unknown systematic errors and the method of least squares michael grabe alternative error model:...
TRANSCRIPT
Unknown systematic errors and the method of least squares
Michael Grabe
alternative error model: true values and biases
2
Quantity to be measured true value
Does metrology exist without a net of true values?
First Principle
Not likely!
3
Impact of true values and biases
in least squares
Gauß-Markoff theorem
Assessment of uncertainties
Traceability
Key Comparisons
4
xAβ
Least squares adjustment
mrmm
r
r
aaa
aaa
aaa
...
............
...
...
21
22221
11211
A
r
...2
1
β
mx
x
x
...2
1
xTraceability
5
xAβ xAβ0
m
1iix
m1
β
n
1lili x
n1
x
Mean of means
averaging is permitted if and only if
the respective true values are identical
m
2
1
x
...
x
x
β
1
...
1
1
6
mass
mg1kg0.25
mg1kg0.75
Adjustment ad hoc ?
7
m
2
1
x
...
x
x
empirical variance-covariance matrix
A different approach
8
m
1iii xwβ
Mean of means
9
xAβ xAβ
m
2
1
x
...
x
x
x
Let the input data be arithmetic means
xAAAβ T1T
00 xAAAβ T1T
10
Gauß-Markoff Theorem
The uncertainties are minimal...
...if the system has been weighted appropriately
11
biases abolish the theorem ...
according to the GUM we should have
rmQE min rmQmin
but we encounter
12
no more test of consistency
how to weight the system to
minimize uncertainties?
Consequences ...
13
more ... and of utmost importance:
reduce measurement uncertainties
weightings
shift estimators and
14
a picture
reduction
before after
shift
true value
kβ
kβ
15
Traceability:
vary the weights by trial and error ...
Assessment of uncertainties
16
Key Comparison
National Standards
1β 2β mβ...
true value
true valuetrue value
...
17
Round Robin
Calibration of a Travelling Standard T
...(1)T (2)T (m)T
(1)β (2)β (m)β...
T
18
Key comparisons do more ...
m1,...,i;βTd (i)i
and the differences
Consider the grand mean
(i)m
1iiTwβ
KCRV
19
m
1jjs,jis,iis,
m
1j
Tijj
2i
Pd
fwf2wf
wswsw2sn
1ntu
i
βuTuu 2(i)2d i
„consistent“ with
and look forid
(i) uβT where
20
Problem:
In some cases the GUM may localize the true value of the travelling standard, in others not ...
when
Should we test (i)T against β
(i)T constributes to β ?
21
Differences between KCRV and individual calibrations
1du
2du
mdu
...(2)T
β
(m)T(1)T
true value
KCRV
22
Individual Calibrations
a horizontal line should intersect each of the uncertainties
...(1)T
(2)T
(m)Ttrue value
23
β KCRV
(1)T
(2)T
...(m)T
true value
KCRV and individual calibrations