1 Leslie Pendrill 060606 Multivariable measurements Quantities & Measurement 2. Regression ...3322 1 0 , + + + = T a T a T a ET) (X F Y=Measured quantities Sought quantities { } {} Z X f : 2 Leslie Pendrill 060606 Multivariable measurements 2. Regression ) (X F Y=y = A.x Advanced Measurement Uncertainty Analysis ( ) y A A A xT Tfit = 1( )12 = A A VToLeast-squares fit: Variance-covariance: 3 Leslie Pendrill 060606 Measurement instruments 2.1 The e.m.f. at a thermocouple junction is 645(10) V at the steam point, 3375(3) V at the zinc point and 9149(49) V 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) V) find a1, a2 and a3. Bentley exercise 2.1 4 Leslie Pendrill 060606 2.1 Answer E(T) = a1T + a2T2 + a3T3 a1 = 5.840 V/ C a2 = 6.37 10-3V/ C a3 = -2.651 10-6 V/ C AT0T1T2T3T0( )2T1( )2T2( )2T3( )2T0( )3T1( )3T2( )3T3( )3

((((((((:=0 200 400 600 800 10002.10702.107E Ebis Txa1a2a3|

\|||.:=yE0E1E2E3|

\||||||.:=M 4 :=Least-squares fit Advanced Measurement Uncertainty Analysis Bentley exercise 2.1 ( ) y A A A xT Tfit = 1Emf (V) T (C) 0 200 400 600 800 100000.0050.01E uE E uE +EbisEbis1T.. 5 Leslie Pendrill 060606 2.1 Answer a1 = 5.83969718 (0) V/ C a2 = 6.36808419 (0) 10-3V/ C a3 = -2.65055967 (0) 10-6 V/ C E(T) = a1T + a2T2 + a3T3 y = A.x R A x y :=Vc ATA ( )1 R R ( ) 1M N |

\ ||. :=Residuals M = 4 N = 3 Advanced Measurement Uncertainty Analysis Bentley exercise 2.1 0 200 400 600 800 100005.1016RT. 6 Leslie Pendrill 060606 Multivariable measurements 2. Regression ) (X F Y=d = A.c c= (AT.W.A)-1.AT.W.d Advanced Measurement Uncertainty Analysis W, i i.1si21i1si2Weighting matrix Measured quantities Sought quantities { } {} Z X f : 7 Leslie Pendrill 060606 Multivariable measurements 2. Regression ) (X F Y=d = A.c c= (AT.W.A)-1.AT.W.d Vc. . .TAWA1.( ).RR1MNvavGroup variance Advanced Measurement Uncertainty Analysis Experimental variance = =102) (1Miiy uMvav 8 Leslie Pendrill 060606 2.1 Answer a1 = 5.8(4) V/ C a2 = 6.3(1.5) 10-3V/ C a3 = -2.6 (1.2) 10-6 V/ C E(T) = a1T + a2T2 + a3T3 M = 4 N = 3 Advanced Measurement Uncertainty Analysis Bentley exercise 2.1 Weighted least-squares fit xfit= (AT.W.A)-1.AT.W.y AT0T1T2T3T0( )2T1( )2T2( )2T3( )2T0( )3T1( )3T2( )3T3( )3

((((((((:=0 200 400 600 800 10002.10702.107E Ebis Txa1a2a3|

\|||.:=yE0E1E2E3|

\||||||.:=M 4 :=0 200 400 600 800 100005.1016RTuE1 105 1 105 3 106 4.9 105 |

\|||||||.V =Wi i ,1uEi( )20M 1 i1uEi( )2=:=