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Canonical transformation

Model

Under some conditions in a multiple-trait model, we may reduce the computational cost of solving the mixed model equations by applying a dimension-reduction method. A canonical transformation is one such approach.

The canonical transformation is a numerical technique for converting correlated variables into uncorrelated variables on an underlying (canonical) scale. With this transformation, an \(N\)-trait model becomes \(N\) independent single-trait models. It is generally cheaper to repeat the single-trait analysis \(N\) times (which can be done in parallel) than to perform a single \(N\)-trait model analysis. You can back-solve the solutions from the canonical scale to the original scale.

There is a set of strict conditions under which the original canonical transformation is applicable. These conditions include: 1) no missing observations; 2) the same model for all traits; and 3) one random effect in addition to the random residual effect. Subsequently, animal-breeding researchers relaxed all of these conditions (with several practical and theoretical restrictions). Furthermore, the canonical transformation was extended to variance component estimation, although some restrictions still apply.

Let \(\mathbf{y}_i\) be a vector of phenotypes for animal \(i\) across \(N\) traits. Assume a simple animal model with two random effects: an additive genetic effect and a residual effect. Without inbreeding, the variance of \(\mathbf{y}_{i}\) is \(\mathbf{G}_0+\mathbf{R}_0\), where \(\mathbf{G}_0\) is the additive genetic covariance matrix and \(\mathbf{R}_0\) is the residual covariance matrix. The canonical transformation determines a matrix \(\mathbf{Q}\) that satisfies the following conditions: \[ \mathbf{Q}\mathbf{R}_0\mathbf{Q}'=\mathbf{I}\quad\text{and}\quad\mathbf{Q}\mathbf{G}_0\mathbf{Q}'=\mathbf{D} \] where \(\mathbf{D}\) is a diagonal matrix. The matrix \(\mathbf{Q}\) is available through the eigenvalue decomposition, shown in the textbook.

Then, we define the transformed observation vector as \(\mathbf{y}_{i}^{*}=\mathbf{Qy}_{i}\). In this way, \(\mathrm{var}(\mathbf{y}_{i}^{*})=\mathbf{D}+\mathbf{I}\), implying that the transformed variables have no correlations among traits. We can use \(\mathbf{y}_{i}^{*}\) as the transformed observations, calculate \(\mathbf{\hat{u}}_{i}^{*}\) from independent single-trait analyses, and then back-solve for \(\mathbf{\hat{u}}_{i}=\mathbf{Q}^{-1}\mathbf{\hat{u}}_{i}^{*}\). See the textbook for details.

BLUPF90+ does not support the canonical transformation. Therefore, to perform this transformation, the user must prepare the transformed data and back-solve the results manually. The textbook describes the procedure.

Files

You should calculate \(\mathbf{Q}\) from \(\mathbf{G}_0\) and \(\mathbf{R}_0\) manually before the analysis. The textbook gives \(\mathbf{Q}\) as \[ \mathbf{Q} = \left[ \begin{array}{rr} 0.1659 & -0.0792 \\ 0.0168 & 0.1755 \end{array} \right] \]

The data set with the transformed observations is shown in the textbook on p. 125 and is available in the GitHub repository (mr71.dat).

1. Animal ID (Calf)
2. Sex (1 = male and 2 = female)
3. Sire ID
4. Dam ID
5. Yearling weight (kg): original scale
6. Fat yield (kg): original scale
7. Yearling weight (kg): transformed scale
8. Fat yield (kg): transformed scale

The pedigree (mr71.ped) is from the textbook.

The parameter file should be the following.

DATAFILE
mr71.dat
NUMBER_OF_TRAITS
2
NUMBER_OF_EFFECTS
2
OBSERVATION(S)
7 8
WEIGHT(S)

EFFECTS:
2 2 2 cross
1 1 8 cross
RANDOM_RESIDUAL VALUES
1 0
0 1
RANDOM_GROUP
2
RANDOM_TYPE
add_animal
FILE
mr71.ped
(CO)VARIANCES
0.3283 0.0000
0.0000 1.3436
OPTION solv_method FSPAK

The covariance components after the transformation include \(\mathbf{I}\) for the residual covariance matrix and \(\mathbf{D}\) for the additive genetic covariance matrix. This parameter file runs two independent single-trait analyses simultaneously. Alternatively, you may run two separate single-trait analyses and combine the results afterward.

Solutions

The results on the canonical scale are identical to those in the textbook (p. 126). You must back-solve the solutions manually to obtain them on the original scale. For example, for the breeding values of animal 1, you need a solution vector \(\mathbf{\hat{u}}_{1}^{*}\) and the inverse of the transformation matrix, \(\mathbf{Q}^{-1}\), to obtain the original-scale solutions as \(\mathbf{\hat{u}}_{1} = \mathbf{Q}^{-1}\mathbf{\hat{u}}_{1}^{*}\).

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