As ultra high-dimensional longitudinal data are becoming ever more apparent in fields such as public health and bioinformatics, developing flexible methods with a sparse model is of high interest. In this setting, the dimension of the covariates can potentially grow exponentially as $\text{exp}(n^{1/2})$ with respect to the number of clusters n. We consider a flexible semiparametric approach, namely, partially linear single-index models, for ultra high-dimensional longitudinal data. Most importantly, we allow not only the partially linear covariates but also the single-index covariates within the unknown flexible function estimated nonparametrically to be ultra high dimensional. Using penalized generalized estimating equations, this approach can capture correlation within subjects, can perform simultaneous variable selection and estimation with a smoothly clipped absolute deviation penalty, and can capture nonlinearity and potentially some interactions among predictors. We establish asymptotic theory for the estimators including the oracle property in ultra high dimension for both the partially linear and nonparametric components, and we present an efficient algorithm to handle the computational challenges. We show the effectiveness of our method and algorithm via a simulation study and a yeast cell cycle gene expression data.

中文翻译：

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超高维半参数纵向数据分析

随着超高维纵向数据在公共卫生和生物信息学等领域变得越来越明显，开发具有稀疏模型的灵活方法备受关注。在这种情况下，协变量的维度可能呈指数增长$\text{经验值}(n^{1/2})$关于簇数 n. 我们考虑了一种灵活的半参数方法，即部分线性单指数模型，用于超高维纵向数据。最重要的是，我们不仅允许部分线性协变量，而且允许未知灵活函数内的单指数协变量以非参数方式估计为超高维。使用惩罚广义估计方程，这种方法可以捕获受试者内部的相关性，可以使用平滑剪裁的绝对偏差惩罚同时执行变量选择和估计，并且可以捕获非线性和预测变量之间潜在的一些相互作用。我们为估计量建立了渐近理论，包括部分线性和非参数分量的超高维预言属性，我们提出了一种有效的算法来处理计算挑战。我们通过模拟研究和酵母细胞周期基因表达数据展示了我们的方法和算法的有效性。