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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
2
arithmetic weighted
Mandel-Paule Power Moderated Mean
Estimators for mean
3
N
i
ixN
x1
ref
1
N
i
i Nxxxu1
2
refref
2 ).1/()()(
Arithmetic mean
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
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
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86
91
96
101
106
111
116
0 1 2 3 4 5
kB
q / g
low uncertainty
Bad: low sample variance
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
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- 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
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
10
63
68
73
78
83
88
93
98
103
0 1 2 3 4 5
kB
q / g
efficient
Good: efficient
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
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
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
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
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
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
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
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
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
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
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)
22
arithmetic weighted
Mandel-Paule Power Moderated Mean
Test of Estimators by Computer Simulation
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
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
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
26
generally applicable method
Outlier identification
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
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
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
30
generally applicable method
Degree of equivalence
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
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
Recommended