[Numpy-discussion] NEP 21: Simplified and explicit advanced indexing (original) (raw)

Eric Wieser wieser.eric+numpy at gmail.com
Mon Jun 25 23:06:42 EDT 2018

Generally +1 on this, but I don’t think we need

To ensure that existing subclasses of ndarray that override indexing do not inadvertently revert to default behavior for indexing attributes, these attribute should have explicit checks that disable them if getitem or setitem has been overriden.

Repeating my proposal from github, I think we should introduce some internal indexing objects - something simple like:

np.core.*class Indexer(object): # importantly not iterable

def __init__(self, value):
    self.value = valueclass OrthogonalIndexer(Indexer): passclass

VectorizedIndexer(Indexer): pass

Keeping the proposed syntax, we’d implement:

This means that subclasses like the following

class LoggingIndexer(np.ndarray): def getitem(self, ind): ret = super().getitem(ind) print("Got an index") return ret

will continue to work without issues. This includes np.ma.MaskedArray and np.memmap, so this already has value internally.

For classes like np.matrix which inspect the index object itself, an error will still be raised from getitem, since it looks nothing like the values normally passed - most likely of the form

TypeError: 'numpy.core.VectorizedIndexer' object does not support indexing TypeError: 'numpy.core.VectorizedIndexer' object is not iterable

This could potentially be caught in oindex.getitem and converted into a more useful error message.

So to summarize the benefits of the above tweaks:

And the costs:

Eric ​

On Mon, 25 Jun 2018 at 14:30 Stephan Hoyer <shoyer at gmail.com> wrote:

Sebastian and I have revised a Numpy Enhancement Proposal that he started three years ago for overhauling NumPy's advanced indexing. We'd now like to present it for official consideration.

Minor inline comments (e.g., typos) can be added to the latest pull request (https://github.com/numpy/numpy/pull/11414/files), but otherwise let's keep discussion on the mailing list. The NumPy website should update shortly with a rendered version ( http://www.numpy.org/neps/nep-0021-advanced-indexing.html), but until then please see the full text below. Cheers, Stephan ========================================= Simplified and explicit advanced indexing ========================================= :Author: Sebastian Berg :Author: Stephan Hoyer <shoyer at google.com> :Status: Draft :Type: Standards Track :Created: 2015-08-27

