parametric RBF regression methods can outperform BayesA when predicting total GVs in the presence of non-additive effects using SNP markers.
In this chapter we will demonstrate the potential of the kriging approaches applied to genomic data: As a novelty, we will suggest the family of Matérn covariance functions to reflect the functional dependency of the observed covariances from the distance of genotypes expressed as Euclidean norm. Based on this model and the assumed covariance function, we will suggest two kriging approaches. Under both models, parameters and hidden variables are estimated via maximum likelihood (ML) and BLUP of the unknowns is established by solving the corresponding linear kriging systems. All predictions can also be implemented in the form of the so-called mixed model equations (Henderson, 1973). The predictive performance of the two models will be compared to a common genomic BLUP as a reference method in a whole-genome simulation study considering various gene-action models.
Furthermore, we will show that in a limiting case the genomic covariance structure proposed byVanRaden(2008) can be considered as a covariance function with corresponding quadratic variogram. Besides we will prove theoretically that predicted GVs are only scaled by a factor if the covariance structures are linearly transformed. Finally, we will discuss further options for a more differentiated modeling using the suggested methodological approach.
3.2 Prediction methods
3.2.1 Kriging
The term kriging stems from the prediction of ore concentrations in deposits and was mainly developed by Matheron (1962, 1963) based on the master’s thesis of Krige (1951). In geostatistics, kriging is nowadays the standard approach whenever spatial prediction of a so-called regionalized variable (Matheron,1989), e.g. temperature, ozone concentration or soil moisture, has to be performed based on a few isolated measurements of the quantity. It is assumed that the regionalized variable is a realization of a random function with a certain covariance structure. Mostly, the latter is given by a parameterized covariance function (Cressie,1993), and the random function is assumed to be Gaussian.
The kriging approach consists of two steps: (i) estimation of the unknown parameters and hidden variables (in particular by ML or REML) and (ii) prediction of the values of the regionalized variables by performing a BLUP, under the auxiliary assumption that the parameter values and hidden variables estimated in the first step are the true ones.
Many variants of the general kriging principle have been discussed (Cressie,1993). The type of kriging is implied by the unbiasedness condition: In “simple kriging” it is assumed that the underlying regionalized variable has zero-mean, whereas in “universal kriging” a linear model for the mean of the underlying regionalized variable is assumed.
3.2.2 The model for polygenic and genomic data
In our further studies, we assume to have q individuals with pedigree information, n of them being genotyped and having phenotype measurements of a certain quantitative trait.
Typically, GVs have to be predicted for individuals that are genotyped, but have no phenotype data.
We use the following model for the given data:
yi=wTi β+zTi u+g(xi) +ei, i= 1, . . . n,
whereyi is a measurement of the phenotype for individuali,β is an f-vector of nuisance location parameters,xi is ap-vector of dummy SNP instance variates (genotype) observed on individuali, andg is an unknown, random function as described below. Let u∼ N(0, σu2A) be a q-vector of additive genetic effects of q individuals where σu2 is the additive genetic variance due to unmarked polygenes, and A is the numerator relationship matrix. The entries of the numerator relationship matrix are twice the coefficients of coancestry between individuals. The vectorswTi andzTi are known incidence vectors;zi is a unit vector with one component being 1 and all the others zero, indicating the respective position in the pedigree.
Let e = (e1, . . . , en)T be the vector of environmental residual effects with e ∼ N(0, σe2I), whereσ2e is the environmental variance.
We assume that{g(xi),xi ∈Rp}is a Gaussian random field (Lifshits,1995) withE(g(xi)) = 0 and covariance structure given by Cov(g(xi),g(xj)) = E(g(xi)g(xj)) = Kν,h,σK(xi,xj), where Kν,h,σK(·,·) is a covariance function depending on parameters ν, h, and σK. Let Kν,h,σK = (Kν,h,σK(xi,xj))1≤i,j≤n be the corresponding covariance matrix.
