Linear Phenotypic Eigen Selection Index Methods

Based on the canonical correlation, on the singular value decomposition (SVD), and on the linear phenotypic selection indices theory, we describe the eigen selection index method (ESIM), the restricted ESIM (RESIM), and the predetermined proportional gain ESIM (PPG-ESIM), which use only phenotypic information to predict the net genetic merit. The ESIM is an unrestricted linear selection index, but the RESIM and PPG-ESIM are linear selection indices that allow null and predetermined restrictions respectively to be imposed on the expected genetic gains of some traits, whereas the rest remain without any restrictions. The aims of the three indices are to predict the unobservable net genetic merit values of the candidates for selection, maximize the selection response, and the accuracy, and provide the breeder with an objective rule for evaluating and selecting several traits simultaneously. Their main characteristics are: they do not require the economic weights to be known, the ﬁ rst multi-trait heritability eigenvector is used as its vector of coef ﬁ cients; and because of the properties associated with eigen analysis, it is possible to use the theory of similar matrices to change the direction and proportion of the expected genetic gain values without affecting the accuracy. We describe the foregoing three indices and validate their theoretical results using real and simulated data.


The Linear Phenotypic Eigen Selection Index Method
The conditions described in Chap. 2 for the linear phenotypic selection index (LPSI) are necessary and sufficient for constructing the linear phenotypic eigen selection index method (ESIM). The ESIM index can be written as is the unknown index vector of coefficients, t is the number of traits, and y 0 ¼ y 1 y 2 Á Á Á y t ½ is a known vector of trait phenotypic values. The objectives of ESIM are: 1. To predict the net genetic merit H ¼ w 0 g, where g 0 ¼ g 1 g 2 . . . g t ½ is the unknown vector of true breeding values for an individual and w 0 ¼ w 1 w 2 . . . w t ½ is a vector of unknown economic weights.

The ESIM Parameters
The theoretical ESIM selection response can be written as where k I is the standardized selection differential (or selection intensity), is the correlation, and w 0 Cb ¼ σ HI the covariance between H and I respectively, σ I ¼ ffiffiffiffiffiffiffiffiffi ffi b 0 Pb p is the standard deviation of I, C is the covariance matrix of the true breeding values (g), and P is the covariance matrix of the trait phenotypic values (y).
In the ESIM, it is assumed that k I and σ H are fixed, and that C and P are known; thus, to maximize Eq. (7.1), it is necessary to maximize ρ 2 HI ¼ w 0 Cb ð Þ 2 w 0 Cw ð Þ b 0 Pb ð Þ with respect to vectors b and w under the restrictions σ 2 H ¼ w 0 Cw, σ 2 I ¼ b 0 Pb, and 0 < σ 2 H , σ 2 I <1, where σ 2 H ¼ w 0 Cw is the variance of H ¼ w 0 g and σ 2 I ¼ b 0 Pb is the variance of I ¼ b 0 y. That is, it is necessary to maximize the function with respect to b, w, μ, and ϕ, where μ and ϕ are Lagrange multipliers. The derivative results of Eq. (7.2) with respect to b, w, μ, and ϕ are: respectively, where Eq. (7.5) denotes the restrictions imposed for maximizing ρ 2 HI . It can be shown that w 0 Cb ¼ ffiffiffiffiffiffiffi μσ 2 Canonical correlation theory describes the associations between two sets of variables (Hotelling 1935(Hotelling , 1936 and searches for linear combinations, called canonical variables, of each of two sets of variables having maximal correlation. The vector of coefficient of these linear combinations is called the canonical vector and the correlations between the canonical variables is called the canonical correlation (Wilms and Croux 2016).
To see how the ESIM and the canonical correlation theory are related, note that vectors y and g (Eq. 7.1) can be ordered in a new vector x as x 0 ¼ y 0 g 0 ½ , whence the covariance matrix of x is P C C C ! . One measure of the association between the jth linear combination of y(I E ¼ b 0 E j y) and the jth linear combination of g(H E ¼ w 0 E j g) is the jth canonical correlation (λ j ) value obtained from equation P À1 C À λ 2 j I b Ej ¼ 0, where b Ej is the jth canonical vector ( j ¼ 1, 2Á Á Á, t) of matrix P À1 C, and w E j ¼ C À1 Pb E j . Thus, in the canonical correlation context, In the ESIM, the first eigenvector (b E 1 ) of matrix P À1 C should be used on I E ¼ b 0 E 1 y; the first eigenvalue (λ 2 1 ) and b E 1 of P À1 C should be used on the ESIM selection response and on the ESIM expected genetic gain per trait, because, in this case, the ESIM has maximum accuracy compared with other indices, such as the LPSI. The latter results in this subsection imply that the sampling statistical properties associated with the canonical correlation theory are also valid for the ESIM.

