Question

A Excercise DATA- For the following vectors, and y, calculate the indicated similarity or distance measures a (3,3,3,3), y = (4.4.4.4) cosine, correlation, Euclidean, Extended Jaccard b. (0,1,0,1)y(1.01.0) cosine, correlation, Euclidean, SMC Jaccard c (0-1, 0, 1),y - (1.0,-1,0) cosine, correlation, Euclidean d. (1,1,0,1,0,1), y = (1, 1, 1, 0, 0, 1) cosine, correlation, SMC, Jaccard e - (2-1, 0, 2, 0,-3), y = (-1,1,-1, 0, 0,-1) cosino, 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 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 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?!

Answer 1

ANSWER:- Calculate the adicated similarity or distance measures :- ca x = (3,3,3,3) ys (4,4,4,4).calculate Cosine, Comelation, Euclidean, Extendend Jaccard. X=(3,3,3,3) ys (4,4,4,4). Cosine :- x.y = 3 *4 + 3*4 + 3 * 4 + 3 * 4. 48 |tx| - o” ( 3* 3 + 2+ 3 + 3^3 + 3*3) = 136 = 6 Ilyah - " (4*4 + 6+4 + 4874 +44) -164 - 8 COSCX,Y) - Xoy) 11X11.1981). = 48 6.8 = 48 = 1 Correlation o- There is no Linear x and y.. Therefore, the relationship between similarity is equal O. Euclidean : a (x; -7,² Euclidean = 713-4)2 +(3+4)*+ (3-w13-43)*2

= ((-17-1)= (-07-1332 - (+1+1+1) 12 = (H)²2 Euclidean - Ž i Jaccard distance i Extendend Jaccaid = Xoy hittillyll-xy = 48 6² +8²-48 = 48 LOO - 4-8 36+64-48 - 0:923 :.5:48 sa (b) x = (0,1,01) y=(1,0710). Cosine, correlation Euclidean, SMC, Jaccard. cosine : EXY cos. (x, y) = XY .07 TENEY? llxWillyll. - 07l + *0+071 + 1*0 Sorro to

Correlatfon Covariance (x,y) oyoy. x represents std deviation to oy represents std devaition ofy cov. (x,y) = 60 € (x; -meanxs) (yy-mean tw) ox-NE (35 - mean (3) mean (x) = 2 = 1/2 mean CY) = ? = 42: cor (x,y) = L T (0-Y2) (1+y) + (1-7)(0-va) (4+1) 760-ya) (1-72) + Ci- y2) (0-y2)) - } [-+(+)+()+ (%) -% (54) - 3M 2 V Yolly) + (0-125+ (1-x} larve - (43)% = 0.57735

oy = my = 0.57735 Correlation - cov.(X;Y) = (1/3) Oxoy [.5773x0.5773) - 0.9998 = -1 (approxi matery) Correlation = -1 Euclidean i ręcxi-y.)? = [(0-12+(1-0)*+ (0-13 +61-0037% +(612 + (0)2 + (-10° +603972 5:41. +0+1+0) % = (2)

Jaccard i- (>= 0,1,0,1) y = (1,0,1,0) .x.y =.o .o. Jaccard = llx! HY || - X.Y 10414+1.414-0 V XL = '0' 10*0+1*1 + 0* D+1*,) V2 = 1414 llyll so(1+1+0+o+it, toto) 52=1.414 Jaceard = 0 (C) X = (0, -1, 0.1) (1,0, -1,0) Cosine, correlation I Euclidean Cosine = x.y = .0*1+ (1) o toto+1 to l/xl/.1/7!! Voto thuj*(-07 (040)+(ti) x vitit o*o 7 (-1146-1) +(040) correlation - Lovariance - 0. => correlation = 0. There is no linear relationship between X and Y. Therefore, the correlation is 0.

Euclidean :- Euclidean = Euclidean - V(x; -4;) :-((-1)+ (1-0)?#(0-1-1)+7(1-0)% - (E1? + (0)2 + (0 +612) = = (1+1+1+1382 (6)%=2 (C) X = (1-4,0,1,0,1) Y= (1,1,1,0,0,1). Cosine, correlation, SMC, Jaccard. cosine :- Los cx, y) = Xoy <3 KỸ = 1 + 1 + 0 + 0 +0 - 1/XV = V12+12+12+12 = 2 hyll = V2+ ² + ² + 1² = 2 cos Criy)

correlation of Mean 여 X양읗. Mean of y = 2%. Cow.(x,y) = ( 1 -s) (I - s) (1-13)(I- 2/3) + (0-21) (-2%) + (1-2%) (0 - 24%) | f (o-21) (-) (-2%)(1-4) 19 위시위용을-부-뷰+) 1부 : 1 sd (3) - (- -(-) + (1-25 1(1- | + (-) + (1-2) 하 = 0.139. = = 0·51639 | oo Cow(x,y) = - Y4 - : 616 osk 39 :

Jaccardo = tu. Fo+fort for f, = 3, fro = 1 fo,-/ Ja 3.19 = 3/5 Jaccard = 3/5 = 0.6 1 +1+3 1 x=(2,-1,0,2,0-3) y=(-1, 1,-1,0,0,-1) Cosine, correlation of = xoy cosine :- Nxllilly ll. - *E1))+((-1)*1) +10+61)+(2+0) + (0*0)+(6-3)* 611) TW2+*E1D760*o) (2+2) (0x0) (-34-3)* V (-1*(+) + 111*1) + ((-1*(-1)) + 10*o) (070) ((-1)*(+)) so

correlation of Mean of x = Mean of y=.g 0 Covariance = 0. Correlation = 0. - Thanking ** Pls like & you ** suppoot - As per Chegg rule I must answer only One question with t sub parts . so pls upload the second question as a new question. Again I help u.