Abstract
AbstractThe fundamentals for Reproducing Kernel Hilbert Spaces (RKHS) regression methods are described in this chapter. We first point out the virtues of RKHS regression methods and why these methods are gaining a lot of acceptance in statistical machine learning. Key elements for the construction of RKHS regression methods are provided, the kernel trick is explained in some detail, and the main kernel functions for building kernels are provided. This chapter explains some loss functions under a fixed model framework with examples of Gaussian, binary, and categorical response variables. We illustrate the use of mixed models with kernels by providing examples for continuous response variables. Practical issues for tuning the kernels are illustrated. We expand the RKHS regression methods under a Bayesian framework with practical examples applied to continuous and categorical response variables and by including in the predictor the main effects of environments, genotypes, and the genotype ×environment interaction. We show examples of multi-trait RKHS regression methods for continuous response variables. Finally, some practical issues of kernel compression methods are provided which are important for reducing the computation cost of implementing conventional RKHS methods.
Highlights
One of the main goals of genetic research is accurate phenotype prediction
Reproducing Kernel Hilbert Spaces (RKHS) regression was one of the earliest statistical machine learning methods suggested for use in plant and animal breeding (Gianola et al 2006; Gianola and van Kaam 2008) for the prediction of complex traits
Support Vector Machines (SVM), which is studied in the chapter, lately it has been shown that any learning algorithm based on distances between objects can be formulated in terms of kernel functions, applying the so-called “kernel trick.”
Summary
One of the main goals of genetic research is accurate phenotype prediction. This goal has largely been achieved for Mendelian diseases with a small number of risk variants (Schrodi et al 2014). These publications have empirically shown equal or better prediction ability of RKHS methods over linear models For this reason, the applications of kernel methods in GS are expected to continue increasing since they can be implemented in current software of genomic prediction and because they are (a) very flexible, (b) easy to interpret, (c) theoretically appealing for accommodating cryptic forms of gene action (Gianola et al 2006; Gianola and van Kaam 2008), (d) these methods can be used with almost any type of information (e.g., covariates, strings, images, and graphs) (de los Campos et al 2010), (e) computation is performed in an n-dimensional space even when the original input information has more columns ( p) than observations (n) avoiding the p ) n problem (de los Campos et al 2010), (f) they provide a new viewpoint whose full potential is still far from our understanding, and (g) they are very attractive due to their computational efficiency, robustness, and stability. We cover the essentials of kernels methods, and with examples, we show the user how to handcraft an algorithm of a kernel for applications in the context of genomic selection
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