Question $5$

Consider an attribute $x$ of a data set that takes the values $\{x_1,x_2,x_3,$dots,$x_9\}($sorted in

increasing order$).$ We apply two methods $($equal width and equal frequency$)$ to discretize the

attribute into $3$ bins. The bins obtained are shown below:

Equal Width: $\{x_1,x_2,x_3\},\{x_4,x_5,x_6,x_7,x_8\},\{x_9\}$

Equal Frequency: $\{x_1,x_2,x_3\},\{x_4,x_5,x_6\},\{x_7,x_8{x}_9\}$

Explain what the effect will be of applying the following transformations on each

discretization method, i$.$e$.,$ whether the elements assigned to each bin can change if you

discretize the attribute after applying the transformations below. Note that $\stackrel{}{x}$ denotes the

average value and ${}_x$ denotes the standard deviation of attribute $x.$

$($a$)xx-\stackrel{}{x}($i$.$e$.,$ if the attribute values are centered$).$

$($b$)x\frac{x-(\stackrel{}{x})}{{}_x}($i$.$e$.,$ if the attribute values are standardized$).$

$($c$)x$exp$(\frac{x-(\stackrel{}{x})}{{}_x})($i$.$e$.,$ if the values are standardized and exponentiated$).$