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Exercise Set #3

Exercise Set #3

FIZ228 - Numerical Analysis
Dr. Emre S. Tasci, Hacettepe University

Fit the Silica FTIR data [Silica FTIR data] for the wavenumber range [900,1500] cm-1:

  1. Using 3 Gaussians

  2. Using 4 Gaussians

  3. Using 5 Gaussians

  4. Using 3 Lorentzians

  5. Using 2 Gaussians & 2 Lorentzians

Calculate the coefficient of determination (\(r^2\)) for each fit.

Information:

A Gaussian characterized by (\(A,\mu,\sigma\)) is formulized as:

\[G(x;A,\mu,\sigma) = A\exp{\left[-\frac{(x-\mu)^2}{2\sigma^2}\right]}\]

whereas, a Lorentzian characterized by (\(A,x_0,\gamma\)) is formulized as:

\[ L(x;A,x_0,\gamma)=A\left[ { \gamma \over (x - x_0)^2 + \gamma^2 } \right]\]

Hints:

  • Once you solve one of the items, it will be pretty straightforward to apply the same routing to the rest.

  • If at first you don’t get any result or error from the curve_fit() or any other fit function you are using, it is most likely due to a bad starting point. Trial & error is a good approach but taking a hint from Cappeletti et al.’s graph is the best one! ;) [See below]

  • It’s always a good idea to separately plot all the components to see if the components make sense (e.g., absorbance can never take negative values!)

As an example for the last one, check the following fit of 5 Gaussians, with r2 = 0.998:

HW3_bad5G_sum.png

Even though it seems very good, here are its components, separately drawn:

HW3_bad5G.png

which doesn’t make any sense as G3 & G5 Gaussians indicate a negative absorption!

Cappeletti et al.’s graph

imgs/04_Cappeletti.png

Reference & Acknowledgement

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Exercise Set #2

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Exercise Set #4