SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Algorithm 1: Discover the summary graph
Data: Data observed at  $t = 1, k + 1, \ldots, \left\lfloor \frac{T}{k} \right\rfloor k + 1$   
Result: The summary graph  
Construct the skeleton of the MAG with [26], orient edges according to Proposition 3.1;  
For every vertex pair  $A, B$ , set  $A * \rightarrow B$  if there are  $A(t) \rightarrow B(t + k)$  and  $A(t + k) \leftrightarrow B(t + k)$  in the MAG. Called the resulted PD-DAG  $G_0$ ; /* Proposition 3.3 */  
Set  $G = G_0$  and iteratively execute:  
(a) For every  $A * \rightarrow B$ , check  
(i) a directed path from  $A$  to  $B$  with length  $l \leq k - 2$ ;  
(ii) a directed path from  $A$  to  $B$  with length  $l \leq k - 1$  and a confounding structure between  $A$  and  $B$  with lengths  $(r \leq k - 2, q \leq k - 2)$ ;  
If  $\exists$  (i) or (ii), pass. Otherwise, set  $A \rightarrow B$ ; /* Proposition 3.3*/  
(b) Randomly pick a dashed edge  $A * \rightarrow B$ , check whether  $A(t) \perp_G B(t + k)|\mathbf{M}(t + 1), \mathbf{S}|$ ;  
If the  $d$ -separation holds, remove  $A * \rightarrow B$ . Otherwise, set  $A \rightarrow B$ ; /* Theorem 3.5*/  
until all edges in  $G$  are solid ones; return  $G$
Remark 4.2. Algorithm 1 can be generalized to cases where the subsampling factor  $k$  is unknown, by skipping the step (a) below line 3.
In this section, we evaluate our method on synthetic data and a real-world application, i.e., discovering causal pathways in Alzheimer's disease.
Compared baselines. Methods that account for subsampling: 1. SVAR-FCI [16] that extended the FCI algorithm to time series and recovered a MAG over observed variables; 2. NG-EM [10] that achieved identifiability with linear non-Gaussianity and used Expectation Maximization (EM) for estimation. Methods that neglect subsampling: 1. Dynotears [7] that extended the score-based Notears [27] algorithm to the time series; 2. PC-GCE [21] that modified the PC algorithm for temporal data with a new information-theoretic conditional independence (CI) test.
Figure 5: Performance of our method and baselines under different subsampling factors (upper row) and sample sizes (lower row).
Metrics. We use the  $\mathrm{F}_1$ -score, precision, and recall, where precision and recall (resp.) measure the accuracy and completeness of identified causal edges, and  $\mathrm{F}_1 := 2 \cdot \frac{\text{precision} \cdot \text{recall}}{\text{precision} + \text{recall}}$ .
Implementation details. The significance level is set to 0.05. For the MAG recovery, we use the FCI algorithm implemented in the causallearn package<sup>4</sup>. Temporal constraints, e.g., causal precedence, time invariance, and self causation are added as the background knowledge.
5.1 Synthetic study
Data generation. We generate radnom summary graphs with the Erdos-Renyi model [28], where the vertex number is set to 5, the probability of each edge is set to 0.3. For each graph, we generate temporal data with the structural equation  $X_{i}(t) = \sum_{j\in \mathbf{PA}_{i}}f_{ij}(X_{j}(t - 1)) + N_{i}$ , where the function  $f_{ij}$  is randomly chosen from  $\{\text{linear}, \sin, \tanh, \sqrt{sqrt}\}$ , the exogenous noise  $N_{i}$  is randomly sampled from  $\{\text{uniform}, \text{gauss}, \exp., \text{gamma}\}$ . We consider different subsampling factors  $k = \{2, 3, 4, 5\}$  and sample sizes  $n = \{600, 800, 1000, 1200\}$ . For each setting<sup>5</sup>, we replicate over 100 random seeds.
Comparison with baselines. Figure 5 shows the performance of our method and baselines under different subsampling factors (upper row) and sample sizes (lower row). As we can see, our method significantly outperforms the others in all settings. Specifically, compared with methods that account for subsampling (SVAR-FCI and NG-EM), our method achieves both higher precision and recall, indicating less detection error and missing edges. This advantage can be attributed to the fact that our method enjoys an identifiability guarantee (v.s. SVAR-FCI) and meanwhile requires no parametric assumption on the structural model (v.s. NG-EM). Compared with methods that ignore subsampling (Dynotears and PC-GCE), our method achieves higher  $\mathrm{F}_1$ -score, precision, and is comparable in recall. This result shows that our method can effectively reduce the spurious detection induced by unobserved time steps. Besides, we can observe that Dynotears and PC-GCE slightly outperform SVAR-FCI and NG-EM, which again demonstrates the necessity of establishing nonparametric identification in subsampling problems.
