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Understanding causal relationships between machines is crucial for fault diagnosis and optimization in manufacturing processes. These datasets also include domain-specific expert knowledge and chronological order information, reflecting the recording order across different machines, which is pivotal for discerning causal relationships within the manufacturing data. However, previous methods for handling missing data in scenarios akin to real-world conditions have not been able to effectively utilize expert knowledge. Conversely, prior methods that can incorporate expert knowledge struggle with datasets that exhibit missing values. Therefore, we propose COKE to construct causal graphs in manufacturing datasets by leveraging expert knowledge and chronological order among sensors without imputing missing data. Utilizing the characteristics of the recipe, we maximize the use of samples with missing values, derive embeddings from intersections with an initial graph that incorporates expert knowledge and chronological order, and create a sensor ordering graph. The graph-generating process has been optimized by an actor-critic architecture to obtain a final graph that has a maximum reward. Experimental evaluations in diverse settings of sensor quantities and missing proportions demonstrate that our approach compared with the benchmark methods shows an average improvement of Moreover, the F1-score improvement can reach The manufacturing industry is increasingly transitioning towards automation and complexity to enhance product quality and variety Liang et al. For instance, semiconductor Yang et al. This complexity poses challenges in understanding the underlying causal mechanisms of production lines and identifying the root causes of system failures and product defects. In this scenario, understanding the causal relationships between machines is crucial. Furthermore, understanding the causal relationships between components can offer preemptive alerts for prospective errors to reduce assembly line downtime Huegle et al. Therefore, employing causal discovery methods in manufacturing data is important in the recent evolution of manufacturing Marazopoulou et al. In the manufacturing process, the datasets include domain-specific expert knowledge to modify or retain certain edges and chronological order capturing the sequence of data recordings across various machines. The expert knowledge and chronological order reflect true interactions and dependencies influenced by the manufacturing process Marazopoulou et al. Expert knowledge distinguishes between causal relationships from mere correlations, while chronological data captures the sequence of events, aiding in identifying how changes or errors propagate through the system. Therefore, it is essential to leverage information when accurately identifying causal relationships within the manufacturing data. Recent research Yang et al. This approach allows the design of a model that can utilize a prior knowledge graph for constructing the causal graph. However, in manufacturing data, the recipes result in datasets with a high proportion of missing values, as shown in Figure 1. The recipes determine which machines and sensors are involved in the production process, and the sequence of machines used in it. The variability in recipes across different product configurations can lead to the occurrence of missing values in the dataset Kwak and Kim, For example, consider a manufacturing facility that produces semiconductors with varying specifications. Certain product configurations may bypass certain machines or sensors, resulting in missing data for those particular sensors Yang et al. As a result, the dataset may exhibit a high missing rate, particularly for sensors associated with less frequently used product configurations. Consequently, before applying a method Yang et al. To avoid bias from imputed data, some previous approaches will pre-define the types of missing data. According to Rubin, , based on the different types of missingness, the underlying missing mechanism can be divided into three basic types: missing at random MAR , missing completely at random MCAR , and missing not at random MNAR. However, the missingness in manufacturing data stems from variations in recipes, leading to unsuitable accurate definitions of the types of missing data. This obstacle complicates the selection of appropriate methods for causal discovery and the optimization of their utilization in data processing. Other methods that do not specify missing data types struggle with incorporating expert knowledge and chronological order information among sensors Morales-Alvarez et al. When constructing a causal graph from manufacturing data, expert knowledge and chronological order information are crucial as they impact the causal relationships between sensors. However, existing methods struggle to address the challenge posed by high proportions of missing values in data with varied recipes. Additionally, methods capable of handling missing data fail to effectively leverage expert knowledge and chronological order information. This approach optimizes the utilization of samples with missing values by recipes in manufacturing data, thereby avoiding the need for data imputation and achieving accurate variable embeddings. Upon obtaining embeddings for each variable sensor , we use