eigen_indices: Linear Phenotypic Eigen Selection Index Methods (Chapter 7)

In selection.index: Analysis of Selection Index in Plant Breeding

Linear Phenotypic Eigen Selection Index Methods (Chapter 7)

Description

Implements the Linear Phenotypic Eigen Selection Index methods from Chapter 7. These methods resolve index coefficients by maximizing the accuracy squared (rho_HI^2) through an eigenvalue problem rather than requiring pre-specified economic weights.

Methods included: - ESIM : Linear Phenotypic Eigen Selection Index (Section 7.1) - RESIM : Linear Phenotypic Restricted Eigen Selection Index (Section 7.2) - PPG-ESIM: Predetermined Proportional Gain Eigen Selection Index (Section 7.3)

All implementations use C++ primitives (math_primitives.cpp) for quadratic forms and symmetric solves, while eigendecompositions use R's eigen() for correctness and compatibility with the existing package architecture.

Mathematical Foundation

Unlike classical LPSI which requires economic weights w, the ESIM family resolves the index vector b_E by maximizing the squared accuracy:

\rho_{HI}^2 = \frac{b'Cb}{b'Pb}

leading to the generalized eigenproblem (\mathbf{P}^{-1}\mathbf{C} - \lambda^2 I)b = 0.

The largest eigenvalue lambda_E^2 equals the maximum achievable index heritability, and the corresponding eigenvector b_E contains the optimal index coefficients.

References

Ceron-Rojas, J. J., & Crossa, J. (2018). Linear Selection Indices in Modern Plant Breeding. Springer International Publishing. Chapter 7.

Ceron-Rojas, J. J., Crossa, J., Sahagun-Castellanos, J., Castillo-Gonzalez, F., & Santacruz-Varela, A. (2006). A selection index method based on eigen analysis. Crop Science, 46(4), 1711-1721.

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