SNR Models for Non-grouped Data

We define a class of SNR models for non-grouped data as

$$y_i=\eta(\mbox{\boldmath$\phi$}, \mbox{\boldmath$f$}; \mbox{\boldmath$t$}_i) + \epsilon_i, ~~~~ i=1, \cdots, n ,$$ (36)

where $y_i$'s are responses; $\mbox{\boldmath$t$}_i=(t_{1i}, \cdots, t_{di})$ is a covariate in a general domain ${\cal T}$; $\eta$ is a known function of $\mbox{\boldmath$t$}_i$ which depends on a vector of parameters $\mbox{\boldmath$\phi$}=(\phi_1,\cdots,\phi_r)^T$ and a vector of unknown non-parametric function $\mbox{\boldmath$f$}=(f_1,\cdots,f_q)^T$; and $\mbox{\boldmath$\epsilon$}=(\epsilon_1,\cdots,\epsilon_n)^T$ are random errors distributed as $\mbox{N} ({\bf0}, \sigma^{2} W^{-1})$. $f_j$'s are modeled using SS ANOVA models as represented in (). We assume that $W$ depends on a parsimonious set of parameters $\mbox{\boldmath$\tau$}$.

For model (), we denote $\mbox{\boldmath$y$}=(y_1,\cdots,y_n)^T$ and $\mbox{\boldmath$\eta$}(\mbox{\boldmath$\phi$},\mbox{\boldmath$f$})=(\eta(\mbox{... ...cdots,\eta(\mbox{\boldmath$\phi$},\mbox{\boldmath$f$};\mbox{\boldmath$t$}_n))^T$. Model () can then be written in the vector form

$$\mbox{\boldmath$y$}= \mbox{\boldmath$\eta$}(\mbox{\boldmath$\phi$},\mbox{\boldmath$f$}) + \mbox{\boldmath$\epsilon$}.$$ (37)