calc_cramers_dist_equal_space: Approximate the Cramer’s distance between a pair of...

In reichlab/covidHubUtils: Utility functions for the COVID-19 forecast hub

Approximate the Cramer’s distance between a pair of distributions F and G that are represented by a collection of equally-spaced quantiles.

Description

Approximate the Cramer’s distance between a pair of distributions F and G that are represented by a collection of equally-spaced quantiles.

Usage

calc_cramers_dist_equal_space(q_F, tau_F, q_G, tau_G, approx_rule)

Arguments

q_F	vector containing the quantiles of F
tau_F	vector containing the probability levels corresponding to the quantiles of F.
q_G	vector containing the quantiles of G
tau_G	vector containing the probability levels corresponding to the quantiles of G.
approx_rule	string specifying which formula to use for approximation. Valid rules are "approximation1" and "approximation2". See Details for more information.

Details

This function requires the two vectors of quantiles to be of equal length. The approximation methods are formulated based on two collections of quantiles corresponding to equally-spaced probability levels. The approximation formula for "approximation1" is

\text{CD}(F,G) \approx \left\{\frac{1}{K(K+1)}\sum^{2K-1}_{i=1}b_i(b_i+1)(q_{i+1}-q_i)\right\}

and the approximation formula for "approximation2" is

\text{CD}(F,G) \approx \left\{\frac{1}{(K+1)^2}\sum^{2K-1}_{i=1}b_i^2(q_{i+1}-q_i)\right\}

where q_i is an element in a vector of an ordered pooled quantiles of q_F and q_G and b_i is an element of a vector of the absolute values of cumulative sums of \mathbf{a}, whose element is 1 if q_i is a quantile of F or -1 if q_i is a quantile of G. The "approximation1" formula reduces to the WIS if G is a point mass, while the "approximation2" formula is a direct approximation of the integral via a step function.

Value

a single value of approximated pairwise Cramér distance between q_F and q_G

Examples

f_vector <- 1:9
tau_F_vector <- tau_G_vector <- seq(0.1, 0.9, 0.1)
g_vector <- seq(2, 18, 2)
calc_cramers_dist_equal_space(f_vector, tau_F_vector, g_vector, tau_G_vector, "approximation1")

reichlab/covidHubUtils documentation built on Feb. 6, 2024, 1:42 p.m.