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Independent Increments - Stochastic Structural Dynamics - Lecture Slides, Slides of Structural Analysis
Main points of this lecture are: Independent Increments, Properties of Processes, Stationary Independent Increments, Functional Equation, Gaussian Random Process, Autocorrelation and Cross Correlation, Rate of Upcrossing of Level, Spectral Moments
Typology: Slides
2012/2013
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Problem solving session-
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Discussion on properties of processes withIndependent increments
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^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Let
stationary increments
We get the functional equation
(^
)^
is the solution.
Given
f t
X^
t
f t
X^
t^
X
f^
t^
s^
X^
t^
s^
X
X^
t^
s^
X^
s^
X^
s^
X
X^
t^
s^
X^
s^
X^
s^
X
X^
t^
X^
X^
s^
X
f t
f^
s
f t
s^
f t
f^
s
f^
t^
ct f^
X^
c
X^
t^
t
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
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^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
Let
Var
Var
Var
Var
Var
Var
Var
Var
Var
stationary increments 1
Var
g t
X^
t
g t
X^
t^
X^
t^
X
g t
s^
X^
t^
s^
X
X^
t^
s^
X^
s^
X^
s^
X
X^
t^
s^
X^
s^
X^
s^
X
X^
t^
X^
X^
s^
X
g t
s^
g t
g s
g t
ct
g^
X
^
^
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^
^
^
^
^
2
2
2
Var
c
X^
t^
t
^
^
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^
^
^
^
^
^
^
^
^
2
2
2
2
Var^2
Var
Var
2COV
,
X
t^
X
s
X
t^
X
s^
X
t^
X
s
X
t^
X
t^
X
s^
X
s
X
t^
X
t^
X
s^
X
s
X
t^
X
t^
X
s^
X
s
X
t^
X
s^
X
t^
X
s
^
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^
^
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^
^
^
^
^
^
^
^
^
^
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^
^
^
^
^
2
2
2
COV
,
1
-Var
+Var
+Var
2
(^
)^
assuming that
2 COV
,^
min
,
X
t^
X
s
X
t^
X
s^
X
t^
X
s
t^
s^
t^
s^
s^
t^
s
X
t^
X
s^
t s
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
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^
^
2
2 2
2
2
2
0
2
2
2
2
2
2
2
2
2
2
4
2
4
4
2
exp -
;-
2
2
exp -
2
2
exp -
2
2
exp -
3
2
2
X
XX
X
X
X
X
X
X
S
d
d d
^
^
^
^
^
^
^
^
^
^
^
^
Spectral moments
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^
^
^
^
^
^
^
^
^
2
2 2
2
2 2
2
2
2
Autocorrelation function
exp -
;-
2
2 1
exp -
exp
2
2
2
=^
exp
2
Recall:
1
X
XX
X
XX
X
m^
n
n^
XX
m^
n
m^
n
S R
i^
d
d^
R
X
t^
X
t^
d
^
^
^
^
^
^
^
^
^
^
^
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^
^
^
^
^
^
^
^
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
exp
2
exp
2
exp
2
exp
2
exp
1
2
XX
X
XX
X
XX
X X
X
R d
R d d
R d
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
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^
^
^
^
^
^
^
^
2
2
2
2
2
2
2
2 3
2
2
2
2
2
2
2
3
2
2
2
2
2
2
2
2
2
2
4
3
exp
exp
exp
exp
XX
X
XX
X X
X
d^
R
d d
R
d
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
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^
^
^
^
^
^
^
^
^
^
^
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2 3
2
2
2
2
2
4
3
3 4
2
2
2
2
2
4
2
6
4
4
exp
exp
exp
exp
exp
XX
X
XX
X
XX
X
XX
X
XX
X
R d
R
d d
R
d d
R
d d
R
d
^
^
^
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^
^
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^
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^
^
^
^
^
^
^
^
^
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-^
-1.
-^
-0.
0
1
2
(^43210) -1 -
tau
acf
processderivative
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^
^
^
^
PDF of time for first crossing of levelFor high levels of crossings we can approximatethe number of times the level is crossed as a Poissonrandom variable.
exp
;^
k
T
P^
N
T^
k^
T^
k
k
n^
t
^
^
^
^
^
^
^
^
^
^
^
2 2
2 2
exp
First passage time
P^
No points in 0 to T
exp
exp
exp
;^
f
X
f
f T
a^
a
T
T^
T^
P^
P^
N^
T
T^
a
P^
t^
T^
T
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
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^
^
^
2
2
2
3
1
2
3
2
2
3
2
2
1
2
1
2
2
2
3
2 1
1
Average rate of peaks above level
,^
[^
exp
exp
]
x
m
t^
S^
x
S
x
x^
erf
dx
S
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
^
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