a decoder to sequence the variables by causal order. We then remove edges from the ordering graph based on these sequenced variables using the initial graph. We employ an actor-critic architecture to optimize our graph generation model, aiming to maximize the reward by minimizing the Bayesian Information Criterion BIC score. Throughout the training, we record the graph in each iteration, ultimately selecting the graph with the highest reward as the final causal graph for the manufacturing dataset. Additionally, on real-world data, it achieves an improvement above Furthermore, through ablation studies, we demonstrate the impact of expert knowledge and chronological order in the graph-constructing process from manufacturing data. Causal graph construction in the presence of missing data requires considering the impact of missing values on causal discovery. Tu et al. They also proposed corrective measures to achieve a more accurate representation of the true causal graph. Qiao et al. However, these constraint-based methods based on the PC algorithm result in ineffectively constructing graphs for data with hundreds of variables because of the exponential growth of conditional sets Le et al. MissDAG Gao et al. In the E-step, the model estimates the data distribution from the observed available data, taking into account the missing values. The M-step then utilizes the causal discovery model to estimate the data and identify the causal graph among variables. However, this method is time-consuming for hundreds of variables and cannot utilize expert knowledge and chronological order in the graph-constructing process. Several methods use machine learning techniques to address the missing values in the dataset. Morales-Alvarez et al. These methods perform end-to-end training on generative adversarial network-based imputation models and causal structure learning models. However, they are unsuitable for manufacturing datasets since they do not account for expert knowledge and chronological order. Advancements in manufacturing processes, automated measurement tools, and real-time data collection have improved data quality, facilitating accurate representation of product conditions Clavijo et al. As a result, there has been significant progress in applying causal discovery techniques within the manufacturing data. Yang et al. It integrates learning multiple causal discovery algorithms to improve the robustness and accuracy of the learned structure. However, this work assumes that sensors on the same machine lack causal relationships, thereby limiting its ability to accurately identify the causal-effect relationships between sensors within the same machine. Marazopoulou et al. However, these methods only consider a few machines in the dataset, as does the work in multi-stage PCB manufacturing Sim et al. While there are many methods for constructing causal graphs from manufacturing data, they face challenges with datasets in high proportions of missing values or limited dimensions, affecting improper application for our problem. Despite the high proportion of missing values in the dataset, we can obtain complete datasets through imputation for causal discovery. Notears Zheng et al. In the manufacturing dataset, the recorded process follows the description in Figure 1. The assumption of the data generation process is followed by. We follow the assumption of Wang et al. Additionally, we assume that all variables are measurable causally sufficient and there are no confounders, which are latent causes of the observed variables in the dataset. To address this challenge, we break down the causal discovery task into two main components: ordering graph generation and variables selection. Firstly, in Section 4. In Section 4. Then, in Section 4. The framework of our proposed method is illustrated in Figure 2. In manufacturing data, since products pass through machines sequentially, chronological order occurs between the machines. Subsequently, we modify the edges based on expert knowledge by removing or adding them accordingly. Each variable has a distinct number of embeddings as each recipe contains varying observed variables, leading to differing observation frequencies in different variables. Therefore, we average all embeddings that consider knowledge from the initial graph for each variable:. During each generative iteration as shown in Figure 3 , we produce one variable until all variables have been generated, and the resulting sequence is defined as the ordering variable set. To prevent the embeddings of previously generated variables from affecting subsequent variable selection, we mask embeddings that have been generated. We utilize an actor-critic framework to train the ordering graph generation process, conceptualizing it as a multi-step decision-making process Wang et al. Preprocessing data using neural networks can enhance the ability to identify better orderings Wang et al. The evolution of the state changes throughout the iterations of the selection of variables. In causal graph construction, we determine a better graph that minimizes the BIC score which considers the noise variances to be equal Yang et al. The reward function we use is :. We then update the actor for optimizing the policy gradient:. We generate data similar to manufacturing data, following the process used in Yang et al. The data generation is the same as the chronological order of