1 457.643 Structural Random Vibrations In-Class Material: Class 02 I. Basic Elements (Contd.)

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

1

457.643 Structural Random Vibrations In-Class Material: Class 02

I. Basic Elements (Contd.)

 Joint characteristic function: alternative complete description to ______ PDF 𝑀_𝑿(𝜽) ≡ E_𝑿{exp[𝑖𝜽^T𝑿]}

= E_𝑿{exp[𝑖(𝜃₁𝑋₁+ 𝜃₂𝑋₂+ ⋯ + 𝜃_𝑛𝑋_𝑛)]}

= ∫ ⋯ ∫ exp[𝑖(𝜃₁𝑥₁+ 𝜃₂𝑥₂+ ⋯ + 𝜃_𝑛𝑥_𝑛)] 𝑓_𝑿(𝒙)𝑑𝒙

 m____variate F______ transform of _____ PDF

Therefore,

𝑓_𝑿(𝒙) =( )^{1 𝑛}∫ ⋯ ∫ exp[−𝑖(𝜃1𝑥₁+ 𝜃₂𝑥₂+ ⋯ + 𝜃_𝑛𝑥_𝑛)] 𝑑𝜽

Can show __________-generating property for the joint characteristic function, i.e.

1 𝑖^{𝑚1+⋯+𝑚𝑛}

𝜕^{𝑚1+⋯+𝑚𝑛}𝑀_𝑿(𝜽)

𝜕𝜃₁^𝑚1⋯ 𝜃_𝑛^𝑚𝑛 |

𝜽=𝟎

= E[𝑋₁^𝑚1𝑋₂^𝑚2⋯ 𝑋_𝑛^𝑚𝑛]

Some observations:

1) Consistency rule: 𝑀_𝑿(𝜃₁, ⋯ , 𝜃_𝑘, 0, ⋯ ,0) =

2) For statistically independent random variables, 𝑀_𝑿(𝜽) =

 Joint log characteristic function

Remember 𝑀_𝑋(𝜃) = E_𝑋[exp(𝑖𝜃𝑋)] = ∫ exp(𝑖𝜃𝑋) 𝑓_−∞^∞ _𝑋(𝑥)𝑑𝑥 Joint log characteristic function 𝐿_𝑿(𝜽) =

Joint cumulant function 𝜅(𝑿) = 1

𝑖^𝑛

𝜕^𝑛𝐿_𝑿(𝜽)

𝜕𝜃₁⋯ 𝜃_𝑛|

𝜽=𝟎

 𝜅(𝑋_𝑖) =

 κ(𝑋𝑖, 𝑋_𝑗) =

 κ(𝑋𝑖, 𝑋_𝑗, 𝑋_𝑘) =

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

2

 Multivariate normal (Gaussian) distribution Joint PDF:

𝑓_𝑿(𝒙) = 1

(2𝜋)^𝑛/2√|det 𝚺|exp [−1

2(𝒙 − 𝑴)^T𝚺⁻¹(𝒙 − 𝑴)]

 completely determined by _____ and ______ order moments

 denoted by 𝑿~𝑁(𝑴, 𝜮)

 can show 𝒀 = 𝑨𝑿 + 𝒃 ~ 𝑁(𝑨𝑴 + 𝒃, 𝑨𝜮𝑨^T)

e.g. 𝑛 = 1, 𝑋~𝑁(𝜇, 𝜎²) 𝑓_𝑋(𝑥) = 1

√2𝜋𝜎exp [−1 2(𝑥 − 𝜇

𝜎 )²] Can show

M_𝐗(𝜽) = exp (𝑖𝑴^T𝜽 −1

2𝜽^T𝚺𝜽) L_X(𝜽) =

 _________ function of 𝜽

 Higher order (𝑛 ≥ ) cumulants are zero

Example: κ(𝑋_𝑖, 𝑋_𝑗) for bivariate normal random variables

Example: Characteristic function of 𝑌 = 𝑋₁+ 𝑋₂+ ⋯ + 𝑋_𝑛 (and PDF?)

