Transcript Regression

Quantities & Measurement
Multivariable measurements
2.
Regression
2
3
E

a
T

a
T

a
T
...
T
,
0 1
2
3 
Y  F(X)
Z
f :X
Measured quantities
© Leslie Pendrill 060606
Sought quantities
1
Advanced Measurement Uncertainty
Analysis
Multivariable measurements
2.
Regression
Y  F(X)
y = A.x
Least-squares fit:
Variance-covariance:
© Leslie Pendrill 060606



1
T
x

A

A

A
y
fit
T


V
 AA
2
T

1
2
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.
© Leslie Pendrill 060606
Bentley exercise 2.1
3
Advanced Measurement Uncertainty
Analysis
2.1 Answer
E(T) = a1T + a2T2 + a3T3
0.01
Emf (V)
E uE
2 10
E uE
.
7
Ebis 0.005
Ebis1
E Ebis
0
02 10 7
0
.
200 200
0
400
400
600 600
T
T
 0

T1
A  
T
 2
T
 3
T0
T12
T22
T32
2
T0 
3
T
 1 

3
T
 2 
T33
3
a1 = 5.840 µV/ °C
© Leslie Pendrill 060606
 a1 
x    a2 
 
 a3 
 E0 
 
 E1 
y   
 E2 
E 
 3
M  4
8001000
800
T
1000
T (C)



1
T
x

A

A

A
y
fit
a2 = 6.37 10-3µV/ °C
T
Least-squares fit
a3 = -2.651 10-6 µV/ °C
Bentley exercise 2.1
4
Advanced Measurement Uncertainty
Analysis
2.1 Answer
E(T) = a1T + a2T2 + a3T3
5 10
16
.
R
0
0
y = A.x
200
400
Vc    A  A 
1
 

a1 = 5.83969718 (0) µV/ °C
800
1000
T
R   Ax  y
T
600
Residuals



( R R) 

M  N 
1
M=4
N=3
a2 = 6.36808419 (0) 10-3µV/ °C
a3 = -2.65055967 (0) 10-6 µV/ °C
© Leslie Pendrill 060606
Bentley exercise 2.1
5
Advanced Measurement Uncertainty
Analysis
Multivariable measurements
2.
Regression
Y  F(X)
d = A.c
c= (AT.W.A)-1.AT.W.d
2
Weighting matrix
Wi, i
1 .
si
1
1
i
si
2
Z
f :X
Measured quantities
© Leslie Pendrill 060606
Sought quantities
6
Advanced Measurement Uncertainty
Analysis
Multivariable measurements
2.
Regression
Y  F(X)
c= (AT.W.A)-1.AT.W.d
d = A.c
1
1
.
.
.
.
VcAW
A
(R
R
)
vav
MN
T
.
Group
variance
© Leslie Pendrill 060606

1
1M
2
vav
 
u
(y
i)
Mi0
Experimental
variance
7
5 10
Advanced Measurement Uncertainty
Analysis
16
2 10
2.1 Answer
R
0
7
E(T) = a1T + a2T2 + a3T3
E Ebis
0
Weighted least-squares fit
0
T
 0

T1
A  
T
 2
T
 3
 0
T12
T22
T32
T
2
 0 
3
T
 1 

3
T2 
T33
T
2 10
7
200 0
T
-1 T.W.y
xfit=
(A
200
400.W.A)
600
400
600
800 .A800
1000 1000
T
3
 a1 
x    a2 
 
 a3 
 E0 
 
 E1 
y   
 E2 
E 
 3
T
 1  1 0 

5 
1  10
V
uE  

6 
3  10


 4 .9  1 0 5 


1
5
W
i i

 uEi 2
M 1

1
 
i  0 u Ei
2
M  4
M=4
a1 = 5.8(4) µV/ °C
N=3
a2 = 6.3(1.5) 10-3µV/ °C
a3 = -2.6 (1.2) 10-6 µV/ °C
© Leslie Pendrill 060606
Bentley exercise 2.1
8