understanding the law of propagation of uncertainties · uncertainty expression relative...
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Understanding the Law of Propagation
of UncertaintiesEmma Woolliams
4 April 2017
The Law of Propagation of
Uncertainties
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Things you might already “know”
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Has a sensitivity coefficient
Adding in quadrature (% or units)
Averages reduce by 1 n
This term is to do
with correlation
Building on that starting point
• The Law of Propagation of Uncertainties
• How many readings you should average
• What correlation is
Has a sensitivity coefficient
Adding in quadrature (% or units)
Averages reduce by
This term is to do with correlation
1 n
2c
2
2
1
1
1 1
2 ,
n
iii
n n
i ji ji j i
u y
fu x
x
f fu x x
x x
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
THE LAW OF PROPAGATION
OF UNCERTAINTIES
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
Sensitivity Coefficients
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
How sensitive is
my result to
this?
How sensitive is
my result to
this?
Do an
experiment
to find out!
Lamp power supply:
What is the sensitivity of the irradiance to the lamp current?
0.985
0.99
0.995
1
1.005
1.01
1.015
250 500 750 1000 1250 1500 1750 2000 2250 2500
Sig
nal ra
tio
wavelength / nm
Ratio 27.21 A / 27.19 A
Measured ratio0.2 A10 %
PTFE Sphere phase transition
6 °C
1.5 %
How sensitive is
my result to
this?
Calculate it from
the
measurement
equation
Analytical sensitivity coefficients
ixi
fc
x
1 2 3, , ,y f x x x
The local slope:
Translating a change in one
parameter to a change in
the other
When it works well
S
light
1V
V
S light darkV V V
S
dark
1V
V
Very simple case
When it works well
2FEL 0 45 cal
s 2use
E dL
d
2
s FEL 0 45 cal s
3
use use use
22
L E d L
d d d
Inverse square law leads to a 2
When it doesn’t work well
• Because you can’t write a relationship
• Because it’s too difficult to differentiate
• Because it’s a program not an equation
How sensitive is
my result to
this?
Model
it!
Radiance through double aperture
system
2 2
1 2-d 2
2 2 2 2 2 2 2 2
1 2 1 2 1 2
2
4
r rg
r r x r r x r r
1A g L
By differentiating …
22 2 2 2 2
1 2 1 2
2 2 2
1 2
4r r d r r
r r d
2 2
1 2-d 2
2 2 2 2 2 2 2 2
1 2 1 2 1 2
2
4
r rg
r r x r r x r r
2 2 1
1 22g r r
2 2 2
1 2 1 2
1
2 2 2
1 2 2 1
2
2 2
1 2
4 21
4 21
4
r r r rg
r
r r r rg
r
r r dg
d
Or by “modelling”…
Sensitivity to atmospheric model choice
Tuz Gölü CEOS comparison
2010
Difference between continental
and desert model lead up to 3%
difference in 𝜌 values
Monte Carlo simulation
Sensitivity Coefficients
How sensitive is
my result to
this?
Experimentally
Mathematically
Numerically
ixi
fc
x
Vary in the lab and see
what happens
Vary in the
model and
see what
happens
All are acceptable
Adding in Quadrature
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Why quadrature?
Uncertainty Error
Describes the spread Difference to the
(unknowable) true
value
It’s very unlikely
that all will be
maximum in
same direction!
Combining
distributions
leads to the
quadrature rule
Uncertainty expression
Relative uncertainty:
5 mW m-2 nm-1 ± 0.2 %
i.e. uncertainty expressed as a percentage
Absolute uncertainty:
5 mW m-2 nm-1 ± 0.01 mW m-2 nm-1 i.e.
uncertainty expressed in the native
measurement units
Uncertainty expression
Relative uncertainty:
5 mW m-2 nm-1 ± 0.2 %
i.e. uncertainty expressed as a percentage
Absolute uncertainty:
5 mW m-2 nm-1 ± 0.01 mW m-2 nm-1 i.e.
uncertainty expressed in the native
measurement units
Adding in quadrature for absolute
models
y BA C 1B
y y
CA
y
2 2 2 2Au y u u uB C
Adding in quadrature for absolute
models
y BA
1A
y
22 2 21Au u By u
1B
y
2 2 2Au y u Bu
What behaves like an absolute model?
Distance = Measured + aperture thickness
Signal = Light signal – Dark signal
Uncertainty expression
Relative uncertainty:
5 mW m-2 nm-1 ± 0.2 %
i.e. uncertainty expressed as a percentage
Absolute uncertainty:
5 mW m-2 nm-1 ± 0.01 mW m-2 nm-1 i.e.
uncertainty expressed in the native
measurement units
Adding in quadrature for relative
modelsy BA
y y
A AB
2 2 2
u y u
Ay
B
B
A u
y y
B BA
2 2
2 2 2y yu y BA
BAu u
Adding in quadrature for relative
modelsy BA
1
BA A
y y
2 2 2
u y u
Ay
B
B
A u
2
y A y
BB B
2 2
2 2 2Ay y
BB
u uA
y u
Adding in quadrature for relative
models
2y A B
2
1y y
BA A
2 2 2
2u y u A B
BA
u
y
3
2 2y A y
B BB
2 2
2 2 22A
y yB
By u u
Au
What behaves like a relative model?
