# Coordinate Descent for Variance-Component Models
## 4 Numerical Results
### Genetic Data  
We now present simulation results when \(\mathbf{Z}_{i}\) are constructed from the [[https://openmendel.github.io/SnpArrays.jl/latest/#Example-data](https://openmendel.github.io/SnpArrays.jl/latest/#Example-data)]([https://openmendel.github.io/SnpArrays.jl/latest/#Example-data](https://openmendel.github.io/SnpArrays.jl/latest/#Example-data)) (accessed on 1 July 2022) mouse single nucleotide polymorphism (SNP) array data set available from the Open Mendel project [21]. The dataset consists of \(\mathbf{Z}\), an \(n\times c\) matrix consisting of \(c\) genetic variants for \(n\) individual mice. For this experiment, \(c\) = 10,200 and \(n=500\). We artificially generate \(m\) different genetic regions by partitioning the columns of \(\mathbf{Z}\) into \(\mathbf{Z}_{i=1,\ldots,m}\) gene matrices, where \(\mathbf{Z}_{i}\in\mathbb{R}^{n\times r}\). Then, we can compose our fixed matrices \(\mathbf{V}_{1},\ldots,\mathbf{V}_{m}\) as,  
\[\mathbf{V}_{i}=\frac{\mathbf{Z}_{i}\mathbf{Z}_{i}^{\top}}{\left\|\sum_{j=1}^{ m}\mathbf{Z}_{j}\mathbf{Z}_{j}^{\top}\right\|_{F}}.\]  
We simulate \(\boldsymbol{\gamma}\) and \(\boldsymbol{y}\) as we did in Section 4.1. In this case, \(\boldsymbol{y}\) mimics a vector of quantitative trait measurements of the \(n\) mice. This data set is well-suited for testing our method when \(m\) is large (\(m>n\)). In these cases, we observe that, when initialized at the same point, the MM and PXI-CD method converge to different solutions, i.e., they may converge to different stationary points. Therefore, we run all algorithms until the relative change \(\left[L(\boldsymbol{\gamma}^{(t+1)})-L(\boldsymbol{\gamma}^{(t+1)})\right]/ \left|\;\left|\;L(\boldsymbol{\gamma}^{(t)})\;\right|+1\right|\) is less than \(10^{-10}\). Since \(r<n\), we implement PXI-CD utilizing the Woodbury identity. Each simulation scenario is replicated 10 times. The mean running time and mean iteration number are reported with the standard error of the mean running time and mean iteration provided in parentheses. The results of the genetic study simulation are provided in Table 7. We observe that PXI-CD outperforms the MM algorithm for all values of \(m\) and \(r\) for this data set in both the number of iterations and running time until convergence. When \(m>n\), we observe that \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Method** & \(m\) & **Iterations** & **Time (s)** & **Objective** \\ \hline PX-CD & 25 & 75.20 (2.88) & 25.74 (1.46) & \(-\)2603.51 \\ PXI-CD & & 152.00 (6.98) & 95.59 (5.12) & \(-\)2603.51 \\ CD & & 38.50 (1.60) & 172.03 (11.70) & \(-\)2603.51 \\ MM & & 3616.90 (513.57) & 254.87 (36.27) & \(-\)2603.51 \\ PX-CD & 50 & 143.70 (6.93) & 108.55 (5.89) & \(-\)2668.99 \\ PXI-CD & & 177.50 (28.72) & 249.73 (39.38) & \(-\)2668.99 \\ CD & & 79.80 (4.59) & 668.65 (40.71) & \(-\)2668.99 \\ MM & & 11,306.30 (1824.79) & 1697.89 (275.36) & \(-\)2668.99 \\ PX-CD & 100 & 304.00 (12.39) & 412.54 (19.01) & \(-\)2731.09 \\ PXI-CD & & 87.40 (2.54) & 230.54 (7.27) & \(-\)2731.09 \\ CD & & 181.80 (7.63) & 2697.20 (150.64) & \(-\)2731.09 \\ MM & & 11,796.10 (1732.0) & 3358.14 (521.92) & \(-\)2731.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The convergence results when \(\mathbf{V}_{i}\) are full-rank, \(n=1000\), and \(\gamma_{0}=10\).