Random Processes - Stochastic Structural Dynamics - Lecture Slides, Slides of Stochastic Processes

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Random processes-

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Review of theory ofRandom processes

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Parameter (time)

State

Ensemble

Notion of a random process

Working definition

:

A random variablethat evolves in time.OrParametered family ofrandom variables.

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Analogy Random variable

Statics

Random process

Dynamics

When to model a quantity as random variableand when to model it as a random process?This is analogous to asking when to model asystem as static and when as dynamic.

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Terminology Evolution in time : Random processesEvolution in space: Random fields Mathematically it is not necessary to maintain this distinction

Stochastic processesStochastic fieldRandom functionsTime series

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^



^



^

^

^



Let

,^

be a random process.

=parameter; values taken by

,^

=state.

For fixed value ofIf^

,^

is a discrete random variable, then

,^

is a random pro

X^

t t^

X^

t

t

X^

t^

X^

t

A scheme for classification of random processes

^

^

^



^



cess with a

discrete state space.If^

,^

is a continuous random variable, then

,^

is a random process with a

continuous state space.If^

takes only discrete values, we say that

,^

is a random proc

X^

t^

X^

t

t^

X^

t

^



ess with

discrete paramters.If^

takes continuous values, we say that

,^

is a random process with

t continuous parameters.

X^

t^ 

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Evolution of wind velocity in space and time^ Other examples^ (a) Road roughness (evolution in space)(b) wave heights (evolution in space and time)(c) Thickness of a cylindrical shell (evolution in an angle)(d) FRF-s evolution in frequency (and spa

ce)

Parameter need not always be time…

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x

y

z

fixed

t

( ) d t                  

( )

: ground displacement

( )

: ground velocity

( )

: ground acceleartion

u^ g g g g g g g g g

t

d t

v^

t w^

t u^

t

v t

v^

t w^

t u^

t

a t

v^

t w^

t ^



^



 



^



^



^



^



 



^



^



^



^



 



^



^

     

Vector random process

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^

^

^

^

^

^

Function

on

Distributi

y

Probabilit

order

Second

x t X x t X P t t x x

PXX

^

^

^

function

density

y

probabilit

order

Second

x

x

t t x x P t t x x

p^

XX

XX

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^

^

^



^



th order Probability Distribution Function

;

n

i^

i

X

i

n - P

x t

P^

X

t^

x

^



^



^



^



^

 ^



^

^

^

^ n

X n

X^

x

x x

t x P

t x n p



   ^

 2 1

~; ~

~; ~

function

density y

probabilit

order th-

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Expectation of a random process

Mean Variance

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AutocovarianceAutocorrelationAutocorrelationcoefficient

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^

^

 

   

^

^

^



^



^



2

1

2

1

2

2

1

2

12

2

1

(^12)

2

1

12

Let

( ) be a random process and consider its 1st and 2nd order pdf-s.

1

1

;^

exp

;

2

2

1

,^

;^

,

2

1

1

exp

2 1

X

X

X

X

XX

X t

x^

m^

t

p^

x t

x

t

t

p^

x^

x^

t^

t

r

x^

m^

x

r



 

   

^



^



 ^



^



^

^

 

 

^



^



^



^



^





^

  ^





^





Gaussian random process

^

^

^

^



^

^

^

^

^

^

^

^

^



2

2

2

1

1

2

2

12

2

1

2

2

1

2

1

1

2

2

1

1

2

2

12

1

2

2

,

;^

;^

;^

;^

,

X^

X^

X^

X^

XX

m^

x^

m^

x^

m

r

x^

x

m^

m^

t^

m^

m^

t^

t^

t^

r^

r^

t^

t

 



^

^

^



^



^



^

^



^



^





^



^



^



^



^



 

 

^

^

^

^



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 

^

 ^



^

 ^



^



^



^

^

^

^

^



1

1

1

1

2

1 2

1

1 2 2 Continuing further, consider

time instants

and

associated random variables

Let the jpdf of

be given by

,^

,^

,^

;^ ,

exp

;^

n i i

n i^

i

n i^

i

XX^

X^

n^

n t

i

n i

n^

t

X^

t

X^

t

p^

x^

x^

x^

t^ t

t

x^

S^

x^

x^

i^

n

S

S















^

^

^

^

^

^

^

^

^

^

^

^

^

^

^



^

^

^

^

^



^



 

1

2

1

2 :^

&^

is positive definite.

is said to be a Gaussian random process if the above form of pdf is true for any

and for any ch

j^

i^

X^

i^

j^

X^

j

t

t

X^

X^

X^

n

t n

X^

t^

m^

t^

X^

t^

m^

t

Note

S

S^

S

m^

t^

m^

t^

m^

t

x^

x^

x^

x

X^

t

n



^

^

^

^

^

 ^

^

 Definition

 

1

oice of

n. t^ i^ i 

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