February 14th, 2019, 03:40 PM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 Principal Components Analysis Question Consider the following set of three-dimensional points D = {(0,2,1), (0,4,3), (1,4,5), (1,8,6), (1,8,10), (4,8,14), (5,9,13)} (a) Find the least squares estimate for z taken as a function of x and y. (b) Find the least squares estimate for x taken as a function of y and z. (c) Find the best fit plane, in term of principal component analysis. My answers (a) Finding the best fit plane for z taken as a function of x and y A + 0B + 2C = 1 A + 0B + 4C = 3 A + 1B + 4C = 5 A + 1B + 8C = 6 A + 1B + 8C = 10 A + 4B + 8C = 14 A + 5B + 9C = 13 LeastSq(A, B, C) = { -0.331, 1.385, 0.877 } New points that lie on the projection plane: x: 0.000 0.000 1.000 1.000 1.000 4.000 5.000 y: 2.000 4.000 4.000 8.000 8.000 8.000 9.000 z: 1.423 3.176 4.561 8.068 8.068 12.222 14.483 (b) Finding the best fit plane for x taken as a function of y and z A + 2B + 1C = 0 A + 4B + 3C = 0 A + 4B + 5C = 1 A + 8B + 6C = 1 A + 8B + 10C = 1 A + 8B + 14C = 4 A + 9B + 13C = 5 LeastSq(A, B, C) = { 1.714, 6.143, 7.429 } New points that lie on the projection plane: x: -0.356 0.156 1.040 0.737 2.505 4.274 3.645 y: 2.000 4.000 4.000 8.000 8.000 8.000 9.000 z: 1.000 3.000 5.000 6.000 10.000 14.000 13.000 (c) Finding the variances of plane(a) and plane(b). Avg(D) = (1.714, 6.143, 7.429) Var(Proj(Avg, Plane(a))): 29.175 Var(Proj(Avg, Plane(b))): 30.515 Plane(a) has smaller variance, therefore Plane(a) is a better fit plane. Are my answers correct at all??? Last edited by zollen; February 14th, 2019 at 03:48 PM. February 14th, 2019, 04:54 PM #2 Senior Member   Joined: Sep 2016 From: USA Posts: 684 Thanks: 459 Math Focus: Dynamical systems, analytic function theory, numerics I get the same plane as you for (a). However, for (b) I get the plane with coefficients: $(-0.4255, -0.1863, 0.4421)$. I'm using MATLAB for the matrix arithmetic so I'm inclined to trust it but you never know I might have made a mistake somehow. In any case double check your work. For (c) I think you have misunderstood what the question is asking (or I have). I don't think it is asking you to determine whether the plane from (a) or (b) is better. I think it is asking you to find the "true" best fit plane, which is going to be whatever plane is generated by the 2 dominant principal components (the left singular vectors corresponding to the 2 largest singular values). For this you should compute a SVD. That is my understanding of what (c) is asking anyway. I would say the question is not specific enough so if you are in doubt ask your instructor. Thanks from zollen February 14th, 2019, 05:06 PM #3 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 You are correct. I made a mistake with question b. Let me work on c with SVD... Thanks for the tips!! EigenVectors(A): x: 0.205 -0.520 0.829 y: 0.586 0.743 0.322 z: 0.784 -0.420 -0.457 The best fit plane is: 0.829x + 0.322y - 0.457z = 0 Last edited by zollen; February 14th, 2019 at 05:29 PM. Tags analysis, components, principal, question Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Sancho Real Analysis 3 January 22nd, 2018 05:46 PM Tooperoo Algebra 3 March 7th, 2013 06:15 PM Hooman Real Analysis 0 February 7th, 2013 04:27 PM Stingery Advanced Statistics 0 June 12th, 2010 06:06 AM project Linear Algebra 0 January 27th, 2009 02:38 AM

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