Estimate Part Mutual Information (PMI) between X and Y given Z

Computes PMI(X; Y | Z), quantifying the direct association between X and Y after removing the influence of Z. Equivalent to conditional mutual information (CMI) as applied to direct network inference.

Usage

pmi(x, y, z, method = c("binning", "knn"), bins = 5, k = 3, ...)

Arguments

x

Numeric vector (first variable).

y

Numeric vector (second variable).

z

Numeric vector, matrix, or data.frame of conditioning variable(s). Each column is treated as a separate conditioning variable.

method

Estimation method: "binning" (default) or "knn".

bins

Number of bins per dimension for binning (default 5). Decrease for higher-dimensional Z to avoid data sparsity.

k

Number of nearest neighbors for the kNN method (default 3).

...

Additional arguments (currently unused).

Value

A numeric PMI value in nats.

Details

PMI is defined as: $$PMI(X;Y|Z) = \sum_{x,y,z} P(x,y,z) \log \frac{P(x,y|z)}{P(x|z)P(y|z)}$$

Two estimators are provided:

• "binning": histogram-based estimator (fast, good for small Z)

• "knn": k-nearest-neighbor estimator (Frenzel-Pompe, robust for higher-dimensional Z)

References

Zhao et al. (2016) PNAS 113(18):5130-5135. Frenzel & Pompe (2007) Phys. Rev. Lett. 99:204101.

Examples

set.seed(42)
x <- rnorm(100)
z <- rnorm(100)
y <- x + z + rnorm(100, 0, 0.5)
# PMI between x and y conditioning on z should be small
pmi(x, y, z)
#>         1 
#> 0.6292994 
# MI (without conditioning) will typically be larger
mi(x, y)
#> [1] 0.7356992