Estimated ESIM Parameters and Their Sampling Properties
The estimated covariance matrix of the true breeding values (C) and that of the trait phenotypic values (P) are denoted as b C and b P respectively; they can be obtained by restricted maximum likelihood using Eqs. (2.22) to (2.24) described in Chap. 2. With matrices b C and b P, we constructed matrix b T ¼ b P À1 b C and equation where t is the number of traits in the ESIM index. Note that b λ 2 Ej is positive only if b P is positive definite (all eigenvalues positive) and b C is positive semidefinite (no negative eigenvalues); in addition, as b P À1 b C is an asymmetric matrix, the values of b b Ej and b λ 2 Ej should be obtained using the singular value decomposition (SVD) theory (Anderson 2003).
Matrix b T is square and asymmetric of order t Â t and rank q minimum ( p, c), where p and c denote the rank of b P À1 and b C respectively; the rank of b T is equal to c only if b C is square and nonsingular. Thus, matrix b T has a maximum of q eigenvalues different from zero (Rao 2002).
> 0 of L 1/2 are uniquely determined, and they are called the singular values of b T. The columns of V 1 are orthonormal vectors called left singular vectors of b T, and the columns of V 2 are called right singular vectors (Watkins 2002).
Estimators b b E 1 and b λ 2 E 1 of the first eigenvector b E 1 and the first eigenvalue λ 2 E 1 respectively are the first column of matrix V 1 and the first diagonal element of matrix L 1/2 . Thus, because b T b T 0 is a symmetric matrix, the maximum likelihood estimators b λ 2 E 1 and b b E 1 in the ESIM context can be obtained from In the asymptotic context, b λ 2 E 1 and b b E 1 are consistent and unbiased estimators (Anderson 2003).
The latter results allow the ESIM index ( E 1 y to be estimated. The estimator of the maximized ESIM selection response and expected genetic gain respectively, whereas the estimator of the maximized ESIM accuracy is b λ E 1 , which should be similar to the estimator of the square root of the maximized ESIM heritability.
In the asymptotic context, the estimator of and, for i 6 ¼ j, the covariance between b b Ei and b b Ej can be written as where n is the number of individuals or genotypes (Anderson 1999). The variance of b b Ej and the covariance between b b Ei and b b Ej depend not only on n, but also on eigenvalues λ 2 Ei and λ 2 Ej . Suppose that λ 2 Ej > λ 2 Ei ; then, when λ 2 Ej is very close to 1, is very close to 0. By the result of Eq. (7.24), the variance of the first eigenvector can be written as Var In the asymptotic context, the jth estimator ( b λ Ej ) of the canonical correlations has normal distribution with expectation E À b λ Ej Á % λ Ej and variance whereas the jth estimator of the square of the canonical correlations b λ 2 Ej has normal distribution with expectation E À b λ 2 Ej Á % λ 2 Ej and variance In addition, for i 6 ¼ j, the correlation between b λ 2 Ej and b λ 2 Ei is zero, i.e., Corr and Brenner 1999;Muirhead 2005). Equation (7.26) implies that under the restrictions σ 2 H ¼ 1 and σ 2 I ¼ 1, the expectation and variance of b respectively. However, obtaining the expectation and variance of b q is more difficult, because in both equations there are two estimators: b σ H and b λ 1 in the first one, and b P and b b E1 in the second one.

Numerical Examples
We compare ESIM efficiency versus LPSI efficiency using a real data set from commercial egg poultry lines obtained from Akbar et al. (1984). The estimated phenotypic ( b P ) and genetic ( b C ) covariance matrices among the rate of lay (RL, number of eggs), age at sexual maturity (SM, days) and egg weight (EW, kg), were , whereas the estimated selection response, expected genetic gain per trait, accuracy, and heritability of the LPSI were b R S ¼ 1:755 it can be shown that the normalization process only affects the estimated LPSI selection response because in that case, b appears in the numerator and denominator of both estimated parameters.
