Transcript Introduction to Offshore Engineering

Statistics and Probability Theory
in
Civil, Surveying and Environmental
Engineering
Prof. Dr. Michael Havbro Faber
Swiss Federal Institute of Technology
ETH Zurich, Switzerland
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Contents of Todays Lecture
•
Presentation on the result of the classroom assessment
•
What is a random variable?
•
The decision context!
•
What are we doing today?
•
Details will follow 
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Lecture
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Subjects
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What is a random variable?
•
Let us consider a very simple structural engineering
problem!
•
We want to design a steel beam – and assume – based on
experience that the design controlling load effect is the
midspan bending moment M
- the design variable being the moment of resistance W of
the cross section
- the load p and the yield stress sy of the beam are
b
associated with uncertainty
p
h
l
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Mid span
cross-section
1
W  bh 2
6
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What is a random variable?
•
The moment capacity of the cross-section RM and the mid
span moment M are calculated as:
RM  W s y
1
pl
4
M mid span moment
M
RM moment capacity of cross section
W moment of resistance
s y yield stress of the steel
1
W  bh 2
6
p load
l length of beam
b
p
h
l
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Mid span
cross-section
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What is a random variable?
•
We can now establish a design equation as:
RM (b, h)  M  0
The engineer must now select W,
or rather b and h such that the
design equation is fulfilled

1
W (b, h)s y  Pl  0
4

1 2
1
bh s y  Pl  0
6
4
But as p and sy are associated with
uncertainty – she/he must take this
uncertainty into account !
p
b
h
l
Mid span
cross-section
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p
What is a random variable?
b
h
l
Mid span
cross-section
•
The uncertainty is accounted for by representing p and sy
in the design equation as two random variables.
P : Normal distributed: N (  P , s P )
S y : Normal distributed: N ( S y , s S y )
The random variable P represents the random variability of
the load p during a period of one year
The random variable Sy represents the random variability of
the steel yield stress sy - produced by an unknown steel
producer.
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What is a random variable?
•
As the load and yield stress are uncertain the design
equation cannot be fulfilled with certainty – independent
on the choice of b and h.
•
However, it can be fulfilled with probability !
•
The beam can be designed such that the probability of
failure is less or equal to a given number – the requirement
to safety.
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What is a random variable?
•
Let us assume that the load and yield stress are given as:
P:
N ( P , s P ) 
Sy:
N (S y ,s S y )  N (370 MPa,15 MPa)
N (100 kN, 20kN)
we can now write the event of failure as:
1 2
1
bh S y  Pl  0
6
4

S  Sy 
3
Pl  0
2bh 2
This is called a
safety margin!
3
Pl  0
2
2bh
let us further assume that l=5000mm and b=50mm
Sy 
•
Let us now determine h such that the annual probability of
failure is equal to 10-3
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What is a random variable?
•
We have already learned that a linear combination of
Normal distributed random variables is also Normal
distributed
The expected value of S is equal to:
3  5000
P
y
2  50  h 2
3  5000
15 106
3
 370 
100 10  370 
2
2  50  h
h2
 S  S 
The variance of S is equal to:
The probability of failure is
now easily determined from
the standard Normal
cumulative distribution
function
 3  5000  2

sP
2 
2

50

h


2
s S2  s S2
y
 0   S ( h) 
Pf ( h)   

