Hoeffding Dependence Coefficient in Integer Form

This is Wassley Hoeffding's 1948 equation for detecting -frequently nonlinear- relationships in data or variables. Wassley Hoeffding was there at the founding of modern non-parametric statistics and Hoeffding's work has inspired decades of additional work. (Evygene Slutsky may have anticipated some of Hoeffding's work on dependence, but cold war, stoic man and little published in the west.)

Hoeffding D = 30 [ (n-2)(n-3)Sum[(Qi-1)(Qi-2)] + Sum[(Ri-1)(Ri-2)(Si-1)(Si-2)] - 2(n-2)Sum[(Ri-2)(Si-2)(Qi-1)] ] / [ n(n-1)(n-2)(n-3)(n-4)]

"H_integer D" = 256 * [ (n-2)(n-3)Sum[(Qi-1)(Qi-2)] + Sum[(Ri-1)(Ri-2)(Si-1)(Si-2)] - 2(n-2)Sum[(Ri-2)(Si-2)(Qi-1)] ]

In this variation of the Dependence calculation, the association is computed with integers by multiplication of the original statistic with n(n-1)(n-2)(n-3)(n-4)(256/30) (n=number of pairs). Higher values have stronger relationships- good for detection of useful models in machine learning or for genetic algorithm fitness assessment especially in non-linear situations.

Min and Max values of the integer form can be computed for a given length of paired comparisions.

Coded in Rust for generic partially ordinal types (compare {"g","e","n","e","r","i","c","s"} and {3.0,1.0,4.0,1.0,5.0,4.0,3.5,6.1}). Yep- directly compare different data types - if they sort they likely can be run through Hoeffding_Integer.

Why?

Why program in RUST? Rust is fast and has taught me better ways to code!

Why integer? A ?fantasy? that integer hoeffding is a step toward GPU computation of Hoeffding's Dependence Coefficient D. And the annoyance that as n gets large (theoretically leading to higher resolution statistics) the floating point math of the denominator Pochhammer factorial makes small progress invisible (if you have n=1587 pairs, a small scramble of progress is divided by the Pochhammer 10003350094863840 thereby likely vanishing in limited floating point decimals.)

What's Hoeffding's Dependence Coefficient good for? Assigning fitness to nonlinear models for genetic algorithms and machine learning.

The math

Please read the hundred lines of mathematics in the comments of main.rs or lib.rs! I've tried to make the statistic's computation possible to verify, follow and understand.

How to use

first add hoeffding_integer to your cargo file from crates.io (and I haven't published in crates.io as of Oct 2021... but soon)

let textdata: Vec<&str> = vec!["a","a","a","b","b","b","c","c","c"]; let numdata: Vec = vec![ 1 , 2 , 3 , 4 , 5 , 5 , 6 , 7 , 8 ];

let hoeffding_dependence_statistic_as_integer:i128 = hoeffding_integer[textdata, numdata]; let dependence_min = hoeffding_integer_minimum( textdata.len() ); let dependence_max = hoeffding_integer_maximum( textdata.len() );

println!("Compare text and numbers: {:?} vs. {:?}",&textdata, &numdata); println!("Hoeffding's dependence coefficient D as integer: {}", hoeffding_dependence_statistic_as_integer ); println!("min and max possible for statistic: {} <--> {}",dependence_min, dependence_max);