Review from previous class
 Error
VS
Uncertainty
 Definitions of Measurement Errors
 Measurement Statement as An Interval Estimate
X  X Best Estimate  U X
@ C % confidence limit
PX Best Estimate  U X  X True  X Best Estimate  U X   C
 How to find bias and random uncertainty?
1
Error VS Uncertainty
ERROR
•
Known for certainty. Correct it.
•
A fixed number. Not a statistical variable.
•
The difference between the ‘true value’ (XTrue) and the measured value Xi.
 i : X i  X True
UNCERTAINTY
•
Not known for certainty. No correction scheme is possible.
•
An uncertainty is a possible value that an error may have.
•
A statistical variable.
•
2
Uncertainty can only be estimated.
Frequency of occurrences
Definitions of Measurement Errors
i
Statistical Experiment:

i
the population distribution
X
X True
Assume that we know
Measurement i
X True
and that we neglect the variable-but-deterministic error,
the followings are the definitions of errors associated with the i th measurement.
i
X
Xi
be the subscript for the i th measurement/realization
Assumption:
Note that throughout, we
shall neglect the variablebut-deterministic error.
Xi
be the measured value of X at reading/sample i.
X True
be the true value of X.
 i : X i  X True     i
be the total measurement error for reading i.
 :  X  X True
be the systematic/bias error, which is fixed/constant over realizations.
i
be the random/precision error for reading i, which randomly varying over realizations.
3
Measurement Statement as An Interval Estimate
X  X Best Estimate  U X

@ C % confidence limit
The above statement is interpreted as an interval estimate:
P [ X Best Estimate  U X 
X True  X Best Estimate  U X ]  C
Specified range with some confidence
Xi
XBest Estimate - UX
XBest Estimate
XBest Estimate + UX
XTrue (not known)
Measurement statement as an interval estimate.
4
Bias VS (Band Width of) Scatter in Data
i

 Width
X True
i
For repeated measurements, roughly:
 Bias:
results in the deviation of the (population) mean from the true value.
 Random:
results in the scatter in data in a set of repeated measurements.
It is viewed and quantified as
 Width
(the width of) the band (of scatter) not absolute position.
5
Today Contents
 Uncertainty Propagation Equation (UPE)
 Design Uncertainty Analysis
 Detailed Uncertainty Analysis
6
Questions: What is the uncertainty of derived quantities?
Measured quantity 1
X 1  U X1
Measured quantity 2
X i  U Xi
@ C%
Derived quantity
Measured quantity j
@ C%
X J  UXJ
@ C%
r  r ( X1,...X i ,..., X J )
r
Given all
 U r @ C%
U X i ‘s, how can we find U r ?
7
UPE : Uncertainty Propagation Equation
In our experiments, we have
DRE :
r  r ( X 1 ,... X i ,..., X J )
Propagated uncertainty can be found from
2
2
U r 
2
 r

 r

 r


U X1   ...  
U X i   ...  
U XJ 
 X 1

 X i

 X J

2
2
J
 r

 
U X i    iU X i ,
i 1  X i
i 1

J
2

2

Physical
quantity of
uncertainties
r
i :
X i
2
2
2
2
2
 X i r   U X i 
 X J r   U X J 
 U r   X 1 r   U X1 
  ...  

  ...  
 
 
 
  
r
r

X
X
r

X
X
r

X
X
  
1  
1 
i  
i 
J  
J 


J
U Xi
X i r
X ii
U
2
2
  UMFXi FU Xi , UMFXi :

,
FU X i :
, FU r : r
r X i
r
Xi
r
i 1
2
Note: all errors are assumed to be independent.
8
UPE: Derived Equation
r (unit)
For r= f(x)
Interested students can find more details in the references given below.
Resultant
uncertainty band
UPE as a relation between variances.
DRE :
dr
dx
r Ur
r  r ( X 1 ,... X i ,..., X J )
r
r Ur
Square of the ath realization
STEP 1:
Measured
uncertainty band
Expand uncertainties as a Taylor series, and focus only uncertainty terms,
ra  1X 1,a  ...   iX i ,a  ...   J X J ,a    iX i ,a ;
J
ra 
2
 J
   iX i ,a
 i 1


  X
J
i 1
 i2
 J
  j X j ,a
 j 1
 

 i :
i 1
[Neglect H.O.T.]





