Salustowicz benchmark (#226)

* Salustowicz benchmark

* added kriging surrogate with different hypermeters

* added kriging surrogate with different hyper parameters

docs/make.jl

@@ -33,7 +33,8 @@ makedocs(
         "Welded beam function" => "welded_beam.md",
         "Branin function" => "BraninFunction.md",
         "Ackley function" => "ackley.md",
-        "Gramacy & Lee Function" => "gramacylee.md"
+        "Gramacy & Lee Function" => "gramacylee.md",
+        "Salustowicz Benchmark" => "Salustowicz.md"
         ]
     "Contributing" => "contributing.md"
     ]

docs/src/Salustowicz.md

@@ -0,0 +1,66 @@
+## 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(x)*sin(x) - 1)``
+
+Let's import these two packages  `Surrogates` and `Plots`:
+
+```@example salustowicz1D
+using Surrogates
+using Plots
+default()
+```
+
+Now, let's define our objective function:
+
+```@example salustowicz1D
+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
+```
+
+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.
+
+```@example salustowicz1D
+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)
+```
+
+Now, let's fit Salustowicz Function with different Surrogates:
+
+```@example salustowicz1D
+InverseDistance = InverseDistanceSurrogate(x, y, lower_bound, upper_bound)
+randomforest_surrogate = RandomForestSurrogate(x ,y ,lower_bound, upper_bound, num_round = 2)
+lobachevsky_surrogate = LobacheskySurrogate(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, randomforest_surrogate.(xs), label="RandomForest", legend=:topright)
+plot!(xs, lobachevsky_surrogate.(xs), label="Lobachesky", legend=:topright)
+```
+
+Not's let's see Kriging Surrogate with different hyper parameter:
+
+```@example salustowicz1D
+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=:top)
+plot!(xs, salustowicz.(xs), label="True function", legend=:top)
+plot!(xs, kriging_surrogate1.(xs), label="kriging_surrogate1", ribbon=p->std_error_at_point(kriging_surrogate1, p), legend=:top)
+plot!(xs, kriging_surrogate2.(xs), label="kriging_surrogate2", ribbon=p->std_error_at_point(kriging_surrogate2, p), legend=:top)
+plot!(xs, kriging_surrogate3.(xs), label="kriging_surrogate3", ribbon=p->std_error_at_point(kriging_surrogate3, p), legend=:top)
+```