Arvind Rajan

Analytical Standard Uncertainty Evaluation Using Mellin Transform

Uncertainty evaluation plays an important role in ensuring that a designed system can indeed achieve its desired performance. There are three standard methods to evaluate the propagation of uncertainty: 1) analytic linear approximation; 2) Monte Carlo (MC) simulation; and 3) analytical methods using mathematical representation of the probability density function (pdf). The analytic linear approximation method is inaccurate for highly nonlinear systems, which limits its application. The MC simulation approach is the most widely used technique, as it is accurate, versatile, and applicable to highly nonlinear systems. However, it does not define the uncertainty of the output in terms of those of its inputs. Therefore, designers who use this method need to resimulate their systems repeatedly for different combinations of input parameters. The most accurate solution can be attained using the analytical method based on pdf. However, it is unfortunately too complex to employ. This paper introduces the use of an analytical standard uncertainty evaluation (ASUE) toolbox that automatically performs the analytical method for multivariate polynomial systems. The backbone of the toolbox is a proposed ASUE framework. This framework enables the analytical process to be automated by replacing the complex mathematical steps in the analytical method with a Mellin transform lookup table and a set of algebraic operations. The ASUE toolbox was specifically designed for engineers and designers and is, therefore, simple to use. It provides the exact solution obtainable using the MC simulation, but with an additional output uncertainty expression as a function of its input parameters. This paper goes on to show how this expression can be used to prevent overdesign and/or suboptimal design solutions. The ASUE framework and toolbox substantially extend current analytical techniques to a much wider range of applications.

Benchmark Test Distributions for Expanded Uncertainty Evaluation Algorithms

Expanded uncertainty estimation is normally required for mission-critical applications, e.g., those involving health and safety. It helps to get a distribution range of the required confidence level for the uncertainty evaluation of a system. There are a number of available techniques to estimate the expanded uncertainty. However, there is currently no commonly accepted benchmark test distribution set adopted to compare the performances of different techniques when they are used to estimate the expanded uncertainty. Without such a common benchmarking platform, the relative reliability of a particular technique in comparison to other techniques can be untrustworthy. To address the shortcoming, this paper proposes a set of analytically derived benchmark test distributions. It goes on to show the benefits of using them by comparing the performance of existing distribution fitting techniques when applied to the moment-based expanded uncertainty evaluation. The most commonly used moment-based distribution fitting techniques, such as Pearson, Tukeys gh, Cornish-Fisher expansion, and extended generalized lambda distributions, are employed as test cases in this paper. The test distribution set proposed in this paper provides a common benchmarking platform for metrologists intending to assess the performance of different expanded uncertainty estimation techniques. Results from the performance comparison would help practitioners to make a better choice of a distribution fitting technique that would best suit their respective systems.

Standard Uncertainty estimation on polynomial regression models

Polynomial regression model is very important in the modeling and characterization of sensors. The uncertainty propagation through the polynomial nonlinearity can only be estimated through numerical simulation or linearization approximation according to the Guide to the expression of Uncertainty in Measurement. This paper developed a general cookbook style guide to derive the analytical expression of uncertainty propagating through the polynomial regression models. The proposed method can be easily incorporated into any computer algebra system for reliable and fast evaluation. Specific expressions are derived explicitly for some of the most commonly used low order polynomial regression models. The framework is applied to a few recently published sensor and measurement system models.

Moments and maximum entropy method for expanded uncertainty estimation in measurements

The normal approximation and Monte Carlo simulation methods are widely used in the metrology to evaluate the expanded uncertainty, whereby the latter method is known to be the most robust and reliable. In some cases, however, (e.g., when the probability distribution is not known a priori) different frameworks may be desired as an alternative to the aforementioned techniques. One of them is commonly used in metrology — it is the moment (or cumulant)-based method. In view of that, and specifically for the scope of the expanded uncertainty estimation, this paper studies the theoretical viability of using high-order moments. It also analyzes the performance of a relatively new parametric distribution fitting technique known as the maximum entropy method. The discussions in the paper substantiate the confident application of the moment-based approach among practitioners. Furthermore, the results from the performance analysis of the maximum entropy method could guide practitioners in selecting a distribution fitting algorithm that best suits their respective systems.

