Propagation of Error / Uncertainty

Propagation of Error / Uncertainty

1. Estimating Propagation Error

Consider a set of observed variables \(x\) with associated errors \(e_x\) (i.e. standard deviation); to find the output error derived from the propagation of input errors in a function / model such as:

\[z = f(x_1, x_2, \cdots, x_n) \]

The following model may be used:

\[e_{z}^{2} = \sum_{i} \left( \frac{\partial f}{\partial x_{i}} \right)^{2} e_{x_i}^{2}+\sum_{i} \sum_{j \neq i} \frac{\partial f}{\partial x_{i}} \frac{\partial f}{\partial x_{j}} e_{x_{i}} e_{x_{j}} r_{i j} \]

where \(r_{ij}\) is the coefficient of correlation between variables \(x_i\) and \(x_j\)

• The formula is exact for linear functions and a reasonable approximation in other cases.

• If variables chosen are uncorrelated, then second term approaches to zero, that is:

  \[e_{z}^{2} = \sum_{i} \left( \frac{\partial f}{\partial x_{i}} \right)^{2} e_{x_i}^{2} \]

2. Marginal Improvement Rate of Propagation Error

If we take the partial derivative of \(e_z\) with respect to \(e_{x_i}\) and ignore the correlation term by assuming the un-correlation, we get:

\[\frac{\partial e_{z}}{\partial e_{x_{i}}}=\left(\frac{\partial f}{\partial_{x_{i}}}\right)^{2} \frac{e_{x_{i}}}{e_{z}} \]

Using these marginal improvement rates and an estimation of the marginal costs of enhancing data accuracy it is possible to determine an optimum improvement budget.

In practice this problem is not easy because of the law of diminishing returns (回报递减, 边际收益递减) (i.e. each further percentage reduction in the error of a variable will tend to cost proportionately more). However, the above serves to deduce two logical rules:

• Concentrate the improvement effort on those variables with a large error (i.e., \(e_{x_i}\));

• Concentrate the effort on the most relevant variables, i.e. those with the largest value of \(\partial f / \partial x_i\) as they have the largest effect on the dependent variable.

Rules of thumb for building or choosing models if choices are available:

1. Avoid inter-correlated variables;

2. Take "addition" where possible;

3. If you can not take "addition", "multiplication" or "division" is the possible choice;

4. Avoid as far as possible take differences or raising variable to power.

3. Example

Assume that model \(z = f(x, y)\) where \(x = 10 \pm 1\) and \(x = 8 \pm 1\). And further assume that \(x\) and \(y\) are mutually independent. Relative errors of independent variables: \(e_x/x = 10 \%\) and \(e_y/y = 12.5 \%\).

Arithmetic Operator	Model	Propagation error: \(\displaystyle e_{z}^{2} = \left( \frac{\partial f }{ \partial x} \right)^{2} e_{x}^{2} + \left( \frac{\partial f }{ \partial y } \right)^{2} e_{y}^{2}\)
Addition	\(z=f(x,y)=x+y\) \(z=10+8=18\)	\(e_{z}=1.4\) \(e_{z}/z=7.8\%\)
Subtraction	\(z=f(x,y)=x-y\) \(z=10-8=2\)	\(e_{z}=1.4\) \(e_{z}/z=70.0\%\)
Multiplication	\(z=f(x,y)=xy\) \(z=10 \times 8=80\)	\(e_{z}=12.8\) \(e_{z}/z=16 \%\)
Division	\(z=f(x,y)=x^2\) \(z=10/8=1.25\)	\(e_{z}=0.2\) \(e_{z}/z=16 \%\)
Power	\(z=f(x,y)=x^2\) \(z=10^2=100\)	\(e_{z}=20.0\) \(e_{z}/z=20 \%\)

References

[1] J. de D. Ortúzar S. and L. G. Willumsen, "3.2.2 The Model Complexity/Data Accuracy Trade-off", in Modelling Transport, Fourth edition. Chichester, West Sussex, United Kingdom: John Wiley & Sons, 2011, p.p. 68-71.

[2] Propagation of uncertainty, wikipedia, 地址.