Comparison of aleatory and epistemic uncertainty modelling,

Transcription

1 Comparison of aleatory and epistemic uncertainty modelling, Presentation at London, April 18-19, 2012 Hans Schjær-Jacobsen Professor, Director RD&I Ballerup, Denmark

2 Agenda 1. Alternative representations of uncertainty 2. Four dialogues on uncertainty Rectangular and triangular representations Non-monotonic functions Correlation 3. Representation and calculation 4. Net present value case 5. Conclusions 2

3 Rectangular representation [a; b] and {µ; σ} Membership function Alternative interpretations distribution 1 h α α-cut h = 1/(b-a) μ = (a+b)/2 σ 2 = (b-a) 2 / a b 3

4 Trapezoidal representation [a; c; d; b] and {µ; σ} Membership function Alternative interpretations distribution 1 h α α-cut h = 2/(b-a+d-c) μ = h[(b 3 -d 3 )/(b-d)-(c 3 -a 3 )/(c-a)]/6 σ 2 = [3(r+2s+t) 4 +6(r 2 +t 2 )(r+2s+t) 2 -(r 2 -t 2 ) 2 ]/[12(r+2s+t)] 2 r s t 0 0 a c d b 4

5 Triangular representation [a; c; b] and {µ; σ} Membership function Alternative interpretations distribution 1 h α α-cut h = 2/(b-a) μ = (a+b+c)/3 σ 2 = (a 2 +b 2 +c 2 -ab-ac-bc)/ a c b 5

6 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]. 6

7 Rectangular and triangular arguments of x(1-x) 2.4 Possibility

8 Function x(1-x) with rectangular argument x = [0; 1] Possibility x(1-x) 8

9 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]. 9

10 Function x(1-x) with triangular argument x = [0; 0,2; 1] Possibility x(1-x)

11 A Dialogue on Uncertainty (3) Model Owner (MO): How should I represent the independent uncertain variables x 1 and x 2 in my model? Uncertainty Specialist (US): What do you know about the variables? MO: They can attain any value between a 1 and b 1 (a 2 and b 2 ). 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. 11

12 Alternative interpretations of a rectangular distribution μ = (a+b)/2 σ 2 = (b-a) 2 /12 Possibility h = 1/(b-a) a = 7 b =

13 Addition of two rectangular distributions, correlation in per cent 0.16 Possibility % % Sum 35 13

14 Subtraction of two rectangular distributions, correlation in per cent 0.16 Possibility % % Difference 10 14

15 Multiplication of two rectangular distributions, correlation in per cent Possibility % % Product

16 Division of two rectangular distributions, correlation in per cent % Possibility % Quotient

17 A Dialogue on Uncertainty (4) Model Owner (MO): How should I represent the independent uncertain variables x 1 and x 2 in my model? Uncertainty Specialist (US): What do you know about the variables? MO: I know that the nominal values are c 1 and c 2. Due to uncertainty any value between a 1 and b 1 (a 2 and < b 2 ) 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. 17

18 Alternative interpretations of a triangular distribution μ = (a+b+c)/3 σ 2 = (a 2 +b 2 +c 2 -ab-ac-bc)/18 Possibility α-cut h = 2/(b-a) a = 7 c = 10 b =

19 Addition of two triangular distributions, correlation in per cent Possibility % % Sum 35 19

20 Subtraction of two triangular distributions, correlation in per cent Possibility 0% % Difference 10 20

21 Multiplication of two triangular distributions, correlation in per cent % Possibility % Product

22 Division of two triangular distributions, correlation in per cent % Possibility % Quotient

23 Two worlds of risk and uncertainty Uncertainty Imprecision Ignorance Lack of knowledge World Possibility Representation and calculation Possibility distributions [a; ; b] Interval arithmetic Global optimisation Statistical nature Randomness Variability distributions {µ; σ} Linear approximation Monte Carlo simulation 23

24 Modelling by probability distributions The actual economic problem is modelled by a function Y of n independent and uncorrelated uncertain variables Y = Y(X 1, X 2,, X n ). Linear approximation Y is approximated by means of a Taylor series Y Y(μ 1,, μ n ) + Y/ X 1 (X 1 -μ 1 ) + Y/ X 2 (X 2 -μ 2 ) + + Y/ X n (X n -μ n ), where Y/ X i is the partial derivative of Y with respect to X i calculated at (μ 1,, μ n ). The expected value is given by E(Y) = μ = Y(μ 1,, μ n ). The variance is approximated by VAR(Y) = σ 2 ( Y/ X 1 ) 2 σ ( Y/ X n ) 2 σ n2. Monte Carlo simulation 24

25 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(X 1, X 2,, X n ). 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. 25

