This chapter discusses the approach used to estimate the uncertainty of a quantity (or subject parameter) that is computed from measurements of two or more attributes or parameters. The multivariate uncertainty analysis procedure consists of the following steps:
4. Develop the Parameter Value Equation 5. Develop the Error Model
6. Develop the Uncertainty Model
7. Identify the Measurement Process Errors 8. Estimate Measurement Process Uncertainties 9. Account for Error Source Correlations 10. Combine Uncertainties
The processes for developing error models and uncertainty models from the parameter value equation are presented. Identifying measurement process errors, estimating their uncertainties and accounting for cross-correlations are also presented. The volume of a cylinder obtained from length and diameter measurements is used to illustrate the concepts and methods of conducting a multivariate uncertainty analysis.
5. 1 Definitions
5.1.1 Coefficient Equation. An equation that expresses the partial derivative of a parameter value equation or module output equation with respect to a selected parameter or error source.
This equation is used to compute the sensitivity coefficient for the selected parameter or error source.
5.1.2 Combined Uncertainty. The uncertainty in the total error of a value of interest.
5.1.3 Component Error. The error in the measurement of a given component of a multivariate measurement. For example, when expressing cylinder volume as a function of length and diameter components, the error in the cylinder volume measurement, εV, can be expressed in terms of the component errors, εLand εD.
5.1.4 Component Uncertainty. The product of the sensitivity coefficient and the standard uncertainty for a component error.
5.1.5 Computed Parameter Value. The parameter value computed on UncertaintyAnalyzer's Multivariate Analysis Screen. Based on user specified adjusted mean values for root variables and the Parameter Value Equation.
5.1.6 Cross-correlation. The correlation between two error sources for two different compo-nents of a multivariate analysis. For example, if the same person (operator) makes measure-ments of both component X and component Y, then there is a cross-correlation between the operator biases for two components.
5.1.7 Compensating Biases. Measuring parameter biases that offset or compensate one another.
For example, if the same measuring parameter is used to measure the inside diameter of a sleeve and the outside diameter of a shaft that fits into the sleeve, any error or bias in the two
measurements will not affect the quality of fit.
5.1.8 Effective Degrees of Freedom. The degrees of freedom for combined uncertainties computed from the Welch-Satterthwaite formula.
5.1.9 Multivariate Measurements. Measurements in which the subject parameter is a computed quantity based on measurements of two or more attributes or parameters.
5.1.10 Nested Variables. Variables that are defined as a function of root variables or other nested variables.
5.1.11 Nested Variables Equations. Equations that define variables in terms of root variables or other nested variables. They are entered after the parameter value equation.
5.1.12 Parameter Value Equation. A mathematical expression that defines the value of a measurement or parameter value in terms of the values of constituent root variables or error sources.
5.1.13 Root Variables. Variables used to compute the parameter value that are not a function of other variables. Root variable information is specified by the user via the Error Source
Worksheets.
5.1.14 Sensitivity Coefficient. For a multivariate analysis, the sensitivity coefficient of a given root variable is the partial derivative of the parameter value equation with respect to a root variable. For a system analysis, the sensitivity coefficient for a given module parameter is the partial derivative of the module output equation with respect to a module parameter.
5.1.15 System Equation. A mathematical expression that defines the value of a quantity in terms of its constituent variables or components.
5.1.16 Total Uncertainty. See Combined Uncertainty.
5.2 Parameter Value Equation
The parameter value equation is a mathematical relationship between the quantity of interest (subject parameter) and the variables or components to be measured. The parameter value equation is also referred to as the system or governing equation.
For example, suppose we wish to know the volume of a cylinder. The parameter value equation for the cylinder volume is given as
2
2 ⎟
⎠
⎜ ⎞
⎝
= ⎛D L
V π (5-1)
where L and D are the cylinder length and diameter, respectively. From this equation, we see that, to determine the cylinder volume, we need to measure length and diameter components.
In UncertaintyAnalyzer, the parameter value is computed based on user specified mean or nominal values for root variables included in the parameter value equation and other nested variables equations.
5.3 Error Model
In any given measurement scenario, each measured quantity is a potential source of error. For example, errors in the length and diameter measurements will contribute to the overall error in the estimation of the cylinder volume. Therefore, the cylinder volume equation can be expressed as D0 = true or nominal cylinder diameter L0 = true or nominal cylinder length
εV = error in the cylinder volume εD= error in the cylinder diameter εL= error in the cylinder length
By rearranging equation (5-2), we obtain an algebraic expression for the cylinder volume error.
( )
be small compared to the other first order terms. Neglecting these terms, the cylinder volume error equation can be expressed in a simpler form.L
Rearranging equation (5-4), we can further simplify the equation for εV.
D
The coefficients for εLand εDin equation (5-5) are actually the partial derivatives of V with respect to L and D.
Therefore, the cylinder volume error model can be expressed as
D
where the partial derivatives are sensitivity coefficients that determine the relative contribution of the errors in length and diameter to the total error.
5.4 Uncertainty Model
Axiom 3 provides the direct link between error and uncertainty that we need to quantify