Outline Outline Independent Increment Processes Independent - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Independent Increment Processes

Independent Increment Processes

• Cross

Cross-

• Correlation & Cross

Correlation & Cross-

• Covariance

Covariance

• Strict

Strict-

• Sense Stationary Process

Sense Stationary Process

• Jointly SSS Processes

Jointly SSS Processes

• Wild

Wild-

• Sense Stationary Process

Sense Stationary Process

• G. Ahmadi

ME 529 - Stochastics

If increments X(t If increments X(t2

2)

) -

• X(t

X(t1

1) and X(t

) and X(t4

4)

) -

• X(t

X(t3

3)

)

• f process X(t) are uncorrelated (or
• f process X(t) are uncorrelated (or

independent) for any t independent) for any t1

1 < t

< t2

2 ≤

≤ t t3

3 < t

< t4

4, then

, then X(t X(t) is a process with uncorrelated (or ) is a process with uncorrelated (or independent) increments. independent) increments. Poisson, Weiner Poisson, Weiner are independent are independent increment increment

• G. Ahmadi

ME 529 - Stochastics

Cross Cross-

• Correlation

Correlation Cross Cross-

• Covariance

Covariance

( ) ( ) ( ) { } ( )

1 2 2 1 2 1

, , t t R t Y t X E t t R

YX XY

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

2 1 2 1 2 2 1 1 2 1

t t t , t R t t Y t t X E t , t C

Y X XY Y X XY

η η η η − = − − =

SLIDE 2

2

• G. Ahmadi

ME 529 - Stochastics

Orthogonal Processes Orthogonal Processes Uncorrelated Processes Uncorrelated Processes Independent Processes Independent Processes – – group of random group of random variables X(t variables X(t1

1), …,

), …, X(t X(tn

n) are independent of the

) are independent of the group Y(t group Y(t1

1’), …, Y(t

’), …, Y(tm

m’):

’):

( )

, 2

1

= t t RXY

( )

, 2

1

= t t CXY

( ) ( ) ( )

m n m n

y y f x x f y y x x f ,..., ,..., ,..., , ,...,

1 1 1 1

=

• G. Ahmadi

ME 529 - Stochastics

Strict Strict-

• Sense Stationary (SSS) Process

Sense Stationary (SSS) Process

Statistics not affected by shift in time origin. Statistics not affected by shift in time origin.

( ) ( )

τ τ + + =

n n n n

t t x x f t t x x f ,..., , ,..., ,..., , ,...,

1 1 1 1

( ) ( )

τ + = t x f t x f ; ;

( ) ( )

x f t x f = ;

First order density is First order density is independent of time independent of time

• G. Ahmadi

ME 529 - Stochastics

Similarly: Similarly:

( ) ( )

2 1 2 1 2 1 2 1

; , , ; , t t x x f t t x x f − = ( ) { }

const t X E = =η

( ) ( ) ( ) { } ( )

2 1 2 1 2 1,

t t R t X t X E t t R − = =

( )

{ }

const t X E = =

2 2

σ

Second order density depends on time difference Second order density depends on time difference

Statistics of Stationary processes Statistics of Stationary processes

• G. Ahmadi

ME 529 - Stochastics

Jointly SSS Processes Jointly SSS Processes

Joint statistics of Joint statistics of X(t X(t) & ) & Y(t Y(t) are the ) are the same as X(t + same as X(t + τ τ), Y(t + ), Y(t + τ τ) )

( ) ( )

2 1 2 1

, , , ; , t t y x f t t y x f

XY XY

− =

( ) ( ) { } ( )

2 1 2 1

t t R t Y t X E

XY

− =

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Wide Wide-

• Sense Stationary (WSS) Process

Sense Stationary (WSS) Process

If mean is independent of time and If mean is independent of time and autocorrelation depends on autocorrelation depends on τ τ = t = t1

1 –

– t t2

2

Jointly WSS Processes X(t) & Y(t) Jointly WSS Processes X(t) & Y(t)

If both If both X(t X(t) & ) & Y(t Y(t) are ) are WSS, and cross WSS, and cross-

• correlation depends on time difference

correlation depends on time difference

( ) { }

const t X E = =η

( ) ( ) { } ( )

τ τ R t X t X E = +

( ) ( ) { } ( )

τ τ

XY

R t Y t X E = +

• G. Ahmadi

ME 529 - Stochastics

Concluding Remarks Concluding Remarks

• Cross

Cross-

• Correlation & Cross

Correlation & Cross-

• Covariance

Covariance

• Strict

Strict-

• Sense Stationary Process

Sense Stationary Process

• Jointly SSS Processes

Jointly SSS Processes

• Wild

Wild-

• Sense Stationary Process

Sense Stationary Process

• G. Ahmadi

ME 529 - Stochastics