Algorithm 1.

Cross-validation procedure for the DDR CCA analysis

1. Step 1: Split the original SM and BM into five roughly equal-sized groups without breaking the families, using four as the training set (Xin, Yin) and the other as the test set (Xout and Yout).

2. Step 2: Apply DDR to the training set and use the rotated principal loadings from the training set to construct the rotated principal components for the test set (${\widetilde{RC}}_X^{out}$ and ${\widetilde{RC}}_Y^{out}$ in Figure 3).

3. Step 3: Feed both DDR-reduced training set from SM and BM ($R{C}_X^{in}$ and $R{C}_Y^{in}$ in Figure 3) to CCA to obtain canonical variables Pin and Qin, canonical weights Ain and Bin and canonical correlations Rin.

4. Step 4: Construct canonical variables for the test set using canonical weights from the training set using Equation (11) $$\begin{array}{l}{\tilde{P}}^{out}=X^{out}{A}^{in},\\ {\tilde{Q}}^{out}=Y^{out}{B}^{in}.\end{array}$$ (11)

Step 5: Calculate correlations between ${\tilde{P}}^{in}$ and ${\tilde{Q}}^{in}$ as the cross-validated canonical correlations ${\tilde{R}}^{in}$.

5. Step 6: Apply permutation testing to test the number of significant canonical variables for Pin and Qin, and ${\tilde{P}}^{in}$ and ${\tilde{Q}}^{in}$ with 10,000 permutations.

6. Step 7: Compute the assessment measurements such as VE and canonical loadings.

7. Step 8: Go to Step 1 to start the next fold.