Calculate the approximated Cramer’s distance between a pair of distributions F and G that are represented by a collection of quantiles using a specific approximation rule.

calc_cramers_dist_one_model_pair(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", "approximation2", "left_sided_riemann", and "trapezoid_riemann". See Details for more information.

Value

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

Details

This function calculate the aprroximated Cramer’s distance between a pair of distributions F and G that are represented by a collection of quantiles using a specified approximation rule. Specifying "approximation1" or "approximation2" as approx_rule requires the two vectors of quantiles to be of equal length. These approximation methods are formulated based on 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.

Specifying "left_sided_riemann" or "trapezoid_riemann" as approx_rule 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 approach 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.

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_one_model_pair(f_vector, tau_F_vector, g_vector, tau_G_vector, "left_sided_riemann")
#> [1] 2.51