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Abstract：
Considering the context of functional data analysis, we developed and applied a new Bayesian approach via the Gibbs sampler to select basis functions for a finite representation of functional data. The proposed methodology uses Bernoulli latent variables to assign zero to some of the basis function coefficients with a positive probability. This procedure allows for an adaptive basis selection since it can determine the number of bases and which ones should be selected to represent functional data. Moreover, the proposed procedure measures the uncertainty of the selection process and can be applied to multiple curves simultaneously. The methodology developed can deal with observed curves that may differ due to experimental error and random individual differences between subjects, which one can observe in a real dataset application involving daily numbers of COVID-19 cases in Brazil. Simulation studies show the main properties of the proposed method, such as its accuracy in estimating the coefficients and the strength of the procedure to find the true set of basis functions. Despite having been developed in the context of functional data analysis, we also compared the proposed model via simulation with the well-established LASSO and Bayesian LASSO, which are methods developed for non-functional data.
摘要：
考虑到功能数据分析的背景，我们通过Gibbs采样器开发并应用了一种新的贝叶斯方法，以选择用于有限表示函数数据的基函数。所提出的方法使用伯努利潜在变量将具有正概率的某些基函数系数分配为零。该过程允许自适应基础选择,因为它可以确定基础的数量以及应该选择哪些来表示功能数据。此外，所提出的程序测量选择过程的不确定性，可以同时应用于多条曲线。开发的方法可以处理由于实验误差和受试者之间的随机个体差异而可能不同的观察曲线，可以在涉及巴西每日COVID-19病例数的真实数据集应用程序中观察到。仿真研究表明了所提出方法的主要性质，例如，它在估计系数方面的准确性以及找到真正的基函数集的过程的强度。尽管是在功能数据分析的背景下开发的，我们还通过仿真将提出的模型与完善的LASSO和贝叶斯LASSO进行了比较，这是针对非功能性数据开发的方法。
