Proposal: Automatic estimation of number of histogram bins for weighted data
Hi all, this is my first post on this mailing list. I'm writing to propose a method for extending the histogram bandwidth estimators to work with weighted data. I originally submitted this proposal to seaborn: https://github.com/mwaskom/seaborn/issues/2710 and mwaskom suggested I take it here. Currently the unweighted auto heuristic is a combination of the Freedman-Diaconis and Sturges estimator. For reference, these rules are as follows: Sturges: return the peak-to-peak ptp=(i.e. x.max() - x.min()) and number of data points total=x.size. Then divide ptp by the log of one plus the number of data points. ptp / log2(total + 2) Freedman-Diaconis: Find the interquartile-range of the data iqr=(np.subtract(*np.percentile(x, [75, 25]))) and the number of data points total=x.size, then apply the formula: 2.0 * iqr * total ** (-1.0 / 3.0). Taking a look at these it seems (please correct me if I'm missing something that makes this not work) that there is a simple extension to weighted data. If we can find a weighted replacement for p2p, total, and iqr, the formulas should work exactly the same in the weighted case. The p2p case seems easy. Even if the data points are weighed, that doesn't change the min and max. Nothing changes here. For total, instead of taking the size of the array (which implicitly assumes each data point has a weight of 1), just sum the weight to get total=weights.sum(). I believe the IQR is also computable in the weighted case. import numpy as np n = 10 rng = np.random.RandomState(12554) x = rng.rand(n) w = rng.rand(n) sorted_idxs = x.argsort() x_sort = x[sorted_idxs] w_sort = w[sorted_idxs] cumtotal = w_sort.cumsum() quantiles = cumtotal / cumtotal[-1] idx2, idx1 = np.searchsorted(quantiles, [0.75, 0.25]) iqr_weighted = x_sort[idx2] - x_sort[idx1] print('iqr_weighted = {!r}'.format(iqr_weighted)) # test this is the roughtly the same for the "unweighted case" # (wont be exactly the same because this method does not have interpolation) w = np.ones_like(x) w_sort = w[sorted_idxs] cumtotal = w_sort.cumsum() quantiles = cumtotal / cumtotal[-1] idx2, idx1 = np.searchsorted(quantiles, [0.75, 0.25]) iqr_weighted = x_sort[idx2] - x_sort[idx1] iqr_unweighted_repo = x_sort[idx2] - x_sort[idx1] print('iqr_unweighted_repo = {!r}'.format(iqr_unweighted_repo)) iqr_unweighted_orig = np.subtract(*np.percentile(x, [75, 25])) print('iqr_unweighted_orig = {!r}'.format(iqr_unweighted_orig)) This quick and dirty method if weighted quantiles give a close result (which is probably fine for a bandwidth estimator): iqr_weighted = 0.21964093625695036 iqr_unweighted_repo = 0.36649977003903755 iqr_unweighted_orig = 0.30888312408540963 And I do see there is an open issue / PR for weighted quantiles/percentiles: https://github.com/numpy/numpy/issues/8935 https://github.com/numpy/numpy/pull/9211 so this code could make use of that after it lands. Lastly, I think the most common case (or at least my case) for using a weighted histogram is to combine multiple histograms. In this case the number of estimated bins might be greater than the number of weighted data points, and a simple min condition on that number and the estimated number of bins should take care of that. Please let me know: thoughts / opinions / ideas on this topic. I did do some searching for related discussion, but I may have missed it, so point me to that if I missed it. Also if the reason this feature does not exist is because there is some theoretical problem with estimating bandwidth for weighted data that I'm unaware of, I'd be interested to learn about that (although I can't see that being the case because these are just heuristics after all, and I have validated that this works well in my own use-cases). -- -Dr. Jon Crall (him)
To me, this feels like it might be a better fit for SciPy or possibly statsmodels (but maybe not since neither have histogram functions anymore).The challenge with weighted estimators is how the weights should be interpreted. Stata covers the most important cases of weights https://www.reed.edu/psychology/stata/gs/tutorials/weights.html. Would these be frequency weights? Stata supports only frequency weights https://www.stata.com/manuals/u11.pdf#u11.1.6weight. Kevin On Sun, Dec 12, 2021 at 9:45 AM Jonathan Crall <erotemic@gmail.com> wrote:
