Prediction of Parkinson disease progression from sparse longitudinal trajectories using multiclass likelihood contrast learning

Background: Biomedical studies increasingly collect repeated measurements, such as clinical ratings, speech measures, gait summaries, and wearable-sensor features, at irregular subject-specific follow-up times. These data are often sparse and unaligned, making them difficult to use with standard classifiers that require fixed-length input vectors. Parkinson's disease provides a motivating example because progression is longitudinal and heterogeneous, but clinical follow-up is rarely observed on a common schedule. Objective: This study develops and evaluates a multiclass likelihood contrast classifier for sparse longitudinal trajectory recovery. The goal is to classify subjects using their observed repeated measurements while avoiding aggressive reduction of trajectories to averages or simple slopes. Methods: The proposed method fits one class-specific mixed-effects model per outcome class and assigns each held-out subject to the class with the largest marginal likelihood score. The main specification uses a quadratic mean trajectory with subject-specific random intercept and random slope terms. Performance is evaluated in a three-class simulation with imbalanced class sizes and in a Parkinson's motor-UPDRS sparse trajectory recovery experiment. In the Parkinson's experiment, dense histories are first used to define mathematical trajectory phenotypes; most observations are then hidden, and models are asked to recover the phenotype from the sparse record. Baselines include a linear mixed-likelihood classifier, functional k-nearest neighbors, random forest, and support vector machine. Results: In the Parkinson's sparse recovery experiment, after approximately 90% of dense observations were hidden, the proposed model achieved accuracy 0.8095, macro-F1 0.8160, MCC 0.7107, and macro-AUC 0.9268. A train-validation-test diagnostic gave similar held-out behavior, with validation macro-F1 0.7980 and test macro-F1 0.8160 for the proposed model. Conclusion: Multiclass likelihood contrasts offer an interpretable and competitive approach for sparse, unaligned longitudinal classification. The Parkinson's analysis should be read as an algorithmic sparse trajectory recovery study rather than clinical diagnostic validation. Future work should evaluate the method with externally assigned clinical outcomes, multivariate longitudinal biomarkers, prospective cohorts, and ordinal generalized mixed-model likelihoods that better match bounded clinical rating scales.