Question

2. Here, we further explore the cosine and correlation measures. a. What is the range of values that are possible for the cosine measure? b. If two objects have a cosine measure of 1, are they identical? Explain. c. What is the relationship of the cosine measure to correlation, if any? (Hint: Look at statistical measures such as mean and standard deviation in cases where cosine and correlation are the same and different.) d. Figure (a) below shows the relationship of the cosine measure to Euclidean distance for 100,000 randomly generated points that have been normalized to have an L2 length of 1. What general observation can you make about the relationship between Euclidean distance and cosine similarity when vectors have an L2 norm of 1? e. Figure (b) shows the relationship of correlation to Euclidean distance for 100,000 randomly generated points that have been standardized to have a mean of o and a standard deviation of 1. What general observation can you make about the relationship between Euclidean distance and correlation when the vectors have been standardized to have a mean of 0 and a standard deviation of 1? Euclidean Distance Eudidoan Disco % 0.2 0.4 0.6 Correlation 0.8 Cosine Similarity (a) Relationship between Euclidean distance and the cosite measure. (b) Relationship between Euclidean distance and correlation 1. For the following vectors, x and y, calculate the indicated similarity or distance measures. a. x=(2, 2, 2, 2), y=(3,3,3,3) cosine, correlation, Euclidean, Extended Jaccard b. x=(0, 1, 0, 1), y=(1,0, 1, 0) cosine, correlation, Euclidean, SMC, Jaccard c. x=(0,-1, 0, 1), y=(1, 0,-1, 0) cosine, correlation, Euclidean d. x-(1, 1, 0, 1, 0, 1), y=(1,1,1,0,0,1) cosine, correlation, SMC, Jaccard c. x=(2,-1, 0, 2,0,-3), y=(-1, 1,-1, 0, 0,-1) cosine, correlation 2. Here, we further explore the cosine and correlation measures. a. What is the range of values that are possible for the cosine measure? b. If two objects have a cosine measure of 1, are they identical? Explain. c. What is the relationship of the cosine measure to correlation, if any? (Hint: Look at statistical measures such as mean and standard deviation in cases where cosine and correlation are the same and different.) d. Figure (a) below shows the relationship of the cosine measure to Euclidean distance for 100,000 randomly generated points that have been normalized to have an L2 length of 1. What general observation can you make about the relationship between Euclidean distance and cosine similarity when vectors have an L2 norm of 1? e. Figure (b) shows the relationship of correlation to Euclidean distance for 100,000 randomly generated points that have been standardized to have a mean of o and a standard deviation of 1. What general observation can you make about the relationship between Euclidean distance and correlation when the vectors have been standardized to have a mean of O and a standard deviation of 1? Cosine Similarly (A) Relationship betwee distance and the cosine me clien (b) Re a ldean