Propagation of Uncertainty Jake Blanchard Spring 2010 Uncertainty
Propagation of Uncertainty Jake Blanchard Spring 2010 Uncertainty Analysis for Engineers 1
Introduction �We’ve discussed single-variable probability distributions �This lets us represent uncertain inputs �But what of variables that depend on these inputs? How do we represent their uncertainty? �Some problems can be done analytically; others can only be done numerically �These slides discuss analytical Uncertainty Analysis for Engineers 2
Functions of 1 Random Variable �Suppose we have Y=g(X) where X is a random input variable �Assume the pdf of X is represented by f x. �If this pdf is discrete, then we can just map pdf of X onto Y �In other words X=g-1(Y) �So fy(Y)=fx[g-1(y)] Uncertainty Analysis for Engineers 3
Example �Consider Y=X 2. �Also, assume discrete pdf of X is as shown below �When X=1, Y=1; X=2, Y=4; X=3, Y=9 0. 4 0. 35 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 0 0 1 2 3 4 5 6 0 5 10 15 20 Uncertainty Analysis for Engineers 25 30 4
Discrete Variables �Example: ◦ Manufacturer incurs warranty charges for system breakdowns ◦ Charge is C for the first breakdown, C 2 for the second failure, and Cx for the xth breakdown (C>1) ◦ Time between failures is exponentially distributed (parameter ), so number of failures in period T is Poisson variate with parameter T ◦ What is distribution for warranty cost for T=1 year Uncertainty Analysis for Engineers 5
Formulation Uncertainty Analysis for Engineers 6
Plots C=2 =1 Uncertainty Analysis for Engineers 7
CDF For Discrete Distributions �If g(x) monotonically increases, then P(Y<y)=P[X<g-1(y)] �If g(x) monotonically decreases, then P(Y<y)=P[X>g-1(y)] �…and, formally, y y x x Uncertainty Analysis for Engineers 8
Another Example �Suppose Y=X 2 and X is Poisson with parameter Uncertainty Analysis for Engineers 9
Continuous Distributions �If fx is continuous, it takes a bit more work Uncertainty Analysis for Engineers 10
Example Normal distribution Mean=0, =1 Uncertainty Analysis for Engineers 11
Example �X is lognormal Normal distribution Uncertainty Analysis for Engineers 12
If g-1(y) is multi-valued… Uncertainty Analysis for Engineers 13
Example (continued) lognormal Uncertainty Analysis for Engineers 14
Example Uncertainty Analysis for Engineers 15
A second example �Suppose we are making strips of sheet metal �If there is a flaw in the sheet, we must discard some material �We want an assessment of how much waste we expect �Assume flaws lie in line segments (of constant length L) making an angle with the sides of the sheet � is uniformly distributed from 0 to Uncertainty Analysis for Engineers 16
Schematic L w Uncertainty Analysis for Engineers 17
Example (continued) �Whenever a flaw is found, we must cut out a segment of width w Uncertainty Analysis for Engineers 18
Example (continued) �g-1 is multivalued < /2 > /2 Uncertainty Analysis for Engineers 19
Results pdf L=1 cdf Uncertainty Analysis for Engineers 20
Functions of Multiple Random Variables �Z=g(X, Y) �For discrete variables �If we have the sum of random variables �Z=X+Y Uncertainty Analysis for Engineers 21
Example �Z=X+Y 0. 7 0. 6 0. 4 0. 3 0. 2 0. 45 0. 1 0. 4 0 0 0. 5 1 1. 5 2 2. 5 3 3. 5 x 0. 35 0. 3 fy fx 0. 5 0. 2 0. 15 0. 1 0. 05 0 0 5 10 15 20 25 30 35 y Uncertainty Analysis for Engineers 22
Analysis X Y Z P Z-rank 1 10 11 . 08 1 1 20 21 . 04 4 1 30 31 . 08 7 2 10 12 . 24 2 2 20 22 . 12 5 2 30 32 . 24 8 3 10 13 . 08 3 3 20 23 . 04 6 3 30 33 . 08 9 Uncertainty Analysis for Engineers 23
Result 0. 3 0. 25 fz 0. 2 0. 15 0. 1 0. 05 0 0 5 10 15 20 25 30 35 Z Uncertainty Analysis for Engineers 24
Example �Z=X+Y 0. 7 0. 6 fx 0. 5 0. 4 0. 3 0. 2 fy 0. 1 0 0 0. 5 1 1. 5 2 x 2. 5 3 0. 45 0. 4 3. 5 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 0 0. 5 1 1. 5 2 2. 5 3 3. 5 4 4. 5 y Uncertainty Analysis for Engineers 25
Analysis X Y Z P Z-rank 1 2 3 . 08 1 1 3 4 . 04 2 1 4 5 . 08 3 2 2 4 . 24 2 2 3 5 . 12 3 2 4 6 . 24 4 3 2 5 . 08 3 3 3 6 . 04 4 3 4 7 . 08 5 Uncertainty Analysis for Engineers 26
Compiled Data z fz 3 . 08 4 . 28 5 . 28 6 . 28 7 . 08 fz 0. 3 0. 25 0. 2 0. 15 0. 1 0. 05 0 0 1 2 3 4 z 5 6 7 8 Uncertainty Analysis for Engineers 27
Example x and y are integers Uncertainty Analysis for Engineers 28
Example (continued) The sum of n independent Poisson processes is Poisson Uncertainty Analysis for Engineers 29
Continuous Variables Uncertainty Analysis for Engineers 30
Continuous Variables Uncertainty Analysis for Engineers 31
Continuous Variables (cont. ) Uncertainty Analysis for Engineers 32
Example Uncertainty Analysis for Engineers 33
In General… �If Z=X+Y and X and Y are normal dist. �Then Z is also normal with Uncertainty Analysis for Engineers 34
Products Uncertainty Analysis for Engineers 35
Example �W, F, E are lognormal Uncertainty Analysis for Engineers 36
Central Limit Theorem �The sum of a large number of individual random components, none of which is dominant, tends to the Gaussian distribution (for large n) Uncertainty Analysis for Engineers 37
Generalization �More than two variables… Uncertainty Analysis for Engineers 38
Moments �Suppose Z=g(X 1, X 2, …, Xn) Uncertainty Analysis for Engineers 39
Moments Uncertainty Analysis for Engineers 40
Moments Uncertainty Analysis for Engineers 41
Approximation Uncertainty Analysis for Engineers 42
Approximation Uncertainty Analysis for Engineers 43
Second Order Approximation Uncertainty Analysis for Engineers 44
Approximation for Multiple Inputs Uncertainty Analysis for Engineers 45
Example �Example 4. 13 �Do exact and then use approximation and compare �Waste Treatment Plant – C=cost, W=weight of waste, F=unit cost factor, E=efficiency coefficient median cov W 2000 ton/y . 2 F $20/ton . 15 E 1. 6 . 125 Uncertainty Analysis for Engineers 46
Solving… Uncertainty Analysis for Engineers 47
Approximation Uncertainty Analysis for Engineers 48
Variance Uncertainty Analysis for Engineers 49
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