RFC: array-agnostic `quantile`

[mdhaber]

I'm working on adding array API support in scipy.stats (scipy/scipy#20544) and one of the the things I'll need is a quantile function. If there is some support for this idea, I'll convert this issue into a proper proposal.

Looks like there is already wide support:

• numpy.quantile
• torch.quantile
• cupy.quantile
• jax.numpy.quantile
• dask.dataframe.DataFrame.quantile
• tfp.stats.quantiles
• xarray.DataArray.quantile

Previous discussions (not much):

• Proposal to standardize basic statistical functions #10 (comment)
• Behavior of reductions on empty arrays should be specified #232 (comment)

There are many conventions for calculating quantiles. Only a few methods would be required by the standard, and if choice of a default is too contentious, perhaps the array-API can consider method to be a required keyword argument, and libraries would be welcome to keep their own default.

[rgommers]

Thanks for the proposal @mdhaber. I don't quite have an opinion yet - I think it in part depends on the situation with methods (see below).

Also - how much do you actually need this? I only count two instances of it being used in SciPy, one of which is a test case. I just did a grep, so I may be missing some dynamic usage perhaps. The one instance is pretty simple, no keyword usage:

https://github.com/scipy/scipy/blob/d55cb95282b5cfab381e373d3c127630c7e915a0/scipy/stats/tests/test_fit.py#L1002

Only a few methods would be required by the standard,

Which ones? Could we get away with only a default method, and hence no method keyword?

and if choice of a default is too contentious, perhaps the array-API can consider method to be a required keyword argument, and libraries would be welcome to keep their own default.

[kgryte]

Related is our previous discussion on median, which we left out due to implementation difficulty in distributed contexts. That stated, Dask does implement median, so may be time to revisit this.

[mdhaber]

Also - how much do you actually need this?

Personally, I don't need it very badly. You're right that quantile/percentile only appear in a few stats functions (stats.bootstrap, stats.iqr, and some distribution fitting code), and it is easy to compute quantiles using existing array-API functions namely (sort). However, when only a few percentiles are needed, it is more efficient to use a partition approach. One could propose adding partition, since it is more general, but I would only use it for quantile.

Which ones? Could we get away with only a default method, and hence no method keyword?
The standard reference is Sample Quantiles in Statistical Packages.

The current variation in sample quantile definitions causes confusion, and so there is a need to standardize the definition of sample quantile across packages and within packages... and we propose that $Q_8$ is the best choice

$Q_8$ corresponds with NumPy's median_unbiased. However, $Q_7$ (linear) is the default in most libraries. It would be unfortunate to have to choose between the two. $Q_5$ (hazen) also satisfies many desirable properties. So I suppose I would suggest having these three in that order of priority.

One wrench I'd like to throw into my own proposal:
The current behavior of np.quantile(a, q, axis) (and other implementations that follow NumPy) is for the quantiles specified by q, a 1D array-like, to be computed for all axis-slices of the data a. This makes it impossible to efficiently compute a different quantile for each axis-slice of a (which is needed, for example, in a vectorized implementation of the BCA-bootstrap).
I would propose following normal broadcasting rules between a and q (possibly with the constraint that q must be 0D or have length 1 along axis), which would meet both needs. A hybrid possibility, which would just be an extension of the existing behavior, would be to allow q to be an N-D array that is broadcastable with a except along axis. In this case, for each pair of corresponding axis-slices a_i and q_i, it would compute all quantiles of a_i specified by q_i.

[kgryte]

I did a bit of digging, with the disclaimer that my digging is incomplete, but I'll try to summarize my initial findings below.

Overview

I took a peak at which APIs were implemented across array libraries, spot-checked whether/how they were implemented, and did a search for which APIs were used in SciPy and sklearn.

median

Usage

• SciPy: low usage; more in tests, than in implementations
• scikit-learn: a reasonable amount of usage

