Measurement models and additive-error observation models

Let $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ be a probability space (Probability measures and probability spaces), and fix integers $m,d\ge 1$.

Measurement model. A measurement model with input dimension $m$ and output dimension $d$ consists of

1. an open set $U\subseteq {\mathbb{R}}^m$, the input domain, and a map $f:U\to {\mathbb{R}}^d$, the model function, which is Borel measurable as a map of measurable spaces $f:(U,\mathcal{B}(U))⟶({\mathbb{R}}^d,\mathcal{B}({\mathbb{R}}^d))$ (A measurable function between measurable spaces, The Borel sigma-algebra of a topological space);
2. a random element $X:\mathrm{\Omega }\to {\mathbb{R}}^m$, the input vector, whose values lie in $U$, that is, $X(\omega )\in U$ for every $\omega \in \mathrm{\Omega }$ (Random elements and real random variables, with target $({\mathbb{R}}^m,\mathcal{B}({\mathbb{R}}^m))$); write $X_U:\mathrm{\Omega }\to U$, $X_U(\omega ):=X(\omega )$, for its corestriction;
3. the output quantity $Y:=f\circ X_U:\mathrm{\Omega }\to {\mathbb{R}}^d$.

The corestriction $X_U$ is measurable for the relative Borel sigma-algebra $\mathcal{B}(U)$: it is generated by the relative open sets $U\cap O$ with $O\subseteq {\mathbb{R}}^m$ open, so every $A\in \mathcal{B}(U)$ has the form $A=U\cap B$ for some $B\in \mathcal{B}({\mathbb{R}}^m)$, and $X_U^{-1}(A)=X^{-1}(B)$ because $X(\omega )\in U$. For every Borel $C\subseteq {\mathbb{R}}^d$, measurability of $f$ then gives $Y^{-1}(C)=X_U^{-1}(f^{-1}(C))\in \mathfrak{F},$ so $Y$ is a random element.

A point $\mu \in U$ is a nominal input of the model. The model function is continuous, of class $C^k$, or totally differentiable at $\mu$ when $f$ has the corresponding property as a map on the open set $U$ ($C^k$ Euclidean maps and diffeomorphisms, The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder); such a regularity hypothesis must be stated whenever a derivative of $f$ is used. The linearization of $f$ at $\mu$ is the affine map $T_{\mu }(x):=f(\mu )+Df(\mu )(x-\mu )$ when $Df(\mu )$ exists.

Additive-error observation model. Let $\mathrm{\Theta }$ be a set, the parameter set; let $Z$ be a set, the apparatus and preparation settings; let $V\subseteq {\mathbb{R}}^d$ be an open set with $0\in V$, the error domain; let $h:\mathrm{\Theta }\times Z\to {\mathbb{R}}^d$ be a map, the prediction map; and let $\epsilon :\mathrm{\Omega }\to {\mathbb{R}}^d$ be a random element with $\epsilon (\omega )\in V$ for every $\omega$, the error. For each $(\theta ,z)\in \mathrm{\Theta }\times Z$ the observation is the random element $Y_{\theta ,z}:=h(\theta ,z)+\epsilon :\mathrm{\Omega }⟶{\mathbb{R}}^d\mathrm{.}$ Well-definedness: for fixed $\theta$ and $z$ the map ${\tau }_{\theta ,z}(x):=h(\theta ,z)+x$ is a translate of ${\mathbb{R}}^d$, hence continuous, so ${\tau }_{\theta ,z}^{-1}(B)$ is Borel for every Borel $B\subseteq {\mathbb{R}}^d$ (A continuous map has Borel preimages of Borel sets); therefore $Y_{\theta ,z}^{-1}(B)={\epsilon }^{-1}({\tau }_{\theta ,z}^{-1}(B))\in \mathfrak{F}$ by the measurability of $\epsilon$, so $Y_{\theta ,z}$ is a random element in the sense of Random elements and real random variables.

Conventions, hypotheses and scope

The following are additional hypotheses and are not part of either definition: that the error has mean zero, $E[\epsilon ]=0$; that $\epsilon$ is square-integrable, $E\mathrm{\mid }\epsilon {\mathrm{\mid }}^2<+\mathrm{\infty }$; that the error is bounded; that a stated probability law for $\epsilon$ is specified; or that the input vector is of the additive form $X=\mu +\epsilon$ with a deterministic nominal input $\mu$. Any of these must be stated, with its exact quantifiers, at each use.

When a model function is merely continuous, continuity and A continuous map has Borel preimages of Borel sets show that its preimages of Borel sets are Borel, so clause 1 holds automatically for such $f$; the explicit measurability in clause 1 keeps the definition meaningful for discontinuous model functions as well, where required.

The model is an input–output relation between random quantities and their images. It contains no unit assignment, no physical identification of $\mathrm{\Theta }$, $Z$ or $U$ with measured quantities, no sampling protocol and no distributional assumption beyond the stated probability space. A physical consumer must supply the units of the coordinates, the preparation and apparatus protocol, and the empirical status of every distributional or moment hypothesis that it invokes.