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Tutorial: SMeasureTest

Create an orbit of the built-in unidirectionally coupled henon2 map system, and a pair of random time series.

```@example SMeasureTest_henon2_rand using CausalityTools, DynamicalSystemsBase, Plots npts, Ttr = 5000, 500 x, y = columns(trajectory(henon2(c_xy = 1.0), npts, Ttr = Ttr)); xr, yr = rand(npts), rand(npts); nothing # hide

Initialise test, specifying embedding dimension, emebdding lag and number of  nearest neighbors.


```julia
ks = 2:10
test = SMeasureTest(m = 4, τ = 1, K = ks)

SMeasureTest{9,Distances.SqEuclidean,Euclidean}(m = 4, τ = 1, K = 2:10, metric = Distances.SqEuclidean(0.0), tree_metric = Euclidean(0.0))

Compute S-measure statistic separately in both directions, both for the random time series, and for the Henon maps. The test will return a vector of length ks.

```@example SMeasureTest_henon2_rand Ss_r_xy = causality(xr, yr, test) Ss_r_yx = causality(yr, xr, test) Ss_henon_xy = causality(x, y, test) Ss_henon_yx = causality(y, x, test); nothing # hide

Plot the results.


```@example SMeasureTest_henon2_rand
plot(xlabel = "# nearest neighbors (k)", ylabel = "S", ylims = (-0.05, 1.05))
plot!(ks, Ss_r_xy,  label = "random uncoupled system (x -> y)", marker = stroke(2), c = :black)
plot!(ks, Ss_r_yx,  label = "random uncoupled system (y -> x)", marker = stroke(2), c = :red)
plot!(ks, Ss_henon_xy, marker = stroke(2), label = "henon unidir (x -> y)")
plot!(ks, Ss_henon_yx, marker = stroke(2), label = "henon unidir (y -> x)")
savefig("figs/SMeasure_random_plus_henon.svg"); nothing # hide

For uncoupled time series, we expect the value of $S$ to be close to zero. For strongly coupled time series, the value of $S$ should be nonzero and approaching 1. This is exactly what we get: for time random time series, the value of $S$ is close to zero and for the Henon maps, it's clearly non-zero.