Comparison of aleatory and epistemic
uncertainty modelling, using @RISK
Presentation at
Palisade 2012 Risk Conference
London, April 18-19, 2012
Hans Schjær-Jacobsen
Professor, Director RD&I
Copenhagen University College of Engineering
Ballerup, Denmark
+45 4480 5030
hsj@ihk.dk
www.ihk.dk
Agenda
1.
2.
•
•
•
3.
4.
5.
Alternative representations of uncertainty
Four dialogues on uncertainty
Rectangular and triangular representations
Non-monotonic functions
Correlation
Representation and calculation
Net present value case
Conclusions
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Rectangular representation [a; b] and {µ; σ}
Membership
function
Probability
distribution
Alternative
interpretations
h
1
α
h = 1/(b-a)
μ = (a+b)/2
σ2 = (b-a)2/12
α-cut
0
a
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b
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0
3
Trapezoidal representation [a; c; d; b] and {µ; σ}
Membership
function
Probability
distribution
Alternative
interpretations
h
1
α
h = 2/(b-a+d-c)
μ = h[(b3-d3)/(b-d)-(c3-a3)/(c-a)]/6
σ2 = [3(r+2s+t)4+6(r2+t2)(r+2s+t)2
-(r2-t2)2]/[12(r+2s+t)]2
α-cut
r
0
a
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s
c
t
d
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b
0
4
Triangular representation [a; c; b] and {µ; σ}
Membership
function
Probability
distribution
Alternative
interpretations
h
1
α
h = 2/(b-a)
μ = (a+b+c)/3
σ2 = (a2+b2+c2-ab-ac-bc)/18
α-cut
0
a
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c
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b
0
5
A Dialogue on Uncertainty (1)
Model Owner (MO): How should I represent the independent uncertain
variable x in my model?
Uncertainty Specialist (US): What do you know about x?
MO: I know that x can attain any value between a and b (a < b).
US: Do you know more about x?
MO: Not really.
US: Then I suggest that x is best represented by a rectangular
possibility distribution (an interval [a; b]).
MO: What is a possibility distribution? I am used to work with
probability distributions.
US: But you do’nt know the probability distribution of x.
MO: No, but in that case I think it is reasonable to assume that x is best
represented by a rectangular probability distribution [a; b].
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Rectangular and triangular arguments of x(1-x)
2.4
1.2
Probability
Possibility
2.0
1.0
1.6
0.8
Probability
1.2
0.6
0.8
0.4
0.4
0.2
0.0
0.0
-0.2
0.0
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0.2
0.4
0.6
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0.8
1.0
1.2
7
Function x(1-x) with rectangular argument x = [0; 1]
35
1.2
Probability
Possibility
30
1.0
25
0.8
20
0.6
15
0.4
10
0.2
5
0
-0.05
0.00
0.05
0.10
0.15
0.20
0.0
0.30
0.25
x(1-x)
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A Dialogue on Uncertainty (2)
Model Owner (MO): How should I represent the independent uncertain
variable x in my model?
Uncertainty Specialist (US): What do you know about x?
MO: I know that the nominal value of x is c. Due to uncertainty x can
attain any value between a and b (a < c < b).
US: Do you know more about x?
MO: Not really.
US: Then I suggest that x is best represented by a triangular possibility
distribution [a; c; b].
MO: What is a possibility distribution? I am used to work with
probability distributions.
US: But you do’nt know the probability distribution of x.
MO: No, but in that case I think it is reasonable to assume that x is best
represented by a triangular probability distribution [a; c; b].
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Function x(1-x) with triangular argument x = [0; 0,2; 1]
35
1.2
Probability
Possibility
30
1.0
25
0.8
20
0.6
15
0.4
10
0.2
5
0
-0.05
0.00
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0.05
0.10
0.15
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0.20
0.25
x(1-x)
0.0
0.30
10
A Dialogue on Uncertainty (3)
Model Owner (MO): How should I represent the independent uncertain
variables x1 and x2 in my model?
Uncertainty Specialist (US): What do you know about the variables?
MO: They can attain any value between a1 and b1 (a2 and b2).
US: Do you know more?
MO: Not really.
US: Then I suggest that they are best represented by rectangular
possibility distributions.
