Consider an attribute X of a data set that takes the values $\{$x$1,$ x$2,,$ x$9\}$

$($sorted in increasing order of magnitude$).$ We apply two methods $($equal

interval width and equal frequency$)$ to discretize the attribute into three

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 will be the effect 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 transformation

function below. Note that X$$ denotes the average value and $\backslash$sigma x denotes

standard deviation of attribute X$.$

The answer to each subquestion is a dictionary with two keys: $$equal width$$

and $$equal freq$.$ The values of each key is a list of two values: a string and

an integer. The string is either $$Change$$ or $$No change$.$ The value of the

integer is chosen among $(1,...,10),$ chosen according to the list below:

$1.$ The transformation leads to an inversion of the original order of values.

$2.$ The distance between xi and xi$+1$ does not change uniformly.

$3.$ The average value X$$ becomes the smallest value post$-$transformation.

$4.$ The relative ordering of points changes

$5.$ The transformation causes negative values to become positive and vice

versa.

$6.$ The transformation results in all values becoming equal.

$3$

$7.$ The distance between xi and xi$+1$ change uniformly.

$8.$ The standard deviation $\backslash$sigma X becomes zero after the transformation.

$9.$ No change in the relative ordering of points

$10.$ The maximum and minimum values of X get swapped after the transformation.

Subquestions to answer:

$($a$)$ X $->$ X $$ x$,($i$.$e$.,$ if the attribute values are centered$).$

$($b$)$ X $->$ X$$X$$

$\backslash$sigma x

$,($i$.$e$.,$ if the attribute values are standardized$).$

$($c$)$ X $->$ exp h

X$$X$$

$\backslash$sigma x

i

$($i$.$e$.,$ if the values are standardized and exponentiated$).$Consider an attribute X of a data set that takes the values $\{$x$1,$ x$2,,$ x$9\}$

$($sorted in increasing order of magnitude$).$ We apply two methods $($equal

interval width and equal frequency$)$ to discretize the attribute into three

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 will be the effect 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 transformation

function below. Note that X$$ denotes the average value and $\backslash$sigma x denotes

standard deviation of attribute X$.$

The answer to each subquestion is a dictionary with two keys: $$equal width$$

and $$equal freq$.$ The values of each key is a list of two values: a string and

an integer. The string is either $$Change$$ or $$No change$.$ The value of the

integer is chosen among $(1,...,10),$ chosen according to the list below:

$1.$ The transformation leads to an inversion of the original order of values.

$2.$ The distance between xi and xi$+1$ does not change uniformly.

$3.$ The average value X$$ becomes the smallest value post$-$transformation.

$4.$ The relative ordering of points changes

$5.$ The transformation causes negative values to become positive and vice

versa.

$6.$ The transformation results in all values becoming equal.

$3$

$7.$ The distance between xi and xi$+1$ change uniformly.

$8.$ The standard deviation $\backslash$sigma X becomes zero after the transformation.

$9.$ No change in the relative ordering of points

$10.$ The maximum and minimum values of X get swapped after the transformation.

Subquestions to answer:

$($a$)$ X $->$ X $$ x$,($i$.$e$.,$ if the attribute values are centered$).$

$($b$)$ X $->$ X$$X$$

$\backslash$sigma x

$,($i$.$e$.,$ if the attribute values are standardized$).$

$($c$)$ X $->$ exp h

X$$X$$

$\backslash$sigma x

i

$($i$.$e$.,$ if the values are standardized and exponentiated$).$