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

3

 Proof of Central Limit Theorem using characteristic functions

Consider 𝑍 = 𝑋₁+ 𝑋₂+ ⋯ + 𝑋_𝑛 where 𝑋_𝑖, 𝑖 = 1, … , 𝑛 are statistically independent, identically distributed (SIID) random variables. Try

𝑍^′= 1

√𝑛∑ (𝑋_𝑖− 𝜇 𝜎 )

𝑛

𝑖=1

where 𝜇 and 𝜎 respectively denote the common mean and standard deviation of 𝑋_𝑖’s.

Let 𝑌_𝑖 =^{𝑋𝑖−𝜇}

𝜎 Then, 𝑍^′= ¹

√𝑛∑^𝑛_𝑖=1𝑌_𝑖

The characteristic function of 𝑍′ is then derived as 𝑀_𝑍^′(𝜃) = E[exp(i𝜃𝑍^′)] = E {exp [i𝜃

√𝑛∑ 𝑌_𝑗

𝑛 𝑗=1

]} = E [∏ exp (i𝜃𝑌_𝑗

√𝑛)

𝑛 𝑗=1

]

= ∏ E [exp (i𝜃𝑌_𝑗

√𝑛)]

𝑛 𝑗=1

= ∏ 𝑀_𝑌𝑗(𝜃

√𝑛)

𝑛 𝑗=1

= [𝑀𝑌(𝜃

√𝑛)]

𝑛

Let us consider the characteristic function of 𝑌, 𝑀_𝑌(𝜃). Note that the mean of 𝑌 is zero and its standard deviation is one. From the moment generating property of the characteristic function, we confirm

𝑑𝑀_𝑌(𝜃) 𝑑𝜃 |

𝜃=0

= iE[𝑌] = 𝑖 ⋅ 𝜇_𝑌= 0

𝑑²𝑀_𝑌(𝜃) 𝑑𝜃² |

𝜃=0

= i²E[𝑌²] = −(𝜎_𝑌²+ 𝜇_𝑌²) = −1

Therefore, the characteristic function 𝑀_𝑌(𝜃) can be constructed by a Taylor series:

𝑀_𝑌(𝜃) = 1 −𝜃²

2 + 𝑜(𝜃²) 𝑀_𝑌(𝜃

√𝑛) = 1 −𝜃²

2𝑛+ 𝑜 (𝜃² 𝑛) 𝑀_𝑍^′(𝜃) = [1 −𝜃²

2𝑛+ 𝑜 (𝜃² 𝑛)]

𝑛

Since lim

n→∞(1 +^𝑥

𝑛)^𝑛= 𝑒^𝑥,

𝑛→∞lim 𝑀_𝑍^′(𝜃) = exp (−𝜃² 2)

The end result is the characteristic function of the standard ______ distribution. Thus, we hereby proved that 𝑍^′ asymptotically follows the standard _______ distribution as 𝑛 → ∞.

Since 𝑍 is a linear function of 𝑍′, 𝑍 also asymptotically follows a _______ distribution.

statistically independent identically distributed

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

4 II. Introduction to Random Process

II-1. Random Process

 Definitions

Random (stochastic) process: {𝑋(𝑡)} or 𝑋(𝑡) (cf. 𝑥(𝑡))

e.g. earthquake ground motion

Definition 1: Random process is an

“e_______” (collection) of possible t____ h______ {𝑥⁽¹⁾(𝑡), 𝑥⁽²⁾(𝑡), … }

Definition 2: “Continuously indexed” r_______ v________, or a family of random variables {𝑋(0), … 𝑋(𝑡_𝑘), … 𝑋(𝑡_𝑚), … }

Note: the concept of random process can be generalized

1) Random field 𝑋(𝑡, 𝑢, 𝑣), e.g. wind pressure at location (𝑢, 𝑣) of the roof at time 𝑡 2) Vector random process:

𝐗(𝑡) = { 𝑋

(𝑡) 𝑋

(𝑡)

⋮ 𝑋

(𝑡)

} e.g. 𝐗

(𝑡) = { 𝑥

(𝑡) 𝑥̇

(𝑡) 𝑥̈

(𝑡) }

3)

Vector random field:

𝐗(𝑡, 𝑢, 𝑣) = {

𝑋

(𝑡, 𝑢, 𝑣) 𝑋

(𝑡, 𝑢, 𝑣)

⋮ 𝑋

(𝑡, 𝑢, 𝑣)

}

Figure credit: http://www.wind.arch.t-

kougei.ac.jp/info_center/windpressure/lowrise/Intro ductionofthedatabase.pdf