Part I : Astro-Stats & Python : Lev-Marq to Markov Chain Monte Carlo and Bootstrapping

Oh, how I am excited to tell you about this script!  I'll briefly reiterate the concepts employed for this exercise and then I'll just jump right into my code and results.

Levenberg-Marquardt (LM)

LM Statistics is very useful for determining the best fitting parameters.  My function, using this statistical method to fit my data, finds the combination of values for the parameters that yields the least \(\chi^2\) after trying many different values for each parameter.  This method requires an initial guess for the parameters.  Consequently, if one's hypothesis is far from the truth, the calculations of this LM method may claim that the best parameters are somewhere completely "out of the ballpark".  This means that one should make a somewhat good guess when inputting your priors (or initial parameters). Refer to my previous post Astro-Stats & Python : Levenberg-Marquardt Statistics for further explanation and an exemplary application of this method.

Markov Chain Monte Carlo (MCMC)

The MCMC method is excellent for finding the confidence interval of each parameter after the best fitting parameters have been found.  From one perspective, this method is also capable of finding the best fitting parameters as well.  However, when the "Random Walk" occurs and my code is checking the Likelihood (for one set of parameters) to see if it should change a certain parameter, it might change that parameter even if the new Likelihood is less than the previous one.  This means that once my MCMC code finds the highest Likelihood--the minimum \(\chi^2\)--and thus finds what is most likely the best parameters, it will continue to "search" around those best parameters to check the gradient of the Likelihoods as the parameters change.  Because it does the search around those values, it may even stumble across another combination of parameter-values that have similar high Likelihoods.  When this happens, it is tough to find which combination of parameters is better than the other.   Essentially, it is great at finding the area around the best fitting parameters but is not superb at pinpointing precisely where the best values are for the parameters.  Refer to my

Astro-Stats & Python : Bootstrapping, Monte Carlo and a Histogram post for a more elaborated description of this methodology and a good example for applying this to data.  (In that example, \(a_0\) and \(a_1\) have very low precision--i.e. large uncertainties--associated with what was calculated to be their best values for the fitting.)

Bootstrapping Statistics

Bootstrapping is any test that utilizes random sampling in which any sampling/outcome can be replaced/repeated.  The randomness component is applied under the Monte Carlo aspect of my work.  The application of this concept of random sampling with replacement is discussed in more detail, once again, in my Astro-Stats & Python : Bootstrapping, Monte Carlo and a Histogram post.  The underlying concept behind this statistical method is that I have created a process that is self-sustaining and thus does not need external input as it produces many outcomes.  This idea of independence and lone-effort alludes to the expression "pulling oneself up by one's own bootstraps".

I think that that was a good way to review on the various things I have been learning before I jump into my next Python script.  Now-a-days, I feel like I am just spitting out new code left and right.  I feel like this course and the Banneker Institute at large is--at a very fast pace--making me substantially more intelligent in Astrophysics research and more productive in Coding.

to be continued...