Conformal Prediction in Dynamic Biological Systems

Uncertainty quantification (UQ) is the process of systematically determining and characterizing the degree of confidence in computational model predictions. In the context of systems biology, especially with dynamic models, UQ is crucial because it addresses the challenges posed by nonlinearity and parameter sensitivity, allowing us to properly understand and extrapolate the behavior of complex biological systems. Here, we focus on dynamic models represented by deterministic nonlinear ordinary differential equations. Many current UQ approaches in this field rely on Bayesian statistical methods. While powerful, these methods often require strong prior specifications and make parametric assumptions that may not always hold in biological systems. Additionally, these methods face challenges in domains where sample sizes are limited, and statistical inference becomes constrained, with computational speed being a bottleneck in large models of biological systems. As an alternative, we propose the use of conformal inference methods, introducing two novel algorithms that, in some instances, offer non-asymptotic guarantees, enhancing robustness and scalability across various applications. We demonstrate the efficacy of our proposed algorithms through several scenarios, highlighting their advantages over traditional Bayesian approaches. The proposed methods show promising results for diverse biological data structures and scenarios, offering a general framework to quantify uncertainty for dynamic models of biological systems.The software for the methodology and the reproduction of the results is available at https://zenodo.org/doi/10.5281/zenodo.13644870.

Uncertainty quantification in metric spaces

This paper introduces a novel uncertainty quantification framework for regression models where the response takes values in a separable metric space, and the predictors are in a Euclidean space. The proposed algorithms can efficiently handle large datasets and are agnostic to the predictive base model used. Furthermore, the algorithms possess asymptotic consistency guarantees and, in some special homoscedastic cases, we provide non-asymptotic guarantees. To illustrate the effectiveness of the proposed uncertainty quantification framework, we use a linear regression model for metric responses (known as the global Fr\'echet model) in various clinical applications related to precision and digital medicine. The different clinical outcomes analyzed are represented as complex statistical objects, including multivariate Euclidean data, Laplacian graphs, and probability distributions.

Deep Learning Framework with Uncertainty Quantification for Survey Data: Assessing and Predicting Diabetes Mellitus Risk in the American Population

Complex survey designs are commonly employed in many medical cohorts. In such scenarios, developing case-specific predictive risk score models that reflect the unique characteristics of the study design is essential. This approach is key to minimizing potential selective biases in results. The objectives of this paper are: (i) To propose a general predictive framework for regression and classification using neural network (NN) modeling, which incorporates survey weights into the estimation process; (ii) To introduce an uncertainty quantification algorithm for model prediction, tailored for data from complex survey designs; (iii) To apply this method in developing robust risk score models to assess the risk of Diabetes Mellitus in the US population, utilizing data from the NHANES 2011-2014 cohort. The theoretical properties of our estimators are designed to ensure minimal bias and the statistical consistency, thereby ensuring that our models yield reliable predictions and contribute novel scientific insights in diabetes research. While focused on diabetes, this NN predictive framework is adaptable to create clinical models for a diverse range of diseases and medical cohorts. The software and the data used in this paper is publicly available on GitHub.

kNN Algorithm for Conditional Mean and Variance Estimation with Automated Uncertainty Quantification and Variable Selection

In this paper, we introduce a kNN-based regression method that synergizes the scalability and adaptability of traditional non-parametric kNN models with a novel variable selection technique. This method focuses on accurately estimating the conditional mean and variance of random response variables, thereby effectively characterizing conditional distributions across diverse scenarios.Our approach incorporates a robust uncertainty quantification mechanism, leveraging our prior estimation work on conditional mean and variance. The employment of kNN ensures scalable computational efficiency in predicting intervals and statistical accuracy in line with optimal non-parametric rates. Additionally, we introduce a new kNN semi-parametric algorithm for estimating ROC curves, accounting for covariates. For selecting the smoothing parameter k, we propose an algorithm with theoretical guarantees.Incorporation of variable selection enhances the performance of the method significantly over conventional kNN techniques in various modeling tasks. We validate the approach through simulations in low, moderate, and high-dimensional covariate spaces. The algorithm's effectiveness is particularly notable in biomedical applications as demonstrated in two case studies. Concluding with a theoretical analysis, we highlight the consistency and convergence rate of our method over traditional kNN models, particularly when the underlying regression model takes values in a low-dimensional space.