26 Independent stochastic variables Intervals Triple estimates {μ; σ} = {μ 1 ; σ 1 } # {μ 2 ; σ 2 } [a; b] = [a 1 ; b 1 ] # [a 2 ; b 2 ] [a; c; b] = [a 1 ; c 1 ; b 1 ] # [a 2 ; c 2 ; b 2 ] Addition μ = μ 1 + μ 2 ; σ 2 = σ σ 2 a = a 1 + a 2 ; b = b 1 + b 2 a = a 1 + a 2 ; c = c 1 + c 2 ; b = b 1 + b 2 Subtraction μ = μ 1 - μ 2 ; σ 2 = σ σ 2 a = a 1 - b 2 ; b = b 1 - a 2 a = a 1 - b 2 ; c = c 1 - c 2 ; b = b 1 - a 2 Multiplication μ = μ 1 μ 2 ; σ 2 σ 12 μ σ 22 μ 1 a = min(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ); b = max(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ) a = min(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ); c = c 1 c 2 ; b = max(a 1 a 2, a 1 b 2, b 1 a 2, b 1 b 2 ) Division μ = μ 1 /μ 2 ; σ 2 σ 2 1 /μ σ 22 μ 2 1 /μ 4 2, if μ 2 0 a = min(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2,); b = max(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2 ), if 0 [a 2 ; b 2 ] a = min(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2,); c = c 1 /c 2 ; b = max(a 1 /b 2, a 1 /a 2, b 1 /b 2, b 1 /a 2 ), if 0 [a 2 ; b 2 ] Table 1. Formulas for basic calculations with alternative representations of uncertain variables. 26

27 Discounted cash flow case Net present value over n periods NPV = a 0 + a 1 (1+r 1 ) -1 + a 2 (1+r 1 ) -1 (1+r 2 ) a n (1+r 1 ) -1 (1+r 2 ) -1 (1+r n ) -1, a i = X i1 X i2 + X i3 + X i4 + + X im, i = 0,,n. a i : net cash flow in i th period r i : rate of interest in i th period 27

28 Interval analysis ($1000) YEAR 0 YEAR 1 YEAR 2 YEAR 3 YEAR 4 Turnover [4.200; 5.200] [12.400; ] [15.900; ] [13.800; ] Margin (%) [44,50; 45,50] % [45,00; 47,00] % [45,50; 48,50] % [44,00; 48,00] % Direct cost [-2.886; ] [-7.755; ] [-9.865; ] [-8.736; ] Margin [-1.869; 2.366] [5.580; 6.672] [7.234; 8.779] [6.072; 7.488] Marketing cost [-1.050; -950] [-1.000; -800] [-975; -700] [-800; -600] [-800; -600] Indirect production cost [-950; -700] [-1.375; ] [-675; -525] [-675; -525] RD&E cost [-3.050; ] [-1.700; ] [-350; -250] [-150; -50] [-150; -50] Operating income [-4.100; ] [-1.781; -534] [2.880; 4.452] [5.609; 7.604] [4.447; 6.313] Investment [-5.100; ] [-2.200; ] [0; 700] Net cash flow NCF [-9.200; ] [-3.981; ] [2.880; 4.452] [5.609; 7.604] [4.447; 7.013] Rate of interest r (%) [8,50; 9,50] % [9,00; 11,00] % [9,50; 12,50] % [10,50; 13,50] % Discounted cash flow DCF [-9.200; ] [-3.669, ] [2.369; 3.764] [4.102; 5.871] [2.865; 4.901] Net present value NPV [-3.532; 3.514] Table 2a. Discounted cash flow analysis by interval analysis (Interval Solver 2000, overall absolute and relative precision 10-6 ). Input variables in shaded cells. 28

29 Uniform probability distributions ($1000) YEAR 0 YEAR 1 YEAR 2 YEAR 3 YEAR 4 Turnover {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} Marketing cost {-1.000; 29} {-900; 58} {-838; 79} {-700; 58} {-700; 58} Indirect production cost {-825; 72} {-1.300; 43} {-600; 43} {-600; 43} 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} {350; 202} Net cash flow NCF {-9.000; 71} {-3.210; 202} {3.657; 257} {6.590; 342} {5.712; 364} Rate of interest r (%) {9,00; 0,29} % {10,00; 0,58} % {11,00; 0,87} % {12,00; 0,87} % Discounted cash flow DCF {-9.000; 71} {-2.945; 184} {3.050; 215} {4.952; 262} {3.832; 251} Net present value NPV {-111; 470} 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}. 29

30 Net present values Monte Carlo: {-111; 468} Input Interval/Fuzzy (double and triple estimates) Stochastic Uniform [-3.532; 3.514] {-111; 470} Triangular [-3.532; 1.317; 3.514] {313; 355} Monte Carlo: {314; 354} 30

31 Relative frequency Membership function Comparisons 0,0012 0,0010 Input: Triangular NPV: Normal µ = 313, σ = 355 (Table 3b) Most possible case (Table 3a) 1,2 1,0 0,0008 0,0006 Input: Uniform NPV: Normal µ = -111, σ = 470 (Table 2b) 0,8 0,6 0,0004 0,4 0,0002 0,0000 Worst case (Table 2a & 3a) Net Present Value ($1000) Best case (Table 2a & 3a) 0,2 0,0 31

32 Conclusions 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! 32

33 Thank You!