Cumulative Jensen-Shannon divergence — cjs

Computes the cumulative Jensen-Shannon distance between two samples.

Usage

cjs_dist(
  x,
  y,
  x_weights = NULL,
  y_weights = NULL,
  metric = TRUE,
  unsigned = TRUE,
  ...
)

Arguments

x: numeric vector of samples from first distribution
y: numeric vector of samples from second distribution
x_weights: numeric vector of weights of first distribution
y_weights: numeric vector of weights of second distribution
metric: Logical; if TRUE, return square-root of CJS
unsigned: Logical; if TRUE then return max of CJS(P(x) || Q(x)) and CJS(P(-x) || Q(-x)). This ensures invariance to transformations such as PCA.
...: unused

Value

distance value based on CJS computation.

Details

The Cumulative Jensen-Shannon distance is a symmetric metric based on the cumulative Jensen-Shannon divergence. The divergence CJS(P || Q) between two cumulative distribution functions P and Q is defined as:

$$CJS(P || Q) = \sum P(x) \log \frac{P(x)}{0.5 (P(x) + Q(x))} + \frac{1}{2 \ln 2} \sum (Q(x) - P(x))$$

The symmetric metric is defined as:

$$CJS_{dist}(P || Q) = \sqrt{CJS(P || Q) + CJS(Q || P)}$$

This has an upper bound of $\sqrt{ \sum (P(x) + Q(x))}$

References

Nguyen H-V., Vreeken J. (2015). Non-parametric Jensen-Shannon Divergence. In: Appice A., Rodrigues P., Santos Costa V., Gama J., Jorge A., Soares C. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2015. Lecture Notes in Computer Science, vol 9285. Springer, Cham. doi:10.1007/978-3-319-23525-7_11

Examples

x <- rnorm(100)
y <- rnorm(100, 2, 2)
cjs_dist(x, y, x_weights = NULL, y_weights = NULL)
#>       cjs 
#> 0.4265529