# Deep Knowledge Tracing Integrating Temporal Causal Inference and PINN
## 2 Related Work
### Temporal Causal Inference  
In real life, the occurrence of most events is not at regular intervals but is discrete. A point process is a powerful modeling tool for event sequences, consisting of a time series [19] of binary events occurring in continuous time. Point processes have been successfully applied in various fields such as social networks, finance, equipment maintenance, and electronic health records. The characteristic feature of point process models is their intensity function. The occurrence times of various types of events \(\varepsilon=1,\ldots,\) E are unevenly distributed. A multivariate point process containing \(E\) type of events can be represented using a counting process \(\{N_{e}\}_{\{\varepsilon=1\}}^{E}\), where \(N_{e}=\{N_{e}(t)|t\in[0,T]\}\). The intensity function shown in Equation (1) can be defined for type \(u\) as the expected instantaneous rate of occurrence of type \(e\) events given the history. Each intensity function captures the instantaneous occurrence rate of a class of events conditioned on historical events. \[\lambda_{e}(t)=\frac{\mathbb{E}(dN_{e}(t)|\mathcal{H}_{t})}{dt} \tag{1}\]  
where \(\mathcal{H}_{t}=\{t_{i},e_{i}|t_{i}<t,e_{i}\in\varepsilon\}\) representing all types of events that occurred before time t.  
Granger causality [20] emphasizes the temporal sequence of events and was originally used to study the dependency structure of multivariate time series. It has also been extended to multi-type event sequences [21]. In short, to test the causality between \(A\) and \(B\), let \(\Omega_{n}\) be the set of all information up to time (\(t=1,2,\ldots,n\)) (including information other than \(A\) and \(B\)), and let \(B_{n}\) be all the information about \(B\) up to time n. Here, both \(B_{n}\) and \(\Omega_{n}\) are multivariate random variables, and \(B_{n}\in\Omega_{n}\). Thus, \(\Omega_{n}-B_{n}\) represents all information up to time \(n\) excluding \(B\). Assume that the following: (1) the present and the past can influence the future, but the future cannot influence the past; (2) \(\Omega_{n}\) does not contain any redundant information, meaning that if a variable \(Z_{n}\) is functionally related to one or more other variables, it should be removed from \(\Omega_{n}\). If \(P(A_{n+1}|\Omega_{n})\
eq P(A_{n+1}|\Omega_{n}-B_{n})\), then variable B is considered a cause of variable \(A\), indicating that \(B_{n}\) contains unique information that influences the occurrence of \(A_{n+1}\). Intuitively, for event sequence data, if including historical events of one type improves the prediction of future events of another type, we say that one type of event has Granger causality with respect to the other type. The Hawkes process [22] assumes that past events can independently and additively influence the occurrence of future events, specifically through a collection of pairwise kernel functions. The Hawkes process is a mathematical model used to model self-exciting processes; it is a counting process that describes a series of events occurring over time. In this process, the occurrence of each event increases the likelihood of the next event happening, while the effect of this stimulation gradually diminishes over time. The core concept of the Hawkes process is the conditional intensity function, it represents the expected frequency of an event occurring near a certain point in time, given all past events. Its mathematical expression is as follows:  
\[\lambda^{*}(t)=\mu+\sum_{t_{i}<t}g(t-t_{i}) \tag{2}\]where \(\lambda^{*}(t)\) is the conditional intensity function, \(\mu\) is the background intensity, \(g(t-t_{i})\) is the triggering kernel, and \(t_{i}\) is the time of the \(i\)-th event. The triggering kernel can take various forms, with one common type being the exponential decay function, which indicates that the stimulating effect of past events on the current time point decreases exponentially as the time interval increases. An event sequence records the occurrence of a specific type of event. Point processes can be used to describe event sequences.