11. For the following vectors, 𝑥 = (0, 1, 0, 1,1) 𝑎𝑛𝑑 𝑦 = (1, 0, 1, 0,1)1. Calculate Simple Matching Coefficient2. Calculate Euclidean3. Calculate Jaccard Coefficient

1. Simple Matching Coefficient (SMC):

SMC is a similarity measure between two binary data objects. It is calculated as the number of matching attributes divided by the total number of attributes.

For vectors x and y, the matching attributes are at positions 3 and 5. So, we have 2 matching attributes. The total number of attributes is 5.

SMC = Number of matching attributes / Total number of attributes = 2 / 5 = 0.4

2. Euclidean Distance:

The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. It is calculated as the square root of the sum of the squares of the differences between the corresponding coordinates of the points.

For vectors x and y, the Euclidean distance is calculated as follows:

Euclidean Distance = sqrt((x1-y1)² + (x2-y2)² + (x3-y3)² + (x4-y4)² + (x5-y5)²) = sqrt((0-1)² + (1-0)² + (0-1)² + (1-0)² + (1-1)²) = sqrt(1 + 1 + 1 + 1 + 0) = sqrt(4) = 2

3. Jaccard Coefficient:

The Jaccard coefficient measures similarity between finite sample sets, and is defined as the size of the intersection divided by the size of the union of the sample sets.

For vectors x and y, the intersection (where both are 1) is at position 5, so the intersection size is 1. The union (where either is 1) is at positions 2, 3, 4 and 5, so the union size is 4.

Jaccard Coefficient = Size of intersection / Size of union = 1 / 4 = 0.25