Table 1 List of frequently used symbols
From: Selection index theory for populations under directional and stabilizing selection
Symbols | Meaning |
|---|---|
Counts | |
O | Number of breeding objectives |
K | Number of traits |
Q | Number of known biallelic QTL |
N | Number of selection candidates |
Information on selection candidates | |
\({\bf{y}}_{i}\) | K-vector with trait values of animal i |
\({\bf{TBV}}_i\) | K-vector with true breeding values of animal i |
\({\bf{EBV}}_i\) | K-vector with estimated breeding values of animal i |
\({\text{MV}}_{ik}\) | True Mendelian sampling variance of the haplotypes transmitted by animal i for trait k |
\(F_i\) | Inbreeding coefficient of animal i |
\(\overline{f}_i\) | Average kinship of animal i with animals of the opposite sex |
\(X_{iq}\) | Allele content of animal i for QTL q |
\(H_{iq}\) | Indicator for heterozygosity of animal i at QTL q |
Parameters related to the total merit | |
\(\text{TM}({\bf{y}}_{i})\) | Total merit of animal i with phenotype vector \({\bf{y}}_{i}\) |
\(\phi (\varvec{\upxi })\) | Profit of a breed whose phenotypic distribution is defined by \(\varvec{\upxi }\) |
\(\phi _{n}(\varvec{\uppi})\) | Expected profit of the breed after n generations of selection under breeding policy \(\varvec{\uppi}\) |
\({\bf{Opt}}_o\) | K-vector with optimum trait values for breeding objective o |
\(\omega _{ok}\) | Weight of trait k for breeding objective o |
Parameters related to index selection | |
\(\varvec{\uppi}\) | Breeding policy that provides the selection index for a given state \({\varvec{{\uptheta}}}\) of the breeding program |
\(\bf{c}\) | Vector with genetic contributions of the selection candidates. |
\(f(\varvec{\uppi})=\tilde{f}_{\varvec{\uppi}}(\bf{c}_{\varvec{\uppi}})\) | Objective function. The function weights the expected profit in future generations. |
\(T_i\) | Aggregate genotype of animal i |
\(I_{ i}\) | Selection index of animal i |
\(T_{ij}\) | True combining ability of male i with female j |
\(C_{ij}\) | Estimated combining ability of male i with female j |
Parameters for characterizing the population | |
\({\varvec{{\uptheta}}_n^{\varvec{\uppi}}}\) | State of the population in generation n |
\({\varvec{\upxi }}_n^{\varvec{\uppi}}\) | Parameter of the phenotypic distribution in generation n |
\(\mu _{nk}\) | Expected mean of trait k in generation n |
\(\sigma _{\text{P}nk}^2\) | Expected phenotypic variance of trait k in generation n |