Abstract -------- NumPy's "advanced" indexing support for indexing arrays with other arrays is one of its most powerful and popular features. Unfortunately, the existing rules for advanced indexing with multiple array indices are typically confusing to both new, and in many cases even old, users of NumPy. Here we propose an overhaul and simplification of advanced indexing, including two new "indexer" attributes oindex and vindex to facilitate explicit indexing. Background ---------- Existing indexing operations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~ NumPy arrays currently support a flexible range of indexing operations: - "Basic" indexing involving only slices, integers, np.newaxis and ellipsis (...), e.g., x[0, :3, np.newaxis] for selecting the first element from the 0th axis, the first three elements from the 1st axis and inserting a new axis of size 1 at the end. Basic indexing always return a view of the indexed array's data. - "Advanced" indexing, also called "fancy" indexing, includes all cases where arrays are indexed by other arrays. Advanced indexing always makes a copy: - "Boolean" indexing by boolean arrays, e.g., x[x > 0] for selecting positive elements. - "Vectorized" indexing by one or more integer arrays, e.g., ``x[[0, 1]]`` for selecting the first two elements along the first axis. With multiple arrays, vectorized indexing uses broadcasting rules to combine indices along multiple dimensions. This allows for producing a result of arbitrary shape with arbitrary elements from the original arrays. - "Mixed" indexing involving any combinations of the other advancing types. This is no more powerful than vectorized indexing, but is sometimes more convenient. For clarity, we will refer to these existing rules as "legacy indexing". This is only a high-level summary; for more details, see NumPy's documentation and and Examples below. Outer indexing ~~~~~~~~~~~~~~ One broadly useful class of indexing operations is not supported: - "Outer" or orthogonal indexing treats one-dimensional arrays equivalently to slices for determining output shapes. The rule for outer indexing is that the result should be equivalent to independently indexing along each dimension with integer or boolean arrays as if both the indexed and indexing arrays were one-dimensional. This form of indexing is familiar to many users of other programming languages such as MATLAB, Fortran and R. The reason why NumPy omits support for outer indexing is that the rules for outer and vectorized conflict. Consider indexing a 2D array by two 1D integer arrays, e.g., x[[0, 1], [0, 1]]: - Outer indexing is equivalent to combining multiple integer indices with itertools.product(). The result in this case is another 2D array with all combinations of indexed elements, e.g., np.array([[x[0, 0], x[0, 1]], [x[1, 0], x[1, 1]]]) - Vectorized indexing is equivalent to combining multiple integer indices with zip(). The result in this case is a 1D array containing the diagonal elements, e.g., np.array([x[0, 0], x[1, 1]]). This difference is a frequent stumbling block for new NumPy users. The outer indexing model is easier to understand, and is a natural generalization of slicing rules. But NumPy instead chose to support vectorized indexing, because it is strictly more powerful. It is always possible to emulate outer indexing by vectorized indexing with the right indices. To make this easier, NumPy includes utility objects and functions such as np.ogrid and np.ix, e.g., x[np.ix([0, 1], [0, 1])]. However, there are no utilities for emulating fully general/mixed outer indexing, which could unambiguously allow for slices, integers, and 1D boolean and integer arrays. Mixed indexing ~~~~~~~~~~~~~~ NumPy's existing rules for combining multiple types of indexing in the same operation are quite complex, involving a number of edge cases. One reason why mixed indexing is particularly confusing is that at first glance the result works deceptively like outer indexing. Returning to our example of a 2D array, both x[:2, [0, 1]] and x[[0, 1], :2] return 2D arrays with axes in the same order as the original array. However, as soon as two or more non-slice objects (including integers) are introduced, vectorized indexing rules apply. The axes introduced by the array indices are at the front, unless all array indices are consecutive, in which case NumPy deduces where the user "expects" them to be. Consider indexing a 3D array arr with shape (X, Y, Z): 1. arr[:, [0, 1], 0] has shape (X, 2). 2. arr[[0, 1], 0, :] has shape (2, Z). 