manufacturing, with directional influence between sensors across different machines, as shown in Figure 1. In our setting, each variable has a different mean value but shares the same covariance matrix, averaging the data of its parent variables as its mean. Additionally, we simulate expert knowledge by randomly selecting 10 edges as mandatory existing edges in the datasets. To simulate product movement across machines according to specified recipes, we randomly select machines to represent scenarios where a product does not pass through a machine, resulting in missing values for all sensors in those selected machines. Achieving precise missing proportions at the machine level is challenging. Our real-world dataset records values following in Figure 1 , for the semiconductor wafer manufacturing process. The dataset consists of 18 machines, sensors, and 75 recipes. All values are continuous, with an average missing rate of Many existing causal discovery methods require complete datasets as input. In Table 1 , these methods are denoted as Imp. Due to the unsuitability of constraint-based methods for high-dimensional datasets, we employ score-based methods such as Notears Zheng et al. For baseline methods, we increase the number of training iterations to ensure convergence, keeping other parameters at their default settings. In causal graph construction, excessive edge predictions yield higher recall scores but lower precision scores, whereas models predicting fewer edges exhibit better precision but lower recall. Both scenarios can lead to inefficiencies in manual edge selection for engineers. Therefore, we choose the harmonic mean of recall and precision, which is the F1-score, as the final evaluation metric for comparison of the baselines and COKE. In total, our evaluation covered twelve datasets, with results presented in Table 1. We made the following observations: 1 COKE exhibits low sensitivity to increasing missingness. MissDAG showed decreases of COKE showed minimal performance drops of 0. Hence, COKE is suitable for scenarios with high proportions of missing values. Notears Imp. Some experiments showed that dropping missing values yields better results than imputation. Compared to GARL, which incorporates expert knowledge and chronological order, Notears and ICA-LiNGAM exhibited significant differences with or without imputation, highlighting the importance of reducing inductive bias by incorporating information. Consequently, dataset variations with different missing ratios resulted in similar datasets. In experiments using Notears Drop. Assuming nonlinear relationships between variables, Table 2 indicates that COKE has the best performance on the F1-score. Our method may not meet precision expectations as MissDAG predicts only nine edges, while our model predicts hundreds, resulting in lower precision. Figure 5 a shows the training times for each model. MissDAG exhibited a longer training time for numerous variables, indicating inefficiency in high-dimensional causal discovery. While COKE requires more time in datasets with fewer variables, it significantly reduces processing time compared to MissDAG in datasets with more variables. The designed reward function proves beneficial for causal graph construction, as evidenced by the positive correlation between the F1 score and the reward. Despite incorporating an incomplete data process, our model consistently enhances the F1-score and achieves convergence. To evaluate the importance of chronological order CO , expert knowledge EK , and the incomplete data process Incomp. Chronological ordering information is crucial for accurate causal graph construction, as indicated by the significant performance drop observed when this aspect is neglected. This results in inferior performance across all 12 settings, failing to achieve even the second-best result in any of them. The impact of the incomplete data process is significant as the number of variables increases. These experiments show the essential roles that chronological order, expert knowledge, and the incomplete data process play in our model, demonstrating their importance in achieving robust results. In this paper, we introduce COKE, a method designed to construct causal graphs by leveraging expert knowledge and chronological order in datasets with high missing values. COKE utilizes manufacturing recipes to maximize the utility of samples with missing values, enabling graph construction without data imputation. By employing the graph attention network, COKE integrates expert knowledge and chronological order into the actor-critic training process. It utilizes the BIC score as a reward function to optimize the graph generation model, eliciting the final causal graph with the maximum reward. Experimental results show that in synthetic data with multiple representative missing proportions and variable counts, COKE outperformed several strong baselines across various settings showcasing the effectiveness of our framework. We believe COKE could flexibly construct causal graphs not only in manufacturing datasets but also in other domains where leveraging knowledge is crucial for establishing accurate causal relationships. In the future, how to build causal graphs using only samples with missing values will be further investigated within this framework, as real-world data often lack complete datasets. Model Precision Recall F1 Notears 0.