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

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Chapter 6 Estimating Parameters From Observational Data Instructor: Prof. Wilson Tang CIVL 181 Modelling Systems with Uncertainties CIVL 181 Modelling Systems with Uncertainties
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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
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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
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Common Distributions and their Parameters Table 6.1, p279
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Common Distributions and their Parameters (Cont’d)
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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.
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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
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? estimating is X is
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This note was uploaded on 04/18/2009 for the course CE 408 taught by Professor Staff during the Spring '00 term at HKUST.

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

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