$(6$pts$)$ Suppose that you have a market basket dataset consisting of $1($purchase$)$ and $0($non$-$

purchase$)$ in which rows are users and columns are items.

$(1)(3$pts$)$ Is it possible to apply the item$-$based collaborative filtering to this dataset for

recommendation? If yes, write the formula to predict $P_{aj}.$

$(2)(3$pts$)$ Is it possible to apply the association rule mining to make recommendations? If

yes, describe how you would like to build a recommender system. $($If you want, you can

use Korean.$)$

Which of the following is not true?

$(1)$ The k$-$means clustering is sensitive to noise objects.

$(2)$ The k$-$medoids clustering is less sensitive to noise objects than the k $-$

means clustering.

$(3)$ The k$-$means algorithm finds new centroids at every iteration.

$(4)$ The PAM algorithm computes a new distance matrix at every iteration. Consider the similarity matrix of four data points $($A$,$ B$,$ C$,$ D$)$ shown below.

Find the optimal clustering result that maximizes the following quantity,

$Z={\displaystyle \underset{k=1}{\overset{3}{}}}{\displaystyle \underset{(ij)in{C}_k}{\overset{?}{}}}s(i,j),$

where $s(i,j)$ is the similarity between object i and $j,$ and $C_k$ indicates the

$k$ th cluster. The object i and $j$ belong to $C_k.$ Notice that the number of

clusters is $3.$ Report the clustering result with its $Z$ value. In order to perform PCA, we obtained the eigenvalues and the eigenvectors

from the covariance matrix of a two$-$dimensional dataset.

$1$ st eigenvector $=(\frac{1}{\sqrt[\phantom{2}]{2}},-\frac{1}{\sqrt[\phantom{2}]{2}}),2$ nd eigenvector $=(\frac{1}{\sqrt[\phantom{2}]{2}},\frac{1}{\sqrt[\phantom{2}]{2}}),1$ st eigenvalue $=1,2$ nd

eigenvalue $=1$

Which of the following would be the contour plot of the original dataset?

$(2)$

$(3)$

$(4)$

PCA was performed based on a dataset. Below shows the first two eigenvector and eigenvalues. Which of the following is not true?

$\backslash$table$[[,$eigenvectors$],[$PC$1,$PC$2],[$x$1,0\mathrm{.}6,-0\mathrm{.}8],[$x$2,0\mathrm{.}8,0\mathrm{.}6],[$x$3,0,0],[$x$4,0,0],[$x$5,0,0],[$x$6,0,0],[$x$7,0,0],[$x$8,0,0],[$x$9,0,0],[$x$10,0,0],[$eigenvalues$,2\mathrm{.}1,0\mathrm{.}4]]$

$(1)$ The original dataset is $10-$dimensional.

$(2)$ The total variance of the original dataset is greater than $2\mathrm{.}5.$

$(3)$ The variance of $x3$ is zero.

$(4)$ The correlation between $x1$ and $x2$ is not zero.