Calculating measurement uncertainties with imc FAMOS

The estimation of measurement uncertainties is an essential part of metrological analyses in order to clarify the significance of the results. With the imc FAMOS analysis software, this can be efficiently calculated and evaluated even for complex algorithms.

This article provides a comprehensive overview of the options and procedures for calculating measurement uncertainties with imc FAMOS.

Basics of measurement uncertainty calculation

Before we look at the specific functions of imc FAMOS, it is important to understand some basic concepts of measurement uncertainty calculation.

The GUM as a guide

The “Guide to the expression of uncertainty in measurement” (GUM) serves as an international guideline for the determination and specification of measurement uncertainties. imc FAMOS is largely based on the GUM specifications, but goes beyond them in some areas in order to meet the practical requirements of test & measurement engineers.

Standard measurement uncertainty

When we talk about measurement uncertainty in this article, we are generally referring to the standard measurement uncertainty. This corresponds to the standard deviation of the measured values and indicates the range around the measured value in which the true value lies with a probability of around 68 % (assuming a Gaussian distribution).

Measurement uncertainty of individual measured values vs. sequences of measured values

The GUM primarily deals with the measurement uncertainty of individual measured values. In practice, however, you often have to deal with entire sequences of measured values or time series. imc FAMOS offers options for analyzing both individual characteristic values and entire data sets resulting from analysis sequences with regard to their measurement uncertainty

Determining the measurement uncertainty of the input data

The first step in calculating measurement uncertainties is to determine the uncertainty of the input data. This can be done in various ways:

Estimation from measurement data

In some cases, the measurement uncertainty can be estimated directly from the available measurement data. An example of this is estimating the measurement uncertainty from the noise band of a temperature curve.

Use of the UncertaintySet() command

In imc FAMOS, the measurement uncertainty of an input signal is set with the `UncertaintySet()` command.

Example: UncertaintySet(Temperature, "Uncertainty", 0.25)

This command assigns a measurement uncertainty of 0.25 °C to the “Temperature” input signal. For time series, this uncertainty is assigned to each individual measured value.

Consideration of specific disturbance variables

imc FAMOS allows specific disturbance variables such as noise, mains hum or offset drift to be taken into account in addition to the GUM. These can also be assigned using the `UncertaintySet()` function.

You can find out more about adding disturbances to data in our whitepaper

Measurement uncertainty and its effects on data analysis – Applications in imc FAMOS

Calculation of measurement uncertainty propagation

Once the input uncertainties have been defined, imc FAMOS can calculate the propagation of these uncertainties using an algorithm. The Monte Carlo method is used for this.

Monte Carlo method in imc FAMOS

The Monte Carlo method is based on adding noise or disturbances to the input data in different variations and running the algorithm several times with the respective noisy data. The deviations in the results are statistically evaluated in order to determine the measurement uncertainty of the results.

Carrying out the Monte Carlo simulation

The number of Monte Carlo trials is defined in imc FAMOS using the function `UNCERTAINTY_LOOP`. A typical procedure is as follows:

1. determination of the input uncertainty

2. loop over M Monte Carlo trials:

a. Adding noise to the input data in a new variation

b. Calculation of the algorithm with noisy data

c. Iterative calculation of the result uncertainty

3. Evaluation of the determined measurement uncertainty

Example: Calculation of the rise time

Here is an example sequence code for calculating the measurement uncertainty of the rise time of a temperature curve:

UncertaintySet(Temperature, "Uncertainty", 0.25)

UNCERTAINTY_LOOP 1000

_Temperature = UncertaintyModify(Temperature)

_Temperature = smo(_Temperature,1)

L1 = mean(cut(_Temperature,5,10))

L2 = mean(cut(_Temperature,30,35))

RiseTime = pos(_Temperature, L1+(L2-L1)*0.9) - pos(_Temperature, L1+(L2-L1)*0.1)

UncertaintyCalc(RiseTime)

End

uc = UncertaintyGet(RiseTime, "Uncertainty")

In this example, an input uncertainty of 0.25 °C is defined and 1000 Monte Carlo tests are carried out. At the end, the calculated measurement uncertainty of the rise time is queried.

You can find out more about the Monte Carlo method in our white paper

Determination of measurement uncertainty in imc FAMOS – Monte Carlo method

Analyzing the measurement uncertainty of measurement sequences

When analyzing measurement sequences or time series, there are some special challenges and possibilities.

