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

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 G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics Poisson Random Variable X( ξ ξ ) Poisson Random Variable X( ) Consider probability experiment of Consider probability experiment of placing points at random on a line. placing points at random on a line. ∞ k k { ( ) } a ( ) a ( ) ∑ − ξ = = − = δ − a a ∈ (t number of points ∈ P X k e f x e x k Let n(t 1 Let n(t 1 , t , t 2 2 ) ) – – number of points (t 1 1 ,t ,t 2 2 ); ); X k ! k ! = k 0 X(t) = n(0,t) is a Poisson random variable X(t) = n(0,t) is a Poisson random variable { } if if = ( ) E X a t λ k { ( ) } t = − i) = = − λ i) t t t P n t , 2 t k e 1 2 1 { } k ! ii) If intervals (t 1 ii) If intervals (t 1 ,t ,t 2 2 ) & (t ) & (t 3 3 ,t ,t 4 4 ) are non ) are non- - σ x = = + 2 2 2 E X a a a overlapping, random variables n(t overlapping, random variables n(t 1 1 ,t ,t 2 2 ) & ) & n(t 3 ,t 4 ) are independent. n(t 3 ,t 4 ) are independent. G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 1

To obtain autocorrelation of X(t), assume t 2 > To obtain autocorrelation of X(t), assume t 2 > λ t X(t) is a Poisson process with parameter λ X(t) is a Poisson process with parameter t { } [ ] t 1 and consider { ( ) ( ) ( ) } { ( ) ( ) } ( ) t 1 and consider − = − 2 E X t X t X t E X t X t E X t 1 2 1 1 2 1 ( ) t λ k { ( ) } t Noting that (0,t 1 ) and (t 1 ,t 2 ) do not overlap, Noting that (0,t 1 ) and (t 1 ,t 2 ) do not overlap, = = − λ P X t k e k ! ) ( ) { ( ) } { ( ) ( ) } ( − = − λ + λ 2 2 E X t E X t X t R t , t t t 1 2 1 1 2 1 1 { ( ) } = λ E X t t [ ] ( ) ( ) or or λ λ − = − λ − λ 2 2 t t t R t , t t t 1 2 1 1 2 1 1 { } ( ) ( ) t ≥ = λ + λ = λ + λ 2 2 2 2 t R t 1 , t t t t E X t t t 2 1 2 1 2 1 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics If the points on the line have non- -uniform uniform If the points on the line have non Similarly: Similarly: ( ) λ (t), λ t density λ (t), λ ∫ t density t must be replaced by : must be replaced by : λ τ τ d ( ) 0 t ≥ = λ + λ 2 t R t 1 , t t t t ⎛ ⎞ k ( ) 2 1 ∫ t λ τ τ 2 1 2 2 ⎜ ⎟ { ( ) } ( ) d ∫ t = λ τ τ ∫ t ( ) ⎝ ⎠ { ( ) } E X t d − λ τ τ d 0 = = P X t k e 0 0 ! k ( ) ( ) = λ + λ 2 R t , t t t min t , t ( ) 1 2 1 2 1 2 ( ) ( ) ( ) ( ) t t min t , t ∫ ∫ ∫ = λ τ τ λ τ τ + λ τ 1 2 1 2 R t 1 , t d d dz 2 1 1 2 2 0 0 0 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 2

