PolyCalAPP

In metrology, the straight-line calibration and the polynomial calibration problem is addressed in the ISO Technical Specifications ISO/TS 28037:2010 and ISO/TS 28038:2018 which are based on the GUM uncertainty framework defined in GUM[3] using the law of propagation of uncertainty.

Here, in the considered calibration experiment, the direct measurements taken by two independent measurement devices are represented by $m$-dimensional vectors

\begin{align}\label{eq1} \mathbf X &= \boldsymbol \mu + E_{\mathcal X} = \boldsymbol \mu + \left(\mathbf X^{A} + \mathbf X^{B} + X^{B}_0 \mathbf 1 \right),\\ \mathbf Y &= \boldsymbol \nu + E_{\mathcal Y} = \boldsymbol \nu + \left(\mathbf Y^{A} + \mathbf Y^{B} + Y^{B}_0 \mathbf 1 \right), \label{eq1b} \end{align}

\[ E \begin{pmatrix} \mathbf X\\ \mathbf Y \end{pmatrix} = \begin{pmatrix} \boldsymbol \mu\\ \boldsymbol \nu \end{pmatrix}, \quad \mathop{\mathrm{Cov}} \begin{pmatrix} \mathbf X\\ \mathbf Y \end{pmatrix} = \mathbf \Sigma = \begin{pmatrix} \mathbf \Sigma_{\mathcal X} & \mathbf \Sigma_{\mathcal X,\mathcal Y}\\ \mathbf \Sigma_{\mathcal Y,\mathcal X} & \mathbf \Sigma_{\mathcal Y} \end{pmatrix} \]

where \(\boldsymbol \mu = \left(\mu_1,\dots,\mu_m\right)'\) and \( \boldsymbol \nu = \left(\nu_1,\dots,\nu_m\right)'\) represent the true unknown values of the measurands expressed in units of the measuring devices. These parameters are functionally related by the polynomial calibration function,

\[ \boldsymbol \nu = \beta_0 \boldsymbol 1 + \beta_1 \boldsymbol \mu + \beta_2 \boldsymbol \mu^2 + \cdots + \beta_p \boldsymbol \mu^p \]

The components of the measurement errors are assumed to be mutually independent zero-mean random variables, fully specified by their probability distributions (here by their characteristic functions)derived by Type A and Type B methods of evaluation.

PolyCal estimates the patrameters of the polynomial calibration function \(\boldsymbol \beta = (\beta_0,\dots,\beta_p)'\) together with their uncertainties (covariance matrix) and their marginal state-of-knowledge distributions. Moreover, the algorithm evaluates the fitted values \(y_{Fit}\) of the calibration function for given \(x_{Fit}\),

\[ y_{Fit} = \beta_0 + \beta_1 x_{Fit}^{} + \beta_2 x_{Fit}^2 +\cdots + \beta_p x_{Fit}^p, \]

together with their uncertainties and the state-of-knowledge distributions. The output from the PolyCal algorithm can be further used to evaluate the results of the future measurements by the calibrated instrument, together with their uncertainties and the state-of-knowledge distributions.

result = PolyCal(x,y)

result = PolyCal(x,y,[],options)

MATLAB Code

x = [4.7, 5.5, 6.5, 7.5, 8.4]';
y = [5.8, 7.9, 10.4, 12.7, 15.2]';
xFit = linspace(5,8,11)';
clear options
options.order = 1;
options.cfXA = {@(t)cf_Normal(0.01*t), ...
    @(t)cf_Normal(0.1*t), ...
    @(t)cf_Normal(0.1*t), ...
    @(t)cf_Student(0.1*t,5), ...
    @(t)cf_RectangularSymmetric(0.2*t)};
options.cfXB = {@(t)cf_RectangularSymmetric(0.1*t), ...
    @(t)cf_RectangularSymmetric(0.1*t), ...
    @(t)cf_TriangularSymmetric(0.1*t), ...
    @(t)cf_TriangularSymmetric(0.1*t), ...
    @(t)cf_RectangularSymmetric(0.2*t)};
options.cfXB0 = [];
options.cfYA = {@(t)cf_Normal(0.15*t), ...
    @(t)cf_Normal(0.2*t), ...
    @(t)cf_Normal(0.2*t), ...
    @(t)cf_Normal(0.3*t), ...
    @(t)cf_Normal(0.3*t)};
options.cfYB = {@(t)cf_RectangularSymmetric(0.2*t), ...
    @(t)cf_RectangularSymmetric(0.2*t), ...
    @(t)cf_TriangularSymmetric(0.3*t), ...
    @(t)cf_TriangularSymmetric(0.3*t), ...
    @(t)cf_RectangularSymmetric(0.3*t)};
options.cfYB0 = @(t)cf_ArcsineSymmetric(0.1*t);
options.tolDiff = 1e-2;
result = PolyCal(x,y,xFit,options);
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MATLAB Code

cfBeta = result.cfBeta;
resultBeta0 = cf2DistGP(cfBeta{1},[],[],options);
figure;
resultBeta1 = cf2DistGP(cfBeta{2},[],[],options);

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