Constant vs. non-constant measurement uncertainty budget

A distinction must be made as to whether the measurement uncertainty remains constant or changes over the entire data set. If the measurement uncertainty budget remains constant, the entire measurement can be characterized with a single value for the measurement uncertainty.

Example: Force measurement with measurement uncertainty dependent on the measured value

In a force measurement with an amplifier, the measurement uncertainty can depend on the measured value. An example of a force-dependent measurement uncertainty:

Uncertainty of measurement (force) = 2N + force * 0.015

In this case, the measurement uncertainty varies over the measuring range.

All-encompassing measurement uncertainty

To reduce a complex measurement uncertainty to a single value, an all-encompassing measurement uncertainty can be calculated. This is calculated as the root mean square of all individual standard deviations:

You can find out more about uncertainty budgets and all-encompassing uncertainties in our whitepaper

Measurement uncertainty of measured value series – Applications in imc FAMOS

Advanced analyses and visualizations

imc FAMOS offers various options for advanced analysis and visualization of measurement uncertainties.

Probability density function

The function `UncertaintyCalc()` enables the calculation of the probability density function (PDF) of the results. The smoothness of the PDF can be used to assess whether the number of Monte Carlo simulations was sufficient.

Detailed analysis of the measurement uncertainty

For complex algorithms, it may be useful to analyze the measurement uncertainty for each individual measurement point of the result data set. This can be realized with the function `UncertaintyCalc()` with appropriate parameters.

Example: Analyzing an amplitude spectrum

Here is an example of the detailed analysis of the measurement uncertainty of an amplitude spectrum:

UncertaintySet(acc, "Uncertainty", 3)

UNCERTAINTY_LOOP 1000

_acc = UncertaintyModify(acc)

AmplitudeSpectrum = AmpSpectrumRMS_1(_acc, 500, 2, 0, 1)

UncertaintyCalc(AmplitudeSpectrum)

End

uc = UncertaintyGet(AmplitudeSpectrum, "Uncertainty")

UncertaintyCalc(AmplitudeSpectrum, 99, 0, "uc")

This code first calculates the overall measurement uncertainty of the amplitude spectrum and then performs a detailed analysis for each individual spectral line.

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Practical applications and benefits of imc FAMOS in the calculation of measurement uncertainties

The calculation of measurement uncertainties with imc FAMOS offers many advantages and possible applications:

Validation of algorithms

The robustness of algorithms can be tested and validated by analyzing the propagation of measurement uncertainties.

Optimization of measurement setups

Detailed knowledge of measurement uncertainties makes it possible to specifically optimize measurement setups and identify weak points.

Quality assurance

In quality assurance, precise knowledge of measurement uncertainties is essential in order to be able to make reliable statements about compliance with tolerances.

Research and development

In research and development, measurement uncertainty analysis helps to understand and continuously improve the limits of measurement methods and algorithms.

You can find an example of improving the robustness of an algorithm in our whitepaper

Measurement uncertainty: Improvements in the algorithm – Applications in imc FAMOS

Conclusion

The calculation of measurement uncertainties with imc FAMOS offers extensive possibilities for the detailed analysis and optimization of measurement methods and evaluation algorithms. Thanks to the combination of user-friendly functions and powerful analysis methods such as Monte Carlo simulation, even complex measurement uncertainty analyses can be carried out efficiently.

However, it is important to note that the quality of the measurement uncertainty calculation depends largely on the correct assessment of the input uncertainties. A careful analysis of the measurement setup and all relevant influencing factors is therefore essential.

Overall, the measurement uncertainty calculation with imc FAMOS is a powerful tool that helps measurement technicians, engineers and scientists to improve the reliability and significance of their measurements and analyses and to make well-founded decisions based on the data obtained.

