Metas.UncLib. Michael Wollensack und Marko Zeier. Metas Seminar Ein Messunsicherheitsrechner für Fortgeschrittene

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1 Federal Department of Justice and Police FDJP Federal Office of Metrology METAS Metas.UncLib Ein Messunsicherheitsrechner für Fortgeschrittene Metas Seminar Michael Wollensack und Marko Zeier

2 Programming today is a race between software engineers striving to build bigger and better idiot-proof programs, and the Universe trying to produce bigger and better idiots. So far, the Universe is winning. Rich Cook 2

3 Outline 1. Introduction 1.1 VNA Tools II Project 1.2 MU according to ISO GUM 1.3 Metas.UncLib 1.4 Linear Uncertainty Propagation 1.5 Large System Modelling 2. Features 3. MATLAB Demo 3

4 1.1 VNA Tools II Project ( FP.060) Goal: Development of a GUM based uncertainty calculator for Vector Network Analysis Duration: 2 years (Feb Jan 2010) METAS RF & MW laboratory: M. Wollensack J. Rüfenacht M. Zeier Partner: Agilent Technologies, Santa Rosa, USA (Largest manufacturer of VNAs and calibration standards) 4

5 1.1 VNA Tools II Overview Metas.UncLib Metas.Instr Metas.UncLib.Core Metas.Instr.VisaExtensions Metas.UncLib.LinProp Metas.Instr.Driver Metas.UncLib.MCProp Metas.UncLib.DB Metas.Vna Metas.VnaTools Metas.Vna.Calibration Metas.VnaTools.Gui Metas.Vna.ErrorModel

6 1.2 MU according to ISO-GUM (scalar) X 1 X 2 X 3 Input quantities u X u X,, 1 2 f Measurement model Y Output quantity f 2 f 2 u Y u X1 u X2 X1 X2 95% Y Expanded measurement uncertainty U Y (1.96) u Y 6

7 1.2 MU according to ISO-GUM (multivariate) corr X 1 X 2 X 3 Correlation between input quantities V X Uncertainty Matrix f 1 f 2 corr Y 1 Y 2 Several output quantities correlated! V J V J T Y f,x X f,x Lin. Uncertainy Propagation Y 1 95% Expanded measurement uncertainty U Y 2 p,0.95 V Y Y 2 Calculations are tedious! 7

8 1.3 Metas.UncLib General Purpose Uncertainty Library It does support multidimensional uncertainty calculation advanced math (Complex, Vector, Matrix) automated linear uncertainty propagation Monte Carlo uncertainty propagation (preliminary) take care of correlations advanced storage / archiving (keeps full information) interfacing with other applications It does NOT help to build a measurement model have a nice graphical interface produce fancy output 8

9 1.4 Linear Uncertainty Propagation (LUP) Based on GUM Tree concept (Blair Hall, IRL/MSL, NZ) Automatic Differentiation Calculate Derivatives (sensitivity coefficients) with machine precision Object oriented implementation Hide complexity from user 9

10 LUP: Automatic Differentiation To calculate sensitivity coefficients X 1 X 2 X X N Measurement model z f x Decomposition in elementary functions 1 z f f f x 2 M Derivatives of elementary functions are explicitly programmed z 1 z Chain rule J J J f,x f,f f,x i i j j Naturally supported by computers because compiler always decomposes equations into elementary functions 10

11 Metas.UncLib Object 1.4 LUP: Metas.UncLib Object New Data Type (contains value, dependencies and sensitivities) value z i X 1 X 2 X X N depends on x1, x3, x4, Jacobian (sensitivities) zi zi zi,,, x1 x3 x4 updated at each computational step Overloaded Operators hide complexity Objects can be used like ordinary numbers Uncertainty information on demand z 1 z

12 Metas.UncLib Object Metas.UncLib Object 1.4 LUP: Correlations Caused by common influences The system takes automatically care of correlations by recognizing common influences value z 1 value z 2 depends on x, x 2 3 depends on x x 1, 2 Jacobian (sensitivities) z x z, x Jacobian (sensitivities) z x z, x

