Outline
Copulas for Uncertainty Analysis
GUM
Refractive index
Shortcomings
Antonio Possolo
GUM SUPPLEMENT 1
Change-of-Variables Formula
Monte Carlo Method
COPULAS
Definition & Illustrations
GUM Example H.2
Jun 21st, 2010
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Refractive Index
Refractive Index — Partial Derivatives
Apex angle α, refractive index
n, immersed in medium with
refractive index m
ƒ̇m (m, φ, ψ, α) =
α
Light enters prism at angle φ,
traverses prism’s body, and
exits at angle ψ
sin(α/ 2)
ϕ
ƒ̇φ (m, φ, ψ, α) =m
ψ
As prism rotates about light’s
entrance point, δ = 2(φ + ψ − α)
decreases, reaches minimum
δM , then increases
n=m
sin(φ + ψ − α/ 2)
ƒ̇ψ (m, φ, ψ, α) =m
sin((α + δM )/ 2)
ƒ̇α (m, φ, ψ, α) = −
sin(α/ 2)
— 3 / 33
cos(φ + ψ − α/ 2)
sin(α/ 2)
cos(φ + ψ − α/ 2)
sin(α/ 2)
m h cos(φ + ψ − α2 )
2
sin( α2 )
+
sin(φ + ψ − α2 ) i
sin( α2 ) tn( α2 )
— 4 / 33
Streamlining Computation
GUM Approximation — Shortcomings
If some first order partial derivatives of ƒ are zero at
values of input quantities, then GUM’s
approximation may be faulty
Computation of standard uncertainty using GUM’s
Taylor approximation is tedious and error-prone,
especially when measurement equation is
non-linear or involves special functions
Radiant power W = κ cos(A)
GUM’s approximation may be poor when ƒ is
markedly non-linear in neighborhood of values of
input quantities
Employ software capable of computing derivatives
Analytically (for example, Maple)
Numerically (for example, R)
Need to study ƒ ’s curvature is extra burden
R is freely available, and its source code is open:
http://www.r-project.org/
If curvature is appreciable and influential,
need higher-order approximation
metRology package (LGC / NIST)
However, examples can be constructed where
first-order approximation is better than second-order
approximation — Wang & Iyer (2005)
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GUM Approximation — More Shortcomings
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GUM Supplement 1 — Problem
Expanded uncertainties and coverage intervals
involve coverage factor whose value depends on
generally unverifiable assumption that output
quantity has Gaussian or Student’s t distribution
Given
Measurement equation Y = ƒ (X1 , . . . , Xn )
Joint probability distribution of X1 , . . . , Xn
(with density φ)
Even when Y ∼ tν , Welch-Satterthwaite formula
often yields inappropriate value for ν
Determine Y’s probability distribution
Example (GUM H.1)
Derive Y’s density analytically
 = S + d − S (1 − δα θ − αS δθ)
— Change-of-variables formula
Welch-Satterthwaite formula yields νeff = 16
Sample Y’s distribution
GUM’s 99 % coverage interval 17 % longer
than it needs to be
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Change of Variables Formula
GUM Supplement 1 — Step 1
Transformation τ = (ƒ , τ2 , . . . , τn )
One-to-one, continuously differentiable
Inverse has non-vanishing Jacobian
Y
Z2
...
Zn
=
=
PROPAGATING DISTRIBUTIONS (MONTE CARLO)
STEP 1: Select probability distribution to describe
state of knowledge about input quantities
ƒ (X1 , . . . , Xn )
τ2 (X1 , . . . , Xn )
=
If X1 , . . . , Xn can be regarded as independent
random variables, then assign probability
distributions to each one of them separately
τn (X1 , . . . , Xn )
Y has probability density ψ
ψ(y) =
Z
Z
...
Z2
φ τ −1 (y, z2 , . . . , zn )
Otherwise, assign multivariate distribution to all of
them together . . . using a copula
In either case, mean and standard deviation of each
Xj ’s distribution should be j and (j )
Zn
| Jτ −1 (y, z2 , . . . , zn )| dz2 . . . dzn
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STEP 4: Assign to (y) (standard uncertainty of
output quantity) the value of the standard deviation
of y1 , . . . , yK
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Shaded region includes
95 % of total area under
curve: footprint on
horizontal axis is a
coverage interval for
measurand with 95%
confidence
12
10
0
yk = ƒ (k1 , . . . , kn ) for k = 1, . . . , K
2
4
STEP 3: Simulate K vectors of values of input
quantities, and compute
8
STEP 2: Choose suitably large number K for size of
sample to generate from probability distribution of
output quantity
Use sample {y1 , . . . , yK }
to estimate probability
density of prism’s
refractive index — most
complete
characterization of
uncertainty
6
PROPAGATING DISTRIBUTIONS (MONTE CARLO)
14
GUM Supplement 1 — Refractive Index (CI)
Probability Density
GUM Supplement 1 — Steps 2–4
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1.35
1.45
1.55
1.65
Refractive Index
— 12 / 33
GUM Example H.2
GUM Example H.2
INPUT QUANTITIES
OUTPUT QUANTITIES
Amplitude V of sinusoidally-alternating potential
difference across electrical circuit’s terminals
Amplitude  of alternating current
Phase-shift angle ϕ of alternating potential
difference relative to alternating current
Resistance R =
V

