ChiMerge $[$Ker$92]$ is a supervised, bottom$-$up $($i$.$e$.,$ merge$-$based$)$ data discretization method. It relies on $\backslash$chi $2$ analysis: Adjacent intervals with the least $\backslash$chi $2$ values are merged together until the chosen stopping criterion satisfies.

$($a$)$ Briefly describe how ChiMerge works.

$($b$)$ Take the IRIS data set, obtained from the University of California$$Irvine Machine

Learning Data Repository $($www$.$ics.uci.edu$/$mlearn$/$MLRepository$.$html$),$ as a data set to be discretized. Perform data discretization for each of the four numeric attributes using the ChiMerge method. $($Let the stopping criteria be: max$-$interval $=6).$ You need to write a small program to do this to avoid clumsy numerical computation. Submit your simple analysis and your test results: split$-$points, final intervals, and the documented source program.

Propose an algorithm, in pseudocode or in your favorite programming language, for the following:

$($a$)$ The automatic generation of a concept hierarchy for nominal data based on the number of distinct values of attributes in the given schema.

$($b$)$ The automatic generation of a concept hierarchy for numeric data based on the equal$-$width partitioning rule.

$($c$)$ The automatic generation of a concept hierarchy for numeric data based on the equal$-$frequency partitioning rule.