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--- readme.blupf90new [2018/08/21 20:33] – shogo
+++ readme.blupf90new [2022/05/03 23:25] – dani
@@ Line 11: / Line 11: @@
 ===== Options =====
 <file>
-OPTION method VCE (default BLUP)
+OPTION method VCE (default BLUP with blupf90 options)
 </file>
-Run airemlf90 (default running blupf90)
+Run airemlf90 for variance component estimation (default running blupf90)
 <file>
 OPTION conv_crit 1d-12
@@ Line 133: / Line 133: @@
 <file>OPTION out_se_covar_function</file>
 Indicate to store in file samples of (co)variances function for postprocessing (histogram, etc.)
+==== Omit A-inverse ====
+    OPTION omit_ainv
+This option prohibits the program from creating $\mathbf{A}^{-1}$.
+It is especially useful for GBLUP.
+For example, if you would like to perform the exact GBLUP, you can put the following options to your parameter file.
+    OPTION omit_ainv
+    OPTION TauOmega 1.0 0.0
+    OPTION AlphaBeta 0.95 0.05
+With the above options, the program doesn't create $\mathbf{A}^{-1}$ but calculates $\tau\mathbf{G}^{-1}-\omega\mathbf{A}_{22}^{-1}$. When the omega ($\omega$) is zero, only $\mathbf{G}^{-1}$ will be included in the equations. $\mathbf{G}$ is blended with $\mathbf{A}_{22}$ as $\alpha\mathbf{G}+\beta\mathbf{A}_{22}$ before the inversion ($\alpha=0.95$ and $\beta=0.05$ in this case).
+=== Details ===
+Assuming a single-trait ssGBLUP, the mixed model equations are as follows.
+\(
+\left[
+\begin{array}{ll}
+\mathbf{X}'\mathbf{X} & \mathbf{X}'\mathbf{Z}\\
+\mathbf{Z}'\mathbf{X} & \mathbf{Z}'\mathbf{Z} + \lambda\mathbf{H}^{-1}
+\end{array}
+\right]
+\left[
+\begin{array}{c}
+\mathbf{\hat{b}}\\
+\mathbf{\hat{u}}
+\end{array}
+\right]
+=
+\left[
+\begin{array}{c}
+\mathbf{X}'\mathbf{y}\\
+\mathbf{Z}'\mathbf{y}
+\end{array}
+\right]
+\)
+where $\mathbf{H}$ is a matrix combining additive genetic relationship matrices and a genomic relationship matrix.
+\(
+\mathbf{H}^{-1}
+=
+\mathbf{A}^{-1}
++
+\left[
+\begin{array}{cc}
+\mathbf{0} & \mathbf{0} \\
+\mathbf{0} & \tau\mathbf{G}^{-1}-\omega\mathbf{A}_{22}^{-1}
+\end{array}
+\right]
+\)
+If we omit $\mathbf{A}^{-1}$ and $\mathbf{A}_{22}^{-1}$, the equations are equivalent to GBLUP.
+GBLUP by BLUPF90 was not so easy because the program creates $\mathbf{A}^{-1}$ by default and there was no way to avoid this behavior.
+The new option removes $\mathbf{A}^{-1}$ from the equations so GBLUP will be easily performed.
 ===== Tricks =====