Implementations

• NumPy implements in terms of partition
• CuPy implements as an element-wise kernel
• PyTorch returns the lower of two medians for an even number of elements, rather than the average (NumPy returns the avg to avoid breaking astropy); to return an average, must call quantile.
• Dask supports median
• TensorFlow does not appear to have a dedicated median API, instead deferring to percentile

quantile

Usage

• SciPy: low usage
• scikit-learn: limited usage; mainly in tests

Implementations

• NumPy implements in terms of partition
• CuPy relies on sort
  • only supports a subset of methods: lower, higher, midpoint, nearest, and linear
• PyTorch supports the same methods as CuPy
• Dask supports in DataFrames, but not arrays
• TensorFlow implements in terms of percentile
  • supports the same methods as CuPy
  • API differs in that you specify the number of quantiles you want

percentile

Usage

• SciPy: low usage
• scikit-learn: a reasonable amount of usage

Implementations

• NumPy implements in terms of quantile
• PyTorch does not support
• CuPy implements in terms of quantile
  • supports only linear, lower, higher, nearest, and midpoint methods
• Dask supports "approximate" percentiles
  • supports same methods as CuPy
• TensorFlow implements in terms of sort
  • supports same methods as CuPy

partition

Implementations

• NumPy: https://numpy.org/doc/stable/reference/generated/numpy.partition.html
• CuPy supports: https://docs.cupy.dev/en/stable/reference/generated/cupy.partition.html
• PyTorch does not support
• Dask does not support
• TensorFlow does not support

[mdhaber]

I reviewed scipy.stats more thoroughly for uses of median, quantile, and/or percentile. The first three functions would benefit from having quantile in the array API; for the rest, median would suffice.

• scipy.stats.bootstrap uses percentile here, but I would be happy to change it to use quantile. It would benefit from a quantile function with improved vectorization as described above.
• scipy.stats.iqr uses percentile here, but I'd be happy to change it to use quantile.
• scipy.stats.epps_singleton_2samp uses iqr here.
• scipy.stats.siegelslopes is currently implemented using Pythran, but I think it can be reasonably fast without Pythran if we wanted to use other backends. (Could Pythran dispatch to other backends in the future?) It uses median here and again outside the loop.
• scipy.stats.theilslopes uses median here.
• scipy.stats.median_test, as one might imagine, uses median here.
• scipy.stats.levene uses median here.
• scipy.stats.fligner uses median here.
• scipy.stats.mstats.sen_seasonal_slopes uses np.ma.median here, but one could argue that this should be implemented in stats with regular np.median.

There are also uses of partition in stats.trim functions (trim_mean, trimboth, trim1).

Most of these functions have something else that would make array API conversion challenging at the moment, but I don't think any have non-starters for array API support. I've omitted uses I don't think will get array API support any time soon (e.g. levy_stable). There are also uses in other sub-packages (e.g. scipy.ndimage.median_filter), but I'm less familiar with their prospects for array API support.

[ogrisel]

Ideally it would be nice to have support for weights and filtering missing values automatically.

NumPy 2 recently added support for weighted percentiles/quantiles via the weights kwarg:

• https://numpy.org/devdocs/reference/generated/numpy.quantile.html#numpy-quantile

Adding support for nan value filtering is being discussed in #621.

However, this is not necessarily easy to handle both weights and missing values (EDIT: actually it mostly means forcing the weight of nan entries to zero before computing the cumsum of weights of the sorted observations). There is a related pull request in NumPy here:

• BUG: weighted nanpercentile, nanquantile and multi-dim q numpy/numpy#26582

In scikit-learn we maintain our own _weighted_percentile helper to deal with this and it's used in several places of our code base.

[lorentzenchr]

In my experience, quantiles are very important in practice and occur all over the places.

Which ones? Could we get away with only a default method, and hence no method keyword?
The standard reference is Sample Quantiles in Statistical Packages.

The current variation in sample quantile definitions causes confusion, and so there is a need to standardize the definition of sample quantile across packages and within packages... and we propose that Q8 is the best choice

Q8 corresponds with NumPy's median_unbiased. However, Q7 (linear) is the default in most libraries. It would be unfortunate to have to choose between the two. Q5 (hazen) also satisfies many desirable properties. So I suppose I would suggest having these three in that order of priority.

I could imagine to let the default method vary (for easier consensus), but each one must have a linear option (most have it already).
For weighted quantiles in particular, method inverted_cdf would be mandatory.

[ogrisel]

For weighted quantiles in particular, method inverted_cdf would be mandatory.

I would also advocate for averaged_inverted_cdf in addition to inverted_cdf, which makes it possible to get a strict equivalence between unweighted quantiles (typically computed with the linear strategy by default) and unit-weighted quantiles on even sample sizes.