Modeling conditional distributions for functional data extends the concept of a mean response in functional regression settings, where vector predictors are paired with functional responses. This extension is challenging because of the nonexistence of well-defined densities, cumulative distributions, or quantile functions in the Hilbert space where the response functions are located. To address this challenge, we simplify the problem by assuming that the response functions are Gaussian processes, which means that the conditional distribution of the responses is determined by conditional mean and conditional covariance. We demonstrate that these quantities can be obtained by applying global and local Fréchet regression, where the local version is more flexible and applicable when the covariate dimension is low and covariates are continuous, while the global version is not subject to these restrictions but is based on the assumption of a more restrictive regression relation. Convergence rates for the proposed estimates are obtained under the framework of M-estimation. The corresponding estimation of conditional distributions is illustrated with simulations and an application to bike-sharing data, where predictors include weather characteristics and responses are bike rental profiles. We also show that our methods are applicable to the challenging problem to study functional fragments. Such data are observed in accelerated longitudinal studies and correspond to functional data observed over short domain segments. We demonstrate the utility of conditional distributions in this context by using the time (age) at which a subject enters the domain of a fragment in addition to other covariates as predictor and the function observed over the domain of the fragment as response.

中文翻译：

---

功能响应的条件分布回归

为功能数据建模条件分布扩展了功能回归设置中平均响应的概念，其中向量预测变量与功能响应配对。这种扩展具有挑战性，因为在响应函数所在的希尔伯特空间中不存在明确定义的密度、累积分布或分位数函数。为了应对这一挑战，我们通过假设响应函数是高斯过程来简化问题，这意味着响应的条件分布由条件均值和条件协方差确定。我们证明了这些量可以通过应用全局和局部 Fréchet 回归来获得，其中局部版本在协变量维度较低且协变量连续时更加灵活和适用，而全局版本不受这些限制，而是基于更严格的回归关系的假设。建议估计的收敛率是在 M 估计的框架下获得的。通过模拟和自行车共享数据的应用说明了条件分布的相应估计，其中预测变量包括天气特征，响应是自行车租赁概况。我们还表明，我们的方法适用于研究功能片段的挑战性问题。这些数据在加速纵向研究中观察到，并且对应于在短域段上观察到的功能数据。