# Pairwise asymmetric inference

CausalityTools.paiFunction
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}
This syntax is deprecated

This syntax is deprecated. It will continue to work for CausalityTools v1.X, but will be removed in CausalityTools v2. See here for updated syntax.

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].
source
• McCracken2014McCracken, James M., and Robert S. Weigel. "Convergent cross-mapping and pairwise asymmetric inference." Physical Review E 90.6 (2014): 062903.
• Sugihara2012Sugihara, George, et al. "Detecting causality in complex ecosystems." Science (2012): 1227079.http://science.sciencemag.org/content/early/2012/09/19/science.1227079
• Luo2015"Questionable causality: Cosmic rays to temperature." Proceedings of the National Academy of Sciences Aug 2015, 112 (34) E4638-E4639; DOI: 10.1073/pnas.1510571112 Ming Luo, Holger Kantz, Ngar-Cheung Lau, Wenwen Huang, Yu Zhou