Mixed membership models are a flexible class of probabilistic data representations used for unsupervised and semi-supervised learning, allowing each observation to partially belong to multiple clusters or features. In this manuscript, we extend the framework of functional mixed membership models to allow for covariate-dependent adjustments. The proposed model utilizes a multivariate Karhunen-Lo\`eve decomposition, which allows for a scalable and flexible model. Within this framework, we establish a set of sufficient conditions ensuring the identifiability of the mean, covariance, and allocation structure up to a permutation of the labels. This manuscript is primarily motivated by studies on functional brain imaging through electroencephalography (EEG) of children with autism spectrum disorder (ASD). Specifically, we are interested in characterizing the heterogeneity of alpha oscillations for typically developing (TD) children and children with ASD. Since alpha oscillations are known to change as children develop, we aim to characterize the heterogeneity of alpha oscillations conditionally on the age of the child. Using the proposed framework, we were able to gain novel information on the developmental trajectories of alpha oscillations for children with ASD and how the developmental trajectories differ between TD children and children with ASD.

翻译：混合隶属度模型是一类灵活的概率数据表示方法，用于无监督和半监督学习，允许每个观测部分属于多个聚类或特征。在本研究中，我们将功能性混合隶属度模型框架扩展至允许协变量依赖的调整。所提出的模型采用多元Karhunen-Loève分解，从而构建了可扩展且灵活的模型。在此框架内，我们建立了一组充分条件，确保均值、协方差及分配结构在标签置换意义下的可识别性。本研究主要受自闭症谱系障碍（ASD）儿童脑电图（EEG）功能成像研究的启发。具体而言，我们关注典型发育（TD）儿童与ASD儿童α振荡的异质性表征。由于已知α振荡会随儿童发育而变化，我们旨在以儿童年龄为条件刻画α振荡的异质性。利用所提出的框架，我们获得了关于ASD儿童α振荡发育轨迹的新信息，并揭示了TD儿童与ASD儿童之间发育轨迹的差异。