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Tutorial

To begin, install this package with 

Pkg.add("FastGaussianProcesses")

and then import via

using FastGaussianProcesses

To construct specific sampling sequences we also need to be 

using QMCGenerators

Kernel 1s Lines

Lattice Kernels

PLOTDIR = joinpath(@__DIR__,"src/assets")
plot_gp_kernel_latticeseqb2_1s_lines(figpath=joinpath(PLOTDIR,"kernels.lines.latticeseqb2.svg"))

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Digital Kernels

plot_gp_kernel_digitalseqb2g_1s_lines(figpath=joinpath(PLOTDIR,"kernels.lines.digitalseqb2g.svg"))

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RBF Kernels

plot_gp_kernel_rbf_1s_lines(figpath=joinpath(PLOTDIR,"kernels.lines.rbf.svg"))

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Kernel 1s Contours and Surfaces

Lattice Kernels

PLOTDIR = joinpath(@__DIR__,"src/assets")
plot_gp_kernel_latticeseqb2_1s_contsurfs(figpath=joinpath(PLOTDIR,"kernels.contsurf.latticeseqb2.svg"))

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Digital Kernels

plot_gp_kernel_digitalseqb2g_1s_contsurfs(figpath=joinpath(PLOTDIR,"kernels.contsurf.digitalseqb2g.svg"))

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RBF Kernels

plot_gp_kernel_rbf_1s_contsurfs(figpath=joinpath(PLOTDIR,"kernels.contsurf.rbf.svg"))

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GPs 1s

function f1s(x::Vector{Float64})
    [   x[1]*sin(10*x[1]), # f^{(0)}
        sin(10*x[1]) + 10*x[1]*cos(10*x[1]) # f^{(1)}
    ]
end

GP Lattice Kernel

gp_1s_latticeseqb2 = FastGaussianProcess(f1s,RandomShift(LatticeSeqB2(1),1,7),2^2;β=[i for i=0:1,j=1:1],verbose=0)
plot_gp_optimization(gp_1s_latticeseqb2,figpath=joinpath(PLOTDIR,"optim.1s.latticeseqb2.svg"))
plot_gp_1s(gp_1s_latticeseqb2;f=f1s,β=[0,1],figpath=joinpath(PLOTDIR,"gp.1s.latticeseqb2.svg"))

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GP Digital Kernel

gp_1s_digitalseqb2g = FastGaussianProcess(f1s,RandomDigitalShift(DigitalSeqB2G(1),1,7),2^2;β=[i for i=0:1,j=1:1],verbose=0)
plot_gp_optimization(gp_1s_digitalseqb2g,figpath=joinpath(PLOTDIR,"optim.1s.digitalseqb2g.svg"))
plot_gp_1s(gp_1s_digitalseqb2g;f=f1s,β=[0,1],figpath=joinpath(PLOTDIR,"gp.1s.digitalseqb2g.svg"))

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GP RBF Kernel

gp_1s_rbf = GaussianProcessRBF(f1s,IIDU01Seq(1,7),2^2;β=[i for i=0:1,j=1:1],verbose=0)
plot_gp_optimization(gp_1s_rbf,figpath=joinpath(PLOTDIR,"optim.1s.rbf.svg"))
plot_gp_1s(gp_1s_rbf;f=f1s,β=[0,1],figpath=joinpath(PLOTDIR,"gp.1s.rbf.svg"))

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GPs 2s

function f2s(x::Vector{Float64})
    [   x[1]*sin(10*x[1]) - x[2]*cos(10*x[2]), # f^{(0,0)},
        sin(10*x[1]) + 10*x[1]*cos(10*x[1]), # f^{(1,0)}
        10*x[2]*sin(10*x[2]) - cos(10*x[2]) # f^{(0,1)}
    ]
end

GP Lattice Kernel

gp_2s_latticeseqb2 = FastGaussianProcess(f2s,RandomShift(LatticeSeqB2(2),1,7),2^8;β=[0 0; 1 0; 0 1],optim_steps=40,verbose=0)
plot_gp_optimization(gp_2s_latticeseqb2,figpath=joinpath(PLOTDIR,"optim.2s.latticeseqb2.svg"))
plot_gp_2s(gp_2s_latticeseqb2;f=f2s,β=[0 0; 1 0; 0 1],figpath=joinpath(PLOTDIR,"gp.2s.latticeseqb2.svg"))

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GP Digital Kernel

gp_2s_digitalseqb2g = FastGaussianProcess(f2s,RandomDigitalShift(DigitalSeqB2G(2),1,7),2^8;β=[0 0; 1 0; 0 1],optim_steps=40,verbose=0)
plot_gp_optimization(gp_2s_digitalseqb2g,figpath=joinpath(PLOTDIR,"optim.2s.digitalseqb2g.svg"))
plot_gp_2s(gp_2s_digitalseqb2g;f=f2s,β=[0 0; 1 0; 0 1],figpath=joinpath(PLOTDIR,"gp.2s.digitalseqb2g.svg"))

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GP RBF Kernel

gp_2s_rbf = GaussianProcessRBF(f2s,IIDU01Seq(2,7),2^8;β=[0 0; 1 0; 0 1],optim_steps=40,verbose=0)
plot_gp_optimization(gp_2s_rbf,figpath=joinpath(PLOTDIR,"optim.2s.rbf.svg"))
plot_gp_2s(gp_2s_rbf;f=f2s,β=[0 0; 1 0; 0 1],figpath=joinpath(PLOTDIR,"gp.2s.rbf.svg"))

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In the following we must be 

using CairoMakie
m = 2; n = 2^m
x = FirstLinear(RandomDigitalShift(DigitalSeqB2G(1),1,11),m)[:,1]
kmat = Matrix{Float64}(undef,2*n,2*n)
for i1=1:n,i2=1:n,β1=0:1,β2=0:1 kmat[β1*n+i1,β2*n+i2] = ((β1+β2)==0)+kernel_digshiftinvar_s1(x[i1],x[i2],β1,β2,4) end 
fig = Figure(resolution=(400,400),backgroundcolor=:transparent)
ax = Axis(fig[1,1],backgroundcolor=:transparent)
heatmap!(ax,1:2*n,2*n:-1:1,kmat,colormap=:julia_colorscheme)#:nipy_spectral) # https://docs.juliahub.com/MakieGallery/Ql23q/0.2.17/generated/colors.html
hidespines!(ax); hidedecorations!(ax); hidexdecorations!(ax,grid = false); hideydecorations!(ax, ticks = false)
save(joinpath(PLOTDIR,"logo.svg"),fig)

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