9  Handling Noise: Optimal Computational Budget Allocation in Spot

This notebook demonstrates how noisy functions can be handled with OCBA by Spot.

9.1 Example: Spot, OCBA, and the Noisy Sphere Function

import numpy as np
from math import inf
from spotPython.fun.objectivefunctions import analytical
from spotPython.spot import spot
from scipy.optimize import shgo
from scipy.optimize import direct
from scipy.optimize import differential_evolution
import matplotlib.pyplot as plt

9.1.1 The Objective Function: Noisy Sphere

The spotPython package provides several classes of objective functions. We will use an analytical objective function with noise, i.e., a function that can be described by a (closed) formula: \[f(x) = x^2 + \epsilon\]

Since sigma is set to 0.1, noise is added to the function:

fun = analytical().fun_sphere
fun_control = {"sigma": 0.1,
               "seed": 123}

A plot illustrates the noise:

x = np.linspace(-1,1,100).reshape(-1,1)
y = fun(x, fun_control=fun_control)
plt.figure()
plt.plot(x,y, "k")
plt.show()

Spot is adopted as follows to cope with noisy functions:

  1. fun_repeats is set to a value larger than 1 (here: 2)
  2. noise is set to true. Therefore, a nugget (Lambda) term is added to the correlation matrix
  3. init size (of the design_control dictionary) is set to a value larger than 1 (here: 2)
spot_1_noisy = spot.Spot(fun=fun,
                   lower = np.array([-1]),
                   upper = np.array([1]),
                   fun_evals = 50,
                   fun_repeats = 2,
                   infill_criterion="ei",
                   noise = True,
                   tolerance_x=0.0,
                   ocba_delta = 1,
                   seed=123,
                   show_models=True,
                   fun_control = fun_control,
                   design_control={"init_size": 3,
                                   "repeats": 2},
                   surrogate_control={"noise": True})
spot_1_noisy.run()

<spotPython.spot.spot.Spot at 0x15ca19900>

9.3 Noise and Surrogates: The Nugget Effect

9.3.1 The Noisy Sphere

9.3.1.1 The Data

We prepare some data first:

import numpy as np
import spotPython
from spotPython.fun.objectivefunctions import analytical
from spotPython.spot import spot
from spotPython.design.spacefilling import spacefilling
from spotPython.build.kriging import Kriging
import matplotlib.pyplot as plt

gen = spacefilling(1)
rng = np.random.RandomState(1)
lower = np.array([-10])
upper = np.array([10])
fun = analytical().fun_sphere
fun_control = {"sigma": 2,
               "seed": 125}
X = gen.scipy_lhd(10, lower=lower, upper = upper)
y = fun(X, fun_control=fun_control)
X_train = X.reshape(-1,1)
y_train = y

A surrogate without nugget is fitted to these data:

S = Kriging(name='kriging',
            seed=123,
            log_level=50,
            n_theta=1,
            noise=False)
S.fit(X_train, y_train)

X_axis = np.linspace(start=-13, stop=13, num=1000).reshape(-1, 1)
mean_prediction, std_prediction, ei = S.predict(X_axis, return_val="all")

plt.scatter(X_train, y_train, label="Observations")
plt.plot(X_axis, mean_prediction, label="mue")
plt.legend()
plt.xlabel("$x$")
plt.ylabel("$f(x)$")
_ = plt.title("Sphere: Gaussian process regression on noisy dataset")

In comparison to the surrogate without nugget, we fit a surrogate with nugget to the data:

S_nug = Kriging(name='kriging',
            seed=123,
            log_level=50,
            n_theta=1,
            noise=True)
S_nug.fit(X_train, y_train)
X_axis = np.linspace(start=-13, stop=13, num=1000).reshape(-1, 1)
mean_prediction, std_prediction, ei = S_nug.predict(X_axis, return_val="all")
plt.scatter(X_train, y_train, label="Observations")
plt.plot(X_axis, mean_prediction, label="mue")
plt.legend()
plt.xlabel("$x$")
plt.ylabel("$f(x)$")
_ = plt.title("Sphere: Gaussian process regression with nugget on noisy dataset")

The value of the nugget term can be extracted from the model as follows:

S.Lambda
S_nug.Lambda
9.088150104540512e-05

We see:

  • the first model S has no nugget,
  • whereas the second model has a nugget value (Lambda) larger than zero.

9.4 Exercises

9.4.1 Noisy fun_cubed

Analyse the effect of noise on the fun_cubed function with the following settings:

fun = analytical().fun_cubed
fun_control = {"sigma": 10,
               "seed": 123}
lower = np.array([-10])
upper = np.array([10])

9.4.2 fun_runge

Analyse the effect of noise on the fun_runge function with the following settings:

lower = np.array([-10])
upper = np.array([10])
fun = analytical().fun_runge
fun_control = {"sigma": 0.25,
               "seed": 123}

9.4.3 fun_forrester

Analyse the effect of noise on the fun_forrester function with the following settings:

lower = np.array([0])
upper = np.array([1])
fun = analytical().fun_forrester
fun_control = {"sigma": 5,
               "seed": 123}

9.4.4 fun_xsin

Analyse the effect of noise on the fun_xsin function with the following settings:

lower = np.array([-1.])
upper = np.array([1.])
fun = analytical().fun_xsin
fun_control = {"sigma": 0.5,
               "seed": 123}
spot_1_noisy.mean_y.shape[0]
18