Just as a quick note, as I suspect at some point the future this will be useful to someone else, as well.
I have a possibly naive question. I don't really understand this particular set of output:
In : import numpy
In : a1 = numpy.random.random((512,512,512)).astype("float32")
In : a1.sum(axis=0).sum(axis=0).sum(axis=0) Out: 67110312.0
In : a1.sum() Out: 16777216.0
I recognize that the intermediate sums may accumulate error differently than a single call to .sum(), but I guess my concern is that it's accumulating a lot faster than I anticipated. (Interesting to note that a1.sum() returns 0.5*512^3, down to the decimal; is it summing up the mean, which should be ~0.5?) However, with a 256^3 array:
In : import numpy
In : a1 = numpy.random.random((256,256,256)).astype("float32")
In : a1.sum(axis=0).sum(axis=0).sum(axis=0) Out: 8389703.0
In : a1.sum() Out: 8389245.0
The errors are much more reasonable. Is there an overflow or something that occurs with the 512^3? These problems all go completely away with a float64 array, but the issue originally showed up when trying to normalize an on-disk float32 array of size 512^3, where the normalization factor was off by a substantial factor (>2x) depending on the mechanism used to sum. My suspicion is that perhaps I have a naive misconception about intermediate steps in summations, or there is a subtlety I'm missing here.
I placed a sample script I used to test this here:
Thanks for any insight anybody can provide,
It's not quite an overflow.
In : from numpy import *
In : x = float32(16777216.0)
In : x + float32(0.9) Out: 16777216.0
You are accumulating your result in a float32. With the a.sum() approach, you eventually hit a level where the next number to add is always less than the relative epsilon of float32 precision. So the result doesn't change. And will never change again as long as you only add one number at a time. Summing along the other axes creates smaller intermediate sums such that you are usually adding together numbers roughly in the same regime as each other, so you don't lose as much precision.
Use a.sum(dtype=np.float64) to use a float64 accumulator.
-- Robert Kern
"I have come to believe that the whole world is an enigma, a harmless enigma that is made terrible by our own mad attempt to interpret it as though it had an underlying truth." -- Umberto Eco
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Hi Robert, Thank you very much for that explanation; that completely makes sense. I didn't know about the dtype for accumulators/operators -- that did just the trick.