Effect of correlation on a weighted
mean
Presentation by:
Christopher Eiø
Background
Weighted mean as key comparison ref value
‹ Correlation sometimes occurs during a key comparison exercise
‹ Weighted mean is calculated using:
N
xw =
‹ Also exists between two or more separate measurements carried out
by the pilot laboratory
∑x u
i
‹ One participant may take its traceability from another
i =1
N
∑
1
i =1
2
( xi )
2
u ( xi )
‹ and the uncertainty is given by:
N
u 2 ( xw ) = ∑
‹ In the past, effects of correlation have often been ignored
i =1
1
u 2 ( xi )
‹ This does not take any correlation into account
Incorporating correlation
Least squares method
X = wxu(X)
‹ Very difficult to express algebraically
‹ Mean can be considered as a ‘line of best fit’ between data points
‹ Data can be expressed in matrix form and a least squares method
used to determine the mean
⎛ x1 ⎞
⎜ ⎟
⎜x ⎟
x=⎜ 2⎟
...
⎜ ⎟
⎜x ⎟
⎝ N⎠
⎛ u 2 ( x1 )
u ( x1 , x 2 )
⎜
u 2 ( x2 )
⎜ u ( x2 , x1 )
V =⎜
...
...
⎜
⎜ u( x , x ) u( x , x )
N
N
1
2
⎝
w = ATV-1
... u ( x1 , x N ) ⎞
⎟
... u ( x 2 , x N ) ⎟
⎟
...
...
⎟
... u 2 ( x N ) ⎟⎠
⎛1⎞
⎜ ⎟
⎜1⎟
A=⎜ ⎟
...
⎜ ⎟
⎜1⎟
⎝ ⎠
u(X) = (ATV-1A)-1
No correlation
Correlation between two participants
⎞
⎟
⎟
⎟
⎟
2
... u ( x N ) ⎟⎠
⎛ u 2 ( x1 )
0
⎜
u 2 ( x2 )
⎜ 0
V =⎜
...
⎜ ...
⎜ 0
0
⎝
...
...
...
0
0
...
⎛B
V = ⎜⎜
⎝0
⎛ u 2 ( x3 ) ...
0 ⎞
⎜
⎟
...
... ⎟
D = ⎜ ...
⎜ 0
... u 2 ( x N ) ⎟⎠
⎝
⎛ u 2 ( x1 ) u ( x1 , x2 ) ⎞
⎟
B = ⎜⎜
2
⎟
⎝ u ( x 2 , x1 ) u ( x 2 ) ⎠
X = wxu(X)
X = xw
Correlation between two participants
u i = u ( xi )
r = r ( x1 , x 2 ) =
Correlation between two participants
r=0
u ( x1 , x 2 )
u ( x1 )u ( x 2 )
r = 1, all uncertainties equal
0
0
0
N
x
u1 x1
u x
+ (1 − r 2 ) 22 + (1 − r 2 )∑ i2
)
u 2 u1 2
u1 u 2
i =3 u i
X =
N
u 1
u
1
1
(1 − r 1 ) 2 + (1 − r 2 ) 2 + (1 − r 2 )∑ 2
u 2 u1
u1 u 2
i =3 u i
N
x
u1 x1
u x
+ (1 − r 2 ) 22 + (1 − r 2 )∑ i2
)
u 2 u1 2
u1 u 2
i =3 u i
X =
N
u 1
u
1
1
(1 − r 1 ) 2 + (1 − r 2 ) 2 + (1 − r 2 )∑ 2
u 2 u1
u1 u 2
i =3 u i
(1 − r
0
u2 (X ) =
(1 − r
0
0
Using l’Hopital’s Rule:
N
x
u1 x1
u x
+ (1 − r 2 ) 22 + (1 − r 2 )∑ i2
)
u 2 u1 2
u1 u 2
i =3 u i
X =
N
u 1
u
1
1
(1 − r 1 ) 2 + (1 − r 2 ) 2 + (1 − r 2 )∑ 2
u 2 u1
u1 u 2
i =3 u i
0
(1 − r
∞
0
1
1
1− r2
N
1
1
x1 + x 2 + ∑ xi
2
2
i =3
N −1
u( X ) =
Example – noise temperature comparison
r=1
u (X ) =
X =
0
Correlation between two participants
2
0
0
0
0
1
N
u 1
u
1 ⎡
1
1 ⎤
(1 − r 1 ) 2 + (1 − r 2 ) 2 + (1 − r 2 )∑ 2 ⎥
2 ⎢
u 2 u1
u1 u 2
1 − r ⎣⎢
i =3 u i ⎦
⎥
0
0⎞
⎟
D ⎟⎠
N
⎡
u1 1
u2 1
1 ⎤
2
⎢(1 − r ) 2 + (1 − r ) 2 + (1 − r )∑ 2 ⎥
u 2 u1
u1 u 2
⎢⎣
i =3 u i ⎥
⎦
With full correlation, the mean appears to disregard all uncorrelated
components and the uncertainty tends to 0.
T1 = 9947 K
u(T1)= 102 K
T2 = 10172 K
u(T2)= 125 K
T3 = 10098 K
u(T3)= 73 K
T4 = 10295 K
u(T4)= 46 K
r(T1, T2) = 0
Unweighted mean:
T = 10128 K, U(T) = 91 K
Weighted mean:
T = 10200 K, U(T) = 70 K
u0
N −1
Example – noise temperature comparison
T1 = 9947 K
u(T1)= 102 K
T1 = 9947 K
u(T1)= 102 K
T2 = 10172 K
u(T2)= 125 K
T2 = 10172 K
u(T2)= 125 K
T3 = 10098 K
u(T3)= 73 K
T3 = 10098 K
u(T3)= 73 K
T4 = 10295 K
u(T4)= 46 K
T4 = 10295 K
u(T4)= 46 K
r(T1, T2) = 0.5
r(T1, T2) = 0.9
Unweighted mean:
T = 10142 K, U(T) = 94 K
Unweighted mean:
T = 10149 K, U(T) = 95 K
Weighted mean:
T = 10207 K, U(T) = 72 K
Weighted mean:
T = 10190 K, U(T) = 73 K
Example – noise temperature comparison
T1 = 9947 K
u(T1)= 102 K
T2 = 10172 K
u(T2)= 125 K
T3 = 10098 K
u(T3)= 73 K
T4 = 10295 K
u(T4)= 46 K
r(T1, T2) = 0.99999999
End
That’s all, folks!
Example – noise temperature comparison
Conclusions
‹ Correlation appears to have some effect on a mean
‹ Effect is not quite understood
‹ Paper to follow in the near future (“Effect of correlation on a
Unweighted mean:
T = 10151 K, U(T) = 95 K
Weighted mean:
T = 8949 K, U(T) = 0.2 K
weighted mean”, M G Cox, C P Eiø)