision has been shown.
G.4 More experiments on Continuous-valued time series
In this section, we conduct more experiments with continuous-valued time series with Gaussian noises.
In Fig 6(a), we test our algorithm with and without utilizing the turning point rule. See lines 13-14 in Algorithm B1 and section E for more information about the turning point rule. Let  $\mathrm{PCMCl}_{\Omega}$  TP
(a) Exponential noise
(b) Binary time series
Figure 4: a)  $F_{1}$  Score, Adjacency Precision, and Adjacency Recall when  $\omega_{\mathrm{max}}$  varies for experiments on continuous-valued time series with Exponential noise, length  $T = \{500,2000,8000\}$ ,  $\tau_{\mathrm{max}} = 5$  and  $n = 5$ . b)  $F_{1}$  Score, Adjacency Precision, and Adjacency Recall when  $\omega_{\mathrm{max}}$  varies for experiments on binary time series with length  $T = \{4000,8000,12000\}$ ,  $\tau_{\mathrm{max}} = 3$  and  $n = 3$ .
(b) Performance when  $\tau_{\mathrm{max}}$  varies
Figure 5:  $\mathrm{PCMCl}_{\Omega}$  is tested on 3-variate binary time series. Every marker corresponds to the average accuracy rate or average running time over 100 trials. a) The accuracy rate of  $\hat{\omega}$  for different time series lengths and different  $\omega_{\mathrm{max}}$ . b)  $F_{1}$  Score, Adjacency Precision, and Adjacency Recall when  $\tau_{\mathrm{max}}$  varies for experiments with time series length  $T = 4000$ ,  $\omega_{\mathrm{max}} = 3$  and  $n = 3$ .
denote the version of  $\mathrm{PCMCl}_{\Omega}$  that the turning point rule is utilized in choosing  $\omega$ .  $\mathrm{PCMCl}_{\Omega}$  non-TP means that the turning point rule is not applied and  $\omega$  is chosen directly according to Lemma D.5.
Fig.6(a) shows that the algorithm  $\mathrm{PCMCl}_{\Omega}$  non-TP and  $\mathrm{PCMCl}_{\Omega}$  TP have similar performance with various  $T$  and  $\omega_{\mathrm{max}}$ . With  $T = 500$ ,  $\mathrm{PCMCl}_{\Omega}$  non-TP yields slightly larger standard errors for those metrics, compared to  $\mathrm{PCMCl}_{\Omega}$  TP. As time length  $T$  increases, the performance of the algorithm  $\mathrm{PCMCl}_{\Omega}$  non-TP has consistently increased and is even slightly better than  $\mathrm{PCMCl}_{\Omega}$  TP.
The consistent performance of  $\mathrm{PCMCl}_{\Omega}$  under different chosen rules of  $\omega$  supports our theoretical result; that is, the correct periodicity leads to the most sparse causal graph.
In Fig 6(b), non-stationary time series are produced instead of semi-stationary ones. Consequently, the causal mechanisms for each univariate time series no longer appear sequentially and periodically. The proposed method performs slightly better in terms of F1 score and precision. However, the recall rate is the worst compared to other baselines.
In Fig 6(c), we conduct experiments in the nonlinear setting. The proposed algorithms  $\mathrm{PCMCI}_{\Omega}$  TP and  $\mathrm{PCMCI}_{\Omega}$  non-TP perform the best.
In Fig.6(d) with  $\omega_{\mathrm{ub}} < \omega_{\mathrm{max}}$ , the performance of the proposed algorithm is significantly worse compared to the scenario where  $\omega_{\mathrm{ub}} > \omega_{\mathrm{max}}$ . However, with  $\omega_{\mathrm{ub}} < \omega_{\mathrm{max}}$ , the proposed algorithm can still detect a less dense graph in comparison to other baselines. Based on these outcomes, it is essential to maintain a slightly higher  $\omega_{\mathrm{ub}}$  without significantly impacting the number of efficient samples utilized in each CI test.
(a) Performance with and without turning point (b) Performance in non-stationary setting without periodicity
(c) Performance in non-linear SCM
Figure 6: Multiple algorithms are tested on 5-variate time series with different time lengths  $T$ . Every line corresponds to a different algorithm. Every marker corresponds to the average performance over 50 trials. In (a), the consistent performance of PCMCI under different chosen rules of  $\omega$  supports our theoretical result; that is, the correct periodicity  $\omega$  leads to the most sparse causal graph. In (b), data sets are in a non-stationary setting without periodicity. In (c), the structural causal model (SCM) is non-linear. In (d), algorithm  $\mathrm{PCMCI}_{\Omega}$  are tested under conditions that  $\omega_{\mathrm{ub}} > \omega_{\mathrm{max}}$  and  $\omega_{\mathrm{ub}} < \omega_{\mathrm{max}}$  respectively.
(d) Performance when  $\omega_{\mathrm{ub}} > \omega_{\mathrm{max}}$  and  $\omega_{\mathrm{ub}} < \omega_{\mathrm{max}}$
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