Definition of Stochastic Processes Definition of Stochastic - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

• Definition of Stochastic Processes

Definition of Stochastic Processes

• 1

1st

st Order Density and Distribution

Order Density and Distribution

• 2

2nd

nd Order Density and Distribution

Order Density and Distribution

• Conditional Density

Conditional Density

• Expected Value and Autocorrelation

Expected Value and Autocorrelation

• Auto

Auto-

• covariance and Variance

covariance and Variance

• Examples of Physical Stochastic

Examples of Physical Stochastic Processes Processes

• G. Ahmadi

ME 529 - Stochastics

Given a random experiment Given a random experiment ℑ ℑ: (S, F, P), to each : (S, F, P), to each ξ ξ we assign time function we assign time function X(t, X(t,ξ ξ) ) t t∈ ∈ (0,T) (0,T) Stochastic process Stochastic process family of time functions, family of time functions, ξ

ξ∈ ∈ S. S.

t ξ1 ξ2 ξ3 ξ4 t1

X(t X(t, ,ξ ξ) )

• G. Ahmadi

ME 529 - Stochastics

Meaning of X(t, Meaning of X(t,ξ ξ) )

• t,

t,ξ ξ vary: family of time vary: family of time functions functions

• t vary: single time

t vary: single time functions functions

• ξ

ξ vary: random variable vary: random variable

• t,

t,ξ ξ fixed: single number fixed: single number Note: ... are n Note: ... are n random variables random variables

( )

ξ , t X

t ξ1 ξ2 ξ3 ξ4 t1

Example of a stochastic process. Sample space has four outcomes.

( ) ξ ,

1 1

t X X =

( ) ξ ,

n n

t X X =

SLIDE 2

2

• G. Ahmadi

ME 529 - Stochastics

Derivative of Air Temp Derivative of Air Temp Over Ocean Over Ocean Derivative of Air Velocity Derivative of Air Velocity

• G. Ahmadi

ME 529 - Stochastics

Alternative definition Alternative definition: Stochastic process is : Stochastic process is a family of random variables X(t a family of random variables X(t1

1), X(t

), X(t2

2),…

),… for t for t ∈ ∈ (0,T). (0,T). i.

• i. Discrete

Discrete – – set is finite/ set is finite/countably countably infinite infinite ii.

• ii. Continuous

Continuous – – set is non set is non-

• countably

countably infinite infinite

• G. Ahmadi

ME 529 - Stochastics

1 1st

st Order Density & Distribution Functions

Order Density & Distribution Functions

( ) ( ) { }

x t X P t x F ≤ = ,

( ) ( )

x t x F t x f ∂ ∂ = ; ;

2 2nd

nd Order Density & Distribution Functions

Order Density & Distribution Functions

( ) ( ) ( ) { }

2 2 1 1 2 1 2 1

, ; , x t X x t X P t t x x F ≤ ∩ ≤ =

( ) ( )

2 1 2 1 2 1 2 2 1 2 1

, ; , , ; , x x t t x x F t t x x f ∂ ∂ ∂ =

• G. Ahmadi

ME 529 - Stochastics

Stratnovich Stratnovich 2 2nd

nd Order Density Function

Order Density Function Properties Properties

( ) ( ) ( ) ( ) ( ) { }

2 2 1 1 2 1 2 1

, ; , x t X x t X E t t x x f − − = δ δ

( ) ( )

1 1 2 1 1

; , ; , t x F t t x F = ∞

( ) ( )

∫

+∞ ∞ −

=

2 2 1 2 1 1 1

, ; , ; dx t t x x f t x f

( ) ( ) ( ) { }

x t X E t ; x f − = δ

Stratnovich Stratnovich 1 1st

st Order Density Function

Order Density Function

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Conditional Density Conditional Density Mean Value Mean Value Autocorrelation Autocorrelation Autocovariance Autocovariance Variance Variance

( ) ( ) ( ) ( )

2 2 2 1 2 1 2 2 1

, , ; , | ; t x f t t x x f x t X t x f = =

( ) ( ) { } ( )

∫

+∞ ∞ −

= = dx t x xf t X E t ; ,ξ η ( ) ( ) ( ) { } ( )

∫

+∞ ∞ −

= =

2 1 2 1 2 1 2 1 2 1 2 1

, ; , , dx dx t t x x f x x t X t X E t t R

( ) ( ) ( ) [ ] ( ) ( ) [ ] { }

2 2 1 1 2 1,

t t X t t X E t t C η η − − =

( ) ( ) ( ) ( )

2 1 2 1 2 1

, , t t t t R t t C η η − =

( )( )

( ) ( ) ( )

t t t R t t C t

t X 2 2

, , η σ − = =

• r
• r
• G. Ahmadi

ME 529 - Stochastics

Sample time & space variations of fluctuation Sample time & space variations of fluctuation velocity components in turbulent near wall flows velocity components in turbulent near wall flows

• G. Ahmadi

ME 529 - Stochastics

Accelerograms Accelerograms of 1940 El Centro Earthquakes

• f 1940 El Centro Earthquakes

Actual Actual Simulated Simulated

• G. Ahmadi

ME 529 - Stochastics

STS STS-

• 41 Z Lift

41 Z Lift-

• Off Acceleration

Off Acceleration Original Original Simulated Simulated