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Strictly increasing, interpolated resampling

# CausalityToolsBase.causality — Method.

causality(source::AbstractUncertainIndexValueDataset, 
    target::AbstractUncertainIndexValueDataset, 
    test::CausalityTest, constraint::SequentialSamplingConstraint, grid::RegularGrid)

Test for a causal influence from source to target using the provided causality test on single draws of source and target that have been generated according to the provided sequential sampling constraint. After interpolating both source and target to the provided regular grid, the causality test is performed on the interpolated data.

Example

First, generate some example data.

N = 300
a₁, a₂, b₁, b₂, ξ₁, ξ₂, C₁₂ = 0.7, 0.1, 0.75, 0.2, 0.3, 0.3, 0.5

D = rand(N, 2) 
for t in 5:N
    D[t,1] = a₁*D[t-1,1] - a₂*D[t-3,1] +                ξ₁*rand(Normal(0, 1))
    D[t,2] = b₁*D[t-1,2] - b₂*D[t-2,2] + C₁₂*D[t-1,1] + ξ₂*rand(Normal(0, 1))
end

Gather time series and add some uncertainties to them:

ts = collect(1:N)
x, y = D[:, 1], D[:, 2]
t = UncertainValue.(Normal.(ts .+ 0.2 .* rand(N), rand(N)))
uvalx = UncertainValue.(Normal.(x, rand()))
uvaly = UncertainValue.(Normal.(y, rand()))

X = UncertainIndexValueDataset(t, uvalx)
Y = UncertainIndexValueDataset(t, uvaly)

Now, resample both the indices and values of X and Y in a strictly increasing manner according to the indices, interpolate both to the same regular grid, and perform a causality test.

n_bins = ceil(Int, N^(1/4))
te_test = VisitationFrequencyTest(binning = RectangularBinning(n_bins), ηs = -5:5)
pa_test = PredictiveAsymmetryTest(predictive_test = te_test)

# Sample a strictly increasing realisation of the indices, interpolate,
# then perform the causality test on the interpolated data.
causality(X, Y, pa_test, StrictlyIncreasing(), RegularGrid(1:1:N))

source