This section aims to show for the cubic spline fitting with any number of basis functions $J$. If one applied the monotone decomposition, it can achieve a better performance.

using MonotoneDecomposition
using Plots
using Random

Firstly, generate random data from Gaussian process

seed = 16
Random.seed!(seed)
x, y, x0, y0 = gen_data(100, 0.5, "SE_1");

[ Info: σ = 0.5, resulting SNR = 0.8134045943613267

the true curve and noisy observations are shown in the following figure

plot(x0, y0, label = "truth")
scatter!(x, y, label = "sample")

compare the performance of cubic spline fitting (CS) and the corresponding fitting with monotone decomposition (MDCS)

Js = 4:50
nJ = length(Js)
errs = zeros(nJ, 2)
Random.seed!(seed)
for i in 1:nJ
    J = Js[i]
    yhat_cs, y0hat_cs = cubic_spline(J)(x, y, x0)
    errs[i, 1] = sum((y0 - y0hat_cs).^2) / length(y0)
    D, _ = cv_mono_decomp_cs(x, y, Js = J:J, ss = 10.0 .^ (-6:0.05:0))
    y0hat_md = predict(D, x0)
    errs[i, 2] = sum((y0 - y0hat_md).^2) / length(y0)
end

the CV optimized J should be

D, _ = cv_mono_decomp_cs(x, y, Js = Js, ss = 10.0 .^ (-6:0.05:0))
D.workspace.J

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save results

serialize("../res/demo/anyJ-seed$seed.sil", [Js, errs, D.workspace.J])

plot the mean squared prediction error (MSPE) along the number of basis function $J$:

plot(Js, errs[:, 1], xlab = "J", ylab = "MSPE", label = "CS", markershape = :circle, legend = :topleft)
plot!(Js, errs[:, 2], label = "MDCS", markershape = :star5)
Plots.vline!([D.workspace.J], ls = :dash, label = "CVCS")

Locally, I will save the figure and present in the paper.

savefig("../res/demo/anylam-seed$seed.pdf")