I would like to know how normalization affects significant digits. Starting with a dataset where features have wildly different scales and number of digits, I'd like to begin with min-max normalization and cut off insignificant digits. Which should I do first? Here's an example with random data.
Case 0: Starting data set.
[[ 24.2890000000000015 32.5955000000000013 81.2960000000000065] [ 6801.0150000000003274 3900.4432000000001608 3300.7890999999999622] [600001.2340000000549480 190001.0450000000128057 670000.6737000000430271]] These were rounded to 5 decimal places but the representation doesn't reflect that perfectly of course.
Applying min-max normalization yields:
[[0.0000000000000000 0.0000123981981898 0.0000850880737021] [0.0101148729339684 0.0057855086963038 0.0048904710297620] [0.8955195417352746 0.2835573914818911 1.0000000000000000]] The starting dataset has insignificant digits, so I assume the normalized set does as well.
Case 1: round to 5 significant digits, then normalize, then use all digits
[[0.00000000000000000000 0.00001239895695263495 0.00008508815926313482] [0.01011486071619691930 0.00578545003402369546 0.00489049221666485161] [0.89551860037505148782 0.28355611685749604334 1.00000000000000000000]] Since the data were first rounded to significant digits, I would think all digits are now significant. It's a difference in scale, not accuracy of measurement?
Case 2: normalize, then round to 5 significant digits
[[0.00000000000000000000 0.00001239800000000000 0.00008508800000000001] [0.01011500000000000073 0.00578549999999999984 0.00489050000000000075] [0.89551999999999998270 0.28355999999999997874 1.00000000000000000000]] But perhaps I'm wrong, and the numbers should be trimmed after normalization?
The differences between the Cases 0 and 1 are much more apparent with thousands of rows and columns and greater differences in scale. They might be small differences but that matters when the range is small...