calc_cramers_dist_unequal_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 unequally-spaced quantiles.

Description

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

Usage

calc_cramers_dist_unequal_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 "left_sided_riemann" and "trapezoid_riemann". See Details for more information.

Details

This function accommodates cases when the lengths of q_F and q_G are not equal and when tau_F and tau_G are not equal. A Riemann sum is used to approximate a pairwise Cramér distance. The approximation formula for "left_sided_riemann" is

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

and the approximation formula for "trapezoid_riemann" is

\text{CD}(F,G) \approx \left\{\frac{1}{(K+1)^2}\sum^{2K-1}_{i=1}\frac{(\tau^F_j-\tau^G_j)^2+(\tau^F_{j+1}-\tau^G_{j+1})^2}{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 \tau^F_j and \tau^G_j are defined as the probability level of a quantile in q_F when q_i comes from F and the probability level of a quantile in q_G when q_i comes from G, respectively.

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(4, 20, 2)
calc_cramers_dist_unequal_space(f_vector, tau_F_vector, g_vector, tau_G_vector, "left_sided_riemann")

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