There exist several methods to measure the impurity of a set of labelled observations, e.g. D = {(x_n, y_n)}^N_n=1, by looking at the object labels, y_n. Two examples of impurity measures are the Gini index and the entropy.

(a) Consider the case where y_n {0, 1} and x_n {i)¹⁰_i=1, for all n = 1,...,N and the task of predicting the label, y, given the object attribute x. Is this a regression problem, a binary classification problem, or a multi-class classification problem? Please justify the answer.

(b) Describe the process of growing a classification tree using the Gini index.

(c) Consider the following set of data points D = {(1,0), (3, 1), (4, 1), (4, 0), (2, 0), (1, 0), (1, 0), (7, 1), (10, 1), (2, 1)}. where, for each observation, (x_n, y_n) D(n = 1,..., 10), x_n is the object attribute and y_n the object label. What is the Gini index, G(D), and the entropy, S(D), of D?

(d) Which of the following thresholds =2, =3, = 6, is preferable for splitting D from a classification perspective? Choose the threshold that minimises

G_split(, D) = |R₁()| G(R₁()) + |R₂()| G(R₂()),

where R₁() = {(x, y) D | x=}, R₂() = {(x, y) D | x > }, and G(A) is the Gini index of the set A.