s
(
h
)
S


9 1012
 3  5000 
3 2
 15  
  20 10   225 
2 
h4
 2  50  h 
2
2
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What is a random variable?
•
Calculating the probability of failure as a function of h we
0  S (h)
get:
Pf (h)  (
)
Pf (h) 1
s S (h)
0.1
0.01
0.001
0.0001
200
h
210
220
230
240
250
260
270
The height of the beam must thus be equal to 258mm!
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The decision context!
•
Why uncertainty modeling?
Uncertain phenomenon
Data
Random variables
Random processes
Model estimation
Probabilistic model
Consequences of events
Probabilities of events
Risks
Decision Making !
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What are we doing today?
•
We have already introduced random variables as a means
of representing uncertainties which we may quantify based
on observations – related to time frames from which we
have experience and observations!
•
In many real problems of decision making we need to take
into account what might happen in the far future –
exceeding the time frames for which we have experience!
- 475 year design earthquake!
- 100 year storm/flood
- 100 year maximum truck load
- etc..
Thus we need to develop models which
can support us in the modeling of extremes
of uncertain/random phenomena !
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What are we doing today?
•
We have already introduced random variables as a means
of representing uncertainties which we may quantify based
on observations.
•
Often we use random variables to represent uncertainties
which do not vary in time:
- Model uncertainties (lack of knowledge)
- Statistical uncertainties (lack of data).
•
Or we use such random variables to represent the random
variations which can be observed within a given reference
period.
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What are we doing today?
Discrete event of flood
•
In many engineering problems we need
to be able to describe the random
variations in time more specifically:
The occurrences of events at random
discrete points in time (rock-fall,
earthquakes, accidents, queues,
failures, etc.)
- Poisson process, exponential and
Gamma distribution
The random values of events occurring
continuously in time (wind pressures,
wave loads, temperatures, etc.)
- Continuous random processes (Normal
process)
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Continuous stress
variations due to waves
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What are we doing today?
Extreme water level
•
However, we are also interested in
modeling extreme events such as:
the maximum value of an uncertain
quantity within a given reference
period
- extreme value distributions
the expected value of the time till
the occurrence of an event
exceeding a certain severity
- return period
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Maximum wave load
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What are we doing today?
•
In summary we will look at:
- Random sequences (Poisson process)
- Waiting time between events (Exponential and Gamma
distributions)
- Continuous random processes (the Normal process)
- Criteria for extrapolation of extremes (stationarity and
ergodicity)
- The maximum value within a reference period (extreme
value distributions)
- Expected value of the time till the occurrence of an event
exceeding a certain severity (return period)
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Random Sequences
•
The Poisson counting process was originally invented by
Poisson
“life is only good for two things:
to do mathematics and to teach it”
(Boyer 1968, p. 569)
Poisson, Siméon-Denis
(1781-1840)
Student of Laplace
Former law clerk
Poisson was originally
interested in applying
probability theory for the
improvement of procedures
of law
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Random Sequences
The Poisson counting process is one of the most commonly
applied families of probability distributions applied in
reliability theory
5
The Poisson process provides a model for representing rare
events – counting the number of events over time
n(t )
4
4
3
3
1
2
2
0
0.1
0.2
0.3
0.4
0.5
0
1
Pn (t )
•
0
0
1
2
3
4
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5
6
7
8
9
10
t
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Random Sequences
•
The Poisson counting process is one of the most commonly
applied families of probability distributions applied in
reliability theory
The process N(t) denoting the number of events in a (time)
interval (t,t+Dt[ is called a Poisson process if the following
conditions are fulfilled:
1) the probability of one event in the interval (t,t+Dt[ is
asymptotically proportional to Dt.
2) the probability of more than one event in the interval
(t,t+Dt[ is a function of higher order of Dt for Dt→0.
3) events in disjoint intervals are mutually independent.
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Random Sequences
•
The Poisson process can be described completely by its
intensity n(t)
1
P(one event in t , t  Dt )
Dt 0 Dt
n (t )  lim
if n(t) = constant, the Poisson process is said to be
homogeneous, otherwise it is inhomogeneous.
The probability of n events in the time interval (0,t[ is:
n
t

 n ( )d 
 t

0


Pn  t  
exp   n ( )d 
n!
 0

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(n t ) n
Pn  t  
exp  n t 
n!
Homogeneous case !
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Random Sequences
•
Early applications include the studies by:
Ladislaus Bortkiewicz (1868-1931)
- horse kick death in the Prussian cavalry
- child suicide
William Sealy Gosset (“Student”) (1876-1937)
- small sample testing of beer productions (Guinness)
RD Clarke
- study of distribution of V1/V2 hits under the London Raid
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Random Sequences
•
The mean value and variance of the random variable
describing the number of events N in a given time
interval (0,t[ are given as:
t
E  N (t )  Var  N (t )  n ( )d
Inhomogeneous case !
E  N (t )  Var  N (t ) n t
Homogeneous case !
0
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Random Sequences
•
The Exponential distribution
The probability of no events (N=0) in a given time
interval (0,t[ is often of special interest in engineering
problems
- no severe storms in 10 years
- no failure of a structure in 100 years
- no earthquake next year
- …….
This probability is directly achieved as:
0
t

