Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 ›...
Transcript of Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 ›...
![Page 1: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/1.jpg)
1
Determination of KCRV, u(KCRV) & DoE
Final Proposal by KCWG to CCRI(II)
Stefaan Pommé
John Keightley
CCRI(II) Meeting, BIPM, 14-16 May 2013
![Page 2: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/2.jpg)
2
arithmetic weighted
Mandel-Paule Power Moderated Mean
Estimators for mean
![Page 3: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/3.jpg)
3
N
i
ixN
x1
ref
1
N
i
i Nxxxu1
2
refref
2 ).1/()()(
Arithmetic mean
![Page 4: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/4.jpg)
4
101
102
103
104
105
106
107
108
109
0 1 2 3 4 5 6 7 8 9 10
kB
q /
g
Good: ignore wrong unc
![Page 5: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/5.jpg)
5
biased mean
large uncertainty
= not “efficient”
63
68
73
78
83
88
93
98
103
0 1 2 3 4 5
kB
q / g
Bad: inefficient
![Page 6: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/6.jpg)
6
86
91
96
101
106
111
116
0 1 2 3 4 5
kB
q / g
low uncertainty
Bad: low sample variance
![Page 7: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/7.jpg)
7
Remedy against low unc
N
i
iN
ii
NN
xxu
Nxu
1
2
1
2
)1(
)(,
1max)(
Calculate uncertainty from maximum of
- propagated sum of stated uncertainties
- sample variance
![Page 8: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/8.jpg)
8
- measurement uncertainty contains no useful
information
- magnitude error on uncertainty is >2x larger than
magnitude due to metrological reasons
Best solution is close to arithmetic mean if
= the “best” with “bad uncertainty data”
= inefficient with “consistent data”
Arithmetic mean
![Page 9: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/9.jpg)
9
statistical weight = reciprocal variance associated with xi
N
i i
i
u
xxux
12ref
2
ref )(
N
i iuxu 12
ref
2
1
)(
1
Weighted mean
![Page 10: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/10.jpg)
10
63
68
73
78
83
88
93
98
103
0 1 2 3 4 5
kB
q / g
efficient
Good: efficient
![Page 11: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/11.jpg)
11
101
102
103
104
105
106
107
108
109
0 1 2 3 4 5 6 7 8 9 10
kB
q / g
biased mean
low uncertainty
discrepant data set
Bad: sensitive to error
![Page 12: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/12.jpg)
12
- measurement uncertainties are correct
- ‘value’ and ‘uncertainty’ outliers are excluded
Best solution is close to weighted mean if
= the “best” with “perfect data”
= sensitive to “low uncertainty” outliers
Weighted mean
![Page 13: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/13.jpg)
13
,)(1
22ref
2
ref
N
i i
i
su
xxux
N
i i suxu 1,22
ref
2
1
)(
1
s2 is artificially added “interlaboratory” variance to make reduced chi = 1
.)(
1
1~
12
2ref
obs
N
i i
i
u
xx
N
Mandel-Paule mean
![Page 14: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/14.jpg)
14
86
91
96
101
106
111
116
0 1 2 3 4 5
kB
q / g
= weighted mean for a consistent data set
Mandel-Paule mean
![Page 15: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/15.jpg)
15
101
102
103
104
105
106
107
108
109
0 1 2 3 4 5 6 7 8 9 10
kB
q / g
≈ arithmetic mean for an extremely discrepant data set
Mandel-Paule mean
![Page 16: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/16.jpg)
16
63
68
73
78
83
88
93
98
103
0 1 2 3 4 5
kB
q / g
= intermediate between arithmetic and weighted
mean for a slightly discrepant data set
Mandel-Paule mean
![Page 17: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/17.jpg)
17
- measurement uncertainties are informative
- ‘value’ and ‘uncertainty’ outliers are symmetric
Best solution is close to Mandel-Paule mean if
= one of the “best” with “imperfect data”
if no tendency to underestimate uncertainty
Mandel-Paule mean
![Page 18: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/18.jpg)
18
S is a typical uncertainty per datum (max arithmetic or M-P unc)
0<a<2 = power reflects level of trust in uncertainties
Power Moderated Mean - PMM
))(),(max( mp
22 xuxuNS
N
iii xwx
1ref
1
222
ref
2 )(
a
a
Ssuxuw ii
![Page 19: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/19.jpg)
19
= Mandel-Paule mean N
73
78
83
88
93
98
103
108
113
0 1 2 3 4
kB
q /
g
‘understated’ uncertainty?