In the ESIM, the sign and proportion of the expected genetic gain values for traits RL, SM, and EW should be in accordance with the breeder's interest. For example, if the breeder's interest is that the expected genetic gain per trait for RL should be positive and negative for SM, the sign and proportion of the values of the first eigenvector should be modified using a linear combination of the estimated first to achieve expected genetic gain per trait values in RL and SM according to the breeder's interest.
The information needed to obtain the estimated ESIM parameters are matrices We need to find the eigenvalues and eigenvectors of equation Matrix V 1 is equal to V 1 ¼ According to the theory of similar matrices (Harville 1997), the estimated maximized ESIM accuracy, b λ E 1 ¼ 0:6782, should not be affected by matrix F. We can compare ESIM efficiency versus LPSI efficiency to predict the net genetic merit using the ratio of the estimated ESIM accuracy b λ E 1 ¼ 0:6782 to Eq. 5.17). According to the latter result, the ESIM is a better predictor of the net genetic merit and its efficiency is 87.3% higher than that of the LPSI for this data set. Now, we compare ESIM efficiency versus LPSI efficiency using the data set described in Sect. 2.8.1 of Chap. 2. From this data set, we ran five phenotypic selection cycles, each with four traits (T 1 , T 2 , T 3 , and T 4 ), 500 genotypes, and four replicates for each genotype. The economic weights for T 1 , T 2 , T 3 , and T 4 were 1, À1, 1, and 1 respectively. In this case, matrix F is an identity matrix of size 4 Â 4 for all five selection cycles. Table 7.1 presents the estimated LPSI, the restricted LPSI (RLPSI), and the predetermined proportional gain LPSI (PPG-LPSI) selection response (the latter two for one, two, and three restrictions) for five simulated selection cycles when their vectors of coefficients are normalized. Table 7.1 also presents the estimated ESIM, the RESIM and the PPG-ESIM selection response for one, two, and three restrictions for five simulated selection cycles. The selection intensity was 10% (k I ¼ 1.755) for all five selection cycles. In this subsection, we compare only LPSI results versus ESIM results. The estimated LPSI selection response when the vector of coefficients was not normalized was described in Chap. 2 (Table 2.4). The averages of the estimated LPSI and ESIM selection responses were 4.70 and 6.31 respectively. Table 7.2 presents the estimated ESIM expected genetic gain per trait, accuracy (b ρ E ), and the values b p E ¼ 100 , expressed as percentages. Table 7.2 also presents the accuracy of the PPG-ESIM and the estimated ratio (b p PE ) of the estimated PPG-ESIM accuracy to the estimated PPG-LPSI accuracy, expressed as percentages, for one, two, and three predetermined restrictions for five simulated selection cycles. In this subsection, we use only the estimated ESIM expected genetic gain per trait and b p E ¼ 100 À b λ E À 1 Á to compare ESIM efficiency versus LPSI efficiency. The estimated LPSI expected genetic gains per trait were presented in Chap. 2, Table 2.4. According to the results shown in Table 2.4, the averages of the estimated LPSI expected genetic gain per trait T1, T2, T3, and T4 for five simulated selection cycles were 7.26, À3.52, 2.78, and 1.58, whereas according to the results of Table 7.2, the averages of the estimated ESIM expected genetic gains per trait were 5.67, À2.67, 1.81, and 2.9 respectively. This means that the estimated LPSI expected genetic gain for traits T1, T2, and T3 was higher than the estimated ESIM expected genetic gain for those traits.
The average of the b p E ¼ 100 À b λ E À 1 Á values was 9.76 for all five selection cycles (Table 7.2). The latter result is not in accordance with the LPSI and ESIM expected genetic gain per trait; however, note that the b p E values are associated with the estimated LPSI and ESIM selection responses (Table 7.1), not with the expected ESIM and LPSI selection responses respectively. Thus, the b p E values indicate that the efficiency of the ESIM and that of the LPSI were very similar because the former was only 9.76% higher than the latter for this data set.
