Despite recent progress in predicting biomarker trajectories from real clinical data, uncertainty in the predictions poses high-stakes risks (e.g., misdiagnosis) that limit their clinical deployment. To enable safe and reliable use of such predictions in healthcare, we introduce a conformal method for uncertainty-calibrated prediction of biomarker trajectories resulting from randomly-timed clinical visits of patients. Our approach extends conformal prediction to the setting of randomly-timed trajectories via a novel nonconformity score that produces prediction bands guaranteed to cover the unknown biomarker trajectories with a user-prescribed probability. We apply our method across a wide range of standard and state-of-the-art predictors for two well-established brain biomarkers of Alzheimer's disease, using neuroimaging data from real clinical studies. We observe that our conformal prediction bands consistently achieve the desired coverage, while also being tighter than baseline prediction bands. To further account for population heterogeneity, we develop group-conditional conformal bands and test their coverage guarantees across various demographic and clinically relevant subpopulations. Moreover, we demonstrate the clinical utility of our conformal bands in identifying subjects at high risk of progression to Alzheimer's disease. Specifically, we introduce an uncertainty-calibrated risk score that enables the identification of 17.5% more high-risk subjects compared to standard risk scores, highlighting the value of uncertainty calibration in real-world clinical decision making. Our code is available at github.com/vatass/ConformalBiomarkerTrajectories.

We begin by reviewing the basics of conformal prediction, in this section.

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The growing importance of intraday electricity trading in Europe calls for improved price forecasting and tailored decision-support tools. In this paper, we propose a novel generative neural network model to generate probabilistic path forecasts for intraday electricity prices and use them to construct effective trading strategies for Germany's continuous-time intraday market. Our method demonstrates competitive performance in terms of statistical evaluation metrics compared to two state-of-the-art statistical benchmark approaches. To further assess its economic value, we consider a realistic fixed-volume trading scenario and propose various strategies for placing market sell orders based on the path forecasts. Among the different trading strategies, the price paths generated by our generative model lead to higher profit gains than the benchmark methods. Our findings highlight the potential of generative machine learning tools in electricity price forecasting and underscore the importance of economic evaluation.

Conformal Prediction (CP) is a popular framework for constructing prediction bands with valid coverage in finite samples, while being free of any distributional assumption. A well-known limitation of conformal prediction is the lack of adaptivity, although several works introduced practically efficient alternate procedures. In this work, we build upon recent ideas that rely on recasting the CP problem as a statistical learning problem, directly targeting coverage and adaptivity. This statistical learning problem is based on reproducible kernel Hilbert spaces (RKHS) and kernel sum-of-squares (SoS) methods. First, we extend previous results with a general representer theorem and exhibit the dual formulation of the learning problem. Crucially, such dual formulation can be solved efficiently by accelerated gradient methods with several hundreds or thousands of samples, unlike previous strategies based on off-the-shelf semidefinite programming algorithms. Second, we introduce a new hyperparameter tuning strategy tailored specifically to target adaptivity through bounds on test-conditional coverage. This strategy, based on the Hilbert-Schmidt Independence Criterion (HSIC), is introduced here to tune kernel lengthscales in our framework, but has broader applicability since it could be used in any CP algorithm where the score function is learned. Finally, extensive experiments are conducted to show how our method compares to related work. All figures can be reproduced with the accompanying code.

This work is concerned with conformal prediction in contemporary applications (including generative AI) where a black-box model has been trained on data that are not accessible to the user. Mirroring split-conformal inference, we design a wrapper around a black-box algorithm which calibrates conformity scores. This calibration is local and proceeds in two stages by first adaptively partitioning the predictor space into groups and then calibrating sectionally group by group. Adaptive partitioning (self-grouping) is achieved by fitting a robust regression tree to the conformity scores on the calibration set. This new tree variant is designed in such a way that adding a single new observation does not change the tree fit with overwhelmingly large probability. This add-one-in robustness property allows us to conclude a finite sample group-conditional coverage guarantee, a refinement of the marginal guarantee. In addition, unlike traditional split-conformal inference, adaptive splitting and within-group calibration yields adaptive bands which can stretch and shrink locally. We demonstrate benefits of local tightening on several simulated as well as real examples using non-parametric regression. Finally, we consider two contemporary classification applications for obtaining uncertainty quantification around GPT-4o predictions. We conformalize skin disease diagnoses based on self-reported symptoms as well as predicted states of U.S. legislators based on summaries of their ideology. We demonstrate substantial local tightening of the uncertainty sets while attaining similar marginal coverage.

Weighted conformal prediction (WCP), a recently proposed framework, provides uncertainty quantification with the flexibility to accommodate different covariate distributions between training and test data. However, it is pointed out in this paper that the effectiveness of WCP heavily relies on the overlap between covariate distributions; insufficient overlap can lead to uninformative prediction intervals. To enhance the informativeness of WCP, we propose two methods for scenarios involving multiple sources with varied covariate distributions. We establish theoretical guarantees for our proposed methods and demonstrate their efficacy through simulations.

This paper presents a new conformal method for generating simultaneous forecasting bands guaranteed to cover the entire path of a new random trajectory with sufficiently high probability. Prompted by the need for dependable uncertainty estimates in motion planning applications where the behavior of diverse objects may be more or less unpredictable, we blend different techniques from online conformal prediction of single and multiple time series, as well as ideas for addressing heteroscedasticity in regression. This solution is both principled, providing precise finite-sample guarantees, and effective, often leading to more informative predictions than prior methods.

Functional data analysis (FDA) (Ramsay and Silverman 2005) is naturally apt to represent and model this kind of data, as it allows preserving their continuous nature, and provides a rigorous mathematical framework. Among the others, Zhou and Pan 2014 analyzed temperature surfaces, presenting two approaches for Functional Principal Component Analysis (FPCA) of functions defined on a non-rectangular domain, Porro-Muñoz et al. 2014 focuses on image processing using FDA, while a novel regularization technique for Gaussian random fields on a rectangular domain has been proposed by Rakêt 2010 and applied to 2D electrophoresis images. Another bivariate smoothing approach in a penalized regression framework has been introduced by Ivanescu and Andrada 2013, allowing for the estimation of functional parameters of two-dimensional functional data. As shown by Gervini 2010, even mortality rates can be interpreted as two-dimensional functional data. Whereas in all the reviewed works functions are assumed to be realization of iid or at least exchangeable random objects, to the best of our knowledge there is no literature focusing on forecasting time-dependent two-dimensional functional data. In this work, we focus on time series of surfaces, representing them as two-dimensional Functional Time Series (FTS).

Uncertainty quantification for prediction is an intriguing problem with significant applications in various fields, such as biomedical science, economic studies, and weather forecasts. Numerous methods are available for constructing prediction intervals, such as quantile regression and conformal predictions, among others. Nevertheless, model misspecification (especially in high-dimension) or sub-optimal constructions can frequently result in biased or unnecessarily-wide prediction intervals. In this paper, we propose a novel and widely applicable technique for aggregating multiple prediction intervals to minimize the average width of the prediction band along with coverage guarantee, called Universally Trainable Optimal Predictive Intervals Aggregation (UTOPIA). The method also allows us to directly construct predictive bands based on elementary basis functions. Our approach is based on linear or convex programming which is easy to implement. All of our proposed methodologies are supported by theoretical guarantees on the coverage probability and optimal average length, which are detailed in this paper. The effectiveness of our approach is convincingly demonstrated by applying it to synthetic data and two real datasets on finance and macroeconomics.