# ch5 - CIVL 181 Modelling Systems with Uncertainties Chapter...

This preview shows pages 1–9. Sign up to view the full content.

Chapter 6 Estimating Parameters From Observational Data Instructor: Prof. Wilson Tang CIVL 181 Modelling Systems with Uncertainties CIVL 181 Modelling Systems with Uncertainties

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Estimating Parameters From Observation Data REAL WORLD “POPULATION” (True Characteristics Unknown) Theoretical Model Sample {x 1 , x 2 , …, x n } Sampling (Experimental Observations) Real Line -∞ < x < ∞ With Distribution f X (x) Random Variable X Inference On f X (x) f X ( x) 2 2 Variance x Mean s σ μ ( 29 - - = = 2 2 1 1 1 x x n s x n x i i Statistical Estimation Role of sampling in statistical inference
Point Estimations of Parameters, e.g. μ , σ 2 , λ , ζ etc. a) Method of moments: equate statistical moments (e.g. mean, variance, skewness etc.) of the model to those of the sample. ( 29 2 2 ˆ , ˆ ; , : normal in e.g. s x N X = = σ μ From Table 6.1 in p250 – 251 See e.g. 6.2 in p. 251 ( 29 ( 29 ( 29 ( 29 [ ] 2 2 2 1 X ar 2 1 exp X E , LN : X lognormal in 2 s e X E V x - = + = ζ λ

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Common Distributions and their Parameters Table 6.1, p279
Common Distributions and their Parameters (Cont’d)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
b) Method of maximum likelihood (p 251-255): Parameter = θ r.v. X with f x (x) Definition: L( θ ) = f X (x 1 , θ ) f X (x 2 , θ ) ⋅ ⋅ ⋅ f X (x n , θ ), where x 1 , x 2 , ⋅ ⋅ ⋅ x n are observed data ( 29 θ θ = θ θ of estimation opt 0 L Physical interpolation – the value of θ such that the likelihood function is maximized (i.e. likelihood of getting these data is maximized ) For practical purpose, the difference between the estimates obtained from these different methods would be small if sample size is sufficiently large.
b)Method of maximum likelihood (Cont’d): λ = 1 λ = 2 λ e - λ x x f X (x) X 1 X 2 Given X 1 λ = 2 more likely Similarly, X 2 λ = 1 more likely Likelihood of λ depends on f X (x i ) and the x i ’s ( 29 ( 29 ( 29 ( 29 λ = λ λ λ λ = θ λ - λ - ˆ 0 d dL e e L 2 1 x x

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
? estimating is X is
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 04/18/2009 for the course CE 408 taught by Professor Staff during the Spring '00 term at HKUST.

### Page1 / 27

ch5 - CIVL 181 Modelling Systems with Uncertainties Chapter...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online