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Multivariable measurements

Quantities & Measurement. Multivariable measurements. 2. Regression. Measured quantities. Sought quantities. Advanced Measurement Uncertainty Analysis. Multivariable measurements. 2. Regression. y = A.x. Least-squares fit:. Variance-covariance:. Measurement instruments.

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Multivariable measurements

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  1. Quantities & Measurement Multivariable measurements 2. Regression Measured quantities Sought quantities

  2. Advanced Measurement Uncertainty Analysis Multivariable measurements 2. Regression y = A.x Least-squares fit: Variance-covariance:

  3. Measurement instruments 2.1 The e.m.f. at a thermocouple junction is 645(10) µV at the steam point, 3375(3) µV at the zinc point and 9149(49) µV at the silver point. Given that the e.m.f.-temperature relationship is of the form: E(T) = a1T + a2T2+ a3T3 (T in °C), cold-point at ice (Eice = 0(10) µV) find a1, a2 and a3. Bentley exercise 2.1

  4. Advanced Measurement Uncertainty Analysis 2.1 Answer E(T) = a1T + a2T2+ a3T3 Emf (V) T (C) Least-squares fit a1 = 5.840 µV/ °C a2 = 6.37 10-3µV/ °C a3 = -2.651 10-6µV/ °C Bentley exercise 2.1

  5. Advanced Measurement Uncertainty Analysis 2.1 Answer E(T) = a1T + a2T2+ a3T3 y = A.x Residuals M = 4 N = 3 a1 = 5.83969718 (0) µV/ °C a2 = 6.36808419 (0) 10-3µV/ °C a3 = -2.65055967 (0) 10-6µV/ °C Bentley exercise 2.1

  6. Advanced Measurement Uncertainty Analysis Multivariable measurements 2. Regression d = A.c c= (AT.W.A)-1.AT.W.d Weighting matrix Measured quantities Sought quantities

  7. Advanced Measurement Uncertainty Analysis Group variance Experimental variance Multivariable measurements 2. Regression c= (AT.W.A)-1.AT.W.d d = A.c

  8. Advanced Measurement Uncertainty Analysis 2.1 Answer E(T) = a1T + a2T2+ a3T3 Weighted least-squares fit xfit= (AT.W.A)-1.AT.W.y M = 4 N = 3 a1 = 5.8(4) µV/ °C a2 = 6.3(1.5) 10-3µV/ °C a3 = -2.6 (1.2) 10-6µV/ °C Bentley exercise 2.1

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