Developing knowledge-driven contemporaneous health index (CHI) that can precisely reflect the underlying patient across the course of the condition's progression holds a unique value, like facilitating a range of clinical decision-making opportunities. This is particularly important for monitoring degenerative condition such as Alzheimer's disease (AD), where the condition of the patient will decay over time. Detecting early symptoms and progression sign, and continuous severity evaluation, are all essential for disease management. While a few methods have been developed in the literature, uncertainty quantification of those health index models has been largely neglected. To ensure the continuity of the care, we should be more explicit about the level of confidence in model outputs. Ideally, decision-makers should be provided with recommendations that are robust in the face of substantial uncertainty about future outcomes. In this paper, we aim at filling this gap by developing an uncertainty quantification based contemporaneous longitudinal index, named UQ-CHI, with a particular focus on continuous patient monitoring of degenerative conditions. Our method is to combine convex optimization and Bayesian learning using the maximum entropy learning (MEL) framework, integrating uncertainty on labels as well. Our methodology also provides closed-form solutions in some important decision making tasks, e.g., such as predicting the label of a new sample. Numerical studies demonstrate the effectiveness of the propose UQ-CHI method in prediction accuracy, monitoring efficacy, and unique advantages if uncertainty quantification is enabled practice.
Let, @math , denote a training set of @math patients. Each measurement @math , is the value of the @math th variable for the @math th subject in a given time @math , where @math is the time index. our goal is, given a training set, convert each measurement @math into an health index @math , which requires a mathematical model of @math . For simplicity, multivariable form of the hypothesis function @math was studies in @cite_13 , i.e., @math , where @math is a vector of weight coefficients that combines the @math variables. The total number of positive and negative samples is shown by @math and @math respectively, i.e., @math and @math . The formulation of the CHI learning framework is shown in below:
@cite_13
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Abstract In this paper, we develop a novel formulation for contemporaneous patient risk monitoring by exploiting the emerging data-rich environment in many healthcare applications, where an abundance of longitudinal data that reflect the degeneration of the health condition can be continuously collected. Our objective, and the developed formulation, is fundamentally different from many existing risk score models for different healthcare applications, which mostly focus on predicting the likelihood of a certain outcome at a pre-specified time. Rather, our formulation translates multivariate longitudinal measurements into a contemporaneous health index (CHI) that captures patient condition changes over the course of progression. Another significant feature of our formulation is that, CHI can be estimated with or without label information, different from other risk score models strictly based on supervised learning. To develop this formulation, we focus on the degenerative disease conditions, for which we could utilize the monotonic progression characteristic (either towards disease or recovery) to learn CHI. Such a domain knowledge leads us to a novel learning formulation, and on top of that, we further generalize this formulation with a capacity to incorporate label information if available. We further develop algorithms to mitigate the challenges associated with the nonsmooth convex optimization problem by first identifying its dual reformulation as a constrained smooth optimization problem, and then, using the block coordinate descent algorithm to iteratively solve the optimization with a derived efficient projection at each iteration. Extensive numerical studies are performed on both synthetic datasets and real-world applications on Alzheimer’s disease and Surgical Site Infection, which demonstrate the utility and efficacy of the proposed method on degenerative conditions that include a wide range of applications.