Consider the data points in the table, taken from a 4- dimensional space. Perform the following tasks for the 21 points of this dataset. I. 2. 3. 4. 5. Use any calculator, Matlab/R type of toolbox, or your own small computer program for the following tasks. Include a screenshot of the Matlab code or your program code used by you. a. b. c. (9) Using Manhattan distance, find the three closest and the three farthest points from the point PIO. Report these closest/farthest points, their coordinates, and their distances from PIO. Show your complete calculations for onlv one pair of data points. (9) Using Cosine similarity, find the three closest and the three farthest points from the point PIO. Report these closest/farthest points, their coordinates and their distances from PIO. Show your complete calculations for onlv one pair of data points. (7) Standardize the values of every variable using linear transformation. For each variable now the values should be in the range [O.. .11. Take any four data points form this table and compute all pair-wise distances for both cases, that is, before normalization and after normalization. List the point-pairs in decreasing order of their pair-wise distances for both cases. Is the ordering the same or it has changed? Comment on why any changes in the rank ordering may have occurred. P4 PIO PII P12 P13 P15 P17 P18 P20 (20) Use Minkowski distance formula and compute the distance between points P 7 and P8 for the cases w 12 3 7 3 6 10 12 7 2 6 5 6 2 8 2 4 1 7 4 5 1 x 5 18 3 14 12 17 4 20 9 14 15 15 11 12 16 9 3 14 16 13 7 6 1 6 14 6 9 2 2 5 11 20 11 4 6 5 17 4 3 12 7 z 1 2 1 2 1 2 2 1 2 2 1 1 2 2 2 2 1 of "r" taking the values, 1, 2, 3, 4, 5, 6, and 10. Report the distance values obtained. Discuss the effect of changing the value of "r" as you see in your results. Perform the following steps in the context of Mahalanobis distance: a. b. c. d. e. (5) Compute and show the covariance matrix "C" of this data set. You can use a toolbox such as Matlab to compute this matrix. (5) Find and show the inverse of the matrix C and call it C-inv. Use Matlab type of toolbox to compute the inverse. (6) Find the Mahalanobis distances between all pairs formed using the points P9, P14, and P16. Show all computations performed for finding these distances. (4) Now find Euclidean distances among all point-pairs of part (c) above. (5) Compare and comment on the distances obtained in parts (c) and (d) above for the point-pairs examined. (15) Consider the attribute W. We want to make three discrete intervals for this attribute. Show how it will be done for the cases of (i) Equal width partition, and (ii) Equal frequency partition. (15) I define a new distance function, for the data points shown in the table above, as follows: = (Wa-Wb)2 - (Xa-Xb)2 + (Ya-Yb) -10*(Za-Zb). Use two distinct point pairs from the table of points above to test if this distance function satisfies all three requirements of being a metric distance function. Show all your work for this verification. What is your intuitive feeling about whether this distance function is a metric or not?