Why Simulation Is the Future of Uncertainty Evaluation |

Quality Digest5 day(s) ago

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Why Simulation Is the Future of Uncertainty Evaluation |

Quality Digest5 day(s) ago

In this article I will show that the conventional method for calculating uncertainty is not always reliable In fact it is generally only exact when the measurement can be represented by a simple linear equation and the input uncertainties are all normally distributed Whenever the measurement is more complex there will be errors in the way uncertainties are combined Using the conventional analytic methods these errors can be difficult to quantify although there are some methods that can be used I will also show how simulation can be a much more reliable approach Uncertainty evaluation requires a mathematical model describing how influences on the measurement combine to produce the measurement result The uncertainty in each influence is then combined with reference to the mathematical model to give the uncertainty in the measurement result This approach enables all available experimental data to be combined with other unobserved sources of uncertainty which we know about by some other means In a previous article I showed how gauge studies can underestimate uncertainty by failing to include these Type-B uncertainties I will start with a simple example of a linear measurement model showing that this gives an exact solution I will then show what happens with more complex models and compare the results with simulations With simulation we can quantify the errors in the analytical approach Uncertainty evaluation for linear models Lets consider the simplest of measurement models with just two influences that add up to give the measurement result These could be the repeatability of the instrument and its calibrated value The measurement result is denoted y and the two influences a and b so we can say that the measurement result is given by y a b Since this is a linear combination if a is increased by one unit then y will increase by one unit This is independent of the value of b The same can be said for a change in b In other words the partial derivatives or sensitivity coefficients are all equal to one Applying the law of propagation of uncertainty the uncertainty of the measurement result is therefore simply given by u2 y u2 a u2 b If u a and u b are normally distributed then u y will also be normally distributed and this will be an exact solution If however u a and u b are uniformly distributed and of equal magnitude then u y will follow a triangular distribution If the inputs are some other combination of distributions then the output distribution cannot readily be determined without simulation Simulating uncertainty I will now briefly explain how measurements are simulated and the results used to prove the accuracy of analytical solutions I will explain the ins and outs of Monte Carlo simulation in a future article In essence simulation involves calculating our measurement function many times but in each case adding some randomly generated error to each influence quantity We then have many simulated measurement results and can calculate the standard deviation of these just as we would if they were actual experimental measurement results In general we can define the measurement as a function of x y f x where x is a row vector of length n containing the value of each influence quantity used to determine the measurement result In order to simulate a measurement we can use a random number generator such as the one available in Excel and most programming languages You can think of a random number generator the same way as rolling dice This is why you will often see dice used to represent simulations The difference is that random number generators are available for all standard probability distributions and they can produce millions of results in a fraction of a second We can therefore produce a column vector r containing m random draws from the standard distribution representing a given influence quantity In order to obtain the actual simulated values of influence quantities we must add their nominal values to their uncertainty multiplied by the random draw from their probability distribution If the uncertainties for each influence quantity are given by a row vector u containing n values then it is possible to generate an m x n matrix X of simulated influence quantities for each of m measurements thus A simulation program would typically loop through each influence quantity assembling X one column at a time Each column in X represents a different influence quantity and each row represents a different simulated measurement For each column r contains a new set of draws from the standard probability distribution assumed for the influence quantity A column vector Y of m simulated measurement results may therefore be calculated by evaluating each row in X Y f x We can then calculate the combined standard uncertainty of the measurement result by simply finding the standard deviation of Y When using analytical methods it would normally be assumed that the combined uncertainty follows a normal or a T-distribution This assumption is then used to find confidence intervals at a given probability level by multiplying the standard uncertainty by a coverage factor to give an expanded uncertainty However when we have simulation results we can find expanded uncertainties without making any assumption about the probability distribution We can directly find confidence intervals by first sorting Y with the measurement results ordered from smallest to largest If we then count 25 percent from the smallest and 25 percent from the largest this gives us the 95-percent intervals In general the proportion of results that must be counted from each end p q is related to the confidence level p by this simple equation p q 1 p 2 If this proportion doesnt exactly fall on a result then there are different ways of interpolating between results Intervals found in this way are generally referred to as quantiles or percentiles The median may therefore also be described as the 50 percentile or the 05 quantile It is relatively easy to convert these quantiles into expanded uncertainties Note that the upper and lower expanded uncertainty may not be the same since the probability distribution is not necessarily symmetrical The expanded uncertainties found in this way are given by Uupper q1-pq median Ulower q1-pq median where q is the quantile at the probability given by its subscript This method is recommended by Supplement 1 to the Guide to the Expression of Uncertainty in Measurement GUM-S1 as a way of validating analytical methods I have changed the notation slightly