GATE DA 2024 | Question 53

MSQ · 2 marks

Consider the following figures representing datasets consisting of two-dimensional features with two classes denoted by circles and squares.

Four scatter plots labelled (i) to (iv), each with filled circles and filled squares on axes marked 1 to 3. In (i) circles are at (1, 3), (2, 3) and (1, 2) and squares at about (2, 1), (3, 1) and (2, 0.3); in (ii) a circle is at (2, 2) and squares at (1, 2), (3, 2), (2, 3) and (2, 1); in (iii) squares are at (1, 3) and (3, 1) and circles at (1, 1) and (3, 3); in (iv) squares are at about (1, 2.2), (1, 1.2) and (2, 2.6) and circles at about (2, 0.5), (3, 1.5) and (3, 0.6).

Which of the following is/are TRUE?

  1. Option A: (i) is linearly separable. (correct)
  2. Option B: (ii) is linearly separable.
  3. Option C: (iii) is linearly separable.
  4. Option D: (iv) is linearly separable. (correct)

Answer

(A), (D)

Explanation

A dataset is linearly separable when a single straight line puts every circle on one side and every square on the other.

  • A holds: in (i) the circles have \(y \ge 2\) and the squares lie around \(y = 1\) or lower, so the line \(y = 1.6\) splits them.
  • B fails: in (ii) the circle at (2, 2) is the midpoint of the squares at (1, 2) and (3, 2). A half-plane holding both squares also holds their midpoint, so the circle cannot be cut off.
  • C fails: (iii) is the XOR pattern. The circles (1, 1), (3, 3) and the squares (1, 3), (3, 1) have the same midpoint (2, 2), which would have to lie on both sides.
  • D holds: in (iv) the squares sit to the upper left and the circles to the lower right, so a slanted line such as \(y = x - 0.3\) works.

Answer: A, D