1) Find some data examples and show examples of calculating

a. Euclidean distance

b. L1 distance

2) Prove or disprove that Euclidean and L1 distance satisfy

a. Positivity

d(x,y) >= 0 for all x and y,

d(x,y) == 0 only if x == y.

b. Symmetry

d(x,y) == d(y,x) for all x and y.

c. Triangle Inequality

d(x,z) <= d(x,y) + d(y,z) for all points x, y, and z

3) Explain why it is not possible or why it is possible to

a. rearrange data so Euclidean distance gives the same meaning as Hamming distance (the number of positions at which the corresponding symbols are different between the two strings of equal length)

b. show that measure d=1-cos(x,y) satisfies positivity, symmetry, and triangle Inequality

4) Draw conclusions about what is important when choosing the distance measure for the evaluation of dissimilarities between data objects.