1© Leslie Pendrill 060606

Multivariable measurements

Quantities & MeasurementQuantities & Measurement

2. Regression

...33

2210, TaTaTaET

)(XFY

Measured quantities Sought quantities

ZXf :

2© Leslie Pendrill 060606

Multivariable measurements

2. Regression

)(XFY

y = A.x

Advanced Measurement Uncertainty Advanced Measurement Uncertainty AnalysisAnalysis

yAAAx TTfit

1

12 AAV T

Least-squares fit:

Variance-covariance:

3© Leslie Pendrill 060606

Measurement instrumentsMeasurement 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© Leslie Pendrill 060606

2.1 Answer E(T) = a1T + a2T2 + a3T3

a1 = 5.840 µV/ °C a2 = 6.37 10-3µV/ °C a3 = -2.651 10-6 µV/ °C

A

T0

T1

T2

T3

T0 2

T1 2

T2 2

T3 2

T0 3

T1 3

T2 3

T3 3

0 200 400 600 800 10002 10

7

0

2 107

E Ebis

T

x

a1

a2

a3

y

E0

E1

E2

E3

M 4

Least-squares fit

Advanced Measurement Uncertainty Advanced Measurement Uncertainty AnalysisAnalysis

Bentley exercise 2.1

yAAAx TTfit

1

Emf (V)

T (C)0 200 400 600 800 1000

0

0.005

0.01

E uE

E uEEbis

Ebis1

T

.

.

5© Leslie Pendrill 060606

2.1 Answer

a1 = 5.83969718 (0) µV/ °C a2 = 6.36808419 (0) 10-3µV/ °C

a3 = -2.65055967 (0) 10-6 µV/ °C

E(T) = a1T + a2T2 + a3T3

y = A.x R A x y

Vc AT

A 1R R( )

1

M N

Residuals

M = 4 N = 3

Advanced Measurement Uncertainty Advanced Measurement Uncertainty AnalysisAnalysis

Bentley exercise 2.1

0 200 400 600 800 1000

0

5 1016

R

T

.

6© Leslie Pendrill 060606

Multivariable measurements

2. Regression

)(XFY

d = A.c

c= (AT.W.A)-1.AT.W.d

Advanced Measurement Uncertainty Advanced Measurement Uncertainty AnalysisAnalysis

W ,i i.1

si

21

i

1

si2

Weighting matrix

Measured quantities Sought quantities

ZXf :

7© Leslie Pendrill 060606

Multivariable measurements

2. Regression

)(XFY

d = A.c c= (AT.W.A)-1.AT.W.d

Vc ...TA W A1

.( ).R R1

M Nvav

Group variance

Advanced Measurement Uncertainty Advanced Measurement Uncertainty AnalysisAnalysis

Experimental variance

1

0

2 )(1 M

iiyu

Mvav

8© Leslie Pendrill 060606

2.1 Answer

a1 = 5.8(4) µV/ °C a2 = 6.3(1.5) 10-3µV/ °C

a3 = -2.6 (1.2) 10-6 µV/ °C

E(T) = a1T + a2T2 + a3T3

M = 4 N = 3

Advanced Measurement Uncertainty Advanced Measurement Uncertainty AnalysisAnalysis

Bentley exercise 2.1

Weighted least-squares fit

xfit= (AT.W.A)-1.AT.W.y

A

T0

T1

T2

T3

T0 2

T1 2

T2 2

T3 2

T0 3

T1 3

T2 3

T3 3

0 200 400 600 800 10002 10

7

0

2 107

E Ebis

T

x

a1

a2

a3

y

E0

E1

E2

E3

M 4

0 200 400 600 800 1000

0

5 1016

R

T

uE

1 105

1 105

3 106

4.9 105

V

Wi i

1

uEi 2

0

M 1

i

1

uEi 2