Maximum LIkelihood Estimation Based MapClassification

[tensor-calculus]

Closes #111 (hopefully)

I wasn't sure how to proceed with the LP solution so I used the Dynamic Programming Approach. Added a new class MaximumLikelihood that accepts an array of standard deviations sigma along with y, the implementation follows the formulation described in:

Mu, L., & Tong, D. (2019). Choropleth Mapping with Uncertainty: A Maximum Likelihood Based Classification Scheme.

Added the above paper to references.bib and added 2 tests to test_mapclassify.py.
There are a lot of comments meant for the sake of understanding and can be removed before merging.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 88.9%. Comparing base (bc27575) to head (df62f83).

Additional details and impacted files

@@           Coverage Diff           @@
##            main    #290     +/-   ##
=======================================
- Coverage   90.2%   88.9%   -1.3%     
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  Files         11      11             
  Lines       1308    1348     +40     
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+ Hits        1180    1198     +18     
- Misses       128     150     +22     

Files with missing lines	Coverage Δ
mapclassify/__init__.py	100.0% <ø> (ø)
mapclassify/classifiers.py	86.8% <100.0%> (-1.8%)	⬇️

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[sjsrey]

Is there any guidance on where sigma values are to be obtained?

[tensor-calculus]

For example, the widely used American Community Survey (ACS) data include sampling uncertainty through published “margin of error” (MOE) estimates.

The uncertainty values were included in the dataset that they were using. If the data is generated by a statistical model then the standard deviation could be calculated directly from the model's own variance estimates i believe.

Ultimately uncertainty values have to come from either the people collecting the data or the ones modeling it, so it falls outside of our scope i guess.

[sjsrey]

This would need to be tested on large n problems for scalability.

[tensor-calculus]

Hmm, agreed.

The reason I used a small dataset is because I could quickly calculate the correct output by hand and compare it with the classifier's output. If I were to use a large dataset I would never know if the output is correct or not, maybe I can try using the same dataset the authors were using and then compare the result with theirs?

P.S. This particular test example was giving same bins with the regular classifiers and MLE classifier so I changed it in one of the later commits and added a better test example that would give different bins to highlight the difference.