Property of Equation Systems in Proportionally Restricted BLUP Using Canonical Transformation

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  • TAKAHASHI Tsuyoshi
    Graduate School of Science and Technology, Niigata University Hyogo Branch, Wagyu Cattle Registry Association
  • IWAISAKI Hiroaki
    Faculty of Agriculture, Niigata University
  • ITOH Yohji
    Office of Biostatistics, Otsuka Parmaceutical Co. Ltd.

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  • 正準変換を利用した比例制限付きBLUPにおける連立方程式系の特徴
  • Property of Equation Systems in Proport

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Abstract

In this paper, we describe how the procedure using canonical transformation for the restricted best linear unbiased prediction with the so-called zero restrictions considered (RBLUP CT) can be applied to the proportionally restricted best linear unbiased prediction of breeding values. Then we examine the property of simultaneous equation systems required in the presented procedure (PRBLUPCT) by investigating the ranks and the numbers of positive eigenvalues of the key matrices. Denoting the numbers of traits in the multi-trait animal model and the restricted traits by q and r, respectively, it is shown that while in the RBLUP CT with zero restrictions the number of the independent equation systems is q-r, that in the PRBLUP CT becomes q-r+1. That is, as r increases, the number of the independent equation systems to be solved decreases, and in the case of all traits restricted proportionally, for instance, the rank and the number of positive eigenvalues of the key matrix become unity and it is needed to solve only one system of the usual BLUP equations for the single transformed variable.

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