1. Chapter 5 Estimating Parameters From Observational Data Instructor: Prof. Wilson Tang CIVL 181 Modelling Systems with Uncertainties

2. Estimating Parameters From Observation Data REAL WORLD “POPULATION” (True Characteristics Unknown) 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) Statistical Estimation Role of sampling in statistical inference

3. Point Estimations of Parameters, e.g. ,  2,,  etc. a)Method of moments: equate statistical moments (e.g. mean, variance, skewness etc.) of the model to those of the sample. From Table 5.1 in pp 224 – 225 See e.g. 5.2 in p. 227

4. Common Distributions and their Parameters

5. Common Distributions and their Parameters (Cont’d)

6. b) Method of maximum likelihood: Parameter =  r.v. X with f x (x) Definition: L(  ) = f X (x 1,  ) f X (x 2,  )  f X (x n,  ), where x 1, x 2,  x n are observed data Physical interpolation – the value of  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.

7. b)Method of maximum likelihood (Cont ’ d): = 1 = 2 e - x x f X (x) X1X1 X2X2 GivenX 1  = 2 more likely Similarly,X 2  = 1 more likely  Likelihood of depends on f X (x i ) and the x i ’s

8. X: ,  What would you expect the value of X to be? As n   Var(X)  Before collections of data, X 1 is a r.v. = X  X is r.v.

9. What is the distribution of X?  n1n1 n2n2 n  n 1 > n 2

10. Confidence interval of  We would like to establish P(? <  < ?) = 0.95 0.95 k 0.025 = 1.96 -1.96 0.025

11. Confidence interval of  (Cont’d) Not a r.v.  confidence interval  /2 1 –  k  /2

12. E5.5 Daily dissolved oxygen (DO) n = 30 observations s = 2.05 mg/l assume =  x = 2.52 mg/l Determine 99% confidence interval of  As confidence level   interval     <>  n   <> 

13. Confidence Interval of  when  is unknown Small f Large f  N(0,1) 0  known

14.  /2 p t  /2,f 0 (for known  case)

15. E5.9 Traffic survey on speed of vehicles. Suppose we would like to determine the mean vehicle velocity to within  2 kph with 99 % confidence. How many vehicles should be observed? Assume  = 3.58 from previous study Scatter 2.58 What if  not known, but sample std. dev. expected to s = 3.58 and desired to be with  2 ?

16. E5.9 (Cont’d) Compare with n  21 for  known

17. Lower confidence limit Upper confidence limit  1 –  kk Not  /2  known  unknown

18. - Similar to estimations of 

20. What about an area? B C D

22. In general, r1r1 r2r2  h 

23. Interval Estimation of  2 sample variance  2 statistics  – confidence level n – no. of sample

24. E 5.13 DO data: n = 30, s 2 = 4.2

25. Estimation of proportions

26. 10 out of 50 specimens do not have pass CBR requirement. E 5.14

27. Review on Chapter 5