J
x - Ux
x
x  Ux
x (unit)
J
i 1 j 1


i 1 j 1
error
r
X i
  i j X i,a X j,a 
J
i ,a X i ,a     i j X i ,a X j ,a 1   ij  ;
J
x x
1,
i j
0,
i j
 ij  
error
[1] Dunn, P. F., 2005, “Measurement and data analysis for engineering and
science,” McGraw-Hill, New York.
[2] Coleman, H. W., Glenn, W., and Steels, Jr., 1998, Experimentation and
uncertainty analysis for engineers, 2nd Edition, Wiley, New York.
9
Decompose the “errors”
STEP 2:
X i ,a   i ,a   i ,a
we have
X i,a X j ,a  i,a  j ,a  i,a  j ,a   i,a  j ,a   i,a  j ,a
The square-equation becomes
ra 
2

  X i,a X i,a    i j X i,a X j,a 1   ij 
J
J
 i2
i 1

i 1 j 1
  

J
2
i
i 1
J
J
i ,a  i ,a

  i ,a  i ,a   i ,a  i ,a   i ,a  i ,a 
  i j  i,a  j,a   i,a  j,a   i,a  j,a   i,a  j,a 1   ij 
J
i 1 j 1
10
STEP 3:
Sum over a, divide by N, take limit N  infinity, we have the relation between variances
 


1 N
1 N  J 2
2
ra  
  i  i ,a  i ,a   i ,a  i ,a   i ,a  i ,a   i ,a  i ,a  
N a 1
N a 1  i 1


1 N  J


N a 1  i 1



   i j  i,a  j ,a   i,a  j ,a   i,a  j ,a   i,a  j ,a 1   ij 
J
j 1
Neglect correlations between two errors
 2 1 N
















 i
i ,a i ,a
i ,a i ,a
i ,a i ,a
i ,a i ,a 
N


i 1 
a 1
J




1 N
 i ,a  j ,a   i ,a  j ,a   i ,a  j ,a   i ,a  j ,a
 i j 1   ij
N
j 1 
a 1

J

J

i 1
 

  i2 2i
1 N
ra 2
: lim

N  N
a 1
 2i
 r2

J
 i2 2i
i 1
 r2
where
    1   
J
J
i
j
ij
i 1 j 1
i  j
   i j
1 N
: lim
 i,a  i,a 
N  N
a 1
 2i



1 N
: lim
  i,a  i,a 
N  N
a 1
are the corresponding variances, and
 i  j


1 N
: lim   i,a  j ,a   i  j  i   j
N  N
a 1
are the corresponding covariances. And,
coefficients.
  i j
 i  j and   i j


1 N
 lim
  i,a  j,a   i j   i   j
N  N
a 1
are the corresponding cross correlation
11
STEP 4: Estimate the population properties with the sample properties.
Since we never know the population properties, we estimate them with the sample
properties.
Replace the above population variances with sample variances
 r2


  i2 2i

  i2 S2i
J
 i2 2i
i 1
 
u c2
S r2

J
i 1
 i2 S 2i
    1   
J
J
i
j
ij
i 1 j 1
    1   S
J
i  j
J
i
i 1 j 1
j
ij
i  j
   i j
 S i j


where S r , S i , S i are the corresponding sample standard deviation, and
S  i  j , S i j are the corresponding sample covariances.
 So far, no assumption regarding the types of error distribution has been made.
12
STEP 5: Multiply by the coverage factor K. Assume Normal distribution for r. Convert to Ur.
S r2


J
i 1

 i2 S 2i
  i2 S2i
    1   S
J
J
i
j
ij
i 1 j 1
i  j

 S i j
To obtain the expanded uncertainty Ur [ISO terminology] at the specified confidence limit C, we
multiply Sr with the coverage factor K.
U r  KS r  Kuc 

It is in choosing K that assumptions regarding the type(s) of the error distributions must be
made.

