Camel function (3 random inputs, scalar output)

In this example, PCE is used to generate a surrogate model for a given set of 2D data.

Six-hump camel function

\[f(x) = \Big(4-2.1x_1^2 + rac{x_1^4}{3} \Big)x_1^2 + x_1x_2 + (-4 + 4x_2^2)x_2^2\]

Description: Dimensions: 2

Input Domain: This function is evaluated on the hypercube \(x_1 \in [-3, 3], x_2 \in [-2, 2]\).

Global minimum: \(f(x^*)=-1.0316,\) at \(x^* = (0.0898, -0.7126)\) and \((-0.0898, 0.7126)\).

Reference: Molga, M., & Smutnicki, C. Test functions for optimization needs (2005). Retrieved June 2013, from http://www.zsd.ict.pwr.wroc.pl/files/docs/functions.pdf.

Import necessary libraries.

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
from matplotlib import cm
from UQpy.distributions import Uniform, JointIndependent
from matplotlib.ticker import LinearLocator, FormatStrFormatter
from UQpy.surrogates import *

Define the function.

def function(x, y):
    return (4 - 2.1 * x ** 2 + x ** 4 / 3) * x ** 2 + x * y + (-4 + 4 * y ** 2) * y ** 2

Create a distribution object, generate samples and evaluate the function at the samples.

np.random.seed(1)

dist_1 = Uniform(loc=-2, scale=4)
dist_2 = Uniform(loc=-1, scale=2)

marg = [dist_1, dist_2]
joint = JointIndependent(marginals=marg)

n_samples = 250
x = joint.rvs(n_samples)
y = function(x[:, 0], x[:, 1])

Visualize the 2D function.

xmin, xmax = -2, 2
ymin, ymax = -1, 1
X1 = np.linspace(xmin, xmax, 50)
X2 = np.linspace(ymin, ymax, 50)
X1_, X2_ = np.meshgrid(X1, X2)  # grid of points
f = function(X1_, X2_)

fig = plt.figure(figsize=(10, 6))
ax = fig.add_subplot(projection='3d')
surf = ax.plot_surface(X1_, X2_, f, rstride=1, cstride=1, cmap='gnuplot2', linewidth=0, antialiased=False)
ax.set_title('True function')
ax.set_xlabel('$x_1$', fontsize=15)
ax.set_ylabel('$x_2$', fontsize=15)
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
ax.view_init(20, 50)
fig.colorbar(surf, shrink=0.5, aspect=7)

plt.show()

Visualize training data.

fig = plt.figure(figsize=(10, 6))
ax = fig.add_subplot(projection='3d')
ax.scatter(x[:, 0], x[:, 1], y, s=20, c='r')

ax.set_title('Training data')
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
ax.view_init(20, 50)
ax.set_xlabel('$x_1$', fontsize=15)
ax.set_ylabel('$x_2$', fontsize=15)
plt.show()

Visualize training data.

max_degree = 6

Compute a PCE using least squares regression.

polynomial_basis = TotalDegreeBasis(joint, max_degree)
least_squares = LeastSquareRegression()
pce = PolynomialChaosExpansion(polynomial_basis=polynomial_basis, regression_method=least_squares)

pce.fit(x,y)

Compute PCE using LASSO regression.

polynomial_basis = TotalDegreeBasis(joint, max_degree)
lasso = LassoRegression()
pce2 = PolynomialChaosExpansion(polynomial_basis=polynomial_basis, regression_method=lasso)

pce2.fit(x,y)

Compute PCE using Ridge regression.

polynomial_basis = TotalDegreeBasis(joint, max_degree)
ridge = RidgeRegression()
pce3 = PolynomialChaosExpansion(polynomial_basis=polynomial_basis, regression_method=ridge)

pce3.fit(x,y)

PCE surrogate is used to predict the behavior of the function at new samples.

n_test_samples = 20000
x_test = joint.rvs(n_test_samples)
y_test = pce.predict(x_test)

Plot PCE prediction.

fig = plt.figure(figsize=(10,6))
ax = fig.add_subplot(projection='3d')
ax.scatter(x_test[:,0], x_test[:,1], y_test, s=1)

ax.set_title('PCE predictor')
ax.zaxis.set_major_locator(LinearLocator(10))
ax.zaxis.set_major_formatter(FormatStrFormatter('%.02f'))
ax.view_init(20,50)
ax.set_xlim(-2,2)
ax.set_ylim(-1,1)
ax.set_xlabel('$x_1$', fontsize=15)
ax.set_ylabel('$x_2$', fontsize=15)
plt.show()

Error Estimation

Validation error.

# validation sample
n_samples = 150
x_val = joint.rvs(n_samples)
y_val = function(x_val[:,0], x_val[:,1])

# PCE predictions
y_pce  = pce.predict(x_val).flatten()
y_pce2 = pce2.predict(x_val).flatten()
y_pce3 = pce3.predict(x_val).flatten()

# mean relative validation errors
error = np.sum(np.abs((y_val - y_pce)/y_val))/n_samples
error2 = np.sum(np.abs((y_val - y_pce2)/y_val))/n_samples
error3 = np.sum(np.abs((y_val - y_pce3)/y_val))/n_samples

print('Mean rel. error, LSTSQ:', error)
print('Mean rel. error, LASSO:', error2)
print('Mean rel. error, Ridge:', error3)

Mean rel. error, LSTSQ: 6.6383407624691066e-15
Mean rel. error, LASSO: 0.013733318222447327
Mean rel. error, Ridge: 0.01399966972099561

Moment Estimation

Returns mean and variance of the PCE surrogate.

n_mc = 1000000
x_mc = joint.rvs(n_mc)
y_mc = function(x_mc[:,0], x_mc[:,1])
mean_mc = np.mean(y_mc)
var_mc = np.var(y_mc)

print('Moments from least squares regression :', pce.get_moments())
print('Moments from LASSO regression :', pce2.get_moments())
print('Moments from Ridge regression :', pce3.get_moments())
print('Moments from Monte Carlo integration: ', mean_mc, var_mc)

Moments from least squares regression : (1.1276190476190477, 1.3807125276839571)
Moments from LASSO regression : (1.1266773278385855, 1.3702252349746957)
Moments from Ridge regression : (1.1265829663089746, 1.368059711959485)
Moments from Monte Carlo integration:  1.127527316332168 1.3805224979467277