Time-varying Hawkes Process

Constructing a knowledge graph has been a valuable strategy for understanding the complex relationships within Electronic Health Records (EHR) features. However, the relationships are dynamic. New diseases like COVID-19 or the introduction of FDA-approved drugs can significantly alter treatment strategies. To address this challenge, we propose a time-varying graphical model to analyze the evolving relationships between features.

Specifically, we introduce the Time-Varying Hawkes Process with the following formula: \[\begin{equation} \lambda_{j}^{(i)}(t) = \mu_{j}^{(\tau_i)} + \sum_k B_{jk}^{(\tau_i)} \int_{t-w}^t d N_k^{(i)}(u) \, , \end{equation}\] where \(N\) is a point process, \(\lambda\) is the infinitesimal probability of the occurrence of feature, and \(\tau_i\) is the calendar year of the patient’s first date of being recorded with a target disease. The key parameter, \(\mu\) is a baseline intensity and \(B\) describes the directional relationships between features.

Here, we exhibit the predicted parameters from our proposed algorithm on the dataset that comprises over seven thousand Multiple Sclerosis patients collected from UPMC. \(B\) and \(mu\) are the key parameters in the Hawkes process. B(MStoDrug) describes the influence of the occurrence of feature MS on the occurrence of the drug code, while B(DrugtoMS) describes the impact of the drug on the occurrence of MS code. cosine(SPPMI) is computed from the SVD-SPPMI using the data from each calendar year and cosine(Encoder), cosine(Decoder) are the cosine similarities output from the proposed algorithm. As \(B^{(\tau)} = U^{(\tau)} (V^{(\tau)})^{\text{T}}\), \(U^{(\tau)}\) can be viewed as decoder embeddings and \(V^{(\tau)}\) can be reviewed as encoder embeddings. The cosine similarities derived from them are cosine(Decoder) and cosine(Encoder) respectively.