Intermediate results. We evaluate the intermediate results of Algorithm 1 (MAG, edges identified by proxies, summary graph) and report the results in Figure 6. As shown, both the recovered MAG and edges identified by proxies have high accuracy under moderate subsampling factors and sample sizes, hence explaining the effectiveness of our algorithm in recovering the summary graph. Meanwhile, we can also observe that the performance slightly decreases when the subsampling factor  $k$  is large. This
4https://github.com/py-why/causal-learn
We also consider different graph scales  $d = \{5, 15, 25, 35, 45\}$ . Please refer to the appendix for details.
Figure 6: Evaluation of intermediate results. We report  $F_{1}$ -score, precision, and recall of the recovered MAG, edges identified by proxies, and the recovered summary graph.
is explained as follows. When  $k$  is large, there are many unobserved steps between two observations, which weakens the correlation pattern in data and therefore breaks our faithfulness assumption.
5.2 Discovering causal pathways in Alzheimer's disease
Background. Alzheimer's disease (AD) is one of the most common neuro-degenerative diseases. In AD, patients suffer from memory loss due to atrophy of memory-related brain regions, such as the Hippocampus [3] and Temporal lobe [29]. A widely accepted explanation for such atrophy is: the disease releases toxic proteins, e.g.,  $A\beta$  [30] and tau [31]; along anatomical pathways, these proteins spread from one brain region to another, eventually leading to atrophy of the whole brain [32]. Recovering these anatomical pathways, i.e., underlying causal mechanisms, will benefit the understanding of AD pathology and inspire potential treatment methods.
Dataset and preprocessing. We consider the Alzheimer's Disease Neuroimaging Initiative (ADNI) dataset [4], in which the imaging data is acquired from structural Magnetic Resonance Imaging (sMRI) scans. We apply the Dartel VBM [33] for preprocessing and the Statistical Parametric Mapping (SPM) [34] for segmenting brain regions. Then, we implement the Automatic Anatomical Labeling (AAL) atlas [35] to partition the whole brain into 90 regions. In total, we use  $n = 558$  subjects with baseline and month-6 follow-up visits enrolled in ADNI-GO/1/2/3 periods.
Results. Figure 7 shows the recovered summary graph over 90 brain regions. For illustration, in Figure 1 (a), we further group the brain regions into 8 meta-regions according to their anatomical structures. We can observe that i) most causal pathways between meta-regions are unidirectional; ii) the identified sources of atrophy are the Hippocampus, Amygdala, and Temporal lobe, which are all early-degenerate regions [3, 29]; iii) the topology order induced from our result, i.e.,  $\{Hip., Amy., Tem., Cin., Ins., Fro., Par., Occ.\}$  coincides with the temporal degeneration order found in [8, 9]. These results constitute an important finding that can be supported by existing studies: the atrophy (releasing of toxic proteins) is sourced from the Hippocampus and gradually propagates to other brain structures along certain anatomical pathways [32, 36].
In contrast, in Figure 1 (b), the causal relations recovered by the Dynotears baseline $^{6}$  are less clear. For example, most of the identified interactions are bidirectional, which may be due to the spurious edges induced by subsampling. Besides, the identified atrophy source is the Frontal lobe, with AD-related regions identified as outcomes, which is inconsistent with clinical studies. These results, from another perspective, show the importance of modeling subsampling in time series causal discovery.
$^6$ Please refer to the appendix for results of the other baselines.
Figure 7: Recovered summary graph in Alzheimer's disease.
In this paper, we propose a causal discovery algorithm for subsampled time series. Our method leverages the recent progress of proximal causal discovery and can achieve complete identifiability without any parametric assumption. The proposed algorithm can outperform baselines on synthetic data and recover reasonable causal pathways in Alzheimer's disease.
Limitation and future work. Our method relies on an accurate test of the conditional independence (CI) and therefore may suffer from low recall when the CI patterns in data are weak. To solve this problem, we will investigate theories on high-efficiency CI tests and pursue a dedicated solution.
This work was supported by National Key R&D Program of China (2022ZD0114900).
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