While it does feel like this might be more scipy-ish than numpy-ish, numpy has an existing histogram method, with existing heuristics for choosing a number of bins automatically, with existing support for weights. What it is lacking is support for weights and a heuristic jointly. This proposal is not a massive new feature for numpy. It is just plugging a hole that exists in the cross product of possible argument combinations for np.histogram. Thank you for the pointer about interpretation of weights. That was something I felt was going to be a nuance of this, but I didn't have the words to describe it. Within pure numpy, I think it should be possible to compute multiple histograms and then aggregate them. That seems to lend itself towards frequency weights, but it seems to me that probability weights would use the same procedure to estimate bandwidth. https://stats.stackexchange.com/questions/354689/are-frequency-weights-and-s... And ultimately, this is just an estimator used as a convenience for programmers. Most real applications will need to define their bins wrt their problem, but I think if it makes sense for numpy to provide a heuristic baseline for un-weighted data, then it is natural to assume it would do so for weighted data as well. On Mon, Dec 13, 2021 at 4:03 AM Kevin Sheppard <kevin.k.sheppard@gmail.com> wrote:
-- -Dr. Jon Crall (him)
For what it's worth, I've looked into this a long time ago. The missing ingredient has always been weighted quantiles. If I'm not mistaken, the interface already exists, but raises an error. I've had it on my back burner to provide an O(n) C implementation of weighted introselect, but never quite got around to it. I think there has been work to add a O(n log n) implementation recently. - Joe On Thu, Dec 23, 2021 at 1:19 PM Jonathan Crall <erotemic@gmail.com> wrote:
Yes, #9211 <https://github.com/numpy/numpy/pull/9211> is the open PR for weighted quantiles. Is this something I should make an issue for on the numpy github? Or is the correct place to discuss it on this mailing list? I'd like to link to this conversation in two other places on github, but that's difficult when discussion is on the mailing list. But if it's more appropriate to talk here, let me know. On Thu, Dec 23, 2021 at 2:29 PM Joseph Fox-Rabinovitz < jfoxrabinovitz@gmail.com> wrote:
-- -Dr. Jon Crall (him)
To me, this feels like it might be a better fit for SciPy or possibly statsmodels (but maybe not since neither have histogram functions anymore).The challenge with weighted estimators is how the weights should be interpreted. Stata covers the most important cases of weights https://www.reed.edu/psychology/stata/gs/tutorials/weights.html. Would these be frequency weights? Stata supports only frequency weights https://www.stata.com/manuals/u11.pdf#u11.1.6weight. Kevin On Sun, Dec 12, 2021 at 9:45 AM Jonathan Crall <erotemic@gmail.com> wrote:
While it does feel like this might be more scipy-ish than numpy-ish, numpy has an existing histogram method, with existing heuristics for choosing a number of bins automatically, with existing support for weights. What it is lacking is support for weights and a heuristic jointly. This proposal is not a massive new feature for numpy. It is just plugging a hole that exists in the cross product of possible argument combinations for np.histogram. Thank you for the pointer about interpretation of weights. That was something I felt was going to be a nuance of this, but I didn't have the words to describe it. Within pure numpy, I think it should be possible to compute multiple histograms and then aggregate them. That seems to lend itself towards frequency weights, but it seems to me that probability weights would use the same procedure to estimate bandwidth. https://stats.stackexchange.com/questions/354689/are-frequency-weights-and-s... And ultimately, this is just an estimator used as a convenience for programmers. Most real applications will need to define their bins wrt their problem, but I think if it makes sense for numpy to provide a heuristic baseline for un-weighted data, then it is natural to assume it would do so for weighted data as well. On Mon, Dec 13, 2021 at 4:03 AM Kevin Sheppard <kevin.k.sheppard@gmail.com> wrote:
-- -Dr. Jon Crall (him)
For what it's worth, I've looked into this a long time ago. The missing ingredient has always been weighted quantiles. If I'm not mistaken, the interface already exists, but raises an error. I've had it on my back burner to provide an O(n) C implementation of weighted introselect, but never quite got around to it. I think there has been work to add a O(n log n) implementation recently. - Joe On Thu, Dec 23, 2021 at 1:19 PM Jonathan Crall <erotemic@gmail.com> wrote:
Yes, #9211 <https://github.com/numpy/numpy/pull/9211> is the open PR for weighted quantiles. Is this something I should make an issue for on the numpy github? Or is the correct place to discuss it on this mailing list? I'd like to link to this conversation in two other places on github, but that's difficult when discussion is on the mailing list. But if it's more appropriate to talk here, let me know. On Thu, Dec 23, 2021 at 2:29 PM Joseph Fox-Rabinovitz < jfoxrabinovitz@gmail.com> wrote:
-- -Dr. Jon Crall (him)
participants (3)
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Jonathan Crall -
Joseph Fox-Rabinovitz -
Kevin Sheppard