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Table 1 Coefficients describing the genotypes’ distributions moments when using the relation \( {\text{E}}\left[ {X_{im}^{{d_{i} }} X_{jm}^{{d_{j} }} \cdots X_{Km}^{{d_{K} }} } \right] = p_{m} \left( {1 - p_{m} } \right)\alpha_{ij \cdots K}^{{d_{i} d_{j} \cdots d_{K} }} - \left[ {p_{m} \left( {1 - p_{m} } \right)} \right]^{2} \gamma_{ij \cdots K}^{{d_{i} d_{j} \cdots d_{K} }} \) from Additional file 1

From: Approximated prediction of genomic selection accuracy when reference and candidate populations are related

E[X im ] \( \alpha_{i}^{1} = 0 \)
\( \gamma_{i}^{1} = 0 \)
E[X 2 im ] \( \alpha_{i}^{2} = 2 \)
\( \gamma_{i}^{2} = 0 \)
E[X 4 im ] \( \alpha_{i}^{4} = 2 \)
\( \gamma_{i}^{4} = 0 \)
E[X im X jm ] \( \alpha_{ij}^{11} = 4\delta_{ 1} + 2\left[ {\delta_{ 2} + \delta_{ 3} + \delta_{ 4} + \delta_{ 5} + \delta_{ 9} + \delta_{ 1 2} } \right] + \delta_{ 10} + \delta_{ 1 1} + \delta_{ 1 3} + \delta_{ 1 4} = 4a_{ij} \)
\( \gamma_{ij}^{ 1 1} = 0 \)
E[X im X 3 jm ] \( \alpha_{ij}^{13} = 16\delta_{1} + 2\left( {\delta_{2} + \delta_{3} ) + 8(\delta_{4} + \delta_{5} } \right) + 2\left( {\delta_{9} + \delta_{12} } \right) + \delta_{10} + \delta_{11} + \delta_{13} + \delta_{14} \)
\( \gamma_{ij}^{13} = 2 4\delta_{ 1} + 1 2\left( {\delta_{ 4} + \delta_{ 5} } \right) \)
E[X 2 im X 2 jm ] \( \alpha_{ij}^{22} = 1 6\delta_{ 1} + 4\left( {\delta_{ 2} + \delta_{ 3} + \delta_{ 4} + \delta_{ 5} } \right) + 2\left( {\delta_{ 9} + \delta_{ 1 2} } \right) + \delta_{ 10} + \delta_{ 1 1} + \delta_{ 1 3} + \delta_{ 1 4} \)
\( \gamma_{ij}^{22} = 4 8\delta_{ 1} + 8\left( {\delta_{ 2} + \delta_{ 3} + \delta_{ 4} + \delta_{ 5} } \right) - 4\delta_{ 1 5} - 1 6\delta_{ 6} - 8\left( {\delta_{ 7} + \delta_{ 8} } \right) \)
  1. δ s are the 15 classical identity states probabilities between two individuals [3335]
  2. Coefficients of expectations \( {\text{E}}\left[ {X_{i} X_{j} X_{k}^{2} } \right] \) and E[X i X j X k X l ] involve IBD status between three or four different individuals and are explained in Additional file 1