Differential Equations Solutions 66

Differential Equations Solutions 66 - . 866521] for the rst...

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76 Chapter 13. Solutions: Case Study: Fitting Exponentials: An Interest in Rates -1.5 -1 -0.5 0 0.5 1 x 10 -6 -1 0 1 2 x 10 -6 change in x 1 change in x 2 -1 -0.5 0 0.5 1 1.5 x 10 -7 -2 -1 0 1 2 x 10 -6 change in w 1 change in w -1 -0.5 0 0.5 1 x 10 -4 0 2 x 10 -4 change in x 1 -1.5 -1 -0.5 0 0.5 1 x 10 -5 -2 0 2 x 10 -4 change in w 1 -0.01 -0.005 0 0.005 0.01 -0.01 -0.005 0 0.005 0.01 change in x 1 -1 -0.5 0 0.5 1 x 10 -3 -0.02 -0.01 0 0.01 0.02 change in w 1 Figure 13.1. Challenge 1, with α = [0 . 3 , 0 . 4] and η =10 6 (top row), η =10 4 (middle row), and η =10 2 (bottom row). On the left, we plot the two components of x x true and on the right w w true . For the harder problem, when the true α =[ 0 . 30 , 0 . 31], the computa- tions with 4 parameters produced [ 0 . 304889 , 2 . 601087] for the ±rst guess and [ 0 . 304889 , 2 . 601087] for the second. The results for 2 parameters were again better but unreliable for the smaller rate constant: [ 0 . 304889 ,
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Unformatted text preview: . 866521] for the rst guess and [ . 304889 , . 866521] for the second. The residuals for each of these ts are plotted in Figures 13.4 and 13.5. From the fact that none of the residuals from our computed solutions for the rst problem resemble white noise, we can note that the solutions are not good approximations to the data. Troubles in the second problem are more dicult to diagnose, since the residual looks rather white. A single exponential function gives a good approx-imation to this data and the second term has very little eect on the residual. This is true to a lesser extent for the rst dataset....
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This note was uploaded on 01/21/2012 for the course MAP 3302 taught by Professor Dr.robin during the Fall '11 term at University of Florida.

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