3. arr[0, :, [0, 1]] has shape (2, Y), not (Y, 2)! These first two cases are intuitive and consistent with outer indexing, but this last case is quite surprising, even to many higly experienced NumPy users. Mixed cases involving multiple array indices are also surprising, and only less problematic because the current behavior is so useless that it is rarely encountered in practice. When a boolean array index is mixed with another boolean or integer array, boolean array is converted to integer array indices (equivalent to np.nonzero()) and then broadcast. For example, indexing a 2D array of size (2, 2) like x[[True, False], [True, False]] produces a 1D vector with shape (1,), not a 2D sub-matrix with shape (1, 1). Mixed indexing seems so tricky that it is tempting to say that it never should be used. However, it is not easy to avoid, because NumPy implicitly adds full slices if there are fewer indices than the full dimensionality of the indexed array. This means that indexing a 2D array like x[[0, 1]]`` is equivalent_ _to_ _``x[[0, 1], :]``. These cases are not surprising, but they constrain the_ _behavior of mixed indexing._ _Indexing in other Python array libraries_ _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~_ _Indexing is a useful and widely recognized mechanism for accessing_ _multi-dimensional array data, so it is no surprise that many other_ _libraries in_ _the scientific Python ecosystem also support array indexing._ _Unfortunately, the full complexity of NumPy's indexing rules mean that it_ _is_ _both challenging and undesirable for other libraries to copy its behavior_ _in all_ _of its nuance. The only full implementation of NumPy-style indexing is_ _NumPy_ _itself. This includes projects like dask.array and h5py, which support_ _*most*_ _types of array indexing in some form, and otherwise attempt to copy_ _NumPy's API_ _exactly._ _Vectorized indexing in particular can be challenging to implement with_ _array_ _storage backends not based on NumPy. In contrast, indexing by 1D arrays_ _along_ _at least one dimension in the style of outer indexing is much more_ _acheivable._ _This has led many libraries (including dask and h5py) to attempt to define_ _a_ _safe subset of NumPy-style indexing that is equivalent to outer indexing,_ _e.g.,_ _by only allowing indexing with an array along at most one dimension._ _However,_ _this is quite challenging to do correctly in a general enough way to be_ _useful._ _For example, the current versions of dask and h5py both handle mixed_ _indexing_ _in case 3 above inconsistently with NumPy. This is quite likely to lead to_ _bugs._ _These inconsistencies, in addition to the broader challenge of implementing_ _every type of indexing logic, make it challenging to write high-level array_ _libraries like xarray or dask.array that can interchangeably index many_ _types of_ _array storage. In contrast, explicit APIs for outer and vectorized_ _indexing in_ _NumPy would provide a model that external libraries could reliably_ _emulate, even_ _if they don't support every type of indexing._ _High level changes_ _------------------_ _Inspired by multiple "indexer" attributes for controlling different types_ _of indexing behavior in pandas, we propose to:_ _1. Introduce ``arr.oindex[indices]`` which allows array indices, but_ _uses outer indexing logic._ _2. Introduce ``arr.vindex[indices]`` which use the current_ _"vectorized"/broadcasted logic but with two differences from_ _legacy indexing:_ _* Boolean indices are not supported. All indices must be integers,_ _integer arrays or slices._ _* The integer index result dimensions are always the first axes_ _of the result array. No transpose is done, even for a single_ _integer array index._ _3. Plain indexing on arrays will start to give warnings and eventually_ _errors in cases where one of the explicit indexers should be preferred:_ _* First, in all cases where legacy and outer indexing would give_ _different results._ _* Later, potentially in all cases involving an integer array._ _These constraints are sufficient for making indexing generally consistent_ _with expectations and providing a less surprising learning curve with_ _``oindex``._ _Note that all things mentioned here apply both for assignment as well as_ _subscription._ _Understanding these details is *not* easy. The Examplessection in the_ _discussion gives code examples._ _And the hopefully easierMotivational Exampleprovides some_ _motivational use-cases for the general ideas and is likely a good start for_ _anyone not intimately familiar with advanced indexing._ _Detailed Description_ _--------------------_ _Proposed rules_ _~~~~~~~~~~~~~~_ _From the three problems noted above some expectations for NumPy can_ _be deduced:_ _1. There should be a prominent outer/orthogonal indexing method such as_ _``arr.oindex[indices]``._ _2. Considering how confusing vectorized/fancy indexing can be, it should_ _be possible to be made more explicitly (e.g. ``arr.vindex[indices]``)._ _3. A new ``arr.vindex[indices]`` method, would not be tied to the_ _confusing transpose rules of fancy indexing, which is for example_ _needed for the simple case of a single advanced index. Thus,_ _no transposing should be done. The axes created by the integer array_ _indices are always inserted