Spectral structure learning for clinical time series

Ivan Lerner; Anita Burgun; Francis Bach

cs.LGFeb 17, 2025

Spectral structure learning for clinical time series

Authors: Ivan Lerner, Anita Burgun, Francis Bach

Organizations: Université Paris Cité, Inria, Inserm, HeKA, F-75015 Paris, France · Service d’informatique biomédicale, AP-HP, Hôpital Georges Pompidou, F-75015 Paris, France · SIERRA, Inria Paris, F-75015 Paris, France

Abstract

We develop and evaluate a structure learning algorithm for clinical time series. Clinical time series are multivariate time series observed in multiple patients and irregularly sampled, challenging existing structure learning algorithms. We assume that our times series are realizations of StructGP, a k-dimensional multi-output or multi-task stationary Gaussian process (GP), with independent patients sharing the same covariance function. StructGP encodes ordered conditional relations between time series, represented in a directed acyclic graph. We implement an adapted NOTEARS algorithm, which based on a differentiable definition of acyclicity, recovers the graph by solving a series of continuous optimization problems. Simulation results show that up to mean degree 3 and 20 tasks, we reach a median recall of 0.93% [IQR, 0.86, 0.97] while keeping a median precision of 0.71% [0.57-0.84], for recovering directed edges. We further show that the regularization path is key to identifying the graph. With StructGP, we proposed a model of time series dependencies, that flexibly adapt to different time series regularity, while enabling us to learn these dependencies from observations.

Figures & tables

Experiment	Number of tasks	Mean degree	Grid search steps	Number of patients
	$k$	$md$	$n_{\lambda}$	$r$
TOY	4	2	256	$50$
EXP1	10	2	50	$[1,\ldots,100]$
EXP2	10	3	$[2,\ldots,512]$	50
EXP3	$[2,\ldots,20]$	$[1,2,3]$	50	50

Table 1: Summary of simulation parameters

Panel of Figure 1 — Figure 1: A toy model with 4 tasks We sample a random graph whose adjacency matrix sparsity pattern encodes ordered conditional relations between time series (a). We recover the parameters and order of variables from observations sampled uniformly and independently between tasks (see section 2.3 ).

Figure 2: Average metrics for an increasing number of patients Reports of ’EXP1’, increasing the number of patients for 10 tasks and 10 observations per task. The predicted graph (purple dots) is compared with a random graph from the same graph distribution (orange dots). The simulated graphs are random graphs with mean degree 2. Error bars represent bootstrap 95% confidence intervals.

$k$	P (Predicted)	P (Random)	R (Predicted)	R (Random)
4	1.00 [0.83-1.00]	0.50 [0.33-0.67]	1.00 [0.83-1.00]	0.50 [0.33-0.67]
10	0.82 [0.69-0.93]	0.14 [0.08-0.25]	0.94 [0.87-1.00]	0.16 [0.07-0.25]
20	0.71 [0.57-0.84]	0.07 [0.04-0.11]	0.93 [0.86-0.97]	0.07 [0.04-0.12]

Table 2: Precision and recall Precision ( P ) and recall ( R ) median and interquartile for varying mean degree and number of task parameters, from 100 replications of ’EXP3’ simulations with

md=3

. False positives include extra links and reversed links. Metrics are computed from models learned (Predicted) and compared with metrics for random graphs of the same distribution as the simulated data (Random).

Figure 3: Average metrics for an increasing number of grid search steps ( $n_{\lambda}$ ) Reports of ’EXP2’, an increasing number of grid search steps for 50 patients, 10 tasks, 10 observations per task, and random graph of mean degree 3. The predicted graph (red dots) is compared with a random graph from the same graph distribution (blue dots), and also with a predicted graph directly fitted from a random initialization with the optimal $\lambda_{*}$ found from grid search (green dots). Error bars represent bootstrap 95% confidence intervals.

Figure 4: Average metrics for an increasing number of tasks Reports of ’EXP3’, an increasing number of tasks for 50 patients, and 10 observations per task. The predicted graph (purple dots) is compared with a random graph from the same graph distribution (orange dots). Error bars represent bootstrap 95% confidence intervals.

Multi-Output Gaussian Processes

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Apr 30, 2026J/K move · Enter open · S save

Differentiable latent structure discovery for interpretable forecasting in clinical time series

Ivan Lerner, Jean Feydy, Alexandre Kalimouttou, Anita Burgun+1

Universit´e Paris Cit´e and Universit´e Sorbonne Paris Nord, INRIA, Inserm, HEKA, Paris, 75015, France. · Department of Medical Informatics, Assistance Publique Hˆopitaux de Paris (AP-HP), Georges Pompidou European Hospital, Paris, France. · SIERRA, Inria Paris, Paris, 75015, France.+2

Irregular Time-Series Modeling Healthcare Causal Structure Learning Longitudinal EHR Modeling Probabilistic Time Series Forecasting Gaussian Process Regression

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Background: Timely, uncertainty-aware forecasting from irregular electronic health records (EHR) can support critical-care decisions, yet most approaches either impute to a grid or sacrifice interpretability. We introduce StructGP, a continuous-time multi-task Gaussian process that couples process convolutions with differentiable structure learning to uncover a sparse, ordered directed acyclic graph (DAG) of inter-variable dependencies while preserving principled uncertainty. We further propose LP-StructGP, which augments StructGP with latent pathways-shared, temporally shifted trajectories inferred via subject-specific coupling filters and a softmax gating mechanism-to capture cross-patient progression patterns. Both models are trained under sparsity and acyclicity constraints (augmented Lagrangian, Adam) using scalable low-rank updates. Results: In simulations, the approach reliably recovers ground-truth graphs (Structural Hamming Distance approaching 0 as cohorts grow) and pathway assignments (high Adjusted Rand Index). On a MIMIC-IV septic shock cohort (n=1,008; norepinephrine, creatinine, mean arterial pressure), StructGP improves short-horizon (6 h) forecasting over independent-task baselines (average RMSE 0.68 [95%CI: 0.63--0.74] vs. 0.88 [0.83-0.94]) and, with 15 additional inputs, markedly outperforms unstructured kernels (0.63 [0.58-0.69] vs. 3.02 [2.85-3.18]) with superior calibration (coverage 0.96 vs. 0.84). On the PhysioNet Challenge (12k patients, 41 variables), StructGP attains competitive accuracy (MAE 3.72e-2) relative to a state-of-the-art graph neural model while maintaining calibrated uncertainty. Conclusion: These results show that structured process convolutions with latent pathways deliver interpretable, scalable, and well-calibrated forecasting for irregular clinical time series.

Abstract

Multi-Output Gaussian Processes

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