Nonparametric Conditional Density Estimation in a High-Dimensional egression Setting

摘要

In some applications (e.g., in cosmology and economics), the regression E[Z vertical bar x] is not adequate to represent the association between a predictor x and a response Z because of multi-modality and asymmetry of f (Z vertical bar x); using the full density instead of a single point estimate can then lead to less bias in subsequent analysis. As of now, there are no effective ways of estimating f (Z vertical bar x) when x represents high-dimensional, complex data. In this article, we propose a new nonparametric estimator of f (Z vertical bar x) that adapts to sparse (low-dimensional) structure in x. By directly expanding f (Z vertical bar x) in the eigenfunctions of a kernel-based operator, we avoid tensor products in high dimensions as well as ratios of estimated densities. Our basis functions are orthogonal with respect to the underlying data distribution, allowing fast implementation and tuning of parameters. We derive rates of convergence and show that the method adapts to the intrinsic dimension of the data. We also demonstrate the effectiveness of the series method on images, spectra, and an application to photometric redshift estimation of galaxies. Supplementary materials for this article are available online.