Chapter 6 Estimating Parameters From Observational Data

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Chapter 6Estimating Parameters From
Observational Data
CIVL 181 Modelling Systems with Uncertainties

• Instructor Prof. Wilson Tang

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Estimating Parameters From Observation Data
REAL WORLD POPULATION (True Characteristics
Unknown)
Role of sampling in statistical inference
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Point Estimations of Parameters, e.g. m, s2, l, z
etc.

• a) Method of moments equate statistical moments
  (e.g. mean, variance, skewness etc.) of the model
  to those of the sample.

See e.g. 6.2 in p. 251
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Common Distributions and their Parameters
Table 6.1, p279
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Common Distributions and their Parameters (Contd)
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b) Method of maximum likelihood (p 251-255)

• Parameter q r.v. X with fx(x)
• Definition L(q) fX(x1,q) fX(x2,q)???fX(xn,q),
  where x1, x2,???xn are observed data

Physical interpolation the value of q 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 (Contd)

Given X1 ? l 2 more likely Similarly, X2 ? l
1 more likely ? Likelihood of l depends on fX(xi)
and the xis
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X m, s
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Confidence interval of m

• We would like to establish P(? lt m lt ?) 0.95

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Confidence interval of m (Contd)
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Example
Determine 99 confidence interval of m.
As confidence level ? ? interval ?
s ? ? ltgt ?
n ? ? ltgt ?
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Confidence Interval of m when s is unknown
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(for known s case)
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E6.13

• 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 s 3.58 from previous study

2.58, From Table A.1
What if s not known, but sample std. dev.
expected to s 3.58 and desired to be with ? 2 ?
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E6.13 (Contd)
Compare with n ? 21 for s known
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- Similar to estimations of m
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What about an area?
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In general,
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Interval Estimation of s2
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Example
DO data n 30, s2 4.2
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Estimation of proportions
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E6.14
10 out of 50 specimens do not have pass CBR
requirement.
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Review on Chapter 6