Probabilistic methods in Open Earth Tools

Ferdinand Diermanse

Kees den Heijer

Bas Hoonhout

2

Open Earth Tools

• Deltares software• Open source• Sharing code for users of matlab, python, R, …• https://publicwiki.deltares.nl/display/OET/OpenEarth

3

Application: probabilities of unwanted events (failure)

Floods (too much)

Droughts (too little)

Contamination (too dirty)

4

Example application: flood risk analysis

Rainfall

Upstream river

Discharge

Sea water

level

Sobek

5

General problem definition

X1

System/model

X2

Xn

.

.

.

Z

“Boundary

conditions”

“system

variable”

6

Notation

X1

X2

Xn

.

.

.

Z

X = (X1, X2, …, Xn)

Z = Z(X)

System/model

7

General problem definition

X1

model

X2

Xn

.

.

.

Z

?

Statistical

analysis

Probabilistic

analysis

complex

Time consuming

8

failure domain: unwanted events

x1

x2

“failure”

Z(x)=0no “failure”

Z(x)>0

Z(x)<0

Wanted: probability of failure, i.e. probability that Z<0

9

Example Z-function

Failure: if water level (h) exceeds crest height (k): Z = k - h

10

Probability functions of x-variables

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x2

f(x 2)

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x1

f(x 1)

11

Correlations need to be included

x2

f(x)

x1

x1

x2

f(x)

Multivariate distribution function

12

Combination of f(x) and Z(x)

x2

x1

f(x)

Z(x)=C*

“failure”

no “failure”

13

Probability of failure

x2

x1

f(x)

fail

0

P

Z

f d

x

x x

Z(x)=0

14

Problem definition

Problem cannot be solved analytically

Probabilistic estimation techniques are required

Evaluation of Z(x) can be very time consuming

fail

0

P

Z

f d

x

x x

15

Probabilistic methods in Open Earth Tools

Crude Monte Carlo

Monte Carlo with importance sampling

First Order Reliability Method (FORM)

Directional sampling

1616

Crude Monte Carlo sampling

X1

X2

Z=0

failureno failure

1. Take N random samples of the x-variables 2. Count the number of samples (M) that lead to “failure” 3. Estimate Pf = M/N

-4 -2 0 2 40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x1

f(x 1)

-4 -3 -2 -1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

x2

f(x 2)

17

Simple example Crude Monte Carlo: ¼ circle

Uniform 0,1f x

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x 2

4

14

18

Samples crude Monte Carlo

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x1

x 2

no failure

failure

19

MC estimate

100

101

102

103

104

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

number of samples

MC

est

imat

e

1-/4

20

New example: smaller probability of failure

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

z=0

u1

u 2

example limit state: Z = 5 - (u1+u

2)

failure probability: 0.00020

-3 -2 -1 0 1 2 30

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

U1;U2

1000 samples

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

z=0

u1

u 2

1000 samples crude MC

21

How many samples required?

100

101

102

103

104

105

106

0

1

2

x 10-4

number of samples

estim

ated

fai

lure

pro

babi

lity

exact

crude MC

22

23

Crude Monte Carlo

• Can handle a large number of random variables

• Number of samples required for a sufficiently accurate estimate is inversely proportional to the probability of failure

• For small failure probabilities, crude MC is not a good choice, especially if each sample brings with it a time consuming computation/simulation

2424

“Smart” MC method 1: importance sampling

x

f(x)h(x)

f xh x

Manipulation of probability denstity function

Allowed with the use of a correction:

Potentially much faster than Crude Monte Carlo

25

-8 -6 -4 -2 0 2 4 6 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

u

prob

abili

ty d

ensi

tyexample strategy: increase standard deviation by a factor 2

f(u)

h(u)

Example strategy: increase variance

26

-5 -4 -3 -2 -1 0 1 2 3 4 5-5

-4

-3

-2

-1

0

1

2

3

4

5

u1

u 21000 samples MC-IS

Samples

27

Convergence of MC estimate

100

101

102

103

104

105

106

0

1

2

3

4

5

6x 10

-4

number of samples

estim

ated

fai

lure

pro

babi

lity

importance sampling; scaling factor 2

exact

crude MCimportance sampling

28

Example strategy 2

-8 -6 -4 -2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

u

prob

abili

ty d

ensi

tyexample strategy: mu=2 sigma=2

f(u)

h(u)