• Most radiometric equations!
• Anything where uncertainties are in %
Gain = Radiance / Signal
2FEL 0 :45 cal
s 2use
E dL
d
s s
FEL FEL
L L
E E
s s
0 :45 0 :45
L L
s s
use use
2L L
d d
2 2 22
s 0 :45 useFEL
s FEL 0 :45 use
2u L u u du E
L E d
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
HOW MANY READINGS
SHOULD BE AVERAGED?
At the end of this module, you should
understand
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• What the ‘other term’ – the correlation bit means
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
Applying the Law of Propagation of
Uncertainties to an average
Averaging three readings
1 2 3M
3
E E EE
M M M
1 2 3
1
3
E E E
E E E
2 2 2
2 2 2 2
M 1 2 3
1 1 1
3 3 3u E u E u E u E
1 2 3 iu E u E u E u E
2
2 2
M
13
3iu E u E
2 2
M 2
3
3iu E u E
2 2
M
1
3iu E u E
2
2
M3
iu Eu E
M3
iu Eu E
Applying the Law of Propagation of
Uncertainties to an average
Averaging n readings
M 1
n
iiE E n
M 1
i
E
E n
2 2
M 1
n
iiu E u E n
2
2 2
M
1iu E n u E
n
2 2
M
1iu E u E
n
2
2
M
iu Eu E
n
M
iu Eu E
n
Average light – average dark
s light, dark,
1 1
1 1N M
i j
i i
V V VN M
dark,
dark,
1, 1, ,M
jV
j
Vc j
V M
light dark
2 2
2 V V
V
u uu
N M
light, dark,
2 2
2 2 2
1 1
1 1i j
N M
V V V
i j
u u uN M
light,
light,
1, 1, , N
iV
i
Vc i
V N
Taking enough readings
Applying the Law of Propagation of
Uncertainties to an average
Averaging three readings
1 2 3M
3
E E EE
M M M
1 2 3
1
3
E E E
E E E
2 2 2
2 2 2 2
M 1 2 3
1 1 1
3 3 3u E u E u E u E
1 2 3 iu E u E u E u E
2
2 2
M
13
3iu E u E
2 2
M 2
3
3iu E u E
2 2
M
1
3iu E u E
2
2
M3
iu Eu E
M3
iu Eu E
What is the uncertainty associated
with each reading?
1 2 3 iu E u E u E u E
Uncertainty
Describes the spread
Standard
deviation?
The dangers of taking a standard deviation
of a small number of readings
Reading numberValu
es a
nd c
um
ula
tive S
t d
ev
The dangers of taking a standard deviation
from a small number of readings
What you should do
Use more readings!
• Take more measurements today
M
iu Eu E
n
Better estimate of
uncertainty
More to average
What you should do
Use more readings!
• Have a facility commissioning phase where you
take more measurements
M
iu Eu E
n
Better estimate of
uncertainty
Today’s!
From standard deviation of 10: iu E
Today two readings, 2n
What you should do
Use more readings!
• Compare data from different days
Even if there is a drift:
e.g Use standard deviation of the
difference
between two readings
What you should do
Use more readings!
• Compare data from different wavelengths
(smooth out the bumps)
What you should do
Increase the estimate
• Increase standard uncertainty
Will be in revised GUM
2
light2
light,mean
1
3
sNu
N N
What you should do
Use more readings!
• Take more measurements today
• Have a facility commissioning phase
• Compare data from different days
• Compare data from different wavelengths
Increase the estimate
• Increased standard uncertainty
KNOWING WHEN TO STOP!
Use more readings!
But only when it’s worth it!
The metre: 1791 - 1799
Pierre-
Françoise-André
Méchain
Jean-Baptiste-
Joseph
Delambre
The Measure of all Things - Ken Alder
The uncertainties don’t keep dropping
forever
ADEV
Line Fit
Lower Bound
Upper Bound
AlaVar 5.2
Allan STD DEV
Produced by AlaVar 5.2
time / s 100 1000 10000
0.0001
0.001
The Allan Deviation – software
ADEV
Lower Bound
Upper Bound
AlaVar 5.2
Allan STD DEV
Produced by AlaVar 5.2
time / s 100 1000
1E-5
0.0001
www.alamath.com
At the end of this module, you should
understand
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
CORRELATION
At the end of this module, you should
understand
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
The Law of Propagation of
Uncertainties
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Truei iE E S R
Correlation: Type A and Type B
methods
Correlation occurs when there is something in common between the
different quantities, or between the measured values being combined.
Type A: From the data Type B: From knowledge
This is where the
correlation comes
from!
Systematic
Effects!
Correlation: Type A and Type B
methods
Correlation occurs when there is something in common between the
different quantities, or between the measured values being combined.
Type A: From the data
1
1,
1
ni i
i X Y
X X Y Yr x y
n s s
, ,u x y u x u y r x y
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Truei iE E S R
Correlation: Type A and Type B
methods
Correlation occurs when there is something in common between the
different parameters, or between the measurements being combined.
