Multivariate Measurement Uncertainty & GUM Supplement 2

The foundational Guide to the Expression of Uncertainty in Measurement (GUM) was formulated around a singular premise: a measurement produces a single scalar output quantity, $Y = f(X_1, X_2, \dots, X_N)$. From this single output, metrologists calculate a combined standard uncertainty $u_c(y)$ and expand it to a one-dimensional interval $[y - U, y + U]$.

However, modern precision calibration and industrial metrology frequently produce multiple simultaneous output quantities. Whether measuring the complex impedance of an RF component (resistance and reactance), locating a feature with a 3D Coordinate Measuring Machine (X, Y, Z), or simultaneously determining the slope and intercept of a calibration curve, evaluating outputs independently fails to capture the physical reality of the measurement.

To address this fundamental limitation, the Joint Committee for Guides in Metrology published JCGM 102:2011 (GUM Supplement 2): Extension to any number of output quantities. This guide formalizes vector-valued measurement models, output covariance propagation via Jacobian sensitivity matrices, and the evaluation of hyper-ellipsoidal coverage regions.

1. The Multi-Output Challenge: Why Single-Output Models Fail

When a measurement system evaluates several parameters at the same time, treating each parameter as an isolated scalar is a critical metrological error. Consider three universal calibration scenarios:

2. The Mathematical Framework of GUM Supplement 2

GUM Supplement 2 formalizes the generalization from a single scalar measurand to an $m$-dimensional vector measurand $\mathbf{Y}$, determined by an $N$-dimensional vector of input quantities $\mathbf{X}$.

The relationship is governed by a vector-valued measurement function $\mathbf{f} = [f_1, f_2, \dots, f_m]^T$:

The Jacobian Sensitivity Matrix ($\mathbf{J}$)

In univariate GUM, sensitivity coefficients are scalars $c_i = \frac{\partial f}{\partial x_i}$. In multivariate GUM, the sensitivity structure is characterized by the Jacobian Matrix $\mathbf{J}$ of dimension $m \times N$:

Each element $J_{ji} = \frac{\partial f_j}{\partial X_i}$ quantifies how sensitively output component $Y_j$ responds to a small perturbation in input component $X_i$.

The Input Covariance Matrix ($\mathbf{U}_{\mathbf{X}}$)

The dispersion and mutual dependencies of all $N$ input quantities are stored in the symmetric $N \times N$ input covariance matrix $\mathbf{U}_{\mathbf{X}}$:

Diagonal terms $u^2(X_i)$ are the input variances, while off-diagonal terms $u(X_i, X_j) = u(X_j, X_i)$ are the covariances evaluated through Type A simultaneous sampling or Type B shared-influence analysis.

3. Hyper-Ellipsoidal Coverage Regions vs. Rectangular Bounding Boxes

In univariate metrology, expanded uncertainty defines a coverage interval $y \pm k \cdot u(y)$. In multivariate metrology, the output uncertainty is defined by a multi-dimensional coverage region $\mathcal{R}_p$ that encloses a specified fraction $p$ (e.g. 95%) of the multivariate probability distribution.

The Mahalanobis Distance & Coverage Hyper-Ellipsoid

Under the multivariate normal distribution (or by central limit theorem for vector sums), the optimal minimum-volume coverage region is an $m$-dimensional hyper-ellipsoid centered at $\mathbf{y}$:

The quadratic form $d_M^2 = (\mathbf{Y} - \mathbf{y})^T \, \mathbf{U}_{\mathbf{Y}}^{-1} \, (\mathbf{Y} - \mathbf{y})$ is the squared Mahalanobis distance. It measures the statistical distance from any point $\mathbf{Y}$ to the center $\mathbf{y}$, weighted by the inverse covariance matrix $\mathbf{U}_{\mathbf{Y}}^{-1}$.

Geometry & Principal Axes via Eigendecomposition

The geometry of the coverage hyper-ellipsoid is obtained directly from the spectral decomposition of the output covariance matrix $\mathbf{U}_{\mathbf{Y}} = \mathbf{V} \mathbf{\Lambda} \mathbf{V}^T$:

• Principal Axis Directions: The eigenvectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_m$ define the orthogonal spatial axes of the hyper-ellipsoid (its rotation in parameter space).
• Semi-Axis Lengths: The length of the $i$-th principal semi-axis is given by:
  $a_i = k_p \sqrt{\lambda_i}$
  where $\lambda_i$ is the $i$-th eigenvalue of $\mathbf{U}_{\mathbf{Y}}$ and $k_p$ is the multivariate coverage factor.
• Hyper-Volume of the Ellipsoid: The total volume enclosed by the coverage region is:
  $V_m = \frac{\pi^{m/2}}{\Gamma\left(\frac{m}{2} + 1\right)} k_p^m \sqrt{\det(\mathbf{U}_{\mathbf{Y}})}$
  Under a Gaussian distribution, the hyper-ellipsoid is mathematically proven to be the region of absolute minimum volume for any specified coverage probability $p$.

4. Coverage Factor Calculation in $m$ Dimensions ($k = \sqrt{\chi^2_m(p)}$)

In standard univariate GUM, metrologists habitually set the coverage factor to $k = 2$ for an approximate 95.45% confidence interval.

In multivariate metrology, setting $k = 2$ is completely invalid. As the dimension $m$ increases, the probability mass inside a fixed radius drops precipitously.