13 1.5 Large System Modelling The system allows complete modelling of a multi-step process with stored intermediate results (e.g. traceability chain) Primary Standards (base inputs) Characterisation Working Standards (UncNum Objects) Influences (base inputs) Calibration Devices Under Test (UncNum Objects) 13

14 Outline 1. Introduction 2. Features 2.1 Math functions 2.2 Linear Algebra 2.3 Uncertainty functions 2.4 Storage / Archiving 2.5 Interfacing.NET / COM 3. MATLAB Demo 14

15 2. Features Metas.UncLib is a general purpose uncertainty library. Metas.UncLib Metas.UncLib.Core Metas.UncLib.LinProp Metas.UncLib.MCProp Metas.UncLib.DB Linear Uncertainty propagation Monte Carlo propagation Complex Math Vector and Matrix Math Linear Algebra Storage / Archiving.NET / COM Interface 15

16 2.1 Math functions A list with the available functions and constants: Sqrt(x) Sin(x) Sinh(x) Real(x) PI Exp(x) Cos(x) Cosh(x) Imag(x) E Log(x) Tan(x) Tanh(x) Abs(x) J Log10(x) Asin(x) Asinh(x) Angle(x) One Log(x, y) Acos(x) Acosh(x) Conj(x) Zero Pow(x, y) Atan(x) Atanh(x) Sign(x) Atan2(x, y) 16

17 2.2 Linear Algebra A list with the available functions: Dot(M1, M2) Lu(M) Det(M) Inv(M) Solve(A, Y) Cholesky(M) LstSqrSolve(A, Y) WeightedLstSqr Solve(A, Y, W) Fft(M) Ifft(M) Matrix multiplication of matrix M1 and M2 LU decomposition of matrix M Determinate of matrix M Matrix inverse of M Solve linear equation system: A * X = Y Cholesky decomposition of matrix M Least square solve of overdetermined linear equation system. Weighted least square solve of overdetermined linear equation system. Fast Fourier transformation Inverse Fast Fourier transformation 17

18 2.3 Uncertainty functions? The available functions to obtain information from a Metas.UncLib object: C# Metas.UncLib MATLAB LinProp or MCProp Returns the value. y.getvalue() get_value(y) Returns the standard uncertainty. y.getstdunc() get_stdunc(y) Returns the inverse degrees of freedom. Returns the sensitivities to the virtual base inputs (with value 0 and uncertainty 1). This is equal to the uncertainties components Returns the sensitivities to the intermediate results. y.getidof() y.getjacobi() y.getjacobi2(x) get_idof(y) get_jacobi(y) get_jacobi2(y,x) Returns the uncertainty component. Returns the correlation matrix. [y1 y2...]. GetCorrelation() Returns the covariance matrix. [y1 y2...]. GetCovariance() y.getunccomponent(x) get_unc_component(y, x) get_correlation([y1 y2...]) get_covariance([y1 y2...]) 18

19 2.4 Storage / Archiving Save computed Metas.UncLib objects. Reload stored Metas.UncLib objects. After loading all information is restored. Metas.UncLib Object - RealNumber - ComplexNumber - RealArray - ComplexArray - RealUncNumber - ComplexUncNumber - RealUncArray - ComplexUncArray Storage XML Binary <?xml version="1.0"?> <unc_number> <value>5</value> <dependency> </dependency> </unc_number> 19

20 2.5 Interfacing.NET / COM Metas.UncLibCE Metas.UncLib Metas.UncLibCOM Metas.UncLib.Matlab Metas.UncLibCE.Core Metas.UncLib.Core Metas.UncLibCOM.Core Metas.UncLibCE.LinProp Metas.UncLib.LinProp Metas.UncLibCOM.LinProp LinProp Metas.UncLibCE.MCProp Metas.UncLib.MCProp Metas.UncLibCOM.MCProp MCProp Metas.UncLib.DB Metas.UncLibCOM.DB LinPropDB Pocket PC Visual Basic C++ Use Metas.UncLib in Smart Phone Visual C# Matlab Matlab scripts and Windows CE Device Visual C++ Excel functions. LabVIEW... Mathematica Office

21 Outline 1. Introduction 2. Features 3. MATLAB Demo 3.1 Right triangle example 3.2 GUM H2 example 3.3 Resistor cube example 3.4 Circle fit example 3.5 Air line characterization example 21