is a Weiner Process (Brownian Motion) Weiner Process (Brownian Motion) ) is a Statistics of Weiner Processes W(t) W(t Statistics of Weiner Processes when: when: { ( ) } = Mean Mean is a normal process with normal process with ) is a E W t 0 i) W(t i) W(t) { } 2 ( ) w ( ) 1 − = α ( ) 1 w Variance ii) Variance 2 ii) = α = 2 + E W t t ; 2 t f w t e F w ; t erf πα α 2 t t Autocorrelation Autocorrelation ii) Independent increment process, i.e. W(t 2 ) - - ii) Independent increment process, i.e. W(t 2 ) W(t 1 ) is independent of W(t 4 ) - - W(t W(t 3 ) if (t 1 ,t 2 ) W(t 1 ) is independent of W(t 4 ) 3 ) if (t 1 ,t 2 ) ⎧ α ≥ ⎫ t t t ( ) { ( ) ( ) } ( ) = = = α ⎨ 2 1 2 ⎬ and (t 3 ,t 4 ) are non- -overlapping overlapping and (t 3 ,t 4 ) are non R t , t E W t W t min t , t α ≥ 1 2 1 2 1 2 ⎩ ⎭ t t t iii) W(0) = 0 1 2 1 iii) W(0) = 0 G. Ahmadi G. Ahmadi ME 529 - Stochastics 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, ( ) n i ( ) dW t = g i n t dt t ∆ t { ( ) } t 0 =U ∆ t = Mean Mean E n t 0 ( ) ( ) = αδ − Autocorrelation Autocorrelation R t 1 , t t t 2 1 2 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 3

Numerical Simulation of White Noise Process iii) The white noise process is then given by Numerical Simulation of White Noise Process iii) The white noise process is then given by ( ) , , ; n varies ( ) , , ; n varies + ∆ = = + 0 = i) i) For a duration For a duration T T (~ 20 s) select a small time (~ 20 s) select a small time n t i t g i 1 , 2 ,..., m 1 n t 0 0 i linearly over each subinterval. Here, t 0 is a linearly over each subinterval. Here, t 0 is a step ∆ ∆ t t (~ 0.01 to 0.05 s) and divide this into (~ 0.01 to 0.05 s) and divide this into step random variable with uniform density over random variable with uniform density over m = T/ ∆ ∆ t t (~ 400 to 2000) subintervals (~ 400 to 2000) subintervals m = T/ ∆ t,0). - ∆ the subinterval (- the subinterval ( t,0). ii) Generate m+1 zero ii) Generate m+1 zero- -mean unit mean unit- -variance variance normally distributed random numbers G 1 normally distributed random numbers G 1 , , Transformation Form Pair of Uniform Transformation Form Pair of Uniform 1 π ⎛ ⎞ …, G m+1 . Multiply these by where S 0 is …, G m+1 . Multiply these by 2 S where S 0 is 2 ⎜ ⎟ 0 Random Variable to Gaussian Random Variable to Gaussian ∆ t ⎝ ⎠ the constant power spectrum of the white the constant power spectrum of the white 1 π ⎛ ⎞ 2 S 2 noise. Evaluate noise. Evaluate = ⎜ ⎟ = − π = − π g 0 G G 2 ln U sin 2 U G 2 ln U cos 2 U i ∆ i ⎝ ⎠ t 2 1 2 1 1 2 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics µ m Sample Absolute Acceleration Responses at Sample White Noise Process, d d p = 0.05 µ m Sample Absolute Acceleration Responses at Sample White Noise Process, p = 0.05 Top of Structure for El Centro 1940 Quake Top of Structure for El Centro 1940 Quake G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 4

If X (t (t 1 ), X (t (t 2 ), …, X(t X(t n ) jointly normal If X 1 ), X 2 ), …, n ) jointly normal � Poisson Random Variables � Poisson Random Variables [ ( ) ] − − η 2 � Non x t st Order Density) � Non- -Uniform Case Uniform Case (1 st Order Density) ( ) 1 (1 ( ) = 2 C t , t f x ; t e ( ) π 2 C t , t � Weiner Process � Weiner Process th Order Density) (n th Order Density) (n ⎧ ⎫ � White Noise Process ( ( ) ) � White Noise Process ( ) 1 1 ∑∑ ( ( ) ) − = − Λ − η − η ⎨ 1 ⎬ f x ,..., x ; t ,..., t exp x t x t 1 n 1 n 1 ij i i j j ( ) n ⎩ ⎭ 2 π Λ i j 2 2 2 � Normal Process � Normal Process [ ] ( ) = Λ Λ = Λ (Matrix of Covariance) C t i t , (Matrix of Covariance) det j (Note: Linear Combinations of normal processes are also normal) (Note: Linear Combinations of normal processes are also normal) G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics 5