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

aaaaaa

...............

...

...

21

22221

11211

A

r

...2

1

β

mx

xx

...2

1

xTraceability

5

xAβ xAβ0

m

1iix

m1β

n

1lili xn

1x

Mean of means

averaging is permitted if and only if the respective true values are identical

m

2

1

x...xx

β

1...11

6

mass

mg1kg0.25

mg1kg0.75

Adjustment ad hoc ?

7

m

2

1

x...xx

empirical variance-covariance matrix

A different approach

8

m

1iii xwβ

Mean of means

9

xAβ xAβ

m

2

1

x...xx

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 ComparisonNational Standards

1β 2β mβ...

true value

true valuetrue value

...

17

Round RobinCalibration 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 ...

whenShould 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 valueKCRV

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