We introduce a novel framework for the classification of functional data supported on non-linear, and possibly random, manifold domains. The motivating application is the identification of subjects with Alzheimer's disease from their cortical surface geometry and associated cortical thickness map. The proposed model is based upon a reformulation of the classification problem into a regularized multivariate functional linear regression model. This allows us to adopt a direct approach to the estimation of the most discriminant direction while controlling for its complexity with appropriate differential regularization. Our approach does not require prior estimation of the covariance structure of the functional predictors, which is computationally not feasible in our application setting. We provide a theoretical analysis of the out-of-sample prediction error of the proposed model and explore the finite sample performance in a simulation setting. We apply the proposed method to a pooled dataset from the Alzheimer's Disease Neuroimaging Initiative and the Parkinson's Progression Markers Initiative, and are able to estimate discriminant directions that capture both cortical geometric and thickness predictive features of Alzheimer's Disease, which are consistent with the existing neuroscience literature.
翻译:我们引入了非线性、也可能是随机的多个领域所支持的功能数据分类的新框架。激励性应用是从其皮层表面几何图和相关皮层厚度图中确定阿尔茨海默氏病患者。拟议模型的基础是将分类问题重新纳入常规化的多变量功能性回归模型。这使我们能够直接估算最矛盾的方向,同时以适当的差异规范来控制其复杂性。我们的方法并不要求事先估算功能预测器的变量结构,这在我们的应用设置中是计算不可行的。我们对拟议模型的外表预测错误进行理论分析,并探索模拟环境中的有限样本性能。我们将拟议方法应用于从阿尔茨海默氏氏疾病神经造影倡议和Parkinson进步标记倡议中汇集的数据集,并能够估算与现有神经科学文献相一致的阿尔茨海默氏病遗传学和厚度预测特征的偏差方向。