in the above explanation I did this so that it follows a more mathematically-standard approach using italic type lower case characters for scalar quantities bold type lower case characters for vectors and bold type upper-case characters for matrices I have therefore used m for the number of Monte Carlo trials in a simulation instead of the M used in GUM-S1 Ive broken this rule by using Y to represent the vector of m function evaluations of y just to make the distinction between these important variables a bit clearer Probability distribution for linear model Now we have a method of validating uncertainty calculations Using simulation lets have another look at the simple measurement model I used at the beginning of this article y a b I ran simulations each containing ten million trials with the following conditions 1 2 normal distributions of similar size 2 2 normal distributions of different sizes 3 2 uniform distributions of the same size 4 2 uniform distributions of different sizes 5 A normal and a uniform distribution The resulting normalized probability distributions are shown below Standardized probability distributions for each simulation Figure 1 It is clear from these plots of the probability distributions that the output is not always a normal distribution In the case of two uniform distributions with the same uncertainty the output is triangular something which could be predicted theoretically However when they are of a different size the combination becomes more difficult to predict and very difficult to calculate expanded uncertainties from This is also true for the combination of a normal and a uniform distribution The differences become more pronounced the further into the tails you go the assumptions are therefore reasonably good for predicting 95-percent confidence intervals but less so for higher confidence levels In the table below you can see how this affects the accuracy of expanded uncertainties The differences are already quite noticeable even for this simple linear function In the next section I look at some more complex but common measurement functions This is when the differences become really dramatic Figure 2 Comparison of standard uncertainty and expanded uncertainty determined by theoretical and simulation methods Nonlinear measurement functions Where the measurement function is nonlinear things become even more complex In this case different orders of analytical solutions are possible In theory increasing the order of the approximation should increase the accuracy of the uncertainty evaluation In practice there may not be a significant improvement I will use two examples to illustrate this the product of three inputs and a cosine function Products are common when correcting measurements for environment and material properties for example a thermal expansion Cosine errors are common when length measurements have small misalignments Let us consider the product first Imagine a correction for thermal expansion The length of the part L 1m the temperature offset which is being corrected for dT 23k and the coefficient of thermal expansion a 23 micrometers per m per k The measurement result for the correction is given by dL L dT a The first-order approximation of uncertainty is given by u dL 2 dT a 2 u L 2 L a 2 u dT 2 L dT 2 u a 2 The first thing to note about this equation for the uncertainty is that it now depends on the nominal values of the input quantities Unlike the linear measurement function the combined uncertainty is affected by the values of inputs not only their uncertainties It is also possible to include second and third-order terms giving the following somewhat complex uncertainty equation u dL 2 dT a 2 u L 2 L a 2 u dT 2 L dT 2 u a 2 dT 2 u L 2 u a 2 a 2 u L 2 u dT 2 L 2 u a 2 u dT 2 u dT 2 u L 2 u a 2 In theory this higher-order model should be considerably more accurate than the first-order model In practice the higher-order terms will rarely make a significant difference to the result This is because the uncertainties are generally considerably larger than the actual values The first-order terms involve only a single uncertainty while the higher-order terms are products of uncertainties Therefore I would normally expect the higher-order terms to be orders of magnitude smaller than the first-order terms Even with relatively high uncertainties in the three input quantities we see no significant difference between the first-order third-order and simulation results for this example The plot below is for standard uncertainties in the length temperature and CTE of 5 mm 1 k and 10 percent respectively Comparison of three uncertainty calculations for thermal expansion Figure 3 In my final example you will see how the analytical method can sometimes give dramatically inaccurate results In this example Ive considered measuring a 200 mm length with a highly accurate instrument that is poorly aligned something typical of a laser interferometer setup The standard uncertainty of the length measurement is 2 micrometers while the alignment which causes cosine error has a standard uncertainty of 1 degree It should be noted that the analytical method for this type of measurement involves calculating sensitivity coefficients as constants using the finite difference method This can be highly sensitive to the size of the difference used In this example using the standard uncertainty for the angle results in a sensitivity approximately half of that when it is calculated using the expanded uncertainty What we see for this measurement is a highly non-normal distribution as seem in the plot below Figure 4 Significant difference between probability distributions for theoretical method assuming a normal distribution and simulation When we look at the expanded uncertainties we can see the significant effect this discrepancy could have on quality decisions The standard uncertainties or 1 sigma values are relatively similar At 95-percent confidence the actual position of confidence limits could be significantly different but the total range of the uncertainty is still similar At 999 percent confidence even the total range is almost double what the theory predicts and the upper bound uncertainty is 35x what is predicted in theory If youre interested in 6-sigma limits then expect these differences to be even greater Figure 5 Table of predicted uncertainties for theoretical method and simulation I hope this article has opened your eyes to the significance of statistical approximations in quality engineering Look out for my upcoming article in which I will show you how to apply simulation in your uncertainty analysis First published Nov 12 2018 on the engineeringcom blog

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