MO: I am used to work with probability distributions.
US: But you do’nt know the probability distributions.
MO: No, but in that case I think that the variables are best represented
by rectangular probability distributions.
US: Do you know about the correlation between the variables?
MO: No, I think I will run a series of correlation coefficients.
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Alternative interpretations of a rectangular distribution
1.2
Probability
0.125
μ = (a+b)/2
σ2 = (b-a)2/12
Possibility
1.0
0.100
0.8
0.075
0.6
h = 1/(b-a)
0.050
0.4
0.025
0.2
0.000
0.0
5
a=7
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b = 16
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12
Addition of two rectangular distributions,
correlation in per cent
0.16
1.2
Probability
Possibility
1.0
0.12
0.8
0%
0.08
0.6
+100%
0.4
0.04
0.2
0.00
0.0
10
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15
20
25
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30
Sum
35
13
Subtraction of two rectangular distributions,
correlation in per cent
0.16
1.2
Possibility
Probability
1.0
0.12
0.8
0%
0.08
0.6
-100%
0.4
0.04
0.2
0.00
0.0
-10
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-5
0
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5
Difference
10
14
Multiplication of two rectangular distributions,
correlation in per cent
0.014
1.2
Probability
Possibility
0.012
1.0
0.010
0.8
0%
0.008
0.6
+100%
0.006
0.4
0.004
0.2
0.002
0.000
0.0
0
Palisade 2012 Risk
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50
100
150
200
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250
Product
300
15
Division of two rectangular distributions,
correlation in per cent
1.8
1.2
Possibility
Probability
1.6
1.0
1.4
0%
1.2
0.8
-100%
1.0
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0
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0.5
1.0
1.5
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2.0
Quotient
2.5
16
A Dialogue on Uncertainty (4)
Model Owner (MO): How should I represent the independent uncertain
variables x1 and x2 in my model?
Uncertainty Specialist (US): What do you know about the variables?
MO: I know that the nominal values are c1 and c2. Due to uncertainty
any value between a1 and b1 (a2 and < b2) may be attained.
US: Do you know more?
MO: Not really.
US: Then I suggest that they are best represented by triangular
possibility distributions.
MO: I am used to work with probability distributions.
US: But you do’nt know the probability distributions.
MO: No, but in that case I think that the variables are best represented
by triangular probability distributions.
US: Do you know about the correlation between the variables?
MO: No, I think I will run a series of correlation coefficients.
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Alternative interpretations of a triangular distribution
1.2
0.25
Probability
μ = (a+b+c)/3
σ2 = (a2+b2+c2-ab-ac-bc)/18
Possibility
1.0
0.20
0.8
0.15
0.6
0.10
α-cut
0.4
0.05
0.2
h = 2/(b-a)
0.00
0.0
5
a=7
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c = 10
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b = 16
18
18
Addition of two triangular distributions,
correlation in per cent
0.16
1.2
Probability
Possibility
1.0
0%
0.12
0.8
+100%
0.08
0.6
0.4
0.04
0.2
0.00
0.0
10
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15
20
25
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Sum
35
19
Subtraction of two triangular distributions,
correlation in per cent
0.16
1.2
Probability
Possibility
1.0
0%
0.12
0.8
-100%
0.08
0.6
0.4
0.04
0.2
0.00
0.0
-10
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-5
0
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5
Difference
10
20
Multiplication of two triangular distributions,
correlation in per cent
0.014
1.2
Probability
Possibility
0.012
0%
1.0
0.010
0.8
+100%
0.008
0.6
0.006
0.4
0.004
0.2
0.002
0.000
0.0
0
Palisade 2012 Risk
Conference
50
100
150
200
Copenhagen University College of
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250
Product
300
21
Division of two triangular distributions,
correlation in per cent
1.8
1.2
Possibility
Probability
1.6
1.0
0%
1.4
1.2
0.8
-100%
1.0
0.6
0.8
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0.0
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0.5
1.0
1.5
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2.0
Quotient
2.5
22
Two worlds of risk and uncertainty
Uncertainty
Imprecision
Ignorance
Lack of knowledge
Statistical nature
Randomness
Variability
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World
Representation
and calculation
Possibility
Possibility distributions
[a; …; b]
Interval arithmetic
Global optimisation
Probability
Probability distributions
{µ; σ}
Linear approximation
Monte Carlo simulation
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Modelling by probability distributions
The actual economic problem is modelled by a function Y of n
independent and uncorrelated uncertain variables Y = Y(X1, X2,…, Xn).