is true only when the denominators of both estimated correlations are the same, as in the linear selection indices described in Chaps. 2-6.  Akbar et al. (1984) indicates an approximation.   Estimated PPG-ESIM accuracy (b ρ P ) and estimated ratio (b ρ P ) of the b ρ P to the estimated accuracy of the PPG-LPSI (data not presented), expressed in percentages (%), for one, two, and three predetermined restrictions for five simulated selection cycles

The Linear Phenotypic Restricted Eigen Selection Index Method
Similar to the RLPSI (see Chap. 2), the objective of the RESIM is to fix r of t (r < t) traits by predicting only the genetic gains of (t À r) of them. Let H ¼ w 0 g be the net genetic merit and I ¼ b 0 y the ESIM index. In Chap. 2, we showed that Cov(I, g) ¼ Cb is the covariance between the breeding value vector (g) and I ¼ b 0 y. Thus, to fix r of t traits, we need r covariances between the linear combinations of g (U 0 g) and where U 0 is a matrix with 1s and 0s (1 indicates that the trait is restricted and 0 that the trait has no restrictions). In the RESIM, it is possible to solve this problem by maximizing with respect to vectors b and w under the restrictions Pb is the variance of I ¼ b 0 y. Also, the RESIM problem can be solved by maximizing b 0 Cb ffiffiffiffiffiffiffi b 0 Pb p (Eq. 7.12) with respect to vectors b only under the restrictions U 0 Cb ¼ 0 and b 0 b ¼ 1, as we did to obtain Eq. (7.13). Both approaches give the same result, but it is easier to work with the second approach than with the first one.

The RESIM Parameters
To obtain the RESIM vector of coefficients that maximizes the RESIM selection response and the expected genetic gain per trait, we need to maximize the function ½ is a vector of Lagrange multipliers. The derivatives of Eq. (7.28a) with respect to b and v 0 can be written as respectively, where Eq. (7.29) denotes the restriction imposed for maximizing Eq. (7.28a). Using algebraic methods on Eq. (7.28b) similar to those used to obtain Eqs. (7.10) and (7.13), we get is the maximized RESIM heritability obtained under the restriction U 0 Cb ¼ 0; h 2 I R is also the square of the maximized correlation between the net genetic merit and This means that Eq. (7.30) can be written as Thus, the optimized RESIM index is I ¼ b 0 R y. The only difference between Eqs. (7.31) and (7.13) is matrix K. Equation (7.31) was obtained by Cerón-Rojas et al. (2008) by maximizing ρ 2 HI (Eq. 7.1) with respect to vectors b and w under the restriction U 0 Cb ¼ 0, b 0 b ¼ 1, w 0 Cw ¼ 1 and b 0 Pb ¼ 1 in a similar manner to the canonical correlation theory. The RESIM expected genetic gain per trait uses the first eigenvector (b R ) of matrix KP À1 C, whereas the RESIM selection response uses b R and the first eigenvalue (λ 2 R ) of matrix KP À1 C. When U 0 is a null matrix, b R ¼ b E (the vector of the ESIM coefficients); thus, the RESIM is more general than the ESIM and includes the ESIM as a particular case.
In the RESIM context, vector w can be obtained (Cerón-Rojas et al. 2008) as where λ R and b R are the square roots of the first eigenvalue (λ 2 R ) and the first eigenvector of matrix KP À1 C respectively; Ψ ¼ CU and H R ¼ w 0 R g be the net genetic merit in the RESIM context; then, because the correlation between I R ¼ b 0 R y and H R ¼ w 0 R g is not affected by scale change, λ R and λ À1 R can be considered proportional constants and then Ψv can be written as Eq. (7.30). Thus, another way of writing Eq. (7.32) is By Eq. (7.33) and the restriction b 0 Ψ ¼ 0, the covariance between can be written as indicates that the covariance between I R and H R (σ H R I R ) is equal to the variance of I R (σ 2 The maximized correlation between I R and H R (or RESIM accuracy) can be written as Hereafter, to simplify the notation, we write Eq. (7.35) as ρ R or λ R .
The maximized selection response (R R ) and expected genetic gain per trait (E R ) of the RESIM can be written as where matrix F was defined earlier, vector b R should be changed by β R in Eqs. (7.36) and (7.37), and in I R ¼ b 0 R y. Equation (7.36) can also be written as is the standard deviation of the variance of H R , and λ R ¼ ρ H R I R is the first canonical correlation between H R ¼ w 0 R g and I R ¼ b 0 R y. When σ H R ¼ 1, λ R is the covariance between H R ¼ w 0 R g and I R ¼ b 0 R y, and then Eq. (7.36) can be written as R R ¼ k I λ R . This last result was presented by Cerón-Rojas et al. (2008) in their original paper.