PMM: a=2
![Page 20: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/20.jpg)
20
N
73
78
83
88
93
98
103
108
113
0 1 2 3 4
kB
q /
g
= closer to arithmetic mean
less weight to this point
PMM: a=1
![Page 21: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/21.jpg)
21
Choice of power a
power reliability of uncertainties
a = 0 uninformative uncertainties
(arithmetic mean)
a = 0 uncertainty variation due to error at least twice
the variation due to metrological reasons
(arithmetic mean)
a = 2-3/N informative uncertainties with a tendency of being
rather underestimated than overestimated
(intermediately weighted mean)
a = 2 informative uncertainties with a modest error;
no specific trend of underestimation
(Mandel-Paule mean)
a = 2 accurately known uncertainties, consistent data
(weighted mean)
![Page 22: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/22.jpg)
22
arithmetic weighted
Mandel-Paule Power Moderated Mean
Test of Estimators by Computer Simulation
![Page 23: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/23.jpg)
23
Efficiency for discrepant data
arithmetic < , M-P < weighted
0 5 10 15 20
1.0
1.1
1.2
1.3
1.4
1.5
AM
M-P
PMM
WM
sta
nd
ard
de
via
tio
n
N
EFFICIENCY
![Page 24: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/24.jpg)
24
Reliability of uncertainty
weighted << M-P < < arithmetic
0 5 10 15 20
0.5
1.0
1.5
2.0
2.5
3.0
RELIABILITY
AM
M-P
PMM
WM
un
ce
rta
inty
/ s
tan
da
rd d
evia
tio
n
N
![Page 25: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/25.jpg)
25
- measurement uncertainties are informative
- uncertainties tend to be understated
- data seem consistent but are not
Best solution is close to PMM if
= one of the “very best” with “imperfect data”
= more realistic uncertainty than Mandel-Paule mean
= more adjustable to quality of data than M-P mean
Power Moderated Mean
![Page 26: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/26.jpg)
26
generally applicable method
Outlier identification
![Page 27: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/27.jpg)
27
CCRI(II) is the final arbiter regarding correcting or
excluding any data from the calculation of the KCRV.
Statistical tools may be used to indicate data
that are extreme.
= a way to protect the KCRV against erroneous data,
data with understated uncertainty, extreme data
asymmetrically disposed to the KRCV
= a way to lower the uncertainty on the KCRV
Outlier identification
![Page 28: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/28.jpg)
28
- valid for any type of mean, using normalised wi
- default k = 2.5
Outlier identification
)1
1)(( )( ref
22 i
iw
xueu
)1
1)(()( ref
22 i
iw
xueu
xi included in mean
xi excluded from mean
|ei| > ku(ei), ei = xi – xref
![Page 29: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/29.jpg)
29
Both options are possible within the method.
→ outlier rejection should be based on technical grounds
Outlier or not?
L
63
68
73
78
83
88
93
98
103
0 1 2 3 4 5
kB
q /
g
L
63
68
73
78
83
88
93
98
103
0 1 2 3 4 5
kB
q /
g
![Page 30: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/30.jpg)
30
generally applicable method
Degree of equivalence
![Page 31: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/31.jpg)
31
- valid for any type of mean
Degree of equivalence
xi included in mean
xi excluded from mean
di = xi – xref, U(di) = 2u(di)
)()21()( ref222 xuuwdu iii
)()( ref222 xuudu ii
![Page 32: Determination of KCRV, u(KCRV) & DoE › cc › CCRI(II) › Allowed › 22 › 2013_05...asymmetrically disposed to the KRCV = a way to lower the uncertainty on the KCRV Outlier identification](https://reader035.fdocuments.in/reader035/viewer/2022070716/5ed9a22a5139c40fce675982/html5/thumbnails/32.jpg)
32
The Power Moderated Mean keeps a fine balance between efficiency and robustness, while providing also a reliable uncertainty.
Outlier identification and degrees of equivalence are readily obtained.
Conclusions