Performance comparison between expanded uncertainty evaluation algorithms

The use of normal approximation to estimate expanded uncertainty has been very widespread; yet this is one of the practices that is being criticized by various quarters for lack of rigor and potentially misleading. Monte Carlo method is probably the only method trusted to generate reliable expanded uncertainty. Unfortunately, Monte Carlo method is not applicable for type-A evaluations. This is one of the challenges faced by current researchers in measurement community. This paper presents the comparison of expanded uncertainty estimation accuracy between Monte Carlo method, normal approximation and four well-known moment based distribution fitting methods. The Cornish-Fisher approximation is found to be consistently better than normal approximation but none of the moment based approach is comparable to Monte Carlo method in terms of accuracy and consistency.

Analytic standard uncertainty evaluation of polynomial in normal/uniform random variables

The standard uncertainty evaluation is very important in instrumentation and measurement industry because it is used to communicate, compare and combine uncertainty generated by various components in a system. The analytical evaluation of uncertainty has been recognized to be important and carries many advantages from theoretical perspective. Due to perceived complexity and feasibility of mathematical operation, the current practice of analytic uncertainty evaluation is confined to linear or linearized measurement equations, although the linearization is not always justifiable. A simple yet exact analytical method to evaluate standard uncertainty for polynomial nonlinearity was proposed by the authors, but the complexity of the method is high due to comprehensive and complete nature of the method. This paper presents a simplified procedure for normal and uniformly distributed random variables by taking advantage of the symmetry and simplicity of the functional forms. These two types of distributions are the most commonly used distributions in uncertainty analysis either through central limit theorem or maximal entropy principle. The effectiveness of the procedures is demonstrated using documented cases.

Moment-Constrained Maximum Entropy Method for Expanded Uncertainty Evaluation

The probability distribution is often sought in engineering for the purpose of expanded uncertainty evaluation and reliability analysis. Although there are various methods available to approximate the distribution, one of the commonly used ones is the method based on statistical moments (or cumulants). Given these parameters, the corresponding solution can be reliably approximated using various algorithms. However, the commonly used algorithms are limited by only four moments and assume that the corresponding distribution is unimodal. Therefore, this paper analyzes the performance of a relatively new and an improved parametric distribution fitting technique known as the moment-constrained maximum entropy method, which overcomes these shortcomings. It is shown that the uncertainty (or reliability) estimation quality of the proposed method improves with the number of moments regardless of the distribution modality. Finally, the paper uses case studies from a lighting retrofit project and an electromagnetic sensor design problem to substantiate the computational efficiency and numerical stability of the moment method in design optimization problems. The results and discussions presented in the paper could guide engineers in employing the maximum entropy method in a manner that best suits their respective systems.

Measured Quantity Value Estimator for Multiplicative Nonlinear Measurement Models

An estimate of a measurand using a nonlinear function of uncorrelated input quantities can be done by either applying the nonlinear function to the means of the input quantities (Method 1) or calculating the mean of a set of values obtained from propagating individual measurement values through the nonlinear function (Method 2). This paper proposes an improvement over the standard Method 2 procedures when the input quantities are assumed to be statistically independent and the nonlinear function has a general sum-of-product form, which covers many common measurement models. This paper shows that the proposed new approach (called Method 2S), if applicable, always produces a mean-squared error smaller than that of the conventional Method 2 procedures. The proposed approach improves the efficiency of Type-A evaluation as well as the Monte Carlo method. It also well complements the mainstream practices in the measurement uncertainty evaluation.

Moment-based measurement uncertainty evaluation for reliability analysis in design optimization

System uncertainties play a major role in reliability analysis performed in design optimization. According to the Guide to the expression of Uncertainty in Measurement, linear approximation or Monte Carlo simulation can be used to perform the reliability analysis. Unfortunately the linear approximation is unreliable for non-linear problems and the Monte Carlo approach is computationally expensive for the iterative process used in design optimization. The current state-of-the-art techniques in design optimization bypass the direct uncertainty evaluation of the system outputs by finding the first point of failure known as the most probable point. Such an approach introduces additional iterations into the optimization framework and hence leads to high computational time in complex reliability-based design optimization problems. To address the shortcoming, this paper uses a novel moment-based approach to evaluate the measurement uncertainty, and then to perform the reliability analysis. This shortens the computational time significantly while allowing for better quality in the final design. The proposed approach was implemented on a real-world problem of designing an aerospike nozzle. The results show that the proposed method achieves the expected high quality of final design with up to 7-fold shorter computational time compared to the current state-of-the-art techniques.