Compute Quantile-Based Similarity Between Weighted Numeric Distributions

`numerical_similarity()` evaluates the distributional similarity between two weighted numeric vectors by comparing their weighted quantile functions across a fine grid.

Usage

numerical_similarity(x, y, wx, wy)

Arguments

x

Numeric vector. Observations from the first distribution.

y

Numeric vector. Observations from the second distribution.

wx

Numeric vector. Sample weights for x.

wy

Numeric vector. Sample weights for y.

Value

A single numeric similarity score in \([0, 1]\), where 1 represents identical weighted quantile distributions and 0 represents complete dissimilarity.

Details

The comparison uses a grid of 1,000 quantiles evaluated via fquantile. Quantile differences are weighted using a Gaussian kernel centered at the median (\(p = 0.5\), \(\sigma = 0.15\)) to emphasize central distributional overlap while deemphasizing extreme tail behavior.

The score is calculated as a normalized relative distance subtracted from 1: $$\text{Similarity} = 1 - \frac{\sum w_i |q_{x,i} - q_{y,i}|}{\sum w_i \frac{|q_{x,i}| + |q_{y,i}|}{2}}$$

See also

scaleNumeric, categorical_similarity