4.6. Monte Carlo Simulation¶
Under the TAB “Results”, the user can start a Monte Carlo simulation for cross-checking the value and the uncertainty of the output quantity y.
The simulation is done for a chosen (large) number of simulations of the measurement. For this purpose, for each of those quantities having been defined in the symbol list under the TAB “Equations” and characterized there as independent (u) input quantities, simulated input values are taken from their correspondent distributions (normal, rectangular or triangular distributions). The underlying individual distributions have been determined under the TAB “Values, uncertainties”. If the Low-Level Applications, (N+x)-rule has been selected there for counting numbers, their associated values are MC sampled according to a Gamma distribution; the values of the counting rates derived from the counting numbers then are also Gamma distributed. If all these input quantities have got a simulated value, a value of the output quantity - “the first simulated value of y” - is calculated according to the equations already defined.
From the many-fold repetition of this step a statistical distribution of the values of the output quantity is obtained from which its best estimate and its associated standard uncertainty are calculated as arithmetic mean and standard deviation, respectively. At present, with this method only quantities having normal, rectangular or triangular distributions can be considered (see Obtaining MC distributions and statistics derived of it in detail).
The great advantage of this method is that partial derivatives with respect to the independent quantities are not needed!
Note: the random generator used here has a period of about \(10^{18}\).
The procedure just described is correct for the case that no correlations exist between the input quantities. If in the TAB “Values, uncertainties”, however, covariances between pairs of correlating quantities have been given, correlated simulated values must be attributed to these quantities.
Important note on this issue: Since the UncertRadio version 0.05 (2007/11) this method of producing correlated variables was completely adapted to methods described in textbooks where matrix methods are used (keyword: “Gaussian distributed random numbers in n dimensions”; S. Brandt, Datenanalyse; V. Blobel and E. Lohrmann, Statistische und numerische Methoden der Datenanalyse; as well as the Draft of the new Supplement 1 of ISO GUM). This can be proven by trying the UncertRadio example project files Kessel-2a-2006.txp and Kessel-2b-2006.txp from the recent publication by Kessel et al. (2006) (see also: Meaning of the TAB “Uncertainty budget”).
Intermediate results of the MC simulation, partially consisting of tables, are now (Version 2.2.11) collected in a separate text file MC_Tables.txt.
Generally, one will find a good agreement between the results from the MC simulation method and from the analytical method. Therefore, the MC method is a relatively easy and elegant alternative to the more extensive analytical procedure.
What can be followed from deviations between the two methods?
If by using the MC method a result is obtained which deviates from that by the analytical method, one could easily conclude that there could be an error somewhere in the analytical procedure. However, this conclusion not always needs to be true!
What the described MC method actually is calculating can be interpreted as “Propagation of distributions”. This means that it in principle the expectation value of the output quantity is estimated, i.e. the following n-fold integral:
Herein, \(F\left( x_{1},x_{2},\ldots,x_{n} \right)\) designates in compact form the equations which are necessary for the calculation of y (i.e. the formula with which the value of y is usually calculated) and \(\varphi_{i}\left( x_{i} \right)\) the probability density functions of the n input quantities \(x_{i}\) characterized as independent. The widths of the \(\varphi_{i}\left( x_{i} \right)\) are determined from their associated measurement uncertainties \(u\left( x_{i} \right)\). An important assumption for the conventional propagation of uncertainties is that the uncertainties of the \(x_{i}\) should be small. In this case, the probability density functions \(\varphi_{i}\left( x_{i} \right)\) approximately become delta functions with the consequence that the n-fold integral reduces to the conventionally calculated value \(F\left( x_{1},x_{2},\ldots,x_{n} \right)\) of the output quantity y. In this sense deviations between both methods may occur with respect to the output quantity and its uncertainty if any of the involved uncertainties “are not small” which might also be if such quantities belong to the denominator of the evaluation equation (non-linearity).
The collection of project files contains examples in which the discussed deviation between both methods is significant:
Project file |
Special feature |
ISO-Example-1a_EN.txp, ISO-Example-1b_EN.txp |
Here, the alpha self-absorption factor f, having a rather broad rectangular distribution and belonging to the denominator, causes this effect. |
Neut ron-Dose-Cox-2006_EN.txp |
The field specific correction K with a significantly broad rectangular distribution has practically the same effect as in the wipe test example above: the resultant distribution is significantly asymmetric. |
Calibration-o f-weight-Cox-2001_EN.txp |
In this example a significantly larger measurement uncertainty results from the MC method. Rectangular distributions are attributed to three of the involved input quantities. |
Wuebbeler-Ex1_EN.txp |
A non-linear model function in combination with large uncertainties of normal distributed input quantities result in an asymmetric distribution of the output quantity. |
Wuebbeler-Ex2_EN.txp |
Rectangular distributed input quantities result in a trapezoidal distribution of the output quantity. |
