# Limits of the Uncertainty Propagation: Examples and Solutions Using the Monte Carlo Method

1. French College of Metrology
1. Martin Müller,
2. Marco Wolf,
3. Dr. Matthias Rösslein and
4. Prof. Walter Gander

Published Online: 3 FEB 2010

DOI: 10.1002/9780470611371.ch53

## Transverse Disciplines in Metrology

#### How to Cite

French College of Metrology (2009) Limits of the Uncertainty Propagation: Examples and Solutions Using the Monte Carlo Method, in Transverse Disciplines in Metrology, ISTE, London, UK. doi: 10.1002/9780470611371.ch53

#### Author Information

1. ETH Zürich, Switzerland; Empa St. Gallen, Switzerland

#### Publication History

1. Published Online: 3 FEB 2010
2. Published Print: 1 JAN 2009

#### ISBN Information

Print ISBN: 9781848210486

Online ISBN: 9780470611371

## SEARCH

### Keywords:

• measurement process;
• monte carlo method;
• analytical solution;
• valued quantities;
• propagation

### Summary

The Guide to the Expression of Uncertainty in Measurement (GUM) [1] provides internationally agreed recommendations for the evaluation of uncertainties. It introduces the Law of Propagation of Uncertainty to determine the combined standard uncertainty. In a number of cases the application of this method can lead to physically senseless results. Therefore the first supplement to the GUM recommends the use of the Monte Carlo Method. We compare these two methods by means of two examples and show in addition how correlations should be handled using the Monte Carlo Method.

The calculation of the distance of a point to the origin is an example that demonstrates how the inappropriate approximation of the model equation by a Taylor series of order one can lead to physical senseless results using the Law of Propagation of Uncertainties. As the result of the Monte Carlo Method is an approximated probability density function it is possible to avoid such misleading results and help therefore to be more accurate in the documentation of the measurement uncertainty.

Building up physical dependencies by the aim of a correlation coefficient can lead to misunderstandings as the correlation coefficient does not describe a causal relationship of quantities. In this paper we distinguish between statistical correlations, i.e. correlations arising out of multiple measurements of two input quantities, and logical correlations, which accrue from the usage of equal influences in different input quantities. As correlations between two quantities can always be described by a number of uncorrelated quantities and two functions that build up the correlated quantities, it is throughout possible to describe statistical correlations by logical correlations. This process demands a deep knowledge about the measurement process and can be extensive, but it is the only way to take the causal relationship implied by the correlation coefficient in a correct manner into account.

With the aid of the software package MUSE it is possible to build up physical dependencies by using identical distributions, i.e. same random values in one Monte Carlo run, in different parts of the modelled measurement system. MUSE also supports the user by allowing a logical structuring of the model equation. Therefore it is becoming easier to model large measurement processes and identify dependencies within that processes.