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Unformatted text preview: EEL 3105 Problems on Least Squares 1. General theory of least squares: Recall the least squares is set up as a set of equations of the form yi 1 xi1 2 xi 2 N xiN ei where i denotes the experiment number, yi is the output of the i‐th experiment, xik is the k‐th input in the i‐th experiment, and ei is the error or noise in the i‐th experiment. There are N inputs and N parameters 1, 2, …, N that we need to estimate. We have data from M experiments, so that I varies from 1, 2, …, M. We find these parameters so that M J ei2 i 1 is minimized. As I showed in class, the solution is obtained as follows. First we define the matrix x11
R 21 xM 1 x12
xM 2 x1N x2 N xMN Note that R is an MxN matrix. Create a column vector y of size Mx1 by stacking the measurements yi in a column. Then the least squares solution for the optimal values parameters 1, 2, …, N arranged as a Nx1 column vector LS is given by LS ( RT R)1 RT y Once the data x, y are obtained, then finding the least squares parameters is an easy computation using matrix operations. Read your class notes and check that this is correct. 2. Results of an experiment to measure resistance led to the following observations with first rwo being voltage (in volts) and second row being current (milliamps): 1 2.01 2 3.99 3 6.02 4 7.96 5 10.04 1.5 2.99 Use least squares to estimate the resistance. 3. Solve Problems 13.16 and 13.17 from the book. 2.5 4.98 3.5 7.1 4.5 9 5.5 10.2 4. Let us return to the set up in the general theory in 1 above. Consider the error vector corresponding to the least squares solution for the parameters: eLS y R LS Calculate the inner product of eLS and the least squares estimate RLS. Show that this inner product is zero. This is called the orthogonality principle: the best estimate is perpendicular to the noise. ...
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This note was uploaded on 12/27/2011 for the course EEL 3105 taught by Professor Boykins during the Fall '10 term at University of Florida.
- Fall '10