Kernel Biclustering algorithm in Hilbert Spaces

Biclustering algorithms partition data and covariates simultaneously, providing new insights in several domains, such as analyzing gene expression to discover new biological functions. This paper develops a new model-free biclustering algorithm in abstract spaces using the notions of energy distance (ED) and the maximum mean discrepancy (MMD) -- two distances between probability distributions capable of handling complex data such as curves or graphs. The proposed method can learn more general and complex cluster shapes than most existing literature approaches, which usually focus on detecting mean and variance differences. Although the biclustering configurations of our approach are constrained to create disjoint structures at the datum and covariate levels, the results are competitive. Our results are similar to state-of-the-art methods in their optimal scenarios, assuming a proper kernel choice, outperforming them when cluster differences are concentrated in higher-order moments. The model's performance has been tested in several situations that involve simulated and real-world datasets. Finally, new theoretical consistency results are established using some tools of the theory of optimal transport.

Neural interval-censored Cox regression with feature selection

The classical Cox model emerged in 1972 promoting breakthroughs in how patient prognosis is quantified using time-to-event analysis in biomedicine. One of the most useful characteristics of the model for practitioners is the interpretability of the variables in the analysis. However, this comes at the price of introducing strong assumptions concerning the functional form of the regression model. To break this gap, this paper aims to exploit the explainability advantages of the classical Cox model in the setting of interval-censoring using a new Lasso neural network that simultaneously selects the most relevant variables while quantifying non-linear relations between predictors and survival times. The gain of the new method is illustrated empirically in an extensive simulation study with examples that involve linear and non-linear ground dependencies. We also demonstrate the performance of our strategy in the analysis of physiological, clinical and accelerometer data from the NHANES 2003-2006 waves to predict the effect of physical activity on the survival of patients. Our method outperforms the prior results in the literature that use the traditional Cox model.

Hypothesis testing for matched pairs with missing data by maximum mean discrepancy: An application to continuous glucose monitoring

A frequent problem in statistical science is how to properly handle missing data in matched paired observations. There is a large body of literature coping with the univariate case. Yet, the ongoing technological progress in measuring biological systems raises the need for addressing more complex data, e.g., graphs, strings and probability distributions, among others. In order to fill this gap, this paper proposes new estimators of the maximum mean discrepancy (MMD) to handle complex matched pairs with missing data. These estimators can detect differences in data distributions under different missingness mechanisms. The validity of this approach is proven and further studied in an extensive simulation study, and results of statistical consistency are provided. Data from continuous glucose monitoring in a longitudinal population-based diabetes study are used to illustrate the application of this approach. By employing the new distributional representations together with cluster analysis, new clinical criteria on how glucose changes vary at the distributional level over five years can be explored.

Glucose values prediction five years ahead with a new framework of missing responses in reproducing kernel Hilbert spaces, and the use of continuous glucose monitoring technology

AEGIS study possesses unique information on longitudinal changes in circulating glucose through continuous glucose monitoring technology (CGM). However, as usual in longitudinal medical studies, there is a significant amount of missing data in the outcome variables. For example, 40 percent of glycosylated hemoglobin (A1C) biomarker data are missing five years ahead. With the purpose to reduce the impact of this issue, this article proposes a new data analysis framework based on learning in reproducing kernel Hilbert spaces (RKHS) with missing responses that allows to capture non-linear relations between variable studies in different supervised modeling tasks. First, we extend the Hilbert-Schmidt dependence measure to test statistical independence in this context introducing a new bootstrap procedure, for which we prove consistency. Next, we adapt or use existing models of variable selection, regression, and conformal inference to obtain new clinical findings about glucose changes five years ahead with the AEGIS data. The most relevant findings are summarized below: i) We identify new factors associated with long-term glucose evolution; ii) We show the clinical sensibility of CGM data to detect changes in glucose metabolism; iii) We can improve clinical interventions based on our algorithms' expected glucose changes according to patients' baseline characteristics.

The FA Quantifier Fuzzification Mechanism: analysis of convergence and efficient implementations

The fuzzy quantification model FA has been identified as one of the best behaved quantification models in several revisions of the field of fuzzy quantification. This model is, to our knowledge, the unique one fulfilling the strict Determiner Fuzzification Scheme axiomatic framework that does not induce the standard min and max operators. The main contribution of this paper is the proof of a convergence result that links this quantification model with the Zadeh's model when the size of the input sets tends to infinite. The convergence proof is, in any case, more general than the convergence to the Zadeh's model, being applicable to any quantitative quantifier. In addition, recent revisions papers have presented some doubts about the existence of suitable computational implementations to evaluate the FA model in practical applications. In order to prove that this model is not only a theoretical approach, we show exact algorithmic solutions for the most common linguistic quantifiers as well as an approximate implementation by means of Monte Carlo. Additionally, we will also give a general overview of the main properties fulfilled by the FA model, as a single compendium integrating the whole set of properties fulfilled by it has not been previously published.