Estimates the information imbalance of two hypothesised linked system measurements using distance ranks.

## 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

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

- target
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.

- alphas
Numeric vector. Range of X scaling parameters bewtween `0` & `1` inclusive. 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)
alphas <- seq(from = 0, to = 1, by = 0.1)
# \donttest{
#if parallelisation desired,
#this can be achieved using the
#below code
cl <- parallel::makeCluster(2)
# }
# \donttest{
doParallel::registerDoParallel(cl)
# }
#Estimate the forward information imbalance
#between spp_2 and spp_5
egII_for <- tuneII(columns = embedX[,-1], target = embedY[,-1],
tau = 1, alphas = alphas, k = 5)
#Estimate the reverse information imbalance
#between spp_2 and spp_5
egII_rev <- tuneII(columns = embedX[,-1], target = embedY[,-1],
tau = 1, alphas = alphas, k = 5)
# \donttest{
parallel::stopCluster(cl)
# }
```