Estimates the information imbalance of two hypothesised linked system measurements for a given scalar (`alpha`).

## Source

Del Tatto, V., Bueti, D. & Laio, A. (2024) Robust inference of causality in high-dimensional dynamical processes from the Information Imbalance of distance ranks. PNAS 121 (19) e2317256121.

## Arguments

- X
Numeric matrix of hypothesised driving variable measurements. If univariate, call `embed_ts(X)` prior to calling `II()`.

- Y
Numeric matrix of hypothesised response variable measurements. If univariate, call `embed_ts(Y)` prior to calling `II()`.

- tau
Numeric. Time lag of information transfer between X and Y.

- alpha
Numeric. Scaling parameter for X. If information imbalance is minimised at an `alpha` > 0, this may be indicative of Granger causality.

- k
Numeric. Number of nearest neighbours when estimating ranks.

- method
String. Distance measure to be used - defaults to `euclidean` but see `?dist` for options.

## Examples

```
#Load the multivariate simulated
#dataset `simTransComms`
data(simTransComms)
#Embed the spp_2 and spp_5 of the third community
embedX <- embed_ts(X = simTransComms$community3[,c("time","spp_2")],
E = 5, tau = 1)
embedY <- embed_ts(X = simTransComms$community3[,c("time","spp_5")],
E = 5, tau = 1)
#Estimate the forward information imbalance
#between spp_2 and spp_5
egII_for <- II(X = embedX[,-1], Y = embedY[,-1],
tau = 1, alpha = 1, k = 5)
#Estimate the reverse information imbalance
#between spp_2 and spp_5
egII_rev <- II(X = embedY[,-1], Y = embedX[,-1],
tau = 1, alpha = 1, k = 5)
```