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2. Measurement Process Characterization
2.5. Uncertainty analysis

2.5.5. Propagation of error considerations

Top-down approach consists of estimating the uncertainty from direct repetitions of the measurement result The approach to uncertainty analysis that has been followed up to this point in the discussion has been what is called a top-down approach. Uncertainty components are estimated from direct repetitions of the measurement result. To contrast this with a propagation of error approach, consider the simple example where we estimate the area of a rectangle from replicate measurements of length and width. The area

$$ area = length \cdot width $$

can be computed from each replicate. The standard deviation of the reported area is estimated directly from the replicates of area.

Advantages of top-down approach This approach has the following advantages:
  • proper treatment of covariances between measurements of length and width
  • proper treatment of unsuspected sources of error that would emerge if measurements covered a range of operating conditions and a sufficiently long time period
  • independence from propagation of error model
Propagation of error approach combines estimates from individual auxiliary measurements The formal propagation of error approach is to compute:
  1. standard deviation from the length measurements
  2. standard deviation from the width measurements
and combine the two into a standard deviation for area using the approximation for products of two variables (ignoring a possible covariance between length and width),

$$ s_{area} = \sqrt{width^2 \cdot s_{length}^2 + length^2 \cdot s_{width}^2} $$

Exact formula Goodman (1960) derived an exact formula for the variance between two products. Given two random variables, \(x\) and \(y\) (correspond to width and length in the above approximate formula), the exact formula for the variance is:
    $$ V(\bar{x} \bar{y}) = \frac{1}{n} \left[ X^2 V(y) + Y^2 V(x) + 2XYE_{11} + 2X\frac{E_{12}}{n} + 2Y\frac{E_{21}}{n} + \frac{V(x) V(y)}{n} + \frac{Cov((\Delta x)^2, (\Delta y)^2) -E_{11}^2 }{n^2} \right] $$
with
  • \(X = E(x)\) and \(Y = E(y)\) (corresponds to width and length, respectively, in the approximate formula)

  • \(V(x)\) is the variance of \(x\) and \(V(y)\) is the variance \(y\) (corresponds to \(s^2\) for width and length, respectively, in the approximate formula)

  • \( E_{ij} = {(\Delta x)^i, (\Delta y)^j}\) where \( \Delta x = x - X \) and \( \Delta y = y - Y \)

  • \( Cov((\Delta x)^2, (\Delta y)^2) = E_{22} - V(x) V(y) \)

To obtain the standard deviation, simply take the square root of the above formula. Also, an estimate of the statistic is obtained by substituting sample estimates for the corresponding population values on the right hand side of the equation.

Approximate formula assumes indpendence The approximate formula assumes that length and width are independent. The exact formula assumes that length and width are not independent.
Disadvantages of propagation of error approach In the ideal case, the propagation of error estimate above will not differ from the estimate made directly from the area measurements. However, in complicated scenarios, they may differ because of:
  • unsuspected covariances
  • disturbances that affect the reported value and not the elementary measurements (usually a result of mis-specification of the model)
  • mistakes in propagating the error through the defining formulas
Propagation of error formula Sometimes the measurement of interest cannot be replicated directly and it is necessary to estimate its uncertainty via propagation of error formulas (Ku). The propagation of error formula for

$$ Y = f(X, Z, \ldots \, ) $$

a function of one or more variables with measurements, \( (X, Z, \ldots \, ) \) gives the following estimate for the standard deviation of \( Y \):

$$ s_y = \sqrt{ \left( \frac{\partial Y}{\partial X} \right)^2 s_x^2 + \left( \frac{\partial Y}{\partial X} \right)^2 s_z^2 + \cdots + \left( \frac{\partial Y}{\partial X} \right) \left( \frac{\partial Y}{\partial Z} \right) s_{xz}^2 + \cdots } $$

where

  • \( s_x \) is the standard deviation of the \( X \) measurements
  • \( s_z \) is the standard deviation of \( Z \) measurements
  • \( s_y \) is the standard deviation of \( Y \) measurements
  • \( \partial Y / \partial X \) is the partial derivative of the function \( Y \) with respect to \( X \), etc.
  • \( s_{xz} \) is the estimated covariance between the \( X,Z \) measurements
Treatment of covariance terms Covariance terms can be difficult to estimate if measurements are not made in pairs. Sometimes, these terms are omitted from the formula. Guidance on when this is acceptable practice is given below:
  1. If the measurements of \(X\), \(Z\) are independent, the associated covariance term is zero.
  2. Generally, reported values of test items from calibration designs have non-zero covariances that must be taken into account if \(Y\) is a summation such as the mass of two weights, or the length of two gage blocks end-to-end, etc.
  3. Practically speaking, covariance terms should be included in the computation only if they have been estimated from sufficient data. See Ku (1966) for guidance on what constitutes sufficient data.
Sensitivity coefficients The partial derivatives are the sensitivity coefficients for the associated components.
Examples of propagation of error analyses Examples of propagation of error that are shown in this chapter are:
Specific formulas Formulas for specific functions can be found in the following sections:
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