The ratio of the index accuracies and the VPE are also valid in the RESIM context. In Eq. (7.34) we showed that the covariance between . This means that the VPE of the RESIM can be written as Statistical properties associated with the ESIM and described in Sect. 7.1.2 are also valid for the RESIM.

Estimating the RESIM Parameters
We can estimate the RESIM parameters in a similar manner to the ESIM parameters in Sect. 7.1.4. With matrices b C and b P, we constructed matrix b R 1 y and the estimator of the maximized RESIM selection response and its expected genetic gain per trait can be denoted as b respectively, whereas the estimator of the maximized RESIM accuracy is b λ R 1 .

Numerical Examples
We compare the RLPSI results with those of the RESIM using the Akbar et al. (1984) data described in Sect. 7.1.5. We restrict the trait RL (number of eggs) in both indices. In Chap. 3, Sect. 3.1.3, we indicated how to construct matrix U 0 and, in Sect.
3.1.4 of the same chapter, we described how to obtain matrix b K ¼ Â I t À b Q Ã for one and two restrictions. Matrix b K is the same for the RLPSI and the RESIM. Thus, in this subsection we omit the steps needed to construct matrices U 0 and b K. First, we estimate the RLPSI parameters. Assume a selection intensity of 10% (k I ¼ 1.755) and a vector of economic weights w 0 ¼ 19:54 À3:56 17:01 ½ . The estimated RLPSI vector of coefficients for one restriction was b b 0 ¼ 0:29 À0:84 5:78 ½ , and the estimated selection response, expected genetic gain per trait, accuracy, and heritability of the RLPSI were b R ¼ 1:755 R j , whence we shall obtain the eigenvalues and eigenvectors that form For one null restriction, matrix b The estimated RLPSI selection response was b R ¼ 53:01 34:25 ¼ 1:55 ; thus, the estimated RESIM selection response was higher than the estimated RLPSI response. In addition, the estimated RLPSI expected genetic gain per trait was b E 0 ¼ 0 À0:71 2:96 ½ , which is the same as the estimated RESIM expected genetic gain per trait.
We can compare RESIM efficiency versus RLPSI efficiency to predict the net genetic merit using the ratio of the estimated RESIM accuracy b λ E 1 ¼ 0:5798 to the Chap. 5,Eq. 5.17). That is, the RESIM is a better predictor of the net genetic merit and its efficiency was 123% higher than the RLPSI efficiency for this data set. Now, we compare RESIM efficiency versus RLPSI efficiency using the simulated data set described in Sect. 2.8.1 of Chap. 2 for five phenotypic selection cycles, each with four traits (T 1 , T 2 , T 3 , and T 4 ), 500 genotypes, and four replicates for each genotype. The economic weights for T 1 , T 2 , T 3 , and T 4 were 1, À1, 1, and 1 respectively. For this data set, matrix F was equal to an identity matrix of size 4 Â 4 for all five selection cycles.
The first and second parts of columns 3, 4, and 5 of Table 7.1 present the estimated RLPSI and RESIM selection responses respectively for one, two, and three null restrictions for five simulated selection cycles, where the selection intensity was 10% (k I ¼ 1.755) for all five selection cycles. The averages of the estimated RLPSI selection response for each null restriction were 4.43, 4.30, and 4.92, whereas the averages of the estimated RESIM selection response were 4.54, 4.42, and 4.38 respectively. These results indicate that the estimated RLPSI selection response was greater than the estimated RESIM selection response only for three null restrictions.
The first part of Table 7.3 presents the estimated RESIM expected genetic gain per trait for one, two, and three restrictions for five simulated selection cycles. The estimated RLPSI expected genetic gains per trait for one, two, and three restrictions are given in Chap. 3 (Table 3.3). According to the results shown in Table 3.3 (Chap. 3), the averages of the estimated RLPSI expected genetic gains per trait for five simulated selection cycles were À2.52, 2.25, and 2.26 for one restriction; 2.84 and 2.65 for two restrictions; and 3.90 for three restrictions. According to the results shown in Table 7.3, the averages of the estimated RESIM expected genetic gains per trait for five simulated selection cycles were À0.43, À0.75, and 3.90 for one restriction; À0.59 and 3.89 for two restrictions; and 3.90 for three restrictions. This means that the RESIM and RLPSI were the same only for three restrictions, whereas for one and two restrictions, the average of the estimated RESIM expected genetic gains per trait was higher than that of the estimated RLPSI expected genetic gains per trait only for trait 4. Figure 7.2 presents the estimated accuracy of the RLPSI and the RESIM for one, two, and three null restrictions for five simulated selection cycles. In all five selection cycles, the estimated RESIM accuracy was greater than the RLPSI accuracy. This means that the RESIM is a better predictor of the net genetic merit than the RLPSI. Additional results associated with the frequency distribution of the estimated RESIM values are presented in Fig. 7.3. Figure 7.3a presents the frequency distribution of the estimated RESIM values with one null restriction for cycle 2, whereas Fig. 7.3b presents the frequency distribution of the estimated RESIM values with two null restrictions for cycle 5; both figures indicate that the estimated RESIM values approach normal distribution.