at the front, even for a single index._ _4. Boolean indexing is conceptionally outer indexing. Broadcasting_ _together with other advanced indices in the manner of legacy_ _indexing is generally not helpful or well defined._ _A user who wishes the "``nonzero``" plus broadcast behaviour can thus_ _be expected to do this manually. Thus, ``vindex`` does not need to_ _support boolean index arrays._ _5. An ``arr.legacyindex`` attribute should be implemented to support_ _legacy indexing. This gives a simple way to update existing codebases_ _using legacy indexing, which will make the deprecation of plain indexing_ _behavior easier. The longer name ``legacyindex`` is intentionally_ _chosen_ _to be explicit and discourage its use in new code._ _6. Plain indexing ``arr[...]`` should return an error for ambiguous cases._ _For the beginning, this probably means cases where ``arr[ind]`` and_ _``arr.oindex[ind]`` return different results give deprecation warnings._ _This includes every use of vectorized indexing with multiple integer_ _arrays._ _Due to the transposing behaviour, this means that``arr[0, :,_ _indexarr]``_ _will be deprecated, but ``arr[:, 0, indexarr]`` will not for the time_ _being._ _7. To ensure that existing subclasses ofndarraythat override indexing_ _do not inadvertently revert to default behavior for indexing attributes,_ _these attribute should have explicit checks that disable them if_ _``_getitem_`` or ``_setitem_`` has been overriden._ _Unlike plain indexing, the new indexing attributes are explicitly aimed_ _at higher dimensional indexing, several additional changes should be_ _implemented:_ _* The indexing attributes will enforce exact dimension and indexing match._ _This means that no implicit ellipsis (``...``) will be added. Unless_ _an ellipsis is present the indexing expression will thus only work for_ _an array with a specific number of dimensions._ _This makes the expression more explicit and safeguards against wrong_ _dimensionality of arrays._ _There should be no implications for "duck typing" compatibility with_ _builtin Python sequences, because Python sequences only support a limited_ _form of "basic indexing" with integers and slices._ _* The current plain indexing allows for the use of non-tuples for_ _multi-dimensional indexing such as ``arr[[slice(None), 2]]``._ _This creates some inconsistencies and thus the indexing attributes_ _should only allow plain python tuples for this purpose._ _(Whether or not this should be the case for plain indexing is a_ _different issue.)_ _* The new attributes should not use getitem to implement setitem,_ _since it is a cludge and not useful for vectorized_ _indexing. (not implemented yet)_ _Open Questions_ _~~~~~~~~~~~~~~_ _* The names ``oindex``, ``vindex`` and ``legacyindex`` are just_ _suggestions at_ _the time of writing this, another name NumPy has used for something like_ _``oindex`` is ``np.ix``. See also below._ _* ``oindex`` and ``vindex`` could always return copies, even when no array_ _operation occurs. One argument for allowing a view return is that this_ _way_ _``oindex`` can be used as a general index replacement._ _However, there is one argument for returning copies. It is possible for_ _``arr.vindex[arrayscalar, ...]``, where ``arrayscalar`` should be_ _a 0-D array but is not, since 0-D arrays tend to be converted._ _Copying always "fixes" this possible inconsistency._ _* The final state to morph plain indexing in is not fixed in this PEP._ _It is for example possible thatarr[index]`` will be equivalent to arr.oindex at some point in the future. Since such a change will take years, it seems unnecessary to make specific decisions at this time. * The proposed changes to plain indexing could be postponed indefinitely or not taken in order to not break or force major fixes to existing code bases. Alternative Names ~~~~~~~~~~~~~~~~~ Possible names suggested (more suggestions will be added). ============== ============ ======== Orthogonal oindex oix Vectorized vindex vix Legacy legacyindex l/findex ============== ============ ======== Subclasses ~~~~~~~~~~ Subclasses are a bit problematic in the light of these changes. There are some possible solutions for this. For most subclasses (those which do not provide _getitem_ or _setitem_) the special attributes should just work. Subclasses that do provide it must be updated accordingly and should preferably not subclass working versions of these attributes. All subclasses will inherit the attributes, however, the implementation of _getitem_ on these attributes should test subclass._getitem_ is ndarray._getitem_. If not, the subclass has special handling for indexing and NotImplementedError should be raised, requiring that the indexing attributes is also explicitly overwritten. Likewise, implementations of _setitem_ should check to see if _setitem_ is overriden. A further question is how to facilitate implementing the