Matlab Exercises with the SVD

Matlab Exercises with the SVD 1. In R3 , project 300 points in R3 (stored in the 300 × 3 array named B) to the subspace given as the the plane x − y − z = 0. Plot the coordinates of the projection in R2 and then plot both the original data and the projected data on the plane in R3 . Here are some notes: • On the class website, download the Matlab data file SphereData.mat and put it in the directory you are currently working in. To get the data into Matlab, in the command window type: load SphereData You should see the variable X in the Workspace window, where X is 300 × 3. • To plot the data you’ve loaded, type plot3(X(:,1),X(:,2),X(:,3),’k-’); into the command window. • Once you have your projected data in a 300 × 3 array Y , you can plot it as well, keeping the original data: hold on; % This keeps all the old data in the figure plot3(Y(:,1),Y(:,2),Y(:,3),’r-’); hold off; 2. As we saw in class, polynomial interpolation (using a degree n polynomial with n + 1 points) can lead to a high degree of oscillations between data points. If there is error in the data, then perhaps we can make a trade-off between a smoother, well-behaved function, but one that does not exactly hit the data points (like the best fitting line we have constructed in the past). We can do this using the SVD. Recall that, if we’re solving the matrix equation: Ac = y for the parameters in c, we can use the pseudoinverse with smaller rank to provide an approximate solution. Recall that, if [U,S,V]=svd(A,’econ’) is the SVD of A, then we can look at the singular values: diag(S) to see how many large values we have. Once we determine the rank r, then we can compute the pseudoinverse using that rank: [U,S,V]=svd(A); plot(diag(S)) c=V(:,1:k)*S(1:k,1:k)*U(:,1:k)’*y; In the Matlab code provided, we will also use polyval to help us evaluate polynomials. In Matlab, if you have some data in a vector x and a polynomial defined by a coefficient vector c so that c1 tn + c2 tn−1 + · · · + cn t + cn+1 1 In Matlab, you can type: polyval(c,x) On the class website, download the file VanderExample.m. Read it over to see if you understand it, then run it in Matlab. 3. Be sure you read over and understand the Matlab commands in the SampleAuthor.m code. Especially the following: • We compute the mean and store it in the vector Ymean. Is this a vector or a matrix? What is its size? • What does the command repmat do? (We’ve seen it a couple times before). • When we project data to a plane, that plane is isomorphic to R2 - What is the isormorphism? Do we compute these coordinates somewhere in the code? Are they plotted somewhere? • When we represent the data in the original plane, that space is a subspace of R19200 . This is also called the “reconstruction” of the data back to its original subspace. Where is this done in the code? 4. Here is the code that solves Exercise 1. There is an error in one of the lines (not a Matlab error, but a linear algebra error). See if you can find it. load SphereData; [U,S,V]=svd([1 -1 -1]) %Check that V has orthonormal columns. V’*V Coords=V(:,2:3)’*X’; figure(1) plot(Coords(1,:),Coords(2,:),’.’); Y=V(:,1:2)*V(:,1:2)’*X’; Y=Y’; %This puts Y as 300 x 3 (instead of 3 x 300) figure(2) plot3(X(:,1),X(:,2),X(:,3),’k-’); hold on; % This keeps all the old data in the figure plot3(Y(:,1),Y(:,2),Y(:,3),’r-’); hold off 2