In multivariate regression, when covariates are numerous, it is often reasonable to assume that only a small number of them has predictive information. In some medical applications for instance, it is believed that only a few genes out of thousands are responsible for cancers. In that case, the aim is not only to propose a good fit, but also to select the relevant covariates (genes). We propose to perform model selection with additive models in high dimensions (sample size and number of covariates). Our approach is computationally efficient thanks to fast wavelet transforms, it does not rely on cross validation, and it solves a convex optimization problem for a prescribed penalty parameter, called the quantile universal threshold. We also propose a second rule based on Stein unbiased risk estimation geared towards prediction. We use Monte Carlo simulations and real data to compare various methods based on false discovery rate (FDR), true positive rate (TPR) and mean squared error. Our approach is the only one to handle high dimensions, and has the best FDR--TPR trade-off.
Bayesian Functional Generalized Additive Models with Sparsely Observed Covariates
McLean等人（2013）最近提出了函数广义可加模型（FGAM），作为常见函数线性模型（FLM）的更灵活的替代方案，用于回归函数协变量的标量。在本文中，我们开发了一个贝叶斯版本的FGAM的高斯误差的情况下，单位链接功能。我们的方法允许函数协变量被稀疏地观察和测量，而McLean et al.（2013）的估计过程要求它们在规则网格上被无噪声地观察。我们认为这两个蒙特卡罗和变分贝叶斯方法拟合FGAM稀疏观测协变量。由于模型后验分布和全条件分布的复杂形式，标准蒙特卡罗和变分贝叶斯算法不能使用。我们使用的策略来处理更新的参数没有封闭形式的完整条件应该是独立的兴趣应用贝叶斯统计学家与非共轭模型。我们的数值研究表明，我们的算法的好处，首先使用标准技术恢复完整的轨迹，然后拟合一个功能回归模型的两步方法。在一个真实的数据分析中，我们的方法被应用到预测在网上拍卖网站eBay上拍卖的物品的收盘价。