Type B: From knowledge
This is where the
correlation comes
from!
Systematic
Effects!
Calculate covariance
Remove covariance
Averaging partially correlated data
Truei iE E S R 1 2 3M
3
E E EE
True 1 2 3M
3 3
3 3 3
E R R RSE
2
2 2
M3
iu Ru E u S
Averaging Partially correlated data
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Truei iE E S R
2 2
i iu E u S u R
2, ;i ju E E u S i j
Averaging partially correlated data
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
1 2 1 3 1 4
2 3 2 4
3 4
, , , , , ,
, , , ,
, ,
u E E u E E u E E
u E E u E E
u E E
Averaging partially correlated data
Truei iE E S R 1 2 3
M3
E E EE
M M M
1 2 3
1
3
E E E
E E E
2, ;i ju E E u S i j
2 2
i iu E u S u R
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
2 2
2 2 2
M
1 13 3
3 3iu E u S u R
1 2 2 3
1 3 21 12
3 3u S
2
213 2
3u S
2 2
2 2 2
M
1 13 6 3
3 3iu E u S u R
2
2 2
M3
iu Ru E u S
Systematic and random effects: Lamp
measured 5 times (continued)
Systematic effects Random effects
Reference calibration Noise
Alignment Lamp current fluctuation
Lamp current setting
Temperature sensitivities
,S u S ,i iR u R
Truei iE E S R
2 2 2
i iu E u S u R
2
2 2
M
iu Ru E u S
n
Averaging partially correlated data
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
1 2 1 3 1 4
2 3 2 4
3 4
, , , , , ,
, , , ,
, ,
u E E u E E u E E
u E E u E E
u E E
Covariance Matrix
21 1 2 1
22 1 2 2
21 2
, ,
, ,
, ,
1 2
1
2
n
E n
n n n
u E u E E u E E
u E E u E u E E
u E E u E E un E
n
U
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
1 2 1 3 1 4
2 3 2 4
3 4
, , , , , ,
, , , ,
, ,
u E E u E E u E E
u E E u E E
u E E
Covariance Matrix Version
1 2y
n
f f f
x x x
C
21 1 2 1
22 1 2 2
21 2
, ,
, ,
, ,
1 2
1
2
n
E n
n n n
u E u E E u E E
u E E u E u E E
u E E u E E un E
n
U
Covariance matrix
with absolute
variance (squared
uncertainty) and
covariance
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
Covariance matrix
21 1 2 1
22 1 2 2
21 2
, ,
, ,
, ,
1 2
1
2
n
E n
n n n
u E u E E u E E
u E E u E u E E
u E E u E E un E
n
U
T 1 1i i i iE E S R s r
2 2 2 2 2
,2 2
( )
( ).
i i i
E ij
i j
E u S u R u s u r i jU
E E u S u s i j
Negative sensitivity coefficients
y BA 1A
y
1
B
y
22 2 21Au u By u
2 2 2Au y u Bu
E.g. Signal = Light - Dark
Negative sensitivity coefficients
y BA 1A
y
1
B
y
2 2 2 2 1 1 ,Bu y u AA u u B
E.g. Signal = Light - Dark
2 2 2 2 ,u y u u AB BA u
Correlation
reduces
uncertainty
Positive sensitivity coefficients
y BA 1A
y
1
B
y
2 2 2 2 1 1 ,BAu y u u u A B
E.g. Distance = measured + thickness
2 2 2 2 ,u y u u AB BA u
Correlation
increases
uncertainty
But I don’t know what the covariance is!
• Think of the worst-case scenario
Does correlation increase or decrease the
uncertainty?
Treat as “all random” and “all systematic”
At the end of this module, you should
understand
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
CONCLUSIONS AND
SUMMARY
At the end of this module, you should
understand
• The Law of Propagation of Uncertainties
• Why the sensitivity coefficient is central to the law
• How to determine the sensitivity coefficients
• What ‘adding in quadrature’ really means
• How many readings you should average
• What the uncertainty associated with an average is
• How you know you have enough readings
• When you’ve taken too many readings
• What correlation is
• What the uncertainties are for averaging partially correlated data
• How to estimate covariance from numerical and experimental data
• How to deal with not knowing what the correlation is
Sensitivity Coefficients
2 1
2 2c
1 1 1
2 ,
n n n
i i ji i ji i j i
f f fu y u x u x x
x x x
How sensitive is
my result to
this?
Experimentally
Mathematically
Numerically
Taking enough readings, but not too
many
M
iu Eu E
n
Better estimate of
uncertainty
More to average
ADEV
Line Fit
Lower Bound
Upper Bound
AlaVar 5.2
Allan STD DEV
Produced by AlaVar 5.2
time / s 100 1000 10000
0.0001
0.001
Allan Deviation
Correlation
Truei iE E S R
Correlation occurs when there is something in common between the
different parameters, or between the measurements being combined.
Type B: From knowledge
This is where the
correlation comes
from!
Systematic
Effects!
Calculate covariance
Remove covariance