## Kriging surrogate tutorial (1D)

Kriging or Gaussian process regression is a method of interpolation for which the interpolated values are modeled by a Gaussian process.

We are going to use a Kriging surrogate to optimize $f(x)=(6x-2)^2sin(12x-4)$. (function from Forrester et al. (2008)).

First of all import Surrogates and Plots.

using Surrogates
using Plots
default()

### Sampling

We choose to sample f in 4 points between 0 and 1 using the sample function. The sampling points are chosen using a Sobol sequence, this can be done by passing SobolSample() to the sample function.

# https://www.sfu.ca/~ssurjano/forretal08.html
# Forrester et al. (2008) Function
f(x) = (6 * x - 2)^2 * sin(12 * x - 4)

n_samples = 4
lower_bound = 0.0
upper_bound = 1.0

xs = lower_bound:0.001:upper_bound

x = sample(n_samples, lower_bound, upper_bound, SobolSample())
y = f.(x)

scatter(x, y, label="Sampled points", xlims=(lower_bound, upper_bound), ylims=(-7, 17))
plot!(xs, f.(xs), label="True function", legend=:top)

### Building a surrogate

With our sampled points we can build the Kriging surrogate using the Kriging function.

kriging_surrogate behaves like an ordinary function which we can simply plot. A nice statistical property of this surrogate is being able to calculate the error of the function at each point, we plot this as a confidence interval using the ribbon argument.

kriging_surrogate = Kriging(x, y, lower_bound, upper_bound);

plot(x, y, seriestype=:scatter, label="Sampled points", xlims=(lower_bound, upper_bound), ylims=(-7, 17), legend=:top)
plot!(xs, f.(xs), label="True function", legend=:top)
plot!(xs, kriging_surrogate.(xs), label="Surrogate function", ribbon=p->std_error_at_point(kriging_surrogate, p), legend=:top)

### Optimizing

Having built a surrogate, we can now use it to search for minima in our original function f.

To optimize using our surrogate we call surrogate_optimize method. We choose to use Stochastic RBF as optimization technique and again Sobol sampling as sampling technique.

@show surrogate_optimize(f, SRBF(), lower_bound, upper_bound, kriging_surrogate, SobolSample())

scatter(x, y, label="Sampled points", ylims=(-7, 7), legend=:top)
plot!(xs, f.(xs), label="True function", legend=:top)
plot!(xs, kriging_surrogate.(xs), label="Surrogate function", ribbon=p->std_error_at_point(kriging_surrogate, p), legend=:top)

## Kriging surrogate tutorial (ND)

First of all let's define the function we are going to build a surrogate for. Notice how its argument is a vector of numbers, one for each coordinate, and its output is a scalar.

function branin(x)
x1=x
x2=x
a=1;
b=5.1/(4*π^2);
c=5/π;
r=6;
s=10;
t=1/(8π);
a*(x2-b*x1+c*x1-r)^2+s*(1-t)*cos(x1)+s
end
branin (generic function with 1 method)

### Sampling

Let's define our bounds, this time we are working in two dimensions. In particular we want our first dimension x to have bounds -5, 10, and 0, 15 for the second dimension. We are taking 50 samples of the space using Sobol Sequences. We then evaluate our function on all of the sampling points.

n_samples = 50
lower_bound = [-5.0, 0.0]
upper_bound = [10.0, 15.0]

xys = sample(n_samples, lower_bound, upper_bound, SobolSample())
zs = branin.(xys);
50-element Vector{Float64}:
75.00113255018026
103.81470883586417
36.907662832568164
19.075240311961004
16.103701095488965
274.94471084491505
66.51180424955137
17.35419862022773
59.81231716867039
150.79644560556488
⋮
185.44436298325337
215.92600883815678
132.5023378487117
20.411667255226202
321.4655470294592
3.03022589857593
8.869947046775998
81.1487816424358
95.73560827088146

### Building a surrogate

Using the sampled points we build the surrogate, the steps are analogous to the 1-dimensional case.

kriging_surrogate = Kriging(xys, zs, lower_bound, upper_bound, p=[2.0, 2.0], theta=[0.03, 0.003])
(::Kriging{Vector{Tuple{Float64, Float64}}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Float64, Vector{Float64}, Float64, Matrix{Float64}}) (generic function with 2 methods)

### Optimizing

With our surrogate we can now search for the minima of the branin function.

Notice how the new sampled points, which were created during the optimization process, are appended to the xys array. This is why its size changes.

size(xys)
(50,)
surrogate_optimize(branin, SRBF(), lower_bound, upper_bound, kriging_surrogate, SobolSample(), maxiters=10)
((-3.1419755513744256, 10.595909001023479), 0.3978894876571015)
size(xys)
(87,)