No abstract

Detecting the presence of genetically modified plants (adventitious presence of unwanted transgenic plants, AP) from outcrossing species such as maize requires a method that lowers laboratory costs without losing precision. Group testing is a procedure in which groups that contain several units (plants) are analysed without having to inspect individual plants, with the purpose of estimating the prevalence of AP in a population at a low cost without losing precision. When pool (group) testing is used to estimate the prevalence of AP (p), there are sampling procedures for calculating a confidence interval (CI); however, they usually do not ensure precision in the estimation ofp. This research proposes a method to determine the number of pools (g), given a pool size (k), that ensures precision in the estimated proportion of AP (that is, it ensures a narrow CI). In addition, the study computes the maximum likelihood estimator ofpunder pool testing and its exact CI, considering the detection limit of the laboratory,d, and the concentration of AP per unit (c). The proposed sample procedure involves two steps: (1) obtain a sample size that guarantees that the mean width of the CI (\overline{w}) is narrower than the desired width (ω); and (2) iteratively increase the sample size until\overline{w}is smaller than the desired width (ω) with a specified degree of certainty (γ). Simulated data were created and tables are presented showing the different possible scenarios that a researcher may encounter. An R program is given and explained that will reproduce the results and make it easy for the researcher to create other scenarios.

The overfitting phenomenon happens when a statistical machine learning model learns very well about the noise as well as the signal that is present in the training data. On the other hand, an underfitted phenomenon occurs when only a few predictors are included in the statistical machine learning model that represents the complete structure of the data pattern poorly. This problem also arises when the training data set is too small and thus an underfitted model does a poor job of fitting the training data and unsatisfactorily predicts new data points. This chapter describes the importance of the trade-off between prediction accuracy and model interpretability, as well as the difference between explanatory and predictive modeling: Explanatory modeling minimizes bias, whereas predictive modeling seeks to minimize the combination of bias and estimation variance. We assess the importance and different methods of cross-validation as well as the importance and strategies of tuning that are key to the successful use of some statistical machine learning methods. We explain the most important metrics for evaluating the prediction performance for continuous, binary, categorical, and count response variables.

This chapter deals with the main theoretical fundamentals and practical issues of using functional regression in the context of genomic prediction. We explain how to represent data in functions by means of basis functions and considered two basis functions: Fourier for periodic or near-periodic data and B-splines for nonperiodic data. We derived the functional regression with a smoothed coefficient function under a fixed model framework and some examples are also provided under this model. A Bayesian version of functional regression is outlined and explained and all details for its implementation in glmnet and BGLR are given. The examples take into account in the predictor the main effects of environments and genotypes and the genotype × environment interaction term. The examples are done with small data sets so that the user can run them on his/her own computer and can understand the implementation process.

In this chapter, we go through the fundamentals of artificial neural networks and deep learning methods. We describe the inspiration for artificial neural networks and how the methods of deep learning are built. We define the activation function and its role in capturing nonlinear patterns in the input data. We explain the universal approximation theorem for understanding the power and limitation of these methods and describe the main topologies of artificial neural networks that play an important role in the successful implementation of these methods. We also describe loss functions (and their penalized versions) and give details about in which circumstances each of them should be used or preferred. In addition to the Ridge, Lasso, and Elastic Net regularization methods, we provide details of the dropout and the early stopping methods. Finally, we provide the backpropagation method and illustrate it with two simple artificial neural networks.

We give a detailed description of random forest and exemplify its use with data from plant breeding and genomic selection. The motivations for using random forest in genomic-enabled prediction are explained. Then we describe the process of building decision trees, which are a key component for building random forest models. We give (1) the random forest algorithm, (2) the main hyperparameters that need to be tuned, and (3) different splitting rules that are key for implementing random forest models for continuous, binary, categorical, and count response variables. In addition, many examples are provided for training random forest models with different types of response variables with plant breeding data. The random forest algorithm for multivariate outcomes is provided and its most popular splitting rules are also explained. In this case, some examples are provided for illustrating its implementation even with mixed outcomes (continuous, binary, and categorical). Final comments about the pros and cons of random forest are provided.

The adoption of machine learning frameworks in areas beyond computer science have been facilitated by the development of user-friendly software tools that do not require an advanced understanding of computer programming. In this paper, we present a new package (sparse kernel methods, SKM) software developed in R language for implementing six (generalized boosted machines, generalized linear models, support vector machines, random forest, Bayesian regression models and deep neural networks) of the most popular supervised machine learning algorithms with the optional use of sparse kernels. The SKM focuses on user simplicity, as it does not try to include all the available machine learning algorithms, but rather the most important aspects of these six algorithms in an easy-to-understand format. Another relevant contribution of this package is a function for the computation of seven different kernels. These are Linear, Polynomial, Sigmoid, Gaussian, Exponential, Arc-Cosine 1 and Arc-Cosine L (with L = 2, 3, … ) and their sparse versions, which allow users to create kernel machines without modifying the statistical machine learning algorithm. It is important to point out that the main contribution of our package resides in the functionality for the computation of the sparse version of seven basic kernels, which is indispensable for reducing computational resources to implement kernel machine learning methods without a significant loss in prediction performance. Performance of the SKM is evaluated in a genome-based prediction framework using both a maize and wheat data set. As such, the use of this package is not restricted to genome prediction problems, and can be used in many different applications.

This chapter provides elements for implementing deep neural networks (deep learning) for continuous outcomes. We give details of the hyperparameters to be tuned in deep neural networks and provide a general guide for doing this task with more probability of success. Then we explain the most popular deep learning frameworks that can be used to implement these models as well as the most popular optimizers available in many software programs for deep learning. Several practical examples with plant breeding data for implementing deep neural networks in the Keras library are outlined. These examples take into account many components in the predictor as well many hyperparameters (hidden layer, number of neurons, learning rate, optimizers, penalization, etc.) for which we also illustrate how the tuning process can be done to increase the probability of a successful application.