Pairwise asymmetric inference

pai(x, y, d, τ; w = 0, correspondence_measure = Statistics.cor) → Float64
pai(x, y, d, τ, bootstrap_method::Symbol; w = 0, correspondence_measure = Statistics.cor,
    method = :segment, L = ceil(Int, (length(x)-d*τ)*0.2), nreps = 100) → Vector{Float64}

Compute the pairwise asymmetric inference (PAI; McCracken, 2014)^{[McCracken2014]} between x and y. Returns the correspondence between original and cross mapped values (the default is ρ = correspondence_measure(y(t), ỹ(t) | M_xy)).

PAI is a modification to Sugihara et al. (2012)'s CCM algorithm^{[Sugihara2012]}, where instead of using completely out-of-sample prediction when trying to predict $y(t)$, values about both variables are included in the embedding used to make predictions. Specifically, PAI computes the correspondence between the values $y(t)$ and the cross-map estimated values $ỹ(t) | M_xy$, where the $\tilde{y}(t)$ are the values estimated using the embedding $M_{xy} = \{ ( x_t, x_{t-\tau}, x_{t-2\tau}, \ldots, x_{t-(d - 1)\tau} ) \}$. *Note: a d+1-dimensional embedding is used, rather than the d-dimensional embedding used for CCM. Like for the CCM algorithm, the Theiler window r indicates how many temporal neighbors of the predictee is to be excluded during the nearest neighbors search (the default r = 0 excludes only the predictee itself, while r = 2 excludes the point itself plus its two nearest neighbors in time).

If bootstrap_method is specified, then nreps different bootstrapped estimates of correspondence_measure(y(t), ỹ(t) | M_x) are returned. The following bootstrap methods are available:

• bootstrap_method = :random selects training sets of length L consisting of randomly selected points from the embedding $M_x$ (time ordering does not matter). This is method 3 from Luo et al. (2015)^[Luo2015], which critiqued the original Sugihara et al. methodology.
• bootstrap_method = :segment selects training sets consisting of time-contiguous segments (each of lenght L) of embedding vectors in $M_x$ (time ordering matters). This is method 2 from Luo et al. (2015)^[Luo2015].