Mixed additive models - AIP Publishing

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θiMi Where Mi = XiX. ⊤ i , i = 1,...,w, and µ = X0β0. For these we will estimate the variance components θ1,...,θw, aswell estimable vectors through the ...
Mixed Additive Models Francisco Carvalho∗,† and Ricardo Covas∗,† ∗

CMA - Centro de Matemática e Aplicações, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal † Escola Superior de Gestão de Tomar, Instituto Politécnico de Tomar, 2300-313 Tomar, Portugal w

w

i=0

i=1

Abstract. We consider mixed models y = ∑ Xi β i with V(y) = ∑ θi Mi Where Mi = Xi X⊤ i , i = 1, . . . , w, and µ = X0 β 0 . For these we will estimate the variance components θ1 , . . . , θw , aswell estimable vectors through the decomposition of the initial model into sub-models y(h), h ∈ Γ, with V (y(h)) = γ (h)Ig(h) h ∈ Γ. Moreover we will consider L extensions of these models, i.e., y˚ = Ly + ε , where L = D (1n1 , . . . , 1nw ) and ε , independent of n

y, has null mean vector and variance covariance matrix θw+1 Iw , where w = ∑ wi . i=1

Keywords: Linear Models, Variance Components, L Extensions PACS: 02.50.-r

INTRODUCTION The mixed model

w

y = ∑ Xi β i i=0

where β 0 is fixed, is additive if the β 1 , . . . , β w have null mean vectors as well as null cross covariance matrices w

V (β i , β i′ ) = 0ci ×c j , i ̸= i′ so that V(y) = ∑ Xi V(β i )X⊤ i . i=1

Following [6], if the V(β i ), i = 1, . . . , w, are positive definite, we can always assume that V(β i ) = θi Ici , i = 1, . . . , w, w

and so V(y) = ∑ θi Mi , where Mi = Xi X⊤ i , i = 1, . . . , w, while the mean vector of y is µ = X0 β 0 . i=1

This is a general form of the linear model which has been approached in [2, 3, 4, 5]. For these models we will estimate the variance components θ1 , . . . , θw , aswell estimable vectors through the decomposition of the initial model into sub-models y(h), h ∈ Γ, with V (y(h)) = γ (h)Ig(h) h ∈ Γ. Moreover we will consider L extensions of these models, i.e., y˚ = Ly + ε , where L = D (1n1 , . . . , 1nw ) and ε , independent of y, has null mean vector and variance covariance matrix θw+1 Iw , n

where w = ∑ wi . The observations of y˚ are grouped into cells with w1 , . . . , wn observations. Besides the case of cells i=1

with fixed effect sizes, we consider the case in which the u1 , . . . , un are realizations of random variables U1 , . . . ,Un . We will single out the case in which these variables are independent Poisson variables with parameters λ1 , . . . , λn truncated by the assumption Ui ≥ 1, i = 1, . . . , n. In the next section we show how to obtain the sub-models and to carry out the estimation of the parameters of the addictive models. Then we consider L extensions, first with fixed dimension cells and then with random observations cells.

International Conference of Numerical Analysis and Applied Mathematics 2015 (ICNAAM 2015) AIP Conf. Proc. 1738, 060008-1–060008-4; doi: 10.1063/1.4951831 Published by AIP Publishing. 978-0-7354-1392-4/$30.00

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SUBMODELS Taking Mi (0) = Mi , i = 1, . . . , w, we have Mi (0) =

k(1)

k(1)

j=1

j=1

∑ b j (1)A j (1)⊤ A j (1) =

∑ b j (1)A j (1)⊤ A j (1)

with Q j (1) = A j (1)⊤ A j (1), j = 1, . . . , k(1), pairwise orthogonal orthogonal projection matrices, POOPM. This is close connected with the existence of Jordan algebras which has been approached, for example, in [1] following the work of Seely in [7, 8, 9, 10] and [11]. From here we get the first order sub-models w

yℓ (1) = Aℓ (1)y = ∑ Xi,ℓ (1)β i ℓ ∈ Γ(1) = {1, . . . , k(1)}, i=0

where, with Xi (0) = Xi , i = 1, . . . , w, we have Xi,ℓ (1) = Aℓ (1)Xi (0)i = 1, . . . , w, ℓ ∈ Γ(1). We now take Mi,ℓ (1) = Xi,ℓ (1)X⊤ i,ℓ i = 1, . . . , w, ℓ ∈ Γ(1), it being enough to see that M1,ℓ (1) = bℓ (1)Igℓ (1) ℓ ∈ Γ(1), where gℓ (1) = rank (Aℓ (1)), ℓ ∈ Γ(1). Moreover we also have, with ℓ(1) ∈ Γ(1) kℓ(1) (2)

M2,ℓ (1) =



bℓ(2) (2)A⊤ ℓ(2) (2)Aℓ(2) (2),

ℓ(2)=1

the components of ℓ(2) being ℓ(1), and, given ℓ(1), ℓ(2) = 1, . . . , kℓ(1) (2), so that ℓ(2) ∈ Γ(2) = U

v1 ∈Γ(1)

{1, . . . , kv2 (2)} .

