Shared regression interface

For training fold 𝒟_train and horizon τ, define the context table

$${𝒞}_{\tau }\left({𝒟}_{\mathrm{train}}\right)=\left\{\right(X_i,{\tilde{Z}}_i\left(\tau \right)):i\in I_{\mathrm{train}}\}.$$

Every backbone receives this same supervised regression problem. Its empirical squared-error representation is

$${ℒ}_{\tau ,b}\left(f\right)=\frac{1}{\left|I_{\mathrm{train}}\right|}\sum _{i\in I_{\mathrm{train}}}{\left\{{\tilde{Z}}_i\right(\tau )-f(X_i\left)\right\}}^2.$$

This equation defines the common task; it does not claim that every backbone updates task-specific weights. In-context backbones condition on the supplied training table, whereas conventional regressors fit dataset-specific parameters.

For backbone 𝒜_b and held-out observation j,

$${\hat{m}}_{\tau ,b}\left(X_j\right)={𝒜}_b\left\{{𝒞}_{\tau }\right({𝒟}_{\mathrm{train}}),X_j\}.$$

Predicted RMST is larger for longer restricted event-free time. Concordance metrics require the risk orientation

$${\hat{r}}_{\tau ,b}\left(X_j\right)=-{\tilde{m}}_{\tau ,b}\left(X_j\right).$$

The package exposes these two orientations directly as predict() and predict_risk().