In this part, PCA has been applied to a matrix $\backslash ($ X $\backslash )$ to get a set of principal component features. The features are divided into two sets:

$-\backslash$textbf$\{\backslash ($ Z$\_1\backslash )\}$: Contains the top principal components $($those explaining the most variance$).$

$-\backslash$textbf$\{\backslash ($ Z$\_2\backslash )\}$: Contains the remaining components $($less important in terms of variance$).$

You$$re asked to identify the most common option among the following:

$1.\backslash$textbf$\{$Option $($a$)\}$: A point with large values in $\backslash ($ Z$\_1\backslash )$ and small values in $\backslash ($ Z$\_2\backslash ).$

$2.\backslash$textbf$\{$Option $($b$)\}$: A point with small values in $\backslash ($ Z$\_1\backslash )$ and small values in $\backslash ($ Z$\_2\backslash ).$

$3.\backslash$textbf$\{$Option $($c$)\}$: A point with large values in $\backslash ($ Z$\_1\backslash )$ and large values in $\backslash ($ Z$\_2\backslash ).$

$4.\backslash$textbf$\{$Option $($d$)\}$: A point with small values in $\backslash ($ Z$\_1\backslash )$ and large values in $\backslash ($ Z$\_2\backslash ).$

$\backslash$textbf$\{$Explanation$\}$:

$-\backslash ($ Z$\_1\backslash )$ contains the principal components explaining the most variance, so it$$s likely to have larger values since these components capture the most variation in the data.

$-\backslash ($ Z$\_2\backslash )$ contains components that explain less variance, so values are typically smaller here.

$\backslash$textbf$\{$Answer$\}$: $\backslash$textbf$\{$Option $($a$)\},$ a point with large values in $\backslash ($ Z$\_1\backslash )$ and small values in $\backslash ($ Z$\_2\backslash ),$ is more common. This is because the first few principal components capture most of the data variance, so they are likely to have larger values compared to the components in $\backslash ($ Z$\_2\backslash ).$