Taking Aℓ = Aℓ (2)Aℓ(1) (1), ℓ ∈ Γ(2), we get the second order sub-models w

yℓ (2) = Aℓ y = ∑ Xi,ℓ (2)β i ℓ ∈ Γ(2), i=0

with Xi,ℓ (2) = Aℓ (2)Xi i = 0, . . . , w, ℓ ∈ Γ(2). We point out that with the Mi,ℓ (2) = Xi,ℓ (2)X⊤ i,ℓ (2)i = 1, . . . , w, ℓ ∈ Γ(2), we get  ℓ(1) ∈ Γ(1)  M1,ℓ(1) (2) = bℓ(1) (1)Igℓ 

M2,ℓ (2) = bℓ (2)Igℓ ℓ ∈ Γ(2), ( ) with δℓ = rank Aℓ ℓ ∈ Γ(2). This can be applied in order to get, for the vectors of Γ = Γ(w) = U

ℓ(w−1)∈Γ(w−1)

w

{1, . . . , k(w)}, the sub-models yℓ (w) = ∑ Xi,ℓ β i ℓ ∈ Γ, where, with Xi = Xi (i), i = 1, . . . , w, and i=0

Aℓ = Aℓ(w) (w) · · · Aℓ(1) (1)ℓ ∈ Γ, we have Xi,ℓ = Aℓ Xi i = 0, . . . , w, ℓ ∈ Γ. ( ) These sub-models have mean vectors µ ℓ = X0,ℓ β 0 ℓ ∈ Γ and variance-covariance matrices Vℓ (θ ) = b⊤ ℓ θ Igℓ ℓ ∈ Γ, where bℓ has components bℓ(1) (1) · · · bℓ(w) (w) and gℓ is the number of rows of Aℓ , ℓ ∈ Γ.

ESTIMATION We now order the vectors of Γ from 1 to m = #[Γ] and represent by Ph and Pch the orthogonal projection matrices on the c c range space Ωh of X0,h and its orthogonal complement on Ω⊥ h , h = 1, . . . , m. With ph = rank(Ph ) and ph = rank (Ph ), ⊤ c c h = 1, . . . , m, when pch > 0, the mean value of Zh = y⊤ h Ph yh is ph bh θ , h = 1, . . . , m. Then, if Z has components Z1 , . . . , Zm and H has row vectors pc1 b1 · · · pcm bm , the mean vectors of Z will be Hθ . Thus we have for θ the least squares estimator, LSE, ( )† θe = H⊤ H H⊤ Z

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where † indicates the Moore-Penrose inverse. ( )† ( )−1 When the column vectors of H are linearly independent, we may replace H⊤ H by H⊤ H in the expression of e θ and this estimator is unbiased. Going over to estimable vectors, it is well known that Ψ = Gβ 0 is estimable if and only if G = UX0 which is equivalent to having Ψ = Uµ . ( )† e = Gβe (θe ), where βe (θe ) = X⊤ V−1 (θe )X0 X⊤ V−1 (θe )y Now we have the generalized least squares estimator, GLSE, Ψ 0

0

0

0

w

with V(θe ) = ∑ θei Mi . i=1

( ( )† )−1 ⊤ −1 e −1 e When the column vectors of X0 are linearly independent we may replace X⊤ V ( θ )X by X V ( θ )X 0 0 0 0 in the expression of βe (θe ) and this estimator is unbiased. 0

L-EXTENSIONS Fixed Cell Dimension w

With y = ∑ Xi β i an addictive model with w observations and L = D (1n1 · · · 1nw ) we take y˚ = Ly + ε where ε has i=0

w

null mean vector and variance covariance matrix θw+1 In with n = ∑ ni and is independent of y. The observations of i=1 ( ) 1 ⊤ y˚ are grouped in w cells corresponding to the components of y. We point out that L† = D n11 1⊤ · · · 1 and that n n ( 1 nw) w y¨ = L† y = y + ε¨ where ε¨ = L† ε has null mean vector and variance covariance matrix θw+1 D n11 · · · n1w . Thus taking  ) (   Xw+1 = D √1n1 · · · √1nw  

β w+1 = D

(√ √ ) n1 · · · nw ε¨

w+1

we can assume y¨ as an additive model y¨ = ∑ Xi β w+1 which is enough to S the sum of sums of squares of residuals i=0

for the w cells. Since

S n−m is an unbiased estimator of θw+1 , we can extend the LSE for θe obtained in the previous section. To do this we proceed as before using y¨ to obtain m sub-models as well as matrix H and vector z. we now add to z an additional component S zm+1 = n−w thus getting vector z˚ and replace H by   H ˚ =  ··· ···  H 0⊤ 1

θew+1 =

( ⊤ )† ⊤ ˚ ˚ H ˚ H ˚ z˚ . to get the LSE estimator θe = H ˚ and we may replace When the column vectors of H are linearly independent so are those of H, ( ⊤ )−1 ˚ ˚ H ˚ H in the expression of θe and this estimator is unbiased. For estimable vectors we get, as before, the GLSE ( )† ˚ e˚ −1 e˚ −1 βe 0 (θe ) = X⊤ X⊤ 0 V(θ ) X0 0 V(θ ) y w+1 ˚ with V(θe ) = ∑ θei Mi , where Mw+1 = Xw+1 X⊤ w+1 = D i=1

(

1 n1

) · · · n1w .

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(

)† ˚ ⊤H ˚ H by

( )† ( )−1 e˚ )−1 X0 by X⊤ V(θe˚ )−1 X0 When the column vectors of X0 are linearly independent we may replace X⊤ V( θ 0 0 ˚ e e in the expression of β 0 (θ ) and this estimator is unbiased.

Random Cell Dimension ˚ does not depend on We now assume the n1 · · · nw to be realizations of random variables N1 · · · Nw . Since matrix H ˚e n, if its column vectors are linearly independent, the conditional estimator given by θ (n) will be unbiased whatever n and so we obtain an unbiased estimator. Moreover if the column vectors of X0 are linearly independent, the conditional estimator βe 0 (n) will, also whatever n, be unbiased so therefore we obtain an unbiased estimator.

ACKNOWLEDGEMENTS Funded by UID/MAT/00297/2013

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