Finally, in Chap. 10 we present the results of comparing the ESIM with the LPSI and the RESIM with the RLPSI for many selection cycles. Such results are similar to those obtained in this chapter.  One null restriction  Two null restrictions  Three null restrictions  T1 T2  T3  T4  T1 T2 T3  T4  T1 T2 T3 T4  1 Three predetermined  restrictions  T1 T2  T3  T4  T1  T2  T3  T4  T1  T2  T3  T4  1 7 The selection intensity was 10% (k I ¼ 1.755) and the vectors of the PPG for each predetermined restriction were d 0

The Linear Phenotypic Predetermined Proportional Gain Eigen Selection Index Method
In a similar manner to the PPG-LPSI (see Chap. 3), in the PPG-ESIM the breeder pre-sets optimal levels (predetermined proportional gains) on certain traits before the selection is carried out.
be the vector of the PPGs (predetermined proportional gains) imposed by the breeder on r traits and assume that μ q is the population mean of the qth trait before selection. The objective of the PPG-ESIM is to change μ q to μ q + d q , where d q is a predetermined change in μ q (in the RESIM, d q ¼ 0, q ¼ 1, 2, Á Á Á, r, where r is the number of PPGs). That is, the PPG-ESIM attempts to make some traits change their expected genetic gain values based on a predetermined level, whereas the rest of the traits remain without restrictions.
The simplest way to solve the foregoing problem is by maximizing the PPG-ESIM heritability under the restriction r is the number of PPGs, d q (q ¼ 1, 2. . ., r) is the qth element of vector d 0 , U 0 is the RLPSI matrix of restrictions of 1s and 0s, and C is the covariance matrix of genotypic values. Matrix D 0 is a Mallard (1972) matrix of PPGs used to impose predetermined restrictions.
The Mallard (1972) matrix of predetermined restrictions can be written as M Kempthorne and Nordskog (1959) matrix of restrictions of 1s and 0s (1 indicates that the trait is restricted, i.e., d q ¼ 0, and 0 that the trait has no restrictions).
To find the PPG-ESIM vector of coefficients that maximizes the PPG-ESIM selection response and expected genetic gain per trait, we can maximize ρ 2 1, as we did to obtain the RESIM vector of coefficients. Both approaches give the same result, but we use the latter approach because it is easier to work with.

The PPG-ESIM Parameters
To obtain the PPG-ESIM vector of coefficients, we need to maximize the function ½ is a vector of Lagrange multipliers. The derivatives of Eq. (7.40) with respect to b and v 0 were: respectively, where Eq. (7.42) denotes the restriction imposed for maximizing Eq. (7.40). By using algebraic methods on Eq. (7.41) similar to those used to obtain Eq. (7.10) we get and b P are the first eigenvalue and the first eigenvector of matrix K P P À1 C respectively. Note that h 2 I P is PPG-ESIM heritability and λ P is the maximum correlation between I P ¼ b 0 P y and H ¼ w 0 g. When D 0 ¼ U 0 , b P ¼ b R (the vector of coefficients of the RESIM), and when U 0 is a null matrix, b P ¼ b E (the vector of coefficients of the ESIM). That is, the PPG-ESIM is more general than the RESIM and the ESIM and includes the latter two indices as particular cases. Matrices K P ¼ [I t À Q P ] and Q P ¼ P À1 ΨD(D 0 Ψ 0 P À1 ΨD) À1 D 0 Ψ 0 are the same as those obtained in the PPG-LPSI (see Chap. 3). Also, vector b P can be transformed as β P ¼ Fb P ; matrix F was defined earlier.
