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 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.

在观察包括函数（通常是时间序列）的情况下，我们考虑对异质数据进行统计分析以进行预测。我们将混合专家 (ME) 的建模作为对数据异质性进行建模以通过矢量观测进行预测的选择框架，扩展到此功能数据分析环境。我们首先提出了一个新的 ME 模型系列，称为功能 ME (FME)，其中预测变量是来自整个函数的潜在噪声观测值。此外，预测器和真实响应的数据生成过程由表示未知分区的隐藏离散变量控制。其次，通过类似 Lasso 的正则化对基础函数参数的导数施加稀疏性，我们提供了称为 iFME 的 FME 模型的稀疏且可解释的函数表示。我们为套索式正则化最大似然参数估计策略开发了专用的期望最大化算法来拟合模型。所提出的模型和算法在模拟场景和两个真实数据集的应用中进行了研究，获得的结果证明了它们在准确捕获复杂非线性关系和对异构回归数据进行聚类方面的性能。