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stk_example_doe01


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 STK_EXAMPLE_DOE01  Examples of two-dimensional designs

 All designs are constructed on the hyper-rectangle BOX = [0; 2] x [0; 4].

 Examples of the following designs are shown:
  a) Regular grid                         --> stk_sampling_regulargrid,
  b) "Maximin" latin hypercube sample     --> stk_sampling_maximinlhs,
  c) RR2-scrambled Halton sequence        --> stk_sampling_halton_rr2,
  d) Uniformly distributed random sample  --> stk_sampling_randunif.



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 STK_EXAMPLE_DOE01  Examples of two-dimensional designs



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stk_example_doe02


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 STK_EXAMPLE_DOE02  "Sequential Maximin" design

 In this example, a two-dimensional space-filling design is progressively
 enriched with new points using a "sequential maximin" approach. More
 precisely, the k-th point X(k, :) is selected to maximize the distance to the
 set of all previously selected points X(1, :), X(2, :), ..., X(k-1, :).

 NOTES:

  * The resulting design is NOT optimal with respect to the maximin criterion
    (separation distance).

  * This procedure is not truly a *sequential* design procedure, since the
    choice of the k-th point X(k, :) does NOT depend on the response at the
    previously selected locations X(i, :), i < k.

 REFERENCE

  [1] Emmanuel Vazquez and Julien Bect, "Sequential search based on kriging:
      convergence analysis of some algorithms", In: ISI - 58th World
      Statistics Congress of the International Statistical Institute (ISI'11),
      Dublin, Ireland, August 21-26, 2011.



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 STK_EXAMPLE_DOE02  "Sequential Maximin" design



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stk_example_doe03


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 STK_EXAMPLE_DOE03  A simple illustration of 1D Bayesian optimization

 Our goal here is to optimize (maximize) the one-dimensional function

    x |--> x * sin (x)

 over the interval [0; 4 * pi].

 A Matern 5/2 prior with known parameters is used.

 Evaluations points are chosen sequentially using the Expected Improvement (EI)
 criterion, starting from an initial design of N0 = 3 points.



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 STK_EXAMPLE_DOE03  A simple illustration of 1D Bayesian optimization



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stk_example_doe04


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 STK_EXAMPLE_DOE04  Probability of misclassification

 The upper panel shows posterior means and variances as usual, and the
 threshold of interest, which is at T = 0.85 (dashed line).

 The lower panel shows the probability of misclassification as a function of x
 (blue curve), i.e., the probability that the actual value of the function is
 not on the same side of the threshold as the prediction (posterior mean).

 We also plot the expected future probability of misclassification (magenta
 curve), should a new evaluation be made at x = 3.

 Note that both probabilities are obtained using stk_pmisclass.



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 STK_EXAMPLE_DOE04  Probability of misclassification



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stk_example_doe05


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 STK_EXAMPLE_DOE05  A simple illustration of 1D Bayesian optimization

 Our goal here is to minimize the one-dimensional function

    x |--> x * sin (x)

 over the interval [0; 4 * pi], using noisy evaluations.

 Evaluations points are chosen sequentially using either AKG criterion
 (default) or the EQI criterion (set SAMPCRIT_NAME to 'EQI');



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 STK_EXAMPLE_DOE05  A simple illustration of 1D Bayesian optimization



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stk_example_doe06


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 STK_EXAMPLE_DOE06  Sequential design for the estimation of an excursion set

 In this example, we consider the problem of estimating the set

    Gamma = { x in X | f(x) > z_crit },

 where z_crit is a given value, and/or its volume.

 In a typical "structural reliability analysis" problem, Gamma would
 represent the failure region of a certain system, and its volume would
 correspond to the probability of failure (assuming a uniform distribution
 for the input).

 A Matern 5/2 prior with known parameters is used for the function f, and
 the evaluations points are chosen sequentially using any of the sampling
 criterion described in [1] (see also [2], section 4.3).

 REFERENCE

   [1] B. Echard, N. Gayton and M. Lemaire (2011).   AK-MCS: an active
       learning reliability method combining Kriging and Monte Carlo
       simulation.  Structural Safety, 33(2), 145-154.

   [2] J. Bect, D. Ginsbourger, L. Li, V. Picheny and E. Vazquez (2012).
       Sequential design of computer experiments for the estimation of a
       probability of failure.  Statistics and Computing, 22(3), 773-793.



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