Let S P ¼ Ψ 0 P À1 Ψ; then, under the assumption When A P is a null matrix, K P P À1 C ¼ KP À1 C (matrix of the RESIM), and if U 0 is a null matrix, K P P À1 C ¼ P À1 C (matrix of the ESIM), this means that Eq. (7.44) is a mathematical equivalent form of matrix K P P À1 C and that Eq. (7.44) does not require matrix D 0 . The easiest way to obtain b P and λ P is to use matrix [I t À P À1 ΨS À1 Ψ 0 ]P À1 C + A P in Eq. (7.43) instead of matrix K P P À1 C.
In the PPG-ESIM context, vector w can be obtained as whence H ¼ w 0 g can be written as H P ¼ w 0 P g. In Eq. (7.45), λ P is the maximum correlation between I P ¼ b 0 P y and H P ¼ w 0 P g, b P is the first eigenvector of matrix K P P À1 C, v P ¼ λ À1 In a similar manner to the RESIM context, we can assume that λ P and λ À1 P are proportionality constants and it can be shown that the covariance between I P ¼ b 0 P y and H P ¼ w 0 P g (σ H P I P ) is equal to the variance of The accuracy of the PPG-ESIM can also be written as Hereafter, to simplify the notation, we write Eq. (7.46) as ρ P or λ P .
Let β P ¼ Fb P be the PPG-ESIM transformed vector of coefficients by matrix F. By Eqs. (7.1) and (7.46), the maximized selection response (R P ) and expected genetic gain per trait (E P ) of the PPG-ESIM can be written as respectively, where ffiffiffiffiffiffiffiffiffiffiffiffiffi β 0 P Pβ P q ¼ σ I P is the standard deviation of the variance of I P ¼ β 0 P y. Equations (7.47) and (7.48) do not require economic weights. When F is an identity matrix, Equation (7.47) can also be written as is the standard deviation of the variance of H P , and λ P is the canonical correlation between H P and I P ¼ β 0 P y. When σ H P ¼ 1, Eq. (7.47) can be written as R P ¼ k I λ P , where λ P is the covariance between I P ¼ b 0 P y and H ¼ w 0 P g. The prediction efficiency of the PPG-ESIM can be obtained in a similar manner to the ESIM and RESIM. The accuracy of the PPG-ESIM (Eq. 7.46) can be used to construct the ratio of index accuracies. The PPG-ESIM mean square error or the VPE can be obtained as Additional properties associated with the ESIM are also valid for the PPG-ESIM.

Estimating PPG-ESIM Parameters
The procedure used to estimate PPG-ESIM parameters is the same as that described for RESIM. Let b C and b P be the estimated matrices of C and P. In the PPG-ESIM context, we use matrix b C to obtain the estimated eigenvalues and eigenvectors of equation Pj should be obtained using SVD (singular value decomposition). According to SVD, we need to solve equation 1, 2, . . ., t). By Eq. (7.51), the estimated PPG-ESIM index ( The estimator of the maximized PPG-ESIM selection response, and its expected genetic gain per trait, can be denoted as q respectively, whereas the estimator of the maximized accuracy of the PPG-ESIM is b λ P 1 .

Numerical Examples
We compare the results of the PPG-LPSI and the PPG-ESIM using the Akbar et al. (1984) data described earlier. We restrict traits RL and SM, on both indices using the PPG vector d 0 ¼ 3 À1 ½ . In Chap. 3, Sect. 3.1.4, we indicated how to construct matrix U 0 and, in Sect. 3.2.4 of the same chapter, we described how to obtain matrix b K P for one and two restrictions. Matrix b K P is the same for the PPG-LPSI and the PPG-ESIM. Thus, we omit the steps for constructing matrices U 0 and b K P . Assume a selection intensity of 10% (k I ¼ 1.755) and that the vector of economic weights is w 0 ¼ 19:54 À3:56 17:01 ½ . The estimated PPG-LPSI vector of coefficients for two predetermined restrictions was b b 0 ¼ 1:70 1:04 2:93 ½ , and its estimated selection response, expected genetic gain per trait, accuracy, and heritability respectively. In this case, b b 0 b b ¼ 12:57; then, the estimated PPG-LPSI selection response using the normalized PPG-LPSI vector of coefficients was b R ¼ 49:02 12:57 ¼ 3:90, whereas the rest of the estimated PPG-LPSI parameters were the same.
