Science. Approaches to measurement uncertainty evaluation. Introduction. for a safer world 28/05/2017. S L R Ellison LGC Limited, Teddington, UK

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1 Approaches to measurement uncertainty evaluation S L R Ellison LGC Limited, Teddington, UK Science for a safer world 1 Introduction Basic principles a reminder Uncertainty from a measurement equation Gradient methods Finite difference approach Kragten s method Simulation methods Monte Carlo simulation (MCS) 1

2 Measurement uncertainty: Basic principles 3 Measurement uncertainty - ISO definition A parameter, associated with the result of a measurement, that characterises the dispersion of the values that could reasonably be attributed to the measurand The part of the result after the ± 2

3 ISO recommendations Uncertainties arise from several contributions Two ways of evaluating uncertainty components statistical (Type A) and otherwise (Type B) should be treated in the same way Expression as standard deviations Combination by.. the usual method for the combination of variances. Multiplied by a (stated) factor if required Implementation - combining uncertainties the usual method u 2 u u2 The uncertainties are: u 1 Uncertainty contributions for the same quantity In the same units Expressed as standard uncertainties 3

4 Expanded uncertainty U u c -> 68% U= 2u c -> 95% U= k.u c Uncertainties in different quantities: Propagation of uncertainty 8 4

5 Example: The effect of temperature on volume Dispense 100ml from a Calibrated volumetric flask (U= 0.2 ml, k=2) allowing for random filling effects (s = 0.1 ml) at a laboratory temperature 20 ±2 C Estimate the uncertainty in dispensed volume at 20 C Example: The effect of temperature on volume Random variation (ml) Calibration (ml) Volume (ml) Temperature ( o C) How does a temperature uncertainty apply? 5

6 Example: The effect of temperature on volume Volume Gradient: V α u(v) coefficient of volume expansion u(t) Temperature u(v) = gradient u(t) The law of propagation of uncertainty x i parameter affecting analytical result y u(x i ) uncertainty in x i u i (y) uncertainty in ydue to uncertainty in x i u i y i xi ( y) = u( x ) 2 i 2 sensitivity coefficient Describes how much the result changes with changes in input 6

7 Numerical methods for uncertainty propagation 13 Why include numerical methods? Simpler than algebra More general than algebraic differentiation Can obtain gradients when differentiation is intractable Result obtained from algorithm rather than equation May be applicable when simplifying assumptions do not apply Uncertainties large not linear Distributions far from Normal 7

8 Finite difference - principle Measurement result y y + y - b a δ x 0 gradient(b) gradient(a) Input parameter x i y x i y+ y 2δ x i δ x δ x y+ y ui( y) u( xi) 2δ x i Compare finite difference with the GUM GUM first order Expression: a/(b - c) Uncertainty budget: x u c u.c a b c y: 1 u(y): Finite Difference Expression: a/(b - c) Uncertainty budget: x u c u.c a b c y: 1 u(y):

9 Kragten s method Measurement result y y + y + 0 u Input parameter x i δu(x x i ) y y0 ( i( uy ) + i ( y) y uy xi) 0 u( δx + Eurachemguide, sec E.2 i) i Compare Kragten with FD Finite Difference Expression: a/(b - c) Uncertainty budget: x u c u.c a b c y: 1 u(y): Kragten Expression: a/(b - c) Uncertainty budget: x u c u.c a b c y: 1 u(y):

10 Exact vs. Numerical y = a/(b-c) Uncertainties: x u a b c Method Exact first order (GUM) Finite difference (0.01u) Standard uncertainty Kragten Why use a less accurate method? Finite difference Kragten 30 δ x 30 PAH (mg/kg) 20 PAH (mg/kg) 20 u(x i ) 10 0 Gradient=0 u i (y) = 0 u(x i ) = 0 mg/kg 10 0 u i (y) 7 mg/kg Detection wavelength (nm) Detection wavelength (nm) 10

11 Finite difference methods compared Finite difference 1 st order Accurate gradient Faithfully reproduces 1 st order GUM uncertainty Simple to calculate Kragten Exact only for linear examples Does not reproduce 1 st order GUM Simple to calculate 1 st order GUM is insufficient for highly non-linear cases Needs 2 nd and higher order Usually adequate for mild nonlinearity May be betterfor highly nonlinear cases Both much simpler than manual differentiation Monte Carlo/simulation methods GUM Supplement 1 (JCGM 101) Eurachem guide: QUAM:

12 Principle of simulation i) The problem x i x j x k + + y = f(x i, x j, x k,.) y u(y) Principle of simulation x i x j x k p(x i ) + + Random draws from p(x i ) y u(y) Estimated distribution for y 12

13 MCS example y= a/(b-c) (999 replicates) Histogram Q-Q plot Frequency Theoretical Quantiles Values of y Sample Quantiles Calculations carried out using metrology ( Exact vs. Numerical y = a/(b-c) Uncertainties: x u a b c Method Exact first order (GUM) Finite difference (0.01u) Standard uncertainty Kragten MCS y= to ;

14 Summary Numerical methods work when used with care Finite difference and Kragten methods are simple to calculate and usually reliable Kragten smethod less like 1 st order but this is often good! Simulation methods show distributions Applicable to non-normal cases MCS (JCGM 101) simple in principle but computer intensive Future guidance will include further methods Notably Bayesian approaches Software Simple algebraic, Kragten, Finite Difference and MCS metrologyversion running under R 14