Linear approximation
Y is approximated by means of a Taylor series
Y
Y(μ1,…, μn) + ∂Y/∂X1·(X1-μ1) + ∂Y/∂X2·(X2-μ2) + … + ∂Y/∂Xn·(Xn-μn),
where ∂Y/∂Xi is the partial derivative of Y with respect to Xi calculated
at (μ1,…, μn).
The expected value is given by
E(Y) = μ = Y(μ1,…, μn).
The variance is approximated by
VAR(Y) = σ2 (∂Y/∂X1)2·σ12 +…+ (∂Y/∂Xn)2·σn2.
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Monte Carlo
simulation
24
Modelling by possibility distributions
i.e. intervals, fuzzy intervals, etc.
The actual economic problem is modelled by a function Y of n uncertain
variables Y = Y(X1, X2,…, Xn).
NB: Function can be arranged in different ways.
In case of intervals
Y is calculated by means of interval arithmetic (only applicable in the
simple case) or global optimisation (applicable in the general case).
In case of triple estimates
Extreme values of Y are calculated as above.
In case of fuzzy intervals
As above, for all α-cuts.
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Independent stochastic variables
Intervals
Triple estimates
{μ; σ} = {μ1; σ1} # {μ2; σ2}
[a; b] = [a1; b1] # [a2; b2]
[a; c; b] = [a1; c1; b1] # [a2; c2; b2]
μ = μ1 + μ2;
a = a 1 + a 2;
σ2 = σ12 + σ22
b = b1 + b2
μ = μ1 - μ2;
a = a1 - b2;
σ2 = σ12 + σ22
b = b1 - a 2
μ = μ1·μ2;
a = min(a1a2, a1b2, b1a2, b1b2);
σ2 σ12·μ22 + σ22·μ12
b = max(a1a2, a1b2, b1a2, b1b2)
μ = μ1/μ2;
a = min(a1/b2, a1/a2, b1/b2, b1/a2,);
σ2 σ12/μ22 + σ22·μ12/μ24,
b = max(a1/b2, a1/a2, b1/b2, b1/a2),
Addition
Subtraction
Multiplication
Division
if μ2 ≠ 0
if 0
[a2; b2]
a = a1 + a2;
c = c1 + c2;
b = b 1 + b2
a = a1 - b2;
c = c1 - c2;
b = b1 - a 2
a = min(a1a2, a1b2, b1a2, b1b2);
c = c1c2;
b = max(a1a2, a1b2, b1a2, b1b2)
a = min(a1/b2, a1/a2, b1/b2, b1/a2,);
c = c1/c2;
b = max(a1/b2, a1/a2, b1/b2, b1/a2),
if 0
[a2; b2]
Table 1. Formulas for basic calculations with alternative representations of uncertain variables.
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Discounted cash flow case
Net present value over n periods
NPV = a0 + a1·(1+r1)-1 + a2·(1+r1)-1·(1+r2)-1
+ …+ an·(1+r1)-1·(1+r2)-1·…·(1+rn)-1,
ai = Xi1·Xi2 + Xi3 + Xi4 +…+ Xim,
i = 0,…,n.
