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Continuous-Time Learning of Probability Distributions: A Case Study in a Digital Trial of Young Children with Type 1 Diabetes

Antonio Álvarez-López, Marcos Matabuena

Abstract

Understanding how biomarker distributions evolve over time is a central challenge in digital health and chronic disease monitoring. In diabetes, changes in the distribution of glucose measurements can reveal patterns of disease progression and treatment response that conventional summary measures miss. Motivated by a 26-week clinical trial comparing the closed-loop insulin delivery system t:slim X2 with standard therapy in children with type 1 diabetes, we propose a probabilistic framework to model the continuous-time evolution of time-indexed distributions using continuous glucose monitoring data (CGM) collected every five minutes. We represent the glucose distribution as a Gaussian mixture, with time-varying mixture weights governed by a neural ODE. We estimate the model parameter using a distribution-matching criterion based on the maximum mean discrepancy. The resulting framework is interpretable, computationally efficient, and sensitive to subtle temporal distributional changes. Applied to CGM trial data, the method detects treatment-related improvements in glucose dynamics that are difficult to capture with traditional analytical approaches.

Continuous-Time Learning of Probability Distributions: A Case Study in a Digital Trial of Young Children with Type 1 Diabetes

Abstract

Understanding how biomarker distributions evolve over time is a central challenge in digital health and chronic disease monitoring. In diabetes, changes in the distribution of glucose measurements can reveal patterns of disease progression and treatment response that conventional summary measures miss. Motivated by a 26-week clinical trial comparing the closed-loop insulin delivery system t:slim X2 with standard therapy in children with type 1 diabetes, we propose a probabilistic framework to model the continuous-time evolution of time-indexed distributions using continuous glucose monitoring data (CGM) collected every five minutes. We represent the glucose distribution as a Gaussian mixture, with time-varying mixture weights governed by a neural ODE. We estimate the model parameter using a distribution-matching criterion based on the maximum mean discrepancy. The resulting framework is interpretable, computationally efficient, and sensitive to subtle temporal distributional changes. Applied to CGM trial data, the method detects treatment-related improvements in glucose dynamics that are difficult to capture with traditional analytical approaches.

Paper Structure

This paper contains 20 sections, 2 theorems, 71 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Case study: closed-loop insulin delivery in young children with type 1 diabetes
  3. Methodology
  4. Background
  5. Our model
  6. Discrete-time MMD fitting
  7. Continuous-time weight evolution
  8. Simulation study
  9. Case Study: CGM Trial
  10. Final remarks
  11. Statistical theory and proofs
  12. Inference based on estimated weight trajectories
  13. Two-sample MMD statistic.
  14. Wild bootstrap calibration.
  15. Relation to permutation calibration.
  16. ...and 5 more sections

Key Result

Theorem A.1

Let $\{f_t\}_{t\in[0,T]}\subset L^1(\mathbb{R}^d)$ be a family of probability densities. Assume: Then for every $\varepsilon>0$ there exist $K\in\mathbb{N}$, $\sigma^2>0$ and centers $\{\mu_s\}_{s\in[K]}\subset\mathbb{R}^d$ such that, for each $t\in[0,T]$, one can choose $\alpha(t)\in\Delta^{K-1}$ with If, in addition, $t\mapsto f_t$ is continuous in $L^1$, then $t\mapsto\alpha(t)$ can be chosen

Figures (11)

  • Figure 1: Individual participant analysis for the bivariate model (glucose and its first derivative) using $K=5$ Gaussian components. Each row corresponds to one participant (IDs 20, 58, and 82). Left: raw CGM time series showing glucose concentration (mg/dL) over the observation period. Middle left: estimated weight trajectories $\alpha(t) = (\alpha_k(t))_{k=1}^5$ learned by the neural ODE, representing the evolution of the mixture proportions over normalized time $t \in [0,1]$. Middle right and right: contours of the fitted bivariate Gaussian mixture density at the initial and final times, summarizing the joint distribution of glucose level and its rate of change.
  • Figure 2: Comparison of weight trajectory dynamics between Treatment (green) and Control (red) groups for the $d=2$ bivariate model with $K=5$ mixture components. Each panel shows the evolution of component weights $\alpha_s(t)$ between weeks 20--26 over normalized time $t \in [0,1]$. Group means are shown as thick dashed lines. The shaded bands represent a statistical envelope around the mean (e.g., between the 5th and 95th percentiles).
  • Figure 3: Predicted glucose density distributions between weeks 20 and 26, comparing Treatment (green) and Control (red) groups for the $d=2$ model with $K=5$ mixture components. The marginal density over glucose concentration and speed is computed by drawing samples from each participant's GMM, estimating the sliced-Wasserstein barycenter in 2D, and converting the barycenter samples into a smooth density via a Gaussian KDE on the grid. The top row corresponds to the Control group and the bottom row to the Treatment group. In both rows, the first column shows the initial distribution at week 20, the second column displays the final distribution at week 26, and the third column presents the difference between the two distributions.
  • Figure 4: Wild Bootstrap MMD test $p$-values comparing Treatment vs. Control groups over time for the $d=2$ model with $K=5$ mixture components. The dashed black line indicates the significance threshold $\alpha = 0.05$. The colors correspond to the different components: blue (component 1), orange (component 2), green (component 3), red (component 4), and violet (component 5).
  • Figure 5: Quantile curves (median and 25%--75% bands) of the change in GMM mixture weights for each of the $K=5$ components over time, relative to their initial value, for Treatment (green) and Control (red) groups. Each panel shows the temporal evolution of a component's weight deviation from baseline.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Remark 3.1
  • Theorem A.1: Universality
  • Theorem A.2: Finite-sample stability
  • Remark A.3
  • proof : Proof of \ref{['thm:uniform_shared_dictionary']}
  • proof : Proof of \ref{['thm:weights_rate']}