Because of the central limit theorem, we assume that the error distribution for r is normal. [The
same argument can be made for all Xi.]

Hence, K corresponds to t value from t distribution at C. We thus have
U r2



J
t2,a / 2
i 1
 
J
i 1
 i2 S 2i
  i2 S 2i



 i2 t ,a / 2 S i 2
 t2,a / 2
  i j 1   ij S  
J
J
i
i 1 j 1
j
 S i j

    i j 1   ij t2,a / 2 S  
  i2 t ,a / 2 Si 2
J
J
i 1 j 1
i
j
 t2,a / 2 S i j

13
Note: The cross covariances between bias errors can be significant
when the two have common error sources. For example, Xi and Xj
may be calibrated from the same standards.
STEP 6:
Further simplification for our purpose.
U r2

 
J

 i2 t ,a / 2 S i 2
    i j 1   ij t2,a / 2 S  

  i2 t ,a / 2 Si 2
i 1
J
J
i
i 1 j 1
j
 t2,a / 2 S i j

For simplicity, we shall assume the followings for this class.

All errors are independent. Hence, all cross covariances vanishes.

We shall consider only the case where all variables have the same degree of freedom .

Hence, the above relation reduces to the equation introduced earlier:
U r2


J
 i2 Bi2
  i2 Pi 2


i 1
U r2
where

Br2
  i2U i2 ,
J
U i2 : Bi2  Pi 2
i 1

Pr2
 2
 Br :




 Pr2 :

  i2 Bi2 
J
i 1
,
  i2 Pi2 
Bi : t ,a / 2 S i ,
Pi : t ,a / 2 Si
J
i 1
S i
is the sample standard deviation for bias error of Xi,
S i
is the sample standard deviation for random error of Xi,
14
Summary: UPE
r 
2
2
2
 r

 r
 r

 
X 1   ...  
X i   ...  
X J

X

X

X
 1

 i

 J
r
;
i 
; X i  U Xi
X i
2
2
2
2
2
J  r
J


   
X i     i X i 

X
i 1 
i 1


i
2
2
 X r   X i 
X
 X 1 r   X 1 
r 
 r 
 
  ...   J

 
  ...   i
   
r 
 r X 1   X 1 
 r X i   X i 
 r X J 
X r
X
X i
;
UMFXi  i
 i i,
FU Xi 
,
U r  r ,
r X i
r
Xi
2
 X J

 XJ
2
2

J

2
   UMFXi2 FU Xi
i 1

r
FU r 
r
1. The fraction uncertainty (FU) or per cent uncertainty (x 100) is defined as
FU Xi 
U Xi
,
Xi
FU r 
Ur
r
It is often convenient to think of uncertainty in terms of fraction or %.
2. The uncertainty magnification factor for a variable Xi, UMFXi,, is defined as
UMFXi 
X i r
r X i
It is clear from the UPE that variables Xi ‘s that have larger UMF relatively (for equal
fraction uncertainty) contribute more to the final uncertainty in r (Ur) than those with 15
smaller UMF.

Important Notes Regarding The Applications of
DRE and UPE
1.
Current measurement process: As mentioned earlier, the DRE must reflect the
current measurement process. Hence, e.g., even though the definition of density is

M
V
In order to apply the UPE suitably, we must write – depending upon our current
measurement process, e.g.,

 ( M , V1 ) 
DRE :
or
DRE :
  
 
 
 ( M ,V1 ,V2 ) 
  


  
2
M
 M 1V11
V1
2
2  V1 




 (1) 


1

M 
 V1 
2  M 
2
2
 V1 



 
 M   V1 
 M 
2
2
M
V1  V2
2
2
2
    V1   V2 (V1  V2 ) 
    V2 
M
M
  M   V1 (V1  V2 ) 
 (1) 










 



2  
2 
 
M
(
V

V
)
V
M
(
V

V
)
 M  

1
2
  1  

1
2
   V2 
2
2
2
2
2
  M   V1    V1   V2    V2 

 
 
 

