We consider the statistical analysis of heterogeneous data for prediction in situations where the observations include functions, typically time series. We extend the modeling with Mixtures-of-Experts (ME), as a framework of choice in modeling heterogeneity in data for prediction with vectorial observations, to this functional data analysis context. We first present a new family of ME models, named functional ME (FME) in which the predictors are potentially noisy observations, from entire functions. Furthermore, the data generating process of the predictor and the real response, is governed by a hidden discrete variable representing an unknown partition. Second, by imposing sparsity on derivatives of the underlying functional parameters via Lasso-like regularizations, we provide sparse and interpretable functional representations of the FME models called iFME. We develop dedicated expectation--maximization algorithms for Lasso-like (EM-Lasso) regularized maximum-likelihood parameter estimation strategies to fit the models. The proposed models and algorithms are studied in simulated scenarios and in applications to two real data sets, and the obtained results demonstrate their performance in accurately capturing complex nonlinear relationships and in clustering the heterogeneous regression data.

翻译：我们针对包含函数型观测（典型如时间序列）的异质性数据预测问题，开展统计分析方法研究。将混合专家模型（Mixture-of-Experts, ME）——该模型作为处理向量型观测数据异质性预测问题的优选框架——拓展至函数型数据分析场景。首先提出名为函数型ME（FME）的新型ME模型族，其中预测变量为完整函数（可能含噪声观测），且预测变量与真实响应的数据生成过程由表示未知划分的隐离散变量控制。其次，通过LASSO型正则化对底层函数参数的导数施加稀疏性约束，构建FME模型的稀疏可解释函数表示（称iFME）。我们开发基于LASSO型（EM-Lasso）正则化极大似然参数估计策略的专用期望最大化算法进行模型拟合。通过模拟场景及两个真实数据集的应用验证，所提模型与算法在准确捕获复杂非线性关系及异质性回归数据聚类方面展现出优异性能。