Dynamical Survival Analysis with Controlled Latent States

TL;DR

This work introduces a dynamic survival analysis framework where the event intensity for each individual is driven by a latent state evolving under a controlled differential equation. It develops two estimators: a neural controlled differential equation (NCDE) based model and a signature-based linearized CoxSig model, with theoretical guarantees including bias-variance analyses for the signature variant. The approach unifies time-varying covariates and static features, offering a scalable alternative to joint models and traditional survival methods, and demonstrates state-of-the-art performance across synthetic and real-world datasets in finance, healthcare, and logistics. The results highlight the signature-based method’s strength in ranking and calibration, while also discussing limitations such as computational scaling with high-dimensional time series and partial performance on low-dimensional data, suggesting directions for future work on competing risks and multimodal data integration.

Abstract

We consider the task of learning individual-specific intensities of counting processes from a set of static variables and irregularly sampled time series. We introduce a novel modelization approach in which the intensity is the solution to a controlled differential equation. We first design a neural estimator by building on neural controlled differential equations. In a second time, we show that our model can be linearized in the signature space under sufficient regularity conditions, yielding a signature-based estimator which we call CoxSig. We provide theoretical learning guarantees for both estimators, before showcasing the performance of our models on a vast array of simulated and real-world datasets from finance, predictive maintenance and food supply chain management.

Figures (28)

• Figure 1: Sample path $x(t)$ of a $3$-dimensional fractional Brownian motion on top, and three signature coefficients $\mathbf{S}^I(x_{[0,t]})$ associated to different words on the bottom.
• Figure 2: On the top, observed time series up to time $t$ in bold colors and true time series in faded colors. When evaluating our models, we fill-forward the last observed value from $t$ on. On the bottom, signatures of the true path (left), of the observed path (center) and difference in $\ell_2$ norm (right) --- $x_{FF}(t)$ denotes the filled-forward time series.
• Figure 3: Time series $\mathbf{X}^i$ of a randomly picked individual on bottom and unobserved SDE $w^i(t)$ on the top. The red star indicates the first hitting time of the threshold value $w_\star = 2.5$.
• Figure 4: Brier score $\delta t \mapsto \textnormal{BS}(t,\delta t)$, evaluated at $t=0.23$, for the partially observed SDE experiment. Confidence intervals indicate 1 standard deviation.
• Figure 5: C-Index (higher is better) on top and Brier score (lower is better) on bottom for hitting time of a partially observed SDE (left), churn prediction (center) and predictive maintenance (right) evaluated at chosen points $(t,\delta t)$. $t$ is chosen as the first decile of the event times i.e. $90 \%$ of the events occur after $t$. Hollow dots indicate outliers, and error bars indicate $80 \%$ of the interquartile range. We report detailed results for numerous points $(t,\delta t)$ in Appendix \ref{['appendix:more_results']}.

Theorems & Definitions (29)

• Lemma 2.1: A bound on the intensity
• Remark 2.2
• Remark 2.3
• Theorem 3.1: Informal Risk Bound for the Signature Model
• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Proposition 1.4: fermanian2021framing, Proposition 4.
• Theorem 1.5
• Lemma 1.6