 
 M   V1  V2   V1   V1  V2   V2 
2
as the case may be.
2
16
2
Important Notes Regarding The Applications of
DRE and UPE
2.
All Xi’s variables that may affect the uncertainties: Note that the writing of the
DRE must include all Xi’s variables that may affect the final uncertainty in r, and not
only the measurands (measure variables).
For example,
a) if it is determined that the uncertainty in the gas constant R will have
little/acceptable (if neglected) effect on the determination of the density from
measured pressure p and measured temperature T, we may write the DRE as
(assume that the perfect gas law is acceptable):
p
DRE :  ( p, T ) 
 p 1 R 1T 1
RT
2
  
   
 
2
 p 
 T 
   
T 
 p
2
b) However, if this is not the case, we must take the uncertainty in the ‘constant’ R
into account and write:
p
DRE :  ( p, T , R) 
 p 1 R 1T 1
RT
2
  
   

2
 p   T 
    
 p T 
2
 R 
 
R
2
17
Important Notes Regarding The Applications of
DRE and UPE
3.
The derived quantities must be alone on the LHS of the DRE. In writing the DRE, if r is to be
determined/derived from other measured quantities (not measured directly), r must be alone on
the LHS of the DRE.
For example, even though
p  RT
a) if pressure p is to be determined/derived from the measured density  and the measured
temperature T, the DRE must be written as
DRE :
p (  , T , R )  RT   1 R 1T 1
2
2
 p 
  
 T 
     
 
p

T


 
 
 R 
 
 R
2
2
b) on the other hand, if the density  is to be determined/derived from the measured pressure p
and the measured temperature T, the DRE must be written as
DRE :
 ( p, T , R ) 
p
 p 1 R 1T 1
RT
2
  
 
 



2
 p 
 T 
  
 
p
T


 
2
 R 


 R 
2
As one can see, the uncertainties are accumulated from measurement results and the two UPE
18
equations are not the same.
Some Useful Notes on UPE
1. UPE of common DRE’s
1.1
r  r  X 1 , X 2 ,..., X J   kX 1a1 X 2a2 ...X J J
a

r 2  U r2  ka1 X 1a1 1 X 2a2 ...X Ja J U X 1   ... 2  ...
2
2
2
U
U 
U 
 r 
     r   a12  X 1   ...  a J2  XJ
r 
 r 
 X1 
 XJ
2
NOTE:



2
The variables with higher power relatively (for equal per cent
uncertainty) contribute more to the final uncertainty in r, Ur.
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Some Useful Notes on UPE
1.2
r  r  X 1 , X 2   aX 1  bX 2

r 2  U r2  aU X 1 2  (b)U X 2 2  aU X 1 2  bU X 2 2
2
2
2
UX2 
U X1 
 r   U r 
2
2

  b 
       a 
r  r 
 aX 1  bX 2 
 aX 1  bX 2 
2
NOTE: Because r is the difference (aX1-bX2) and Ur depends upon the division by the
difference, care must be taken when attempting to find ‘small difference (X1-X2)
between large quantities.’
For example, the difference in heights is desired by measurements of X1
and X2 separately: X1 = 100 + 1 cm and X2 = 99 + 1 cm. As one can see, although
the uncertainties in X1 and X2 are each of only ~ 1%, the final uncertainty in r =
(X1 - X2) can be as large as ~ 200%.
This is one of the reasons why sometimes differential type instruments
(measure the difference directly) are desired in such situation.
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General VS Detailed Uncertainty Analysis
General Uncertainty Analysis
Detailed Uncertainty Analysis
No distinction is made between bias
and random uncertainties, U
Decompose uncertainty into bias and
random uncertainties, U  B and P
Elemental error sources
1
i
1
i
Measurement system i
(or any variables that can affect U of r.)
X1
Xi
X1
Xi
U1
Ui
B1 P1
Bi Pi
U1  B12  P12
U i  Bi2  Pi 2
r  r ( X 1 ,... X i ,..., X J )
r  r ( X 1 ,... X i ,..., X J )
Measurement statement i
(or any variables that can affect U of r.)
X i  U i @C %
DRE
UPE
r
Ur
r
Statement for r
Br Pr
Ur 
Br2
 Pr2
r  U r @C %
21
Design-Stage Uncertainty

Design-stage uncertainty analysis refers to an initial analysis performed prior to
the measurement.