n
(

)
d



 t

0


P0  t  
exp   n ( )d 
0!
 0

 t

 exp   n ( )d 
 0

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P0 t   exp  n t 
Homogeneous case !
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Random Sequences
•
The probability distribution function of the (waiting) time
till the first event T1 is now easily derived recognizing
that the probability of T1 >t is equal to P0(t) we get:
Homogeneous case !
FT1 (t1 )  1  exp(n t )
FT1 (t1 )  1  P0 (t1 )
t
 1  exp( n ( )d )
0
5
Exponential cumulative distribution
n(t )
4
4
3
3
Exponential probability density
1
2
2
0
0
0.1
0.2
0.3
0.5
0.4
Pn (t )
1
0
0
1
2
3
4
5
6
7
8
9
10
t
fT1 (t1 )  n exp(n t )
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Random Sequences
The Exponential probability density and cumulative
distribution functions
n 2
fT (t )
2.5
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2
1.5
1
0.5
0
0
1
2
3
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4
FT (t )
fT (t )
FT (t )
t
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Random Sequences
•
The exponential distribution is frequently applied in the
modeling of waiting times
-
time till failure
time till next earthquake
time till traffic accident
….
fT1 (t1 )  n exp(n t )
The expected value and variance of an exponentially
distributed random variable T1 are:
E T1   Var T1   1/n
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Random Sequences
•
Sometimes also the time T till the n’th event is of interest in
engineering modeling:
- repair events
- flood events
- arrival of cars at a roadway crossing
If Ti, i=1,2,..n are independent exponentially distributed
waiting times, then the sum T i.e.:
T  T1  T2  ...  Tn1  Tn
follows a Gamma distribution:
fT (t ) 
n (n t )( n-1) exp(n t )
(n  1)!
This follows from repeated use
of the result of the distribution
of the sum of two random variables
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Random Sequences
The Gamma probability density function
n 2
fT (t ) 2.5
n=1
n=4
2
1.5
Exponential
1
0.5
Gamma
t
0
0
0.5
1
1.5
2
2.5
3 3.5
4 4.5
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Random Processes
Continuous random processes
A continuous random process is a random process which
has realizations continuously over time and for which the
realizations belong to a continuous sample space.
Water level
•
30
29
28
27
26
25
24
23
22
21
20
Variations of;
water levels
wind speed
rain fall
.
.
.
0
10
20
30
40
50
60
70
80
90
100
Time (days)
Realization of continuous scalar valued random process
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Random Processes
•
Continuous random processes
The mean value of the possible realizations of a random
process is given as:

 X (t )  E  X (t )   x f X ( x ; t )dx

Function of time !
The correlation between realizations at any two points in
time is given as:
RXX (t1, t2 )  E  X (t1 ) X (t2 ) 
 
 x
1
x2 f XX ( x1, x2 ; t1, t2 )dx1dx2
 
Auto-correlation function – refers to a scalar valued random process
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Random Processes
•
Continuous random processes
The auto-covariance function is defined as:
C XX (t1 , t2 )  E ( X (t1 )   X (t1 ))( X (t2 )   X (t2 )) 
 

  (x  
1
X
(t1 )) ( x2   X (t2 )) f XX ( x1, x2 ; t1, t2 )dx1dx2
 
for t1=t2=t the auto-covariance function becomes the
covariance function:
s X2 ( t )  C XX ( t ,t )  RXX ( t ,t )   X2 ( t )
s X (t )
Standard deviation function
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Random Processes
•
Continuous random processes
A vector valued random process is a random process
with two or more components:
X(t )  ( X 1 (t ), X 2 (t ),.., X n (t ))T
with covariance functions:
C X i X j (t1 , t2 ) 
i j
auto-covariance functions
E ( X i (t1 )   X i (t1 ))( X j (t2 )   X j (t2 )) 


