Surrogate Modeling Preliminaries

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Surrogate Modeling - Preliminaries

• K Sudhakar Amitay Isaacs
• Centre for Aerospace Systems Design Engineering
• Department of Aerospace Engineering
• Indian Institute of Technology
• Mumbai 400 076

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Surrogate!

• Y F(x) is the true model
• Any or all of following are true
• Computationally intense
• Noisy response
• GUI driven. Not available for integration
• Y f(x) is surrogate of F(x)
• stands-in , fills-in , substitutes for F(x)
• Faithful to F(x) within prescribed limits
• Global surrogate
• Surrogate within move limits

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Surrogate Modeling

• Mathematical preliminaries
• Least Square Methods
• Design Of Experiments (DOE)
• Response Surface Method (RSM)
• Design Analysis of Computer Experiments (DACE)

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Mathematical Preliminaries
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Random Numbers - Discrete
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Random Numbers - Discrete
Eg. Dice
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Random Numbers - Continuous
x, dx
Probability Density Function (PDF) Probability,
px x 0 Probability, px, dx p(x) dx
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Random Numbers - Continuous
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Probability Cumulative Density Functions
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PDF
CDF
P
x

• Probability that value ? x
• P(x)
• P(-?) 0, P(?) 1

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Normal Distribution

• N(?, ?) normally distributed random number with
  
• mean ? standard deviation ?
• p( ? - ? lt x lt ? ?) 0.683
• p( ? - 2? lt x lt ? 2?) 0.954
• p( ? - 3? lt x lt ? 3?) 0.997

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Notations

• R(?, ?) random no
• mean ?, std dev ?
• RN (?, ?) random no. Normally distributed
• mean ?, std dev ?
• RU(X1?X2) random no. Uniformly distributed
• X1 ? X ? X2
• RU(0?1) ? ? 0.5, ? 0.28868)

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RU(0, 1) R(0.5, 0.28868)
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Central Limit Theorem ?

• Average of large number (n) of random numbers,
  all from same distribution R?, ? will tend to
  be RN ?, ?/?n
• Error in any experiment is assumed to be due to
  large number of factors and thus expected to be
  random RN?, ?

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Frequency of Average of 10 RU(0, 1)
Y RN(0.5, 0.28868/?10) RN(0.5,
0.0913) ??
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Sample Mean Variance

• Consider x RN(?, ?). We have n samples of x

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Sampling Distribution of Mean

• t is a random number having Student-t
  distribution with parameter ? n 1
• Symmetric, zero mean,
• As ? ? ? Rt ? RN(0, 1)

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Student t Distribution
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Sampling Distribution of Mean

• RN(?, ?) ? x
• Is RN(?, ?) ? RN(?1, ?) ?
• Take n sample of x
• Compute t from n samples
• If t gt t0.005
• We have a sample of such low probability!
• RN (?, ?) ? RN (?1, ?) OR A rare event?
• Note
• 1) t? are tabulated for various ?
• 2) Applicable even if x is R(?, ?) - any
  symmetric bell type pdf

? n-1
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An Example

• CFD Code predicts CL 0.49, claimed to be exact
• Wind tunnel tests are known to be unbiased
  estimators of CL, with an error ? RN(0,?)
• 5 wind tunnel tests are conducted giving

t (0.541 0.49) / (s/?n) 4.75 t0.005
4.6 for ? 4
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Sampling Distribution of Variance

• c is a random number having Chi-Square
  distribution with parameter, ? n 1
• Not symmetric

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Chi-Square Distribution
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Sampling Distribution of Variance

• RN(?, ?) ? x
• Is RN(?, ?) ? RN(?, ?1) ?
• Consider n sample of x
• Compute c from n samples
• If c gt c0.01
• We have a sample of such low probability!
• RN (?, ?) ? RN (?, ?1) OR A rare event?
• Note
• 1) c? are tabulated for various ?
• 2) Applicable even if x is R(?, ?) - any
  symmetric bell type pdf

?
c
c?
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Sampling Distribution of Variance

• are from 2 independent samples
  of size n1 and n2 of 2 distributions
• with same variance
• F is a random number of F-distribution with 2
  parameters, ?1 n1 1 and ?2 n2 1
• Not symmetric

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F- Distribution
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Sampling Distribution of Variance

• Did 2 samples come from same variance pdf?

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Some Operations
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Operations (Contd.)
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Multi-Variate Normal Distribution
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End of Surrogate Modeling - Preliminaries