Suppose we want to assign points to whichever of the points (0,0) or (100,40) is nearer. Depending on whether we use the L1 or L2 norm, a point (x,y) could be clustered with a different one of these two points. For this problem, you should work out the conditions under which a point will be assigned to (0,0) when the L1 norm is used, but assigned to (100,40) when the L2 norm is used. Identify one of those points from the list below.
(54,8)
(56,13)
(56,15)
(50,18)
54,8
To determine the conditions under which a point will be assigned to (0,0) when the L1 norm is used, but assigned to (100,40) when the L2 norm is used, we need to compare the distances from the point to each of the two points using both norms.
The L1 norm, also known as the Manhattan distance or the taxicab distance, is calculated as the sum of the absolute differences between the coordinates:
L1 norm distance = |x - x1| + |y - y1|.
The L2 norm, also known as the Euclidean distance, is calculated as the square root of the sum of the squared differences between the coordinates:
L2 norm distance = sqrt((x - x1)^2 + (y - y1)^2).
Let's calculate the distances from each of the given points to (0,0) and (100,40) using both norms and see which point satisfies the condition mentioned.
For point (54,8):
L1 norm distance to (0,0) = |54 - 0| + |8 - 0| = 54 + 8 = 62.
L2 norm distance to (0,0) = sqrt((54 - 0)^2 + (8 - 0)^2) = sqrt(2916 + 64) = sqrt(2980) ≈ 54.63.
L1 norm distance to (100,40) = |54 - 100| + |8 - 40| = 46 + 32 = 78.
L2 norm distance to (100,40) = sqrt((54 - 100)^2 + (8 - 40)^2) = sqrt(1936 + 1024) = sqrt(2960) ≈ 54.39.
For point (56,13):
L1 norm distance to (0,0) = |56 - 0| + |13 - 0| = 56 + 13 = 69.
L2 norm distance to (0,0) = sqrt((56 - 0)^2 + (13 - 0)^2) = sqrt(3136 + 169) = sqrt(3305) ≈ 57.47.
L1 norm distance to (100,40) = |56 - 100| + |13 - 40| = 44 + 27 = 71.
L2 norm distance to (100,40) = sqrt((56 - 100)^2 + (13 - 40)^2) = sqrt(1936 + 729) = sqrt(2665) ≈ 51.62.
For point (56,15):
L1 norm distance to (0,0) = |56 - 0| + |15 - 0| = 56 + 15 = 71.
L2 norm distance to (0,0) = sqrt((56 - 0)^2 + (15 - 0)^2) = sqrt(3136 + 225) = sqrt(3361) ≈ 57.98.
L1 norm distance to (100,40) = |56 - 100| + |15 - 40| = 44 + 25 = 69.
L2 norm distance to (100,40) = sqrt((56 - 100)^2 + (15 - 40)^2) = sqrt(1936 + 625) = sqrt(2561) ≈ 50.61.
For point (50,18):
L1 norm distance to (0,0) = |50 - 0| + |18 - 0| = 50 + 18 = 68.
L2 norm distance to (0,0) = sqrt((50 - 0)^2 + (18 - 0)^2) = sqrt(2500 + 324) = sqrt(2824) ≈ 53.13.
L1 norm distance to (100,40) = |50 - 100| + |18 - 40| = 50 + 22 = 72.
L2 norm distance to (100,40) = sqrt((50 - 100)^2 + (18 - 40)^2) = sqrt(2500 + 484) = sqrt(2984) ≈ 54.61.
From the calculations, we can see that point (56,15) satisfies the condition mentioned. When the L1 norm is used, (56,15) will be assigned to (0,0), but when the L2 norm is used, it will be assigned to (100,40).