Finite nonlinear observation designs, local identifiability, and weighted information

Let $p\ge 1$, let $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$ be open, and let $\mathcal{X}$ be a finite nonempty set of experimental conditions. For each $x\in \mathcal{X}$, let $q_x\ge 1$, let $F_x:\mathrm{\Theta }\to {\mathbb{R}}^{q_x}$ be $C^1$ in the sense of Continuously differentiable maps, local inverses, and local diffeomorphisms, and let $W_x\in {\mathbb{R}}^{q_x\times q_x}$ be a symmetric positive-definite weight matrix, meaning $z^T{W}_{x}z>0$ for every nonzero $z\in {\mathbb{R}}^{q_x}$.

An approximate design is a function $\xi :\mathcal{X}\to [0,1]$ with ${\sum }_{x\in \mathcal{X}}{\xi }_x=1$, as in Approximate designs and linear-model optimality criteria. Its support is $\mathrm{supp}(\xi )=\{x\in \mathcal{X}:{\xi }_x>0\}$. Define the joint observation map by $F_{\xi }(\theta )=(F_x(\theta ){)}_{x\in \mathrm{supp}(\xi )},$ where the finite product is ordered by any fixed listing of $\mathrm{supp}(\xi )$. Its stacked sensitivity Jacobian at $\theta$ is the vertical concatenation $J_{\xi }(\theta )$ of the matrices $J_x(\theta )=D{F}_x(\theta )$ for $x\in \mathrm{supp}(\xi )$. The associated weighted information matrix is $M_{\theta }(\xi )={\sum }_{x\in \mathcal{X}}{\xi }_x{J}_x(\theta {)}^T{W}_x{J}_x(\theta )\mathrm{.}$ The matrix is symmetric positive semidefinite because each summand has nonnegative quadratic form.

Because the support is finite, $F_{\xi }$ is $C^1$ with derivative $J_{\xi }(\theta )$: the finite collection of component first-order remainders has a stacked Euclidean norm that is still $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$, and the stacked derivative is continuous because each $D{F}_x$ is continuous.

For a fixed design $\xi$, the observation map is locally identifiable at ${\theta }_0\in \mathrm{\Theta }$ if it is injective on some open neighborhood of ${\theta }_0$ in $\mathrm{\Theta }$. It is globally identifiable on $\mathrm{\Theta }$ if it is injective on all of $\mathrm{\Theta }$. A matrix $J:{\mathbb{R}}^p\to {\mathbb{R}}^m$ has full column rank when $Jv=0$ implies $v=0$. Its smallest singular value is ${\sigma }_{\min }(J)={\inf }_{\mathrm{\parallel }v{\mathrm{\parallel }}_2=1}\mathrm{\parallel }Jv{\mathrm{\parallel }}_2$; thus ${\sigma }_{\min }(J)\ge \mu$ means $\mathrm{\parallel }Jv{\mathrm{\parallel }}_2\ge \mu \mathrm{\parallel }v{\mathrm{\parallel }}_2$ for every $v$.

The approximate design uses real weights. An exact $n$-run design instead restricts weights to multiples $n_x\mathrm{/}n$ with integer counts, and is a separate finite allocation problem in Approximate designs and linear-model optimality criteria.