From the propagation of uncertainties to the propagation of distributions

Transcription

1 From the propagation of uncertainties to the propagation of distributions Carlo Carobbi Dipartimento di Elettronica e Telecomunicazioni Università degli Studi di Firenze Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

2 The law of propagation of uncertainties Y = f (X 1, X 2,, X N ), mathematical model N independent input quantities X i One output quantity Y x 1, x 2,, x N best estimates u(x 1 ), u(x 2 ),, u(x N ) standard uncertainties δf δf δf uy ( ) = ux ( 1) + ux ( 2) ux ( N ) δx1 δx2 δxn u(y) standard uncertainty of the output quantity δf/δxi sensitivity coefficients (partial derivatives evaluated at the best estimates of the input quantities) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

3 Propagation of uncertainties Y = f(x) is a linear or quasi-linear function of X X = (X 1, X 2,, X N ) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

4 Linearity The law is valid if the non-linearity is negligible N δf f X X X f x x x + X x (,,..., ) (,,..., ) ( ) 1 2 N 1 2 N i i i = 1 δ Xi The law can be readily modified for correlated input quantities (if linearity applies) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

5 Linearity in log-units In most cases the model is linear in log-units Linear units Log-units e = v af l r c E = V + AF + L v = vr lamn l V = V c r + Lamn + Lc 2 vr 2 2 p = lac l P = V 10log( ) c r + Lac + Lc R R lc Z Z e = p g E = P + G+ Lc D+ 10log d 4π 4 π Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio r c

6 Non-linearity In some cases the effects of non-linearity cannot be easily dealt with: Correction for distance in radiated emissions in semi-anechoic chambers when the measured field is not maximum Correction for AMN impedance when DUT impedance is not larger than AMN impedance and/or AMN impedance tolerance is not small Correction for mismatch if source and load mismatch are not small v v e 1/ d e f, ( h, h, d) r zˆ zˆ amn d v HV tx rx Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio r zˆ zˆ amn amn + zˆ 1+Γ Γ cos( θ ) v 1/ 1 2Γ Γ cos( θ ) + ( Γ Γ ) 2 r S L r S L S L d

7 Central limit theorem Central limit theorem applies when: the model is linear or quasi linear Y = c0 + c1x1+ c2x cnxn input quantities are independent c i u(x i ) have comparable magnitude, then Y is normal with expected value y and standard deviation u(y), where y = c0 + c1x1+ c2x cnxn [ ] [ ] [ ] N N uy ( ) = cux ( ) + cux ( ) cux ( ) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

8 Root-sum-of-the-square The rule of combination of uncertainties is the root-sum-of-the-square [ ] [ ] [ ] N N uy ( ) = cux ( ) + cux ( ) cux ( ) It may happen that one non normal contribution is dominant, especially when N is small (say 2, 3, 4) Then the pdf of Y is not normal being nearly that of the dominant contribution Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

9 Summary If the model is not linear The law of propagation of uncertainties does not apply The central limit theorem does not apply If the model is linear and few non normal contributions are dominant The law of propagation of uncertainties applies The central limit theorem does not apply Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

10 Non-linear model and/or dominant contribution The GUM Supplement I is of help in these cases Implementation of Monte Carlo numerical technique Propagation of distributions (to distinguish from propagation of standard uncertainties) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

11 Propagation of distributions Y = f(x) may be linear or non linear g x1 (ξ 1 ), g x2 (ξ 2 ), g x3 (ξ 3 ) pdfs of the input quantities, g Y (η) pdf of the output quantity Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

12 Sampling of Y by random number generation for X 1, X 2,, X N A sample of Y is calculated from a sample of X A sample of each input quantity is generated by using a random number generator Y (1) = f(x 1(1), X 2(1),X 3(1) ) Y (2) = f(x 1(2), X 2(2),X 3(2) ) Y (M) = f(x 1(M), X 2(M),X 3(M) ) The histogram of Y is readily obtained Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

13 A-priori pdfs for input quantities (1 of 2) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

14 A-priori pdfs for input quantities (2 of 2) Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

15 Conclusions GUM Supplement I provides an easy to implement numerical approach to the evaluation of measurement uncertainty Numerical computation needed when we suspect that the output quantity follows a non-normal distribution Guidance is offered for achieving a pre-defined numerical accuracy and for calculating coverage intervals in the case of asymmetric pdf of the output quantity Resort to numerical computation in EMC testing expected to be limited to few special cases (see slide 6) However wide range of measurements performed by a laboratory (testing, internal calibrations): it is important to be aware that these tools are available Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio

16 From the propagation of uncertainties to the propagation of distributions Carlo Carobbi Dipartimento di Elettronica e Telecomunicazioni Università degli Studi di Firenze Istituto di Fisica Applicata "Nello Carrara" - 3 Luglio