i j
cross-covariance functions
The correlation coefficient function is defined as:
  X i (t1 ), X j (t2 )  
C X i X j (t1 , t2 )
s X (t1 )  s X (t2 )
i
j
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Random Processes
•
Normal or Gauss process
A random process X(t) is said to be Normal if:
for any set;
X(t1), X(t2),…,X(tj)
the joint probability distribution of X(t1), X(t2),…,X(tj)
is the Normal distribution.
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Random Processes
•
Stationarity and ergodicity
A random process is said to be strictly stationary if all its
moments are invariant to a shift in time.
A random process is said to be weakly stationary if the
first two moments i.e. the mean value function and the
auto-correlation function are invariant to a shift in time
 X (t )  cst
RXX (t1 , t2 )  f (t2  t1 )
Weakly stationary
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Random Processes
•
Stationarity and ergodicity
- A random process is said to be strictly ergodic if it is
strictly stationary and in addition all its moments may
be determined on the basis of one realization of the process.
- A random process is said to be weakly ergodic if it is weakly
stationary and in addition its first two moments may be
determined on the basis of one realization of the process.
•
The assumptions in regard to stationarity and ergodicity are
often very useful in engineering applications.
- If a random process is ergodic we can extrapolate
probabilistic models of extreme events within short reference
periods to any longer reference period.
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Extreme Value Distributions
In engineering we are often interested in extreme values
i.e. the smallest or the largest value of a certain quantity
within a certain time interval e.g.:
The largest earthquake in 1 year
The highest wave in a winter season
The largest rainfall in 100 years
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Extreme Value Distributions
We could also be interested in the smallest or the largest
value of a certain quantity within a certain volume or area
unit e.g.:
The largest concentration of pesticides in a volume of
soil
The weakest link in a chain
The smallest thickness of concrete cover
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Extreme Value Distributions
Observed monthly
extremes
Observed annual
extremes
Observed 5-year
extremes
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Extreme Value Distributions
If the extremes within the period T of an ergodic random
process X(t) are independent and follow the distribution:
FXmax
,T ( x)
Then the extremes of the same process within the period:
n T
will follow the distribution:
max
X ,nT
F
( x)   F
max
X ,T
( x) 
n
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Extreme Value Distributions
Extreme Type I – Gumbel Max
When the upper tail of the probability density function falls
off exponentially (exponential, Normal and Gamma
distribution) then the maximum in the time interval T is
said to be Type I extreme distributed
f Xmax
,T ( x)   exp( ( x  u)  exp( ( x  u)))
FXmax
,T ( x)  exp( exp( ( x  u)))
X
max
T
sX
max
T

0.577216
u u




 6
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For increasing time
intervals the variance
is constant but the mean
value increases as:
X
max
nT
  X max
T
6
 s X max ln(n)
T

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Extreme Value Distributions
Extreme Type II – Frechet Max
When a probability density function is downwards limited
at zero and upwards falls off in the form
1 k
FX ( x )  1   ( )
x
then the maximum in the time interval T is said to be Type
II extreme distributed
Mean value and
standard deviation
k
u
FXmax
(
x
)

exp(

  )
,T
 x
k u
f Xmax
(
x
)

 
,T
u  x
k 1
k
u
exp(  )
 x
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X
max
T
s X2
max
T
1
 u(1  )
k
2
1 

 u 2 (1  )   2 (1  )
k
k 

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Extreme Value Distributions
Extreme Type III – Weibull Min
When a probability density function is downwards limited
at e and the lower tail falls off towards e in the form
F ( x)  c( x  e ) k
then the minimum in the time interval T is said to be Type
III extreme distributed
k


x

e


min


FX ,T ( x)  1  exp  

 u e  


k  x e 
f Xmin
(
x
)



,T
u e u e 
k 1
  x  e k 

exp  
  u  e  


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Mean value and
standard deviation
X
min
T
s X2
min
T
1
 e  (u  e )(1  )
k
2
1 

 (u  e ) 2 (1  )   2 (1  )
k
k 

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Return Period
The return period for extreme events TR may be defined as:
TR  n  T 
1
(1  FXmax
,T ( x ))
Example:
Let us assume that - according to the cumulative
probability distribution of the annual maximum traffic load
- the annual probability that a truck load larger than 100
ton is equal to 0.02 – then the return period of such heavy
truck events is:
1
1
TR  n  T 
n
 50 years
0.02
1 0.02
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