# Exploring Data Augmentation Algorithm to Improve Genomic Prediction of Top-Ranking Cultivars
## 2 Materials and Methods
### Statistical Machine Learning Models
#### 2.2.2 Data Augmentation (Augmented Algorithms A)  
Various data augmentation algorithms exist, though our focus will center on the mixup method, as introduced in the paper titled \"mixup: Beyond Empirical Risk Minimization\" by [PERSON] et al. [19], which provides a domain-agnostic approach to the improvement in machine learning models. Using this algorithm, synthetic data are generated through the application of the following mathematical formulae:  
\[\widetilde{\mathbf{x}}=\lambda\mathbf{x}_{i}+(1-\lambda)\mathbf{x}_{j}. \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Dataset** & **Gen** & **Markers** & **Env** & **Traits** & **Trait Name** \\ \hline Maize\_1 & 1000 & 4085 & 11 & 4 & Days to tassel, anthesis silking interval, plant height and ear height \\ \hline Maize\_2 & 1000 & 4085 & 11 & 4 & Days to tassel, anthesis silking interval, plant height and ear height \\ \hline Maize\_3 & 1000 & 4085 & 11 & 4 & Days to tassel, anthesis silking interval, plant height and ear height \\ \hline Maize\_4 & 999 & 4085 & 11 & 4 & Days to tassel, anthesis silking interval, plant height and ear height \\ \hline Soybean\_1 & 1044 & 1810 & 8 & 6 & Plant height, R8, planting, maturity, lodging and grain yield \\ \hline Soybean\_2 & 691 & 1808 & 8 & 6 & Plant height, R8, planting, maturity, lodging and grain yield \\ \hline \hline \end{tabular}
\end{table}
Table 1: Dataset descriptions. Gen denotes genotypes; Env denotes environments.  
where \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\) are rows (vectors) of lines \(i\) and \(j\) with their corresponding marker information; each vector is of length \(p\).  
\[\widetilde{y}=\lambda y_{i}+(1-\lambda)y_{j}. \tag{3}\]  
where \(y_{i}\) and \(y_{j}\) are scalars of the phenotypic (BLUEs) response variable of lines \(i\) and \(j\). \((\mathbf{x}_{i},y_{i})\) and \((\mathbf{x}_{j},y_{j})\) represent two randomly selected lines from our training dataset, with \(\lambda\in[0,1]\) denoting a mixing coefficient. In this research, we used \(\lambda=0.5\). Essentially, mixup enriches the training distribution by incorporating the inherent principle that linear interpolations of feature vectors should correspond to linear interpolations of each one's target values. Notably, the implementation of mixup is highly efficient, requiring only a minimal amount of code and incurring in minimal computational overhead. It is essential to note that in our methodology, synthetic data were exclusively generated from the top 20% of lines in the training set, and models using augmented data were trained using solely the top 20% of lines in the training set, along with the resultant synthetic data generated from the top 20% of those lines. By the top 20% of lines, we mean those lines in the training set with phenotypic values (response variable) larger than the quantile 80% of the response variable. When training the GBLUP model specified in Equation (1) using augmented data, the results were labeled as \"A = Augmented,\" whereas using the original training set for training was denoted as \"C = Conventional.\" The \(\mathbf{G}\) required in the GBLUP model (Equation (1)) was computed with the augmented inputs for the augmented approach and with the original markers (inputs) for the conventional approach.  
It is important to point out that this data augmentation method (mixup) has had a significant impact on the machine learning community. By providing a simple yet effective data augmentation technique, mixup has improved model generalization, robustness, and performance across various applications and domains. Its adaptability and effectiveness have led to widespread adoption and numerous extensions, cementing its place as a fundamental tool in the machine learning practitioner's toolkit [20; 21; 22; 23]. Due to its simplicity, the mixup method (see Equations (2) and (3)) can be applied with conventional machine learning methods and even deep learning methods with complex architectures.