The family of Matérn covariance functions: For the covariance structure we suggest to use the so-called family of Matérn covariance functions, which was introduced byMatérn(1960) andHandcock & Wallis (1994), and which is defined by
Cov(g(xi),g(xj)) =Kν,h,σK(xi,xj) =σK2 · 21−ν Γ(ν)
√
2νkxi−xjk/hνKν
√2νkxi−xjk/h.
Here,k · kis the Euclidean norm,ν >0 is a smoothness parameter,his a scale parameter,σ2K is the variance parameter andKν(·) is a modified Bessel function of the second kind of order ν (Abramowitz & Stegun, 1984). The Matérn function is isotropic, in that Cov(g(xi),g(xj)) only depends on the Euclidean norm of the separation vectorxi−xj.
Matérn covariance functions build a very general class of covariance functions including special cases like the exponential (ν = 1/2) and the Gaussian (ν=∞) covariance function, the ones that have also been used byPiepho (2009). If the smoothness parameter ν is of the formm+ 1/2, where mis an integer, the Matérn function factorizes into the product of an exponential function and a polynomial of degreem,cf. Table 3.1 and Figure3.1. The best fitting parameter valueν is determined through the model-fitting approaches described below.
In matrix notation, the statistical model is
y=Wβ+Zu+g(X) +e, (3.1) where W = (wT1, . . . ,wTn)T is an (n×f)- and Z= (zT1, . . . ,zTn)T is an (n×q)-incidence matrix andg(X) = (g(x1), . . . , g(xn))T. Finally, we assume that the random vectors u,e
3.2 Prediction methods 21
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.2 0.4 0.6 0.8 1.0
Matern covariance functions K with h = 1, σK2=1
Euclidean distance
covariance
Figure 3.1: Matérn covariance functions forh= 1, σ2K = 1 and different values of ν. From top to bottomν=∞,10,2.5,1.5,0.5.
Table 3.1: Special cases of Matérn covariance functions
ν h Kν,h,σK(xi,xj) Exponential 0.5 1 σ2K·exp(−kxi−xjk)
1.5 1 σ2K·exp(−√
3kxi−xjk)·1 +√
3kxi−xjk 2.5 1 σ2K·exp(−√
5kxi−xjk)·1 +√
5kxi−xjk+53kxi−xjk2 Gaussian ∞ 1 exp(−12kxi−xjk2)
and g(X) are independent.
3.2.3 Two kriging approaches and a reference model
We consider two models to predict the total genetic valuezT0u+g(x0) of a certain genotyped individual indexed by 0. This individual belongs to the set of q individuals, but it does not have to be phenotyped. The models differ in the size of the sets of quantities that are estimated in the first kriging step and subsequently used for predictions.
Universal Kriging: Modeling of y: We exploit the fact that y has a multivariate normal distribution,
y∼ N(Wβ, σu2ZAZT +Kν,h,σK +σe2I),
and estimate the parametersβ, σu, σe, ν, h andσK by maximizing the loglikelihood of the corresponding density function.
Then, we perform a best linear unbiased prediction ofg(x0) andzT0u,i.e. we apply the BLUP principle: To obtain ˆg(x0) we minimize
E(ˆg(x0)−g(x0))2 −→ min!
with the linear predictor ˆg(x0) = aTgy under the condition agTW = 0. This approach is called “universal kriging” in other areas of research (Cressie,1993). In fact, the condition assuresaTgWβ= 0 and thereforeEg(x0) = 0 =agTWβ=Eˆg(x0),i.e.gˆ(x0) is unbiased. Let K0 = (Kν,h,σK(x1,x0), . . . , Kν,h,σK(xn,x0))T. The approach results in the following kriging system of equations:
"
W σ2uZAZT +Kν,h,σK +σe2I
0 WT
#
·
"
λ ag
#
=
"
K0 0
# .
Note that this linear system does not depend onβ. Analogously,zT0u can be predicted by the universal kriging estimatorzdT0u=aTuy, whereau satisfies
"
W σ2uZAZT +Kν,h,σK +σe2I
0 WT
#
·
"
λ au
#
=
"
σu2ZAz0 0
# ,
and one getszdT0u+ ˆg(x0) as BLUP ofzT0u+g(x0).