special attributes. Also there is the weird functionality where _setitem_ calls _getitem_ for non-advanced indices. It might be good to avoid it for the new attributes, but on the other hand, that may make it even more confusing. To facilitate implementations we could provide functions similar to operator.itemgetter and operator.setitem for the attributes. Possibly a mixin could be provided to help implementation. These improvements are not essential to the initial implementation, so they are saved for future work. Implementation -------------- Implementation would start with writing special indexing objects available through arr.oindex, arr.vindex, and arr.legacyindex to allow these indexing operations. Also, we would need to start to deprecate those plain index operations which are not ambiguous. Furthermore, the NumPy code base will need to use the new attributes and tests will have to be adapted. Backward compatibility ---------------------- As a new feature, no backward compatibility issues with the new vindex and oindex attributes would arise. To facilitate backwards compatibility as much as possible, we expect a long deprecation cycle for legacy indexing behavior and propose the new legacyindex attribute. Some forward compatibility issues with subclasses that do not specifically implement the new methods may arise. Alternatives ------------ NumPy may not choose to offer these different type of indexing methods, or choose to only offer them through specific functions instead of the proposed notation above. We don't think that new functions are a good alternative, because indexing notation [] offer some syntactic advantages in Python (i.e., direct creation of slice objects) compared to functions. A more reasonable alternative would be write new wrapper objects for alternative indexing with functions rather than methods (e.g., np.oindex(arr)[indices] instead of arr.oindex[indices]). Functionally, this would be equivalent, but indexing is such a common operation that we think it is important to minimize syntax and worth implementing it directly on ndarray objects themselves. Indexing attributes also define a clear interface that is easier for alternative array implementations to copy, nonwithstanding ongoing efforts to make it easier to override NumPy functions [2]. Discussion ---------- The original discussion about vectorized vs outer/orthogonal indexing arose on the NumPy mailing list: * https://mail.python.org/pipermail/numpy-discussion/2015-April/072550.html Some discussion can be found on the original pull request for this NEP: * https://github.com/numpy/numpy/pull/6256 Python implementations of the indexing operations can be found at: * https://github.com/numpy/numpy/pull/5749 * https://gist.github.com/shoyer/c700193625347eb68fee4d1f0dc8c0c8 Examples ~~~~~~~~ Since the various kinds of indexing is hard to grasp in many cases, these examples hopefully give some more insights. Note that they are all in terms of shape. In the examples, all original dimensions have 5 or more elements, advanced indexing inserts smaller dimensions. These examples may be hard to grasp without working knowledge of advanced indexing as of NumPy 1.9. Example array:: >>> arr = np.ones((5, 6, 7, 8)) Legacy fancy indexing --------------------- Note that the same result can be achieved with arr.legacyindex, but the "future error" will still work in this case. Single index is transposed (this is the same for all indexing types):: >>> arr[[0], ...].shape (1, 6, 7, 8) >>> arr[:, [0], ...].shape (5, 1, 7, 8) Multiple indices are transposed if consecutive:: >>> arr[:, [0], [0], :].shape # future error (5, 1, 8) >>> arr[:, [0], :, [0]].shape # future error (1, 5, 7) It is important to note that a scalar is integer array index in this sense (and gets broadcasted with the other advanced index):: >>> arr[:, [0], 0, :].shape (5, 1, 8) >>> arr[:, [0], :, 0].shape # future error (scalar is "fancy") (1, 5, 7) Single boolean index can act on multiple dimensions (especially the whole array). It has to match (as of 1.10. a deprecation warning) the dimensions. The boolean index is otherwise identical to (multiple consecutive) integer array indices:: >>> # Create boolean index with one True value for the last two dimensions: >>> bindx = np.zeros((7, 8), dtype=np.bool) >>> bindx[0, 0] = True >>> arr[:, 0, bindx].shape (5, 1) >>> arr[0, :, bindx].shape (1, 6) The combination with anything that is not a scalar is confusing, e.g.:: >>> arr[[0], :, bindx].shape # bindx result broadcasts with [0] (1, 6) >>> arr[:, [0, 1], bindx].shape # IndexError Outer indexing -------------- Multiple indices are "orthogonal" and their result axes are inserted at the same place (they are not broadcasted):: >>> arr.oindex[:, [0], [0, 1], :].shape (5, 1, 2, 8) >>> arr.oindex[:, [0], :, [0, 1]].shape (5, 1, 7, 2) >>> arr.oindex[:, [0], 0, :].shape (5, 1, 8) >>> arr.oindex[:, [0], :, 0].shape (5, 1, 7) Boