{ } k ! ii) If intervals (t 1 ii) If intervals (t 1 ,t ,t 2 2 ) - - PowerPoint PPT Presentation

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1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Poisson Random Variable

Poisson Random Variable

• Non

Non-

• Uniform Case

Uniform Case

• Weiner Process

Weiner Process

• White Noise Process

White Noise Process

• Normal Process

Normal Process

• G. Ahmadi

ME 529 - Stochastics

Poisson Random Variable X( Poisson Random Variable X(ξ ξ) )

( ) { }

! k a e k X P

k a −

= = ξ

( ) ( )

∑

∞ = −

− = !

k k a X

k x k a e x f δ

{ }

a X E =

{ }

a a X E + =

2 2

a

x = 2

σ

• G. Ahmadi

ME 529 - Stochastics

Consider probability experiment of Consider probability experiment of placing points at random on a line. placing points at random on a line. Let n(t Let n(t1

1, t

, t2

2)

) – – number of points number of points ∈ ∈ (t (t1

1,t

,t2

2);

); X(t) = n(0,t) is a Poisson random variable X(t) = n(0,t) is a Poisson random variable if if i) i) ii) If intervals (t ii) If intervals (t1

1,t

,t2

2) & (t

) & (t3

3,t

,t4

4) are non

) are non-

• verlapping, random variables n(t
• verlapping, random variables n(t1

1,t

,t2

2) &

) & n(t n(t3

3,t

,t4

4) are independent.

) are independent.

( ) { } ( )

! , 2

1

k t e k t t n P

k t λ λ −

= =

1 2

t t t − =

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• G. Ahmadi

ME 529 - Stochastics

X(t) is a Poisson process with parameter X(t) is a Poisson process with parameter λ λt t

( ) { } ( )

! k t e k t X P

k t λ λ −

= = ( ) { }

t t X E λ =

( )

{ }

t t t X E λ λ + =

2 2 2

• G. Ahmadi

ME 529 - Stochastics

Noting that (0,t Noting that (0,t1

1) and (t

) and (t1

1,t

,t2

2) do not overlap,

) do not overlap,

• r
• r

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

)

1 2 1 2 2 1 1 2 1

, t t t t R t X t X E t X E λ λ + − = −

( ) [ ] ( )

1 2 1 2 2 1 1 2 1

, t t t t R t t t λ λ λ λ − − = −

( )

1 2 1 2 2 1,

t t t t t R λ λ + =

1 2

t t ≥

To obtain autocorrelation of X(t), assume t To obtain autocorrelation of X(t), assume t2

2 >

> t t1

1 and consider

and consider

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

{ }

1 2 2 1 1 2 1

t X E t X t X E t X t X t X E − = −

• G. Ahmadi

ME 529 - Stochastics

( )

2 2 1 2 2 1,

t t t t t R λ λ + =

( ) ( )

2 1 2 1 2 2 1

, min , t t t t t t R λ λ + =

1 2

t t ≥

Similarly: Similarly:

• G. Ahmadi

ME 529 - Stochastics

If the points on the line have non If the points on the line have non-

• uniform

uniform density density λ λ(t), (t), λ λt t must be replaced by : must be replaced by :

( )

∫

t

d τ τ λ

( ) { }

( )

( )

! k d e k t X P

k t d

t

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∫ = =

∫

−

τ τ λ

τ τ λ

( ) { } ( )

∫

=

t

d t X E τ τ λ

( ) ( ) ( ) ( )

( )

∫ ∫ ∫

+ =

2 1 2 1

, min 2 2 1 1 2 1, t t t t

dz d d t t R τ λ τ τ λ τ τ λ

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• G. Ahmadi

ME 529 - Stochastics

W(t W(t) ) is a is a Weiner Process (Brownian Motion) Weiner Process (Brownian Motion) when: when: i) i) W(t W(t) ) is a is a normal process with normal process with ii) ii) ii) Independent increment process, i.e. W(t ii) Independent increment process, i.e. W(t2

2)

) -

• W(t

W(t1

1) is independent of W(t

) is independent of W(t4

4)

) -

• W(t

W(t3

3) if (t

) if (t1

1,t

,t2

2)

) and (t and (t3

3,t

,t4

4) are non

) are non-

• overlapping
• verlapping

iii) W(0) = 0 iii) W(0) = 0

( )

t w

e t t w f

α

πα

2

2

2 1 ;

−

= ( )

t w erf t w F α + = 2 1 ;

• G. Ahmadi

ME 529 - Stochastics

( ) { }

= t W E

( )

{ }

t t W E α =

2

( ) ( ) ( ) { } ( )

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

, min , t t t t t t t t t W t W E t t R α α α = ⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≥ ≥ = =

Statistics of Weiner Processes Statistics of Weiner Processes

Mean Mean Variance Variance Autocorrelation Autocorrelation

• G. Ahmadi

ME 529 - Stochastics

A White Noise Process is the derivative of a A White Noise Process is the derivative of a Weiner Process. That is, Weiner Process. That is,

( ) ( )

dt t dW t n =

( ) { }

= t n E

( ) ( )

2 1 2 1,

t t t t R − = αδ

Mean Mean Autocorrelation Autocorrelation

• G. Ahmadi

ME 529 - Stochastics

t ni ∆t t0 =U∆t gi

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• G. Ahmadi

ME 529 - Stochastics

Numerical Simulation of White Noise Process Numerical Simulation of White Noise Process i) i) For a duration For a duration T T (~ 20 s) select a small time (~ 20 s) select a small time step step ∆ ∆t t (~ 0.01 to 0.05 s) and divide this into (~ 0.01 to 0.05 s) and divide this into m = T/ m = T/∆ ∆t t (~ 400 to 2000) subintervals (~ 400 to 2000) subintervals ii) ii) Generate m+1 zero Generate m+1 zero-

• mean unit

mean unit-

• variance

variance normally distributed random numbers G normally distributed random numbers G1

1,

, …, G …, Gm+1

m+1. Multiply these by

. Multiply these by where S where S0

0 is

is the constant power spectrum of the white the constant power spectrum of the white

• noise. Evaluate
• noise. Evaluate

2 1

2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆t S π

i i

G t S g

2 1

2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∆ = π

• G. Ahmadi

ME 529 - Stochastics

iii) The white noise process is then given by iii) The white noise process is then given by , , ; n varies , , ; n varies linearly over each subinterval. Here, t linearly over each subinterval. Here, t0

0 is a

is a random variable with uniform density over random variable with uniform density over the subinterval ( the subinterval (-

• ∆

∆t,0). t,0).

( )

i

g t i t n = ∆ + 1 ,..., 2 , 1 + = m i

( )

0 =

t n

Transformation Form Pair of Uniform Transformation Form Pair of Uniform Random Variable to Gaussian Random Variable to Gaussian

2 1 1

U 2 cos U ln 2 G π − =

2 1 2

U 2 sin U ln 2 G π − =

• G. Ahmadi

ME 529 - Stochastics

Sample White Noise Process, Sample White Noise Process, d dp

p = 0.05

= 0.05 µ µm m

• G. Ahmadi

ME 529 - Stochastics

Sample Absolute Acceleration Responses at Sample Absolute Acceleration Responses at Top of Structure for El Centro 1940 Quake Top of Structure for El Centro 1940 Quake

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• G. Ahmadi

ME 529 - Stochastics

( ) ( )

( ) [ ] ( )

t t C t x

e t t C t x f

, 2

2

, 2 1 ;

η

π

− −

=

( ) ( ) ( ) ( )

( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − − − =

∑∑

− i j j j i i ij n n n

t x t x exp t ,..., t ; x ,..., x f η η Λ Λ π

1 2 1 2 1 1

2 1 2 1

( ) [ ]

j i t

t C , = Λ Λ = Λ det

If X If X (t (t1

1), X

), X (t (t2

2), …,

), …, X(t X(tn

n) jointly normal

) jointly normal

(1 (1st

st Order Density)

Order Density) (n (nth

th Order Density)

Order Density) (Matrix of Covariance) (Matrix of Covariance) (Note: Linear Combinations of normal processes are also normal) (Note: Linear Combinations of normal processes are also normal)

• G. Ahmadi

ME 529 - Stochastics

• Poisson Random Variables

Poisson Random Variables

• Non

Non-

• Uniform Case

Uniform Case

• Weiner Process

Weiner Process

• White Noise Process

White Noise Process

• Normal Process

Normal Process

• G. Ahmadi

ME 529 - Stochastics