Calibration data and calibration rules

Let $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$ be a probability space and let $p,d\ge 1$ be integers. A calibration family is a family $(h_c{)}_{c\in C},h_c:\mathrm{\Theta }⟶{\mathbb{R}}^d,$ of prediction maps on a nonempty parameter set $\mathrm{\Theta }\subseteq {\mathbb{R}}^p$, in fixed numerical coordinates, indexed by a nonempty set $C$ of apparatus settings (Measurement models and additive-error observation models).

A calibration datum for this family consists of

1. a finite family $({\theta }^{(r)}{)}_{1\le r\le n}$, $n\ge 1$, of reference inputs, either fixed values ${\theta }^{(r)}\in \mathrm{\Theta }$ or maps ${\theta }^{(r)}:\mathrm{\Omega }\to \mathrm{\Theta }$, according to the branch specified in clause 4, representing values supplied by a reference standard under stated conditions;
2. a set $Y\subseteq ({\mathbb{R}}^d{)}^n$ of admissible reference observations and the observed vector $y=(y_1,\dots ,y_n)\in Y$;
3. a calibration rule, that is, a map $\mathrm{\Gamma }:D⟶C,(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})⟼c,D\subseteq Y\times {\mathrm{\Theta }}^n;$ the nominal reference tuple ${\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}=({\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}{)}_{r=1}^n$ belongs to ${\mathrm{\Theta }}^n$ and $(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})\in D$;
4. either, in the fixed-value branch, a stated bound $\mathrm{\mid }{\theta }^{(r)}-{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}\mathrm{\mid }\le {\delta }_r$ with ${\delta }_r\ge 0$ for each $r$, or, in the random-input branch, a Borel random element $\eta :\mathrm{\Omega }\to ({\mathbb{R}}^p{)}^n$ of deviations with a stated joint law (Random elements and real random variables), where ${\eta }_r:\mathrm{\Omega }\to {\mathbb{R}}^p$ is its $r$th component and ${\theta }^{(r)}(\omega ):={\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}^{(r)}+{\eta }_r(\omega )\in \mathrm{\Theta }$ for every $r$ and every $\omega \in \mathrm{\Omega }$. The bound uses the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$ on the fixed coordinates; deviations need not themselves belong to $\mathrm{\Theta }$.

Given the datum, the calibrated setting is $c:=\mathrm{\Gamma }(y,{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}})$ and the calibrated prediction map is $h_c$.

Well-definedness and conventions

The rule $\mathrm{\Gamma }$ and the sets $C$, $\mathrm{\Theta }$, $Y$ are data: nothing in the definition requires $\mathrm{\Gamma }$ to be injective, continuous, measurable, or to return an apparatus setting for which the calibrated prediction is accurate. The reference inputs are values of a physical standard; their uncertainty is recorded in clause 4 and must be carried into any quantitative conclusion about $h_c$ by the argument that uses it. The definition states the structure of a calibration only; properties such as consistency, unbiasedness or attainment of a stated accuracy require separate hypotheses and proofs, and no such property is asserted here.

Reference inputs are explicit arguments of the rule: a fixed-reference rule ${\mathrm{\Gamma }}_{{\theta }_{\mathrm{r}\mathrm{e}\mathrm{f}}}(y)$ is only a slice of $\mathrm{\Gamma }$. This mathematical definition attaches no units to the coordinates. A physical consumer must specify one unit and, when a norm compares different coordinates, one positive reference scale per coordinate; subtraction and addition then operate on numerical coordinates in those fixed units.