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Salustowicz Benchmark Function

The true underlying function HyGP had to approximate is the 1D Salustowicz function. The function can be evaluated in the given domain: $x \in [0, 10]$.

The Salustowicz benchmark function is as follows:

$f(x) = e^{-x} x^3 \cos(x) \sin(x) (\cos(x) \sin^2(x) - 1)$

Let's import these two packages Surrogates and Plots:

using Surrogates
using Plots

Now, let's define our objective function:

function salustowicz(x)
    term1 = 2.72^(-x) * x^3 * cos(x) * sin(x)
    term2 = (cos(x) * sin(x) * sin(x) - 1)
    y = term1 * term2
end
salustowicz (generic function with 1 method)

Let's sample f in 30 points between 0 and 10 using the sample function. The sampling points are chosen using a Sobol Sample, this can be done by passing SobolSample() to the sample function.

n_samples = 30
lower_bound = 0
upper_bound = 10
num_round = 2
x = sample(n_samples, lower_bound, upper_bound, SobolSample())
y = salustowicz.(x)
xs = lower_bound:0.001:upper_bound
scatter(x, y, label = "Sampled points", xlims = (lower_bound, upper_bound), legend = :top)
plot!(xs, salustowicz.(xs), label = "True function", legend = :top)
Example block output

Now, let's fit the Salustowicz function with different surrogates:

InverseDistance = InverseDistanceSurrogate(x, y, lower_bound, upper_bound)
lobachevsky_surrogate = LobachevskySurrogate(
    x, y, lower_bound, upper_bound, alpha = 2.0, n = 6)
scatter(
    x, y, label = "Sampled points", xlims = (lower_bound, upper_bound), legend = :topright)
plot!(xs, salustowicz.(xs), label = "True function", legend = :topright)
plot!(xs, InverseDistance.(xs), label = "InverseDistanceSurrogate", legend = :topright)
plot!(xs, lobachevsky_surrogate.(xs), label = "Lobachevsky", legend = :topright)
Example block output

Not's let's see Kriging Surrogate with different hyper parameter:

kriging_surrogate1 = Kriging(x, y, lower_bound, upper_bound, p = 0.9);
kriging_surrogate2 = Kriging(x, y, lower_bound, upper_bound, p = 1.5);
kriging_surrogate3 = Kriging(x, y, lower_bound, upper_bound, p = 1.9);
scatter(
    x, y, label = "Sampled points", xlims = (lower_bound, upper_bound), legend = :topright)
plot!(xs, salustowicz.(xs), label = "True function", legend = :topright)
plot!(xs, kriging_surrogate1.(xs), label = "kriging_surrogate1",
    ribbon = p -> std_error_at_point(kriging_surrogate1, p), legend = :topright)
plot!(xs, kriging_surrogate2.(xs), label = "kriging_surrogate2",
    ribbon = p -> std_error_at_point(kriging_surrogate2, p), legend = :topright)
plot!(xs, kriging_surrogate3.(xs), label = "kriging_surrogate3",
    ribbon = p -> std_error_at_point(kriging_surrogate3, p), legend = :topright)
Example block output