An Improvement on the Standard Linear Uncertainty Quantification Using a LeastSquares Method
 Heejin Cho^{1}Email authorView ORCID ID profile,
 Rogelio Luck^{1} and
 James W. Stevens^{2}
https://doi.org/10.1186/s4046701500419
© Cho et al. 2015
Received: 2 September 2015
Accepted: 17 November 2015
Published: 2 December 2015
Abstract
Linear uncertainty analysis based on a first order Taylor series expansion, described in ASME PTC (Performance Test Code) 19.1 “Test Uncertainty” and the ISO Guide for the “Expression of Uncertainty in Measurement,” has been the most widely technique used both in industry and academia. A common approach in linear uncertainty analysis is to use local derivative information as a measure of the sensitivity needed to calculate the uncertainty percentage contribution (UPC) and uncertainty magnification factors (UMF) due to each independent variable in the measurement/process being examined. The derivative information is typically obtained by either taking the symbolic partial derivative of an analytical expression or the numerical derivative based on central difference techniques. This paper demonstrates that linear multivariable regression is better suited to obtain sensitivity coefficients that are representative of the behavior of the data reduction equations over the region of interest. A main advantage of the proposed approach is the possibility of extending the range, within a fixed tolerance level, for which the linear approximation technique is valid. Three practical examples are presented in this paper to demonstrate the effectiveness of the proposed leastsquares method.
Keywords
Introduction
The topic of estimation of experimental uncertainty is covered in a wide variety of forums. The American Society of Mechanical Engineers publishes an uncertainty standard as part of the performance test codes: ASME PTC 19.11998 “Test Uncertainty” [1]. The International Organization for Standardization (ISO) also publishes a guide on uncertainty calculation and terminology entitled “Guide to the Expression of Uncertainty in Measurement” [2]. These two approaches are compared by Steele et al. [3]. Most textbooks on experimental measurements include a section on uncertainty propagation as well (for example, Refs. [4–6]). Some textbooks specialize in uncertainty [7, 8]. The technical literature also has numerous treatments of uncertainty estimation and propagation in specific applications (for example, Refs. [9–12]). Although there are more sophisticated uncertainty quantification methods, including Monte Carlo [13], Bayesian [14], Latin square sampling techniques [15, 16], by far ASME PTC 19.11998 “Test Uncertainty” standard [1] is the most widely adopted in the current industrial applications. A main goal of this paper is to provide a simple improvement to the practical method provided by the ASME standard.
Background
Total Uncertainty and Covariance Matrix
Covariance Matrix Based on Uncorrelated Bias Errors
Gaussian multivariate probability density function
Development
Geometrical Interpretations
Roberts et al. [19] suggest that the uniform space approach requires fewer function evaluations compared to the uniform probability approach to obtain comparable results for 1D cases. Therefore, uniformspace geometry is used to describe the uncertainty region in this paper. An easy way to generate a uniformspace grid is to inscribe the elliptical area within a rectangular area. Furthermore, to generate a large number of samples for an increased accuracy of the uncertainty results, an efficient sampling technique, such as Latin hypercube sampling technique [15, 16], can be employed to reduce computational cost.
LeastSquares Approach
Figure 4 shows that the leastsquares approach yields larger truncation errors near the high probability region, but improves the truncation errors over the entire the interval of interest leading to a better estimate of the 95 % confidence interval. Although Fig. 4 illustrates a conceptual comparison, the results of the examples represented in the following section support this idea.
Piecewise approach
The piecewise approach is a method used to estimate probability distribution by sorting discrete probabilities cumulatively. This method was described in detail for one dimension in [20]. The extension to multiple dimensions can be used to estimate accurate values for the confidence interval in cases where the exact solution is not available.
Examples
Three simple examples will be presented to illustrate the leastsquares approach to uncertainty estimation.
Numerical values of parameters used to find uncertainty
(Unit: cm)  

b (Base)  h (Height)  
Mean (nominal value)  5  3 
Bias uncertainty  0.5  
Bias standard deviation  0.255 (=0.5/1.96)  
Correlation in bias  1  
Precision standard deviation  0.2  0.3 
Bias and precision covariance matrices are determined by Eqs. (4) and (5).
\( {C}_B=\left[\begin{array}{cc}\hfill {0.255}^2\hfill & \hfill (1)\cdot (0.255)\cdot (0.255)\hfill \\ {}\hfill (1)\cdot (0.255)\cdot (0.255)\hfill & \hfill {0.255}^2\hfill \end{array}\right] \) and \( {C}_P=\left(\begin{array}{cc}\hfill {0.2}^2\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill {0.3}^2\hfill \end{array}\right) \)
The matrices Λ and S are then determined from eigenvalue decomposition of the covariance matrix:\( \varLambda =\left[\begin{array}{cc}\hfill 0.06\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0.2\hfill \end{array}\right] \) and \( S=\left[\begin{array}{cc}\hfill 0.824\hfill & \hfill 0.566\hfill \\ {}\hfill 0.566\hfill & \hfill 0.824\hfill \end{array}\right] \)
The results of the above method are compared to a more accurate solution obtained by the piecewise approach which sections the input XY probability region into a thousand points.
(convection to the probe = radiation from the probe)
 T _{g} :

gas temperature (K)
 T _{t} :

thermocouple temperature (K)
 T _{w} :

pipewall temperature (K)
 ε :

emissivity of the thermocouple
 σ :

StefanBoltzmann constant (2.043 × 10^{−7} kJ/hrm^{2}K^{4})
 h :

convective heat transfer coefficient (kJ/hrm^{2}K).
Nominal, bias, and precision values of each variable
T _{g} (Temp. gas)  T _{t} (Temp. thermocouple)  T _{w} (Temp. pipewall)  ε (Emissivity)  σ (StefanBoltzmann constant)  

Mean (nominal value)  838 K  811 K  672 K  0.55  2.043 × 10^{−7} kJ/hrm^{2}K^{4} 
Precision standard deviation  8 K  7 K  5 K  N/A  N/A 
Bias standard deviation  4 K  3 K  N/A  N/A 
The comparison of the results in Table 3 shows that the confidence interval of the convective heat transfer coefficient (h) for the leastsquares approach provides a better approximation to the more realistic (but much more computationally intensive) piecewise approach than the Taylor series approach. Furthermore, the narrow uncertainty interval produced by the Taylor series approximation can lead to a “false sense of security” regarding the numerical value of the heat transfer coefficient.
Conclusions
A leastsquares approach to linear uncertainty analysis has been described and illustrated. This approach can provide improved results over ordinary uncertainty propagation using a first order Taylor series approximation by minimizing the truncation errors in the linear approximation of the equation being analyzed. A drawback of this approach is that there is no explicit formula to find the sensitivity coefficients. However, in many instances the sensitivity coefficients are obtained through numerical derivatives. In such cases, there is little or no additional computational effort in obtaining the leastsquares solution. This paper also shows a simple way to obtain the covariance matrix used in the uncertainty analysis. In many engineering applications, it is cumbersome to determine the correlation coefficients of the bias errors (ρ), i.e., reasonable engineering judgment is required. Therefore, the authors recommend using covariance matrix expressed in terms of uncorrelated bias errors as shown in the third example. The results in the examples illustrate the advantages of using the leastsquares approach.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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