By Maksimov V. I.
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Additional info for A Boundary Control Problem for a Nonlinear Parabolic Equation
1 for individual observations indexed by t into vectors and matrices. , for the Autism data in Chapter 6, Xti might be replaced by AGEti , the t-th age at which child i is measured). 2), Yi represents a vector of continuous responses for the i-th subject. 1): ⎛ ⎞ Y1i ⎜ Y2i ⎟ ⎜ ⎟ Yi = ⎜ . ⎟ ⎝ .. ⎠ Yni i Note that the number of elements, ni , in the vector Yi may vary from one subject to another. 2) is an ni × p design matrix, which represents the known values of the p covariates, X (1) , . . , X (p) , for each of the ni observations collected on the i-th subject: ⎛ (1) ⎞ (2) (p) X1i X1i · · · X1i ⎜ (1) (2) (p) ⎟ ⎜ X2i X2i · · · X2i ⎟ ⎜ Xi = ⎜ .
The n ×p design matrix X is obtained by stacking all Xi matrices vertically as well. In two-level models or models with nested random eﬀects, the Z matrix is a block-diagonal matrix, with blocks on the diagonal deﬁned by the Zi matrices. The u vector stacks all ui vectors vertically, and the vector ε stacks all εi vectors vertically. The G matrix is a block-diagonal matrix representing the variance-covariance matrix for all random eﬀects (not just those associated with a single subject i), with blocks on the diagonal deﬁned by the D matrix.
4) in the next section using matrix notation. We assume that for a given subject, the residuals are independent of the random eﬀects. 1) can be combined into vectors and matrices, and the LMM can be speciﬁed more eﬃciently using matrix notation as shown in the next section. Specifying an LMM in matrix notation also simpliﬁes the presentation of estimation and hypothesis tests in the context of LMMs. 1 for individual observations indexed by t into vectors and matrices. , for the Autism data in Chapter 6, Xti might be replaced by AGEti , the t-th age at which child i is measured).
A Boundary Control Problem for a Nonlinear Parabolic Equation by Maksimov V. I.