It is used to assist in the selection of equipment and procedures based on their
relative performance and cost.

In the design stage, we consider only sources of uncertainty in general. Only
uncertainty caused by a measurement system’s resolution and estimated
intrinsic errors are considered.

The analysis can tell us the minimum uncertainty in the measured value that
would result from the measurement.
Calibration (Measurement) Process
Standard
Instrument
Manufacturer or in-house calibrator
U X ,d  U B2  U P2 @ C %
Bias uncertainty of current
measurement process
Current
Measurement
Process
Experimenter / User
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Why Detailed Uncertainty Analysis?
Detailed Uncertainty Analysis
•
Detailed uncertainty analysis helps us in the interpretation of measured data.
•
Detailed uncertainty analysis can point out to the potential point of improvement
in our experiment, for example, number of sampling, etc.
•
Decompose uncertainty into bias and random uncertainties in order to analyze
each component effectively.
U X2  BX2  PX2
where
•
U r2  Br2  Pr2
U X , Ur
=
the overall uncertainty
BX , Br
=
the bias uncertainty
PX , Pr
=
the random uncertainty
•
Bias uncertainty can be reduced by calibration, not repeated measurements.
•
Random uncertainty can be reduced by repeated measurements.
23
Scheme for Detailed Uncertainty Analysis (for Single
Test)
Elemental error sources
1
i
Measurement system i
(or any variables that can affect U of r.)
X1
B1
Xi
P1
Bi Pi
r  r ( X 1 ,... X i ,..., X J )
Measurement statement i
(or any variables that can affect U of r.)
X i  U i @C %
DRE
Statement for r
r
Br Pr
r  U r @C %
24
STEP 1. Estimate Bias Uncertainty
STEP 1: Identify elemental error sources
1
Standard one is instrument uncertainty but
there are more.
i
Manufacturer data are interpreted as @ 95%.
X1
Xi
B1
Bi
STEP 2: Combine elemental errors using RSS.
Estimate it @ C%.
Bi  Bi2,1  Bi2, 2  ... @ C %
Bi  t ,a / 2 S Bi @ C %
J
UPE :
Br2

   i2 Bi2

(if given as standard deviation)
STEP 3: Write down the UPE for bias uncertainty
i 1
Br
STEP 4: Find Br using UPE.
25
STEP 2. Estimate Random Uncertainty (with
available measured data)
In case measured data are already available.
1
i
X1
Xi
P1
Pi
STEP 1: Determine the random uncertainty from
the statistics of measured data.
PX  t ,a / 2 S X @ C %
PX  t ,a / 2
J
UPE :
Pr2

   i2 Pi 2

SX
N
@ C%
Single sample (SX
from auxiliary test.)
Multiple N sample (SX
from current test.)
STEP 2: Write down the UPE for random uncertainty
i 1
Pr
STEP 4: Find Pr using UPE.
26
STEP 3. Combine Bias and Random Uncertainties
1
i
X1
Xi
U1  B12  P12
U i  Bi2  Pi 2
Measurement statement i
(or any variables that can affect U of r.)
X i  U i @C %
r  r ( X 1 ,... X i ,..., X J )
r
U r  Br2  Pr2
Statement for r
r  U r @C %
27
Scheme for Detailed Uncertainty Analysis (for Multiple
Test, M tests)

Similar to that of single test.

However, since there are multiple r,
 it is recommended to use statistics of the tests, and not UPE, to
estimate the random uncertainty Pr :
Pr  t ,a / 2

Sr
@ C%
M
For bias uncertainty, the same method as for single test can be used.
28
Reference
 Interested students can find further information in the references given.
 Or, the NIST website http://physics.nist.gov/cuu/Uncertainty/index.html
 The relevant standard is
ISO: Guide to the Expression of Uncertainty in Measurement
often called ISO GUM.
 NIST also has technical notes that are free
http://physics.nist.gov/Pubs/guidelines/TN1297/tn1297s.pdf
http://physics.nist.gov/Document/tn1297.pdf
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