Gaussian process assumption

We assume the dense representations of patients’ EHRs (i.e. patient embeddings) over time follow a Gaussian process:

Since the feature embeddings V are engineered to approximately follow a multivariate normal distribution as described in the Producing feature embeddings section of the Supplementary Materials, it is reasonable to assume Xi,t to be a Gaussian process over time t. We further specify the mean and covariance functions μi(t) and Σit respectively. For some parameters θGP={μ0,μ1,μ2,μ3,μ4,μ5,μH,μYH,σk,αk,τk,ρkl,k=1,⋯,p;l=1,⋯,p}, we assume:

In summary, we assume that patient i’s expected embedding at time t, μi(t), is a function of Yit, Hi, and t. We assume that the marginal variance of embedding component k can be represented by some baseline σk2 scaled by Hi. We denote the correlation between embedding components k and l as ρkl, which we assume to be constant over time. Between timepoints, we employ a first-order univariate autoregressive (AR(1)) kernel structure such that the residual at t, ϵi,t,k=Xi,t,k-E(Xi,t,k|Yi,Hi), is a linear function of its preceding value ϵi,t-1,k with autocorrelation coefficient τk:

r∈[0,1] is an autoregression regularization hyperparameter separately tuned via fivefold cross-validation maximizing the AUROC of Yi,t predictions: r=0 ignores intertemporal correlation while r=1 denotes undampened autoregression. We chose first-degree autoregression over higher-degree models due to computational ease and mitigation of overfitting. We provide a sensitivity analysis with respect to the choice of k-fold cross-validation in Supplementary Fig. ^S2 that demonstrates no significant effect of k on predictive accuracy.