Mixed Model Equations (MME).In the animal breeding context it is well-known that a BLUP-approach for the modely=Wβ+Zu+g(X) +eis equivalent to solving the MME
WTW WTZ WT ZTW ZTZ+σσ2e2
uA−1 ZT
W Z I+σe2K−1ν,h,σ
K
·
βˆ
ˆ u g[(X)
=
WTy
ZTy y
(3.2)
for given variance components estimatede.g.by ML. For a derivation of the MME from the kriging system compare section2.2orDempfle (1982).
Simple Kriging: Joint modeling ofy, u and g(X): In the second approach we model the hidden variablesu andg(X) explicitly and consider the joint density functionfy,u,g ofy,u andg(X) which equals
fy,u,g(X)(y,u,g(X)) =fy|u,g(X)(y)·fu(u)·fg(g(X))
=fe(y−Wβ−Zu−g(X))·fu(u)·fg(g(X))
3.2 Prediction methods 23
=c·exp−1 2 ·
1
σe2ky−Wβ−Zu−g(X)k2
·exp−1 2·
1
σ2uuTA−1u
·exp−1
2·hg(X)TK−1ν,h,σ
Kg(X)i with
c−1 = (2π)n+q/2σen·σuq(detA)1/2·(detKν,h,σK)1/2.
Here, we have to estimate the parameters β, σu, σe, ν, h, σK and the hidden variablesu and g(X). Note that in this approach we consider u andg(X) to be parameters that have to be estimated via ML in the first kriging step. Therefore, we maximize the loglikelihoodJ of the density functionfy,u,g,i.e. we maximize
J = log(c)−1 2 ·
1
σe2ky−Wβ−Zu−g(X)k2+ 1
σ2uuTA−1u+g(X)TK−1ν,h,σ
Kg(X) (3.3) with respect to β,u andg(X). Taking the derivatives with respect toβ,u andg(X) leads to the linear system given in eq. (3.2) which yields estimators for β,u and g(X). When using these estimates in eq. (3.3), the value of J depends only onσu, σe, ν, hand σK. Thus, J can be maximized numerically with respect to these parameters, leading to estimates forβ, σu, σe, ν, h, σK,uand g(X). According to the kriging philosophy, we now assume the values of the estimators (especially the value of the estimator for g(X)) to be the true ones, and g(x0) is predicted via ˆg(x0) =agTg(X) by the BLUP principle. That is, we minimize
E(ˆg(x0)−g(x0))2 −→ min!
with the linear estimator
ˆ
g(x0) =aTgg(X).
This approach is called “simple kriging” (Cressie, 1990, 1993; Chilès & Delfiner, 1999). Note that ˆg(x0) is always unbiased. The solution is
ˆ
g(x0) =KT0K−1ν,h,σ
Kg(X). (3.4)
Finally, the predicted GV is given byg(x\0) +zT0u= ˆg(x0) +zT0uˆ, where ˆu is the estimator obtained in the iterative procedure described above.
Reference model (genomic BLUP): This approach performs a genomic BLUP based on the model
y=Wβ+Zu+ ˜Xg+e, which leads to the kriging system
"
W σ2uZAZT +σg2XG˜ X˜T +σe2I
0 WT
#
·
"
λ a
#
=
"
σu2ZAz0+σ2gXG˜ x˜0 0
#
and predictingzT0u\+ ˜x0Tg =aTy.
Here,β,e∼ N(0,σ2eI),u∼ N(0, σu2A),WandZare defined as in the previous approaches.
The vector g ∼ N(0, σg2G) is multivariate normal with G being a genomic relationship matrix calculated by using the approach of VanRaden (2008). (For the definition of the genomic relationship matrix see the formulas in section 3.6.) The matrix ˜X is a known incidence matrix whose rows consist of unit vectors with one component being 1 and all the others zero, indicating the respective position in theg-vector. Variance components for this model are estimated via ML.