Class 03 ( 1, 1; 2, 2): 2nd order joint PDF

Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering junhosong@snu.ac.kr

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457.643 Structural Random Vibrations In-Class Material: Class 03

II-1. Random Process (contd.)

 “Average” of random process

(a) “Ensemble” average: average over the ensemble

E[𝑋(𝑡)] = lim

→

+ + ⋯ +

𝑛 = 𝑑𝑥 (b) “Temporal” average (for a specific time history)

〈𝑋(𝑡)〉 =

𝑥(𝑡)𝑑𝑡

 Temporal average is another r_____ v______

 Specification of a random process (a) By probabilistic distribution function

 𝑓_{( )}(𝑥, 𝑡): 1^st order “m_________” PDF

 𝑓_{( ) ( )}

(𝑥

₁

, 𝑡

₁

; 𝑥

₂

, 𝑡

₂

): 2

^nd

order joint PDF

 ⋮

 𝑓_{( )⋯ ( )}

(𝑥

1

, 𝑡

1

; ⋯ ; 𝑥

𝑛

, 𝑡

𝑛

): n

^th

order joint PDF

Theoretically, need the ____^th order joint PDF for complete description of a random process

(b) By characteristic function

 𝑀 _{( )}(𝜃, 𝑡): 1^st order characteristic function

 ⋮

 𝑀 _{( )⋯ ( )}

(𝜃

1

, 𝑡

1

; ⋯ ; 𝜃

𝑛

, 𝑡

𝑛

): n

^th

order joint characteristic function

(c) By moment functions (i.e. partial descriptors)

 most common (because of lack of i________)

 E[𝑋(t)] = 𝜇 (𝑡) or 𝜇(𝑡): _______ function

 E[𝑋(𝑡 )𝑋(𝑡 )] = 𝜙 (𝑡 , 𝑡 ) or 𝜙(𝑡 , 𝑡 ): auto ___________ function

 E{[𝑋(𝑡 ) − 𝜇(𝑡 )][𝑋(𝑡 ) − 𝜇(𝑡 )]} = : auto _________ function (d) By a function of random variables

Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering junhosong@snu.ac.kr

2

 𝑋(𝑡) = 𝐴𝑡 + 𝐵

 𝑋(𝑡) = ∑ 𝐴 cos (𝜔 𝑡 + Θ )

(e) Others: random pulses, log-characteristic function, cumulants, ARMA, etc.

 First & second order moment functions

E[𝑋(t)] = 𝜇 (𝑡) or 𝜇(t): (E________) mean function

E[𝑋(𝑡 )𝑋(𝑡 )] = 𝜙 (𝑡 , 𝑡 ) or 𝜙(𝑡 , 𝑡 ): Auto-correlation function

⋮

E{[𝑋(𝑡 ) − 𝜇(𝑡 )][𝑋(𝑡 ) − 𝜇(𝑡 )]} = 𝜙 (𝑡 , 𝑡 ) − 𝜇(𝑡 )𝜇(𝑡 )

= 𝜅 (𝑡 , 𝑡 ) : Auto-covariance function

𝜎 (𝑡) = √ : Standard deviation function ρ (𝑡 , 𝑡 ) =

: Auto-correlation-coefficient function

Example: 77 force time histories during “digging” tasks and their moment functions Note:

If 𝜇 (𝑡) = 0 (zero-mean process), 𝜙 (𝑡 , 𝑡 ) 𝜅 (𝑡 , 𝑡 )

One can transform a random process to a zero-mean process by 𝑌(𝑡) = 𝑋(𝑡) −

Why?

0 1 2 3 4 5 6 7 8 9 10

-15 -10 -5 0 5 10 15 20

Dig Zone

Time (secs)

Force

Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering junhosong@snu.ac.kr

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 For a complex-valued random process, 𝜙 (𝑡 , 𝑡 ) = E[𝑋(𝑡 )𝑋^∗(𝑡 )]

𝜅 (𝑡 , 𝑡 ) = E[(𝑋(𝑡 ) − 𝜇(𝑡 ))(𝑋^∗(𝑡 ) − 𝜇^∗(𝑡 )]

Note that 𝜙 (𝑡, 𝑡) and 𝜅 (𝑡, 𝑡) are always ______-valued.

 More than one random process involved

𝜙 (𝑡 , 𝑡 ) = E[𝑋(𝑡 )𝑌^∗(𝑡 )] : ______ correlation function

𝜅 (𝑡 , 𝑡 ) = E[ 𝑋(𝑡 ) − 𝜇(𝑡 ) 𝑋^∗(𝑡 ) − 𝜇^∗(𝑡 ) ] : ______ covariance function 𝜌 (𝑡 , 𝑡 ) =

: _______ correlation coefficient function

 Importance of 1^st and 2^nd order moment functions

1) Most of the time, 1^st and 2^nd order moment functions are all one can get from data 2) For Gaussian, 1^st and 2^nd order moment functions are all you need for a complete

description.

3) Using Chebyshev bounds, one can get upper bound estimate on the probability using moments

P(|𝑍| > 𝑏) ≤E[|𝑍| ] 𝑏 e.g. 𝑐 = 2, 𝑍 = 𝑋 − 𝜇

P(|𝑋 − 𝜇 | > 𝑏) ≤𝐸[|𝑋 − 𝜇 | ] 𝑏 =

 Five important properties of 𝜙 (𝑡 , 𝑡 ) and 𝜅 (𝑡 , 𝑡 )

1) “Hermitian” (“Symmetric” for a real random process) 𝜙 (𝑡 , 𝑡 ) =

𝜅 (𝑡 , 𝑡 ) =

2) Boundedness

Schwarz inequality |E[𝑋𝑌]| ≤ E[𝑋 ]E[𝑌 ] Thus, |𝜙 (𝑡 , 𝑡 )| ≤ 𝜙 ( , )𝜙 ( , )

Seoul National University Instructor: Junho Song Dept. of Civil and Environmental Engineering junhosong@snu.ac.kr

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Also, |𝜙 (𝑡 , 𝑡 )| ≤ 𝜙 ( , )𝜙 ( , )

Similarly, |𝜅 (𝑡 , 𝑡 )| ≤ 𝜅 ( , )𝜅 ( , ) = σ ( )𝜎 ( ) Note:

If E[𝑋 (𝑡)] is bounded (< ∞) for ∀𝑡,

| (𝑡, 𝑠)| < ∞

If σ (𝑡) is bounded (< ∞) for ∀𝑡,

| (𝑡, 𝑠)| < ∞

𝑋(𝑡) is a “_______ ________” random process

if _________ is always finite

(Check L&S p.121. Later we will confirm that this means PSD exists) 3) Non-negative Definiteness

For an arbitrary function ℎ(𝑡),

𝜙 𝑡 , 𝑡 ℎ(𝑡 )ℎ^∗ 𝑡 ≥

Proof:

(LHS) = {ℎ(𝑡 ) ⋯ ℎ(𝑡 )} 𝜙 𝑡 , 𝑡

× {ℎ^∗(𝑡 ) ⋯ ℎ^∗(𝑡 )}

= 𝐡 E[𝑿𝑿 ]𝐡^∗

= E[𝐡 𝑿𝑿 𝐡^∗]

= E[𝑌𝑌^∗]

= E[ ] 0

Why is this property important?

Fourier transform of non-negative definite function is ____________

(Lin 1967, p.42 – Bochner’s theorem)

Note: However,

𝜙 (𝑡 , 𝑡 ): NOT non-negative definite

∵ 𝐸[XY] can be _________

∴ Cross PSD can be __________

Lin, Y.K. (1967) Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, NY.