ai: net cash flow in i’th period
ri: rate of interest in i’th period
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Interval analysis
($1000)
YEAR 0
YEAR 1
YEAR 2
YEAR 3
YEAR 4
[4.200; 5.200]
[12.400; 14.100]
[15.900; 18.100]
[13.800; 15.600]
Margin (%)
[44,50; 45,50] %
[45,00; 47,00] %
[45,50; 48,50] %
[44,00; 48,00] %
Direct cost
[-2.886; -2.289]
[-7.755; -6.572]
[-9.865; -8.188]
[-8.736; -7.176]
Margin
[-1.869; 2.366]
[5.580; 6.672]
[7.234; 8.779]
[6.072; 7.488]
[-1.000; -800]
[-975; -700]
[-800; -600]
[-800; -600]
[-950; -700]
[-1.375; -1.225]
[-675; -525]
[-675; -525]
Turnover
Marketing cost
[-1.050; -950]
Indirect production cost
RD&E cost
[-3.050; -2.950]
[-1.700; -1.400]
[-350; -250]
[-150; -50]
[-150; -50]
Operating income
[-4.100; -3.900]
[-1.781; -534]
[2.880; 4.452]
[5.609; 7.604]
[4.447; 6.313]
Investment
[-5.100; -4.900]
[-2.200; -1.900]
Net cash flow NCF
[-9.200; -8.800]
[-3.981; -2.434]
[2.880; 4.452]
[5.609; 7.604]
[4.447; 7.013]
[8,50; 9,50] %
[9,00; 11,00] %
[9,50; 12,50] %
[10,50; 13,50] %
[-3.669, -2.223]
[2.369; 3.764]
[4.102; 5.871]
[2.865; 4.901]
Rate of interest r (%)
Discounted cash flow DCF
[-9.200; -8.800]
Net present value NPV
[-3.532; 3.514]
[0; 700]
Table 2a. Discounted cash flow analysis by interval analysis (Interval Solver 2000, overall absolute and relative precision 10-6). Input variables in shaded cells.
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Uniform probability distributions
($1000)
YEAR 0
YEAR 1
YEAR 2
YEAR 3
YEAR 4
{4.700; 289}
{13.250; 491}
{17.000; 635}
{14.700; 520}
Margin (%)
{45,00; 0,29} %
{46,00; 0,58} %
{47,00; 0,87} %
{46,00; 1,15} %
Direct cost
{-2.585; 160}
{-7.155; 276}
{-9.010; 368}
{-7.938; 328}
Margin
{2.115; 131}
{6.095; 239}
{7.990; 333}
{6.762; 293}
{-900; 58}
{-838; 79}
{-700; 58}
{-700; 58}
{-825; 72}
{-1.300; 43}
{-600; 43}
{-600; 43}
Turnover
Marketing cost
{-1.000; 29}
Indirect production cost
RD&E cost
{-3.000; 29}
{-1.550; 87}
{-300; 29}
{-100; 29}
{-100; 29}
Operating income
{-4.000; 41}
{-1.160; 182}
{3.657; 257}
{6.590; 342}
{5.362; 303}
Investment
{-5.000; 58}
{-2.050; 87}
Net cash flow NCF
{-9.000; 71}
{-3.210; 202}
{3.657; 257}
{6.590; 342}
{5.712; 364}
{9,00; 0,29} %
{10,00; 0,58} %
{11,00; 0,87} %
{12,00; 0,87} %
{-2.945; 184}
{3.050; 215}
{4.952; 262}
{3.832; 251}
Rate of interest r (%)
Discounted cash flow DCF
{-9.000; 71}
Net present value NPV
{-111; 470}
{350; 202}
Table 2b. Discounted cash flow analysis by stochastic variables, formulas (11) - (19). Input variables (shaded cells) are derived from uniform probability
distributions corresponding to the interval input variables in Table 2a, however converted to the form {μ; σ}.
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Net present values
Input
Interval/Fuzzy
(double and triple estimates)
Monte Carlo:
{-111; 468}
Stochastic
Uniform
[-3.532; 3.514]
{-111; 470}
Triangular
[-3.532; 1.317; 3.514]
{313; 355}
Monte Carlo:
{314; 354}
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Comparisons
Relative frequency
0,0010
0,0008
0,0006
Input: Triangular
NPV: Normal
µ = 313, σ = 355
(Table 3b)
1,2
1,0
0,8
Input: Uniform
NPV: Normal
µ = -111, σ = 470
(Table 2b)
0,6
0,0004
0,4
0,0002
Worst case
(Table 2a & 3a)
0,2
Best case
(Table 2a & 3a)
0,0000
0,0
-3000
-2000
-1000
0
1000
2000
3000
Net Present Value ($1000)
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Membership function
Most possible case
(Table 3a)
0,0012
Conclusions
• Probability and possibility representations are different!
• If no knowledge of probability distribution is available, use
possibility representation
• If additional statistical knowledge is available use
probability distribution…
• … but be aware of under estimation of risk!
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Thank You!