cos ϕ
Reactance X =
V

sin ϕ
Impedance’s magnitude Z =
V

V, , and ϕ are correlated
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Problem & Solutions
— 14 / 33
Copulas are not Cupolas
PROBLEM
Given probability distributions for input quantities,
and correlations between them, how does one
manufacture a joint probability distribution
consistent with those margins and correlations?
SOLUTIONS
There are infinitely many joint distributions that are
consistent with given marginal distributions and
correlations
Copulas join univariate probability distributions into
multivariate distributions and impose dependence
structure
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Manufacturer mentioned solely to
acknowledge image source, with
no implied recommendation or endorsement by NIST that cupola portrayed is the best available for any
particular purpose
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Copula — Definition
Wassily Hoeffding, 1914–1991
A copula is the cumulative distribution function of a
multivariate distribution on the unit hypercube all
of whose margins are uniform
Clayton copula
inducing Kendall’s
τ = 0.6
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Sklar’s (1959) Theorem
Copula — Example: Definition
Example (Gaussian Copula on Beta Margins)
If H is CDF of multivariate distribution whose
margins have CDFs F1 , . . . , Fn , then there exists
copula C such that
H(1 , . . . , n ) = C F1 (1 ), . . . , Fn (n )
U ∼ BETA(3, 5) has CDF F
V ∼ BETA(4, 2) has CDF G
If F1 , . . . , Fn are continuous, then C is unique
ρ = cor(U, V) = 0.7
C(p1 , . . . , pn ) = H F1−1 (p1 ), . . . , Fn−1 (pn )
Copula C such that
C(p, q) = ρ F −1 (p), G−1 (q) for 0 ¶ p, q ¶ 1
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Copula — Example: Joint Density
Influential Copulas
0.9
Joint distributions with same Gaussian margins and
same correlation
Clayton Copula
v
15
15
0.3
0.5
0.7
Gaussian Copula
0.5
0.7
10
0.9
u
0
0.3
0
0.1
5
5
0.1
10
v
u
−10
0
10
20
30
40
50
−10
0
10
20
30
— 21 / 33
GUM Example H.2
40
50
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GUM Example H.2 — Data
OUTPUT QUANTITIES
Resistance R =
V

cos ϕ
Reactance X =
V

sin ϕ
Impedance’s magnitude Z =
V (V)
5.007
4.994
5.005
4.990
4.999
V

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 (mA)
19.663
19.639
19.640
19.685
19.678
ϕ (rad)
1.045 6
1.043 8
1.046 8
1.042 8
1.043 3
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GUM Example H.2 — Evaluations
GUM Example H.2 — GUM Supplement 1
Input quantity values estimated by averages V, ,
and ϕ, of sets of five indications
CHALLENGES
Quality of estimates of (V), (), (ϕ), cor(V, ),
cor(V, ϕ), and cor(, ϕ) limited by very small number
of indications they are based on
Uncertainties and correlations of input quantities
assessed by Type A evaluations
p
(V) = SD(5.007, 4.994, 5.005, 4.990, 4.999)/ 5
Similarly for () and (ϕ)
cor(V, ), cor(V, ϕ), and cor(, ϕ) estimated by
correlations between paired sets of five indications
Marginal distributions must be chosen for V,  and
ϕ, and then linked using a copula that reproduces
correlations between them
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William Gosset, 1876–1937
GUM Example H.2 — GUM Supplement 1
SOLUTION
Employ Student t4 distributions for
V − μV  − μ
ϕ − μϕ
,
, and
(V)
()
(ϕ)
Tune Gaussian copula using correlation supplicant
Apply copula
To sample correlation matrix
To use sampled correlation matrix in turn
to sample output quantities
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Correlation Supplicant
GUM Example H.2 — GUM Supplement 1
RESULTS
Copula parameter θ determines dependence
structure, in particular correlation matrix ρ(θ)
GAUSSIAN COPULA WITH GAUSSIAN MARGINS
Example
AVE

U99%
STUDENT: θ comprises ν and 
CLAYTON: C(p, q) = p−θ + q−θ − 1
−1/ θ
, θ ¾ −1
R (Ω)
127.7
0.0711
0.18
X (Ω)
219.8
0.296
0.76
Z (Ω)
254.3
0.236
0.61
GAUSSIAN COPULA WITH STUDENT MARGINS
Correlation Supplicant selects θ that minimizes
difference between ρ and ρ(θ)
AVE
Minimization with respect to θ done under constraint
that ρ(θ) must be bona fide correlation matrix

U99%
R (Ω)
127.7
0.0828
0.29
X (Ω)
219.8
0.289
0.93
Z (Ω)
254.3
0.232
0.75
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GUM Example H.2 — Results
GUM Example H.2 — Limitation
GAU−GAU
GAU−STU
All usual copulas fail to capture fact that joint
distribution of (R, X, Z) is degenerate
127.5
128.5
218.5
219.5
221.5
253.0
R
128.5
255.5
Z
However, this is largely irrelevant in this case
because relative standard uncertainties are very
small — all less than 0.1 %
253.5
●
99%
95%
252.5
253.5
●
252.5
218.5
127.5
Z 2 = R2 + X 2
255.0
Z
255.5
254.5
Z
220.5
X
219.5
●
126.5
254.0
X
221.5
R
220.5
254.5
126.5
126.5
127.5
R
128.5
218.5
219.5
220.5
221.5
X
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Conclusions & Reference
When input quantities are correlated, and their joint
distribution is not determined otherwise, copula
must be used to apply Monte Carlo Method of GUM
Supplement 1
Many different copulas can reproduce given
correlations, yet lead to possibly very different
uncertainty evaluations for output quantity
A. Possolo (2010) Copulas for uncertainty analysis
Metrologia, 47 (3): 262–271
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