In the PPG-ESIM, we need matrix b C to obtain the eigenvalues and eigenvectors of respectively. The estimated PPG-LPSI selection response was b R ¼ 49:02 12:57 ¼ 3:90, which means that the estimated PPG-ESIM selection response was greater than the estimated PPG-LPSI response. We compared PPG-ESIM efficiency versus LPSI efficiency to predict the net genetic merit using the ratio of the estimated PPG-ESIM accuracy ( b λ P 1 ¼ 0:6859) to PPG-LPSI accuracy (b ρ ¼ 0:24), i.e., b λ P 1 b ρ ¼ 0:6859 0:24 ¼ 2:858 or, in percentage terms, b p P ¼ 100 2:858 À 1 ð Þ¼185:80. Then, the PPG-ESIM was a better predictor of the net genetic merit and its efficiency was 185.80% higher than that of the PPG-LPSI for this data set. Now, we compare PPG-ESIM efficiency versus PPG-LPSI efficiency using the data set described in Sect. 2.8.1 of Chap. 2 for five phenotypic selection cycles, each with four traits (T 1 , T 2 , T 3 , and T 4 ), 500 genotypes, and four replicates for each genotype. The economic weights for T 1 , T 2 , T 3 , and T 4 were 1, À1, 1, and 1 respectively. For this data set, matrix F was an identity matrix of size 4 Â 4 for all five selection cycles.
The first and second parts of columns 6, 7, and 8 in Table 7.1 present the estimated PPG-LPSI and PPG-ESIM selection responses for one, two, and three predetermined restrictions for five simulated selection cycles. The selection intensity was 10% (k I ¼ 1.755) and the vectors of PPG for each predetermined restriction were d 0 1 ¼ 7, d 0 2 ¼ 7 À3 ½ , and d 0 3 ¼ 7 À3 5 ½ respectively, for all five selection cycles. The estimated PPG-LPSI selection response when the vector of coefficients was not normalized was presented in Chap. 3 (Table 3.5). The averages of the estimated PPG-LPSI selection response for each predetermined restriction were 4.70, 4.91, and 3.14, whereas the averages of the estimated PPG-ESIM selection response were 6.31, 6.28, and 6.75 respectively. These results indicate that the estimated PPG-ESIM selection response was greater than the estimated PPG-LPSI selection response for all predetermined restrictions.
The second part of Table 7.2 presents the estimated PPG-ESIM accuracy (b ρ P ) and the ratio of b ρ P to the estimated PPG-LPSI accuracy (b ρ ), expressed in percentage terms, b p P ¼ 100 À b λ P À 1 Á , where b λ P ¼ b ρ P =b ρ, for one, two, and three predetermined restrictions for five simulated selection cycles. The estimated PPG-LPSI accuracies were presented in Chap. 3 (Table 3.6). The average estimated PPG-ESIM efficiency for each restriction was 9.76%, 11.71%, and 29.03% greater than the PPG-LPSI efficiency for this data set in all five selection cycles.
The second part of Table 7.3 presents the estimated PPG-ESIM expected genetic gain per trait for one, two, and three predetermined restrictions for five simulated selection cycles. The estimated PPG-LPSI expected genetic gains per trait for one, two, and three predetermined restrictions were presented in Chap. 3, Table 3.5, where it can be seen that the averages of the estimated PPG-LPSI expected genetic gains per trait for five simulated selection cycles were 6.85, À3.25, 2.62 and 1.48 for one restriction; 6.93, À2.97, 2.65 and 1.45 for two restrictions; and 5.20, À2.23, 3.72 and 1.43 for three restrictions, whereas for the same set of restrictions, the averages of the estimated PPG-ESIM expected genetic gain per trait were 5.67, À2.67, 1.81, and 2.97 for one restriction; 5.89, À2.52, 2.04, and 2.83 for two restrictions; and 5.71, À2.45, 4.08, and 0.82 for three restrictions (Table 7.3). Because the vectors of predetermined proportional gains for each predetermined restriction were d 0 1 ¼ 7, d 0 2 ¼ 7 À3 ½ , and d 0 3 ¼ 7 À3 5 ½ , the averages of the estimated PPG-LPSI expected genetic gains per trait were closer than those of the estimated PPG-ESIM expected genetic gains per trait for one and two predetermined restrictions, whereas for three restrictions, the results of both selection indices were similar.