29

Samples

-3 -2 -1 0 1 2 3 4 5 6 7-3

-2

-1

0

1

2

3

4

5

6

7

u1

u 2

1000 samples MC-IS

30

Convergence of MC estimate

100

101

102

103

104

105

106

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

-4

number of samples

estim

ated

fai

lure

pro

babi

lity

exact

crude MC

importance sampling

31

Monte Carlo with importance sampling

• Potentially much faster than Crude Monte Carlo

• Proper choice of h(x) is crucial

• Therefore: Proper system knowledge is crucial

32

FORM

Design point: most likely combination leading to failure

33

x

u

F(x)

real world variable X

transformed normallydistributed variable u

(u) = F(x)

f(x)

(u )

(u)

Method is executed with standard normally distributed variables

34

Probability density independent normal values

Probability density decreases

away from origin

35

example

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4 -10-9-8

-7

-6

-6

-5

-5

-4

-4

-3

-3

-2

-2

-2

-1

-1

-1

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

44

55

5

5 5 5

u

v

0.8 1.25Z u v

u en v standard

normally distributed

36

Design point

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4 -10-9-8

-7

-6

-6

-5

-5

-4

-4

-3

-3

-2

-2

-2

-1

-1

-1

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

44

55

5

5 5 5

u

v

Z=0 & shortest distance to origin

37

Start iterative procedure

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4 -10-9

-8-7

-6-5

-5

-4

-4

-3

-3

-2

-2

-1

-1

-1

0

0

0

1

1

1

2

2

2

2

3

3

3

3

44

4

4

55

5 5 5

u1

u 2

38

Estimation of derivatives

0.4 0.6 0.8 1 1.2 1.4 1.60.4

0.6

0.8

1

1.2

1.4

1.6

u

v

3.5

4

4.5

u

v

39

Resulting tangent

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

55

5

5 5 5

u

v

40

Linearisation of Z-function based on tangent

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

-10

0

1

1

1

2

2

2

3

3

4

u1

u 2

41

First estimate of design point

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4 -1

0

0

1

1

1

2

2

2

3

3

4

u1

u 2

42

3D view: Z-function

43

3D view: linearisation of Z-function

44

Smaller steps to prevent “accidents” (relaxation)

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

-1

0

0

1

1

1

2

2

2

3

3

4

u1

u 2

45

2nd iteration step

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

55

5

5 5 5

u

v

46

Linearisation in 2nd iteration step

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4

u

v

47

3D view

48

All iteration steps

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4 -10-9-8-7-6

-6

-5

-5

-4

-4

-3

-3

-2

-2

-2

-1

-1

-1

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

4

4

4

44

55

55 5 5

u

v

49

-value of design point in standard normal space

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

3.5

4 -10-9-8

-7

-6

-6

-5

-5

-4

-4

-3

-3

-2

-2

-2

-1

-1

-1

0

0

0

1

1

1

1

2

2

2

2

3

3

3

3

44

4

44

55

5

5 5 5

u

v

||||

Pfail

50

-values in design point

Limit state(Z = 0)

Design point

u1

u2 Z < 0

β

u2,d = -α2β

u1,d = -α1β

51

FORM

• Very fast method

• Risk: iteration method does not converge, or converges to the wrong design point

52

Directional sampling

Z=0Selected directions

Z-function evaluations

u1

u2

1

2

34

Z<0

Z>0

0

53

Search along 1 direction

z

0

1

2

4

3

54

Resume

Crude Monte Carlo (MC)

Monte Carlo with importance sampling (MC-IS)

First Order Reliability Method (FORM)

Directional Sampling (DS)

Towards the exercises

56

Generic problem statement

x2

x1

f(x)

fail

0

P

Z

f d

x

x x

Z(x)=0

57

Generic problem statement

fail

0

P

Z

f d

x

x x

1. Probability functions, f(x): P -> X

2. Z-function: X -> Z