15  Kriging with Varying Correlation-p

This chapter illustrates the difference between Kriging models with varying p. The difference is illustrated with the help of the spotpython package.

15.1 Example: Spot Surrogate and the 2-dim Sphere Function

import numpy as np
from math import inf
from spotpython.fun.objectivefunctions import analytical
from spotpython.spot import spot
from spotpython.utils.init import fun_control_init, surrogate_control_init
PREFIX="015"

15.1.1 The Objective Function: 2-dim Sphere

  • The spotpython package provides several classes of objective functions.
  • We will use an analytical objective function, i.e., a function that can be described by a (closed) formula: \[f(x, y) = x^2 + y^2\]
  • The size of the lower bound vector determines the problem dimension.
  • Here we will use np.array([-1, -1]), i.e., a two-dim function.
fun = analytical().fun_sphere
fun_control = fun_control_init(PREFIX=PREFIX,
                               lower = np.array([-1, -1]),
                               upper = np.array([1, 1]))
  • Although the default spot surrogate model is an isotropic Kriging model, we will explicitly set the theta parameter to a value of 1 for both dimensions. This is done to illustrate the difference between isotropic and anisotropic Kriging models.
surrogate_control=surrogate_control_init(n_p=1,
                                         p_val=2.0,)
spot_2 = spot.Spot(fun=fun,
                   fun_control=fun_control,
                   surrogate_control=surrogate_control)

spot_2.run()
spotpython tuning: 1.5904060546935205e-05 [#######---] 73.33% 
spotpython tuning: 1.5904060546935205e-05 [########--] 80.00% 
spotpython tuning: 1.5904060546935205e-05 [#########-] 86.67% 
spotpython tuning: 1.5904060546935205e-05 [#########-] 93.33% 
spotpython tuning: 1.4886245843852055e-05 [##########] 100.00% Done...

<spotpython.spot.spot.Spot at 0x36d227920>

15.1.2 Results

spot_2.print_results()
min y: 1.4886245843852055e-05
x0: -0.003850983818095356
x1: -0.00023700100552480994
[['x0', -0.003850983818095356], ['x1', -0.00023700100552480994]]
spot_2.plot_progress(log_y=True)

spot_2.surrogate.plot()

15.2 Example With Modified p

  • We can use set p to a value other than 2 to obtain a different Kriging model.
surrogate_control = surrogate_control_init(n_p=1,
                                           p_val=1.0)
spot_2_p1= spot.Spot(fun=fun,
                    fun_control=fun_control,
                    surrogate_control=surrogate_control)
spot_2_p1.run()
spotpython tuning: 1.5904060546935205e-05 [#######---] 73.33% 
spotpython tuning: 1.5904060546935205e-05 [########--] 80.00% 
spotpython tuning: 1.5904060546935205e-05 [#########-] 86.67% 
spotpython tuning: 1.5904060546935205e-05 [#########-] 93.33% 
spotpython tuning: 1.4886245843852055e-05 [##########] 100.00% Done...

<spotpython.spot.spot.Spot at 0x371187b30>
  • The search progress of the optimization with the anisotropic model can be visualized:
spot_2_p1.plot_progress(log_y=True)

spot_2_p1.print_results()
min y: 1.4886245843852055e-05
x0: -0.003850983818095356
x1: -0.00023700100552480994
[['x0', -0.003850983818095356], ['x1', -0.00023700100552480994]]
spot_2_p1.surrogate.plot()

15.2.1 Taking a Look at the p Values

15.2.1.1 p Values from the spot Model

  • We can check, which p values the spot model has used:
  • The p values from the surrogate can be printed as follows:
spot_2_p1.surrogate.p
array([1.])
  • Since the surrogate from the isotropic setting was stored as spot_2, we can also take a look at the theta value from this model:
spot_2.surrogate.p
array([2.])

15.3 Optimization of the p Values

surrogate_control = surrogate_control_init(n_p=1,
                                           optim_p=True)
spot_2_pm= spot.Spot(fun=fun,
                    fun_control=fun_control,
                    surrogate_control=surrogate_control)
spot_2_pm.run()
spotpython tuning: 2.0901803896651035e-05 [#######---] 73.33% 
spotpython tuning: 2.0901803896651035e-05 [########--] 80.00% 
spotpython tuning: 2.0901803896651035e-05 [#########-] 86.67% 
spotpython tuning: 2.0901803896651035e-05 [#########-] 93.33% 
spotpython tuning: 9.862120336177949e-06 [##########] 100.00% Done...

<spotpython.spot.spot.Spot at 0x37075a9c0>
spot_2_pm.plot_progress(log_y=True)

spot_2_pm.print_results()
min y: 9.862120336177949e-06
x0: -0.002974641180991332
x1: -0.0010067920244660482
[['x0', -0.002974641180991332], ['x1', -0.0010067920244660482]]
spot_2_pm.surrogate.plot()

spot_2_pm.surrogate.p
[1.661981929889647]

15.4 Optimization of Multiple p Values

surrogate_control = surrogate_control_init(n_p=2,
                                           optim_p=True)
spot_2_pmo= spot.Spot(fun=fun,
                    fun_control=fun_control,
                    surrogate_control=surrogate_control)
spot_2_pmo.run()
spotpython tuning: 1.90602728663205e-05 [#######---] 73.33% 
spotpython tuning: 1.90602728663205e-05 [########--] 80.00% 
spotpython tuning: 1.90602728663205e-05 [#########-] 86.67% 
spotpython tuning: 1.90602728663205e-05 [#########-] 93.33% 
spotpython tuning: 5.741913594811217e-06 [##########] 100.00% Done...

<spotpython.spot.spot.Spot at 0x37142bfe0>
spot_2_pmo.plot_progress(log_y=True)

spot_2_pmo.print_results()
min y: 5.741913594811217e-06
x0: -0.002070805151396577
x1: -0.001205686368737997
[['x0', -0.002070805151396577], ['x1', -0.001205686368737997]]
spot_2_pmo.surrogate.plot()

spot_2_pmo.surrogate.p
[1.353513055328678, 1.8723495904158025]

15.5 Exercises

15.5.1 fun_branin

  • Describe the function.
    • The input dimension is 2. The search range is \(-5 \leq x_1 \leq 10\) and \(0 \leq x_2 \leq 15\).
  • Compare the results from spotpython runs with different options for p.
  • Modify the termination criterion: instead of the number of evaluations (which is specified via fun_evals), the time should be used as the termination criterion. This can be done as follows (max_time=1 specifies a run time of one minute):
fun_evals=inf,
max_time=1,

15.5.2 fun_sin_cos

  • Describe the function.
    • The input dimension is 2. The search range is \(-2\pi \leq x_1 \leq 2\pi\) and \(-2\pi \leq x_2 \leq 2\pi\).
  • Compare the results from spotpython run a) with isotropic and b) anisotropic surrogate models.
  • Modify the termination criterion (max_time instead of fun_evals) as described for fun_branin.

15.5.3 fun_runge

  • Describe the function.
    • The input dimension is 2. The search range is \(-5 \leq x_1 \leq 5\) and \(-5 \leq x_2 \leq 5\).
  • Compare the results from spotpython runs with different options for p.
  • Modify the termination criterion (max_time instead of fun_evals) as described for fun_branin.

15.5.4 fun_wingwt

  • Describe the function.
    • The input dimension is 10. The search ranges are between 0 and 1 (values are mapped internally to their natural bounds).
  • Compare the results from spotpython runs with different options for p.
  • Modify the termination criterion (max_time instead of fun_evals) as described for fun_branin.

15.6 Jupyter Notebook

Note