We have summarized the most important advantages of imc FAMOS for the calculation of measurement uncertainties here:

Processing large data sets: imc FAMOS is specially designed for analyzing time series with thousands or millions of measured values. This enables efficient processing of extensive measurement data.
Versatile analysis options: With imc FAMOS, the effects of measurement uncertainties can be analyzed for almost any algorithm written as a FAMOS sequence. This offers great flexibility when investigating different scenarios.
Monte Carlo method: imc FAMOS uses the Monte Carlo method to estimate the propagation of measurement uncertainties. This method enables a robust statistical evaluation of the results.
Adaptation to measurement technology requirements: Going beyond the GUM (Guide to the expression of uncertainty in measurement), imc FAMOS offers the option of taking into account disturbance variables relevant to measurement engineers, such as noise, mains hum or offset drift.
Advanced analyses: The UncertaintyCalc() function offers additional analyses, such as the calculation of the probability density function, in order to better understand and interpret the determined measurement uncertainty.
Flexible input of the measurement uncertainty: With the UncertaintySet() function, the measurement uncertainty for input signals can be easily defined, both for individual measured values and for entire time series.
Customizable number of Monte Carlo trials: Using the UNCERTAINTY_LOOP function, the number of Monte Carlo simulations can be freely selected to achieve the desired accuracy.
Calculation of the overall measurement uncertainty: imc FAMOS offers the possibility to calculate an overall measurement uncertainty for an entire data set, which simplifies the characterization of complex measurement series.
Detailed analysis of individual measurement points: In addition to the all-encompassing measurement uncertainty, imc FAMOS also allows you to examine the measurement uncertainty for each individual measurement point of a result data set.
Testing the robustness of algorithms: By simulating various interference influences, users can check and optimize the reliability of their evaluation algorithms.

These advantages make imc FAMOS a powerful tool for calculating and analyzing measurement uncertainties, especially for complex algorithms and demanding metrological applications.

FAQ: Measurement uncertainty mistakes and explanations in relation to imc FAMOS

Is the specification of a numerical value and physical unit sufficient for a complete measurement result?

No, this is not sufficient. Each measured value must also include a measurement uncertainty. The GUM (Guide to the expression of uncertainty in measurement) provides possible types of presentation in its paragraph “Reporting uncertainty”.

How can the measurement uncertainty be set in imc FAMOS?

In imc FAMOS, the measurement uncertainty of an input signal is defined with the command `UncertaintySet()`. Example: `UncertaintySet(Temperature, “Uncertainty”, 0.25)` assigns a measurement uncertainty of 0.25°C to the input signal “Temperature”.

Can imc FAMOS also take specific disturbance variables into account when calculating measurement uncertainty?

Yes, imc FAMOS allows specific disturbance variables such as noise, mains hum or offset drift to be taken into account in addition to the GUM. These can also be assigned via the `UncertaintySet()` function.

How does imc FAMOS calculate the propagation of measurement uncertainties?

imc FAMOS uses the Monte Carlo method to calculate the propagation of measurement uncertainties. The input data is turned slightly noisy in differing ways and the algorithm is run several times with this noisy data.

Is an exact measurement possible with an incremental encoder?

No, no measurement is exact. Even with an incremental encoder, there are inaccuracies at the start and end of the measurement, which can only be accurate to within a single pulse.

Is the expanded measurement uncertainty always twice the standard measurement uncertainty?

No, this only applies under certain conditions. The expanded measurement uncertainty is the product of the standard measurement uncertainty with a factor greater than one. This factor depends on the underlying distribution and the desired coverage probability.

How can the measurement uncertainty for complex result data sets be calculated in imc FAMOS?

imc FAMOS offers the option of calculating an all-encompassing measurement uncertainty for an entire result data set. This is done by calculating a quadratic mean of all individual standard deviations.

Is it possible to analyze the measurement uncertainty for each individual measurement point of a result data set in imc FAMOS?

Yes, imc FAMOS allows a detailed analysis of the measurement uncertainty for each individual measurement point of a result data set. This can be realized with the function `UncertaintyCalc()` with appropriate parameters.

How many Monte Carlo tests are necessary in imc FAMOS for an accurate measurement uncertainty calculation?

The number of Monte Carlo tests required depends on the desired accuracy. Basically, the higher the number of trials, the more accurate the result. In imc FAMOS, the number of trials can be defined using the function `UNCERTAINTY_LOOP`.

Can imc FAMOS handle non-constant measurement uncertainty budgets?

Yes, imc FAMOS can also handle measurement uncertainty budgets that are not constant. In such cases, the software offers the option of performing detailed analyses for each individual measurement point or calculating an all-encompassing measurement uncertainty for the entire data set.