Computes the cumulative Jensen-Shannon distance between two samples.

cjs_dist(
x,
y,
x_weights = rep(1/length(x), length(x)),
y_weights = rep(1/length(y), length(y)),
...
)

## Arguments

x numeric vector of samples from first distribution numeric vector of samples from second distribution numeric vector of weights of first distribution numeric vector of weights of second distribution 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