22 3. MATLAB Demo 22

23 3.1 Right triangle Cathetus a = 3, u(a) = 0.3 Cathetus b = 4, u(b) = 0.4 What s the value and uncertainty of the hypotenuse c? What s the value and uncertainty of the perimeter U? c U a, b, c a b c 2 2 a, b a b u a a A a, b ab 2 u b b What s the value and uncertainty of the area A? What s the correlation between A and U? 23

24 3.2 GUM H2 example 24

25 3.2.1 GUM H2 example Input quantities 25

26 3.2.2 Matlab Definition of the inputs unc meas = [ e ; e ; e ; e ; e ]; input_values = mean(meas, 1); input_covar = cov(meas)./size(meas, 1); inputs = unc(input_values, input_covar); v = inputs(1); i = inputs(2); phi = inputs(3); 26

27 3.2.3 GUM H2 example Output quantities 27

28 3.2.4 Matlab Compute the outputs r = v./i.*cos(phi); x = v./i.*sin(phi); z = v./i; outputs = [r x z]; output_values = get_value(outputs) output_stdunc = get_stdunc(outputs) output_corr = get_correlation(outputs) 28

29 3.2.5 Matlab Output quantities output_values = output_stdunc = output_corr =

30 3.3 Resistor cube example What s the equivalent resistor of the cube between the two red junctions? And the uncertainty of it? R3 I R1 R2 R1 R2 R3 I1 I2 I3 R8 R9 R6 R7 U=1 R4 R5 R6 R7 R8 R9 R4 R5 I4 I5 I6 I7 I8 I9 R10 R12 R11 R10 R11 R12 I10 I11 I12 I U 1 R I I R U ( 12 1) (12 12) (12 1) (12 1) (12 12) (12 1 ) 30

31 3.4 Circle fit example Where is the center of the circle? What is the uncertainty? Equation for a circle: Im i 0 R i 1,, n Re i Re 0 2Im i Im 0 R 0 a b c 0 R Re i Linearization: 2 i Circle fit equation system: Apd 2Re 2 Re A 2Re 1 2Im 1 2Im 2Im And how to compute the uncertainty? The analytic solution is too complicated. Too much data for Monte Carlo. Metas.UncLib is the ideal solution! 31 0 a bi 2 n R 2 n c a over determined b 2 a p b c d n 2

32 Im Circle fit example Matlab unc %% Circle points f = 1.9; % (GHz) 0.01 circle fit, f = 1.90 GHz V = diag([6.9e-4^2, 6.9e-4^2]); s1 = unc(-8.43e E-3*i, V); s2 = unc(-6.99e E-3*i, V); s3 = unc(-3.73e E-2*i, V); s4 = unc( 3.00E E-2*i, V); s5 = unc( 1.14E E-3*i, V); s6 = unc( 6.92E E-3*i, V); s7 = unc(-7.75e e-3*i, V); s8 = unc(-5.76e E-2*i, V); s9 = unc(-1.31e E-2*i, V); s = [s1 s2 s3 s4 s5 s6 s7 s8 s9].'; %% Circle fit A = [2.*real(s) 2.*imag(s) ones(size(s))]; d = abs(s).^2; p = A\d; S0 = p(1) + i*p(2); R = sqrt(p(3) + abs(s0).^2); Re S0.value = e-005i S0.covariance = e e e e

33 3.5 Air line characterization example S-Parameter of an air line in function of mechanical data. outer conductor inner conductor a b outer conductor full air line length Pos. 0 Pos. 1 Pos. 2 Pos. 3 Pos. i Pos. N a0 b0 a1 b1 a2 b2 a3 b3 ai bi an bn 2-Port air line section 1 2-Port air line section 2 2-Port air line section 3 2-Port air line section i 2-Port air line section N 2-Port air line 33

34 Get Metas.UncLib now! To get Metas.UncLib please contact Michael Wollensack It s distributed in two setup files (*.msi). One for Metas.UncLib the other one for Metas.UncLib.Matlab To use it from Matlab you need at least Version 2006b. 34

35 Thank you for your attention The VNA Tools II Team in action J. Rüfenacht M. Wollensack M. Zeier 35

36 Questions??????????? 36