Multivariate Coverage Factor Reference Table

The table below demonstrates the "dimensionality tax": as dimension $m$ expands, $k$ must increase substantially to maintain a 95% coverage level. Notice the collapse in true coverage if someone mistakenly applies the univariate rule $k = 2$:

For a 2D measurand, the exact cumulative probability within radius $k$ is given analytically by $P(d_M \le k) = 1 - e^{-k^2/2}$. If one sets $k = 2$, the coverage is $1 - e^{-2} = 86.47\%$, leaving over 13.5% of valid calibration results outside the stated interval!

5. Comprehensive Step-by-Step Worked Example: Polar-to-Cartesian Spatial Metrology

To make the matrix arithmetic completely concrete, let us work through an exact metrology problem: a Polar Laser Tracker / Coordinate Measuring System locating a precision fiducial target on an aerospace structure.

Step 1: Calculate Nominal Output Estimates ($\mathbf{y}$)

The vector measurement model is:

Substituting the estimates $r = 100.000\text{ mm}$ and $\theta = 30^\circ$ ($\cos 30^\circ = \frac{\sqrt{3}}{2} \approx 0.866025$, $\sin 30^\circ = 0.500000$):

Step 2: Construct the Input Covariance Matrix ($\mathbf{U}_{\mathbf{X}}$)

Since the inputs are uncorrelated, $\mathbf{U}_{\mathbf{X}}$ is diagonal:

Step 3: Evaluate the Analytical Jacobian Sensitivity Matrix ($\mathbf{J}$)

Taking partial derivatives of each output with respect to each input:

Evaluating numerically at $r = 100$ and $\theta = 30^\circ$:

Step 4: Perform Step-by-Step Matrix Multiplication ($\mathbf{U}_{\mathbf{Y}} = \mathbf{J} \mathbf{U}_{\mathbf{X}} \mathbf{J}^T$)

First, multiply $\mathbf{A} = \mathbf{J} \mathbf{U}_{\mathbf{X}}$:

Next, post-multiply by $\mathbf{J}^T = \begin{bmatrix} 0.866025 & 0.500000 \\ -50.0000 & 86.6025 \end{bmatrix}$ to obtain $\mathbf{U}_{\mathbf{Y}}$:

Assembling the complete output covariance matrix $\mathbf{U}_{\mathbf{Y}}$:

Step 5: Output Uncertainties & Correlation Analysis

The correlation coefficient is:

Although the raw sensor inputs ($r$ and $\theta$) were 100% uncorrelated, the coordinate transformation induced a massive negative correlation ($r = -0.85$). When $X$ shifts positive due to angular deviation, $Y$ must shift negative!

Step 6: Eigendecomposition & Coverage Ellipse Geometry

We find the eigenvalues of $\mathbf{U}_{\mathbf{Y}}$ by solving the characteristic equation $\det(\mathbf{U}_{\mathbf{Y}} - \lambda \mathbf{I}) = 0$:

Where:

Solving the quadratic equation gives the two principal variances:

Step 7: Determine the 95% Coverage Factor & Ellipse Dimensions

For dimension $m = 2$ at coverage probability $p = 0.95$, we look up the chi-square quantile:

The semi-major axis $a$ and semi-minor axis $b$ of the 95% coverage ellipse are:

6. Multivariate Monte Carlo Method (JCGM 102 Clause 7)

When the measurement model $\mathbf{f}(\mathbf{X})$ is strongly non-linear (e.g. high angular uncertainties over large distances) or when input distributions are non-normal (e.g. asymmetric U-shaped RF phase errors), the first-order Taylor series approximation $\mathbf{U}_{\mathbf{Y}} = \mathbf{J} \mathbf{U}_{\mathbf{X}} \mathbf{J}^T$ becomes inadequate.

JCGM 102 Section 7 formalizes the Multivariate Monte Carlo Method (MCM):

1. Sample Generation: Generate $M$ independent sample vectors $\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_M$ (typically $M = 10^5$ to $10^6$) from the assigned multivariate joint probability density function $g_{\mathbf{X}}(\boldsymbol{\xi})$ of the inputs.
2. Vector Model Evaluation: For each drawn vector $\mathbf{x}_k$, evaluate the vector measurement model:
   $\mathbf{y}_k = \mathbf{f}(\mathbf{x}_k), \quad k = 1, 2, \dots, M$
3. Sample Mean and Covariance: Evaluate the sample average vector $\bar{\mathbf{y}}$ and sample output covariance matrix $\mathbf{S}_{\mathbf{y}}$:
   $\bar{\mathbf{y}} = \frac{1}{M} \sum_{k=1}^M \mathbf{y}_k, \qquad \mathbf{S}_{\mathbf{y}} = \frac{1}{M-1} \sum_{k=1}^M (\mathbf{y}_k - \bar{\mathbf{y}})(\mathbf{y}_k - \bar{\mathbf{y}})^T$
4. Coverage Region Construction: Compute the Mahalanobis distance $d_{M,k}^2$ for all $M$ points relative to $\bar{\mathbf{y}}$ and $\mathbf{S}_{\mathbf{y}}$. Sort the distances in ascending order, and define the empirical coverage region boundary at the $p \cdot M$-th index.
5. Validation of the Linear Formulation: Compare the coverage factor and region bounds obtained via MCM against the linear Jacobian propagation. If the difference is smaller than the laboratory's numerical tolerance criterion $\delta$, the linear GUM S2 propagation is validated.

7. ISO/IEC 17025 Accreditation & Reporting Guidelines

For laboratories accredited under ISO/IEC 17025:2017, reporting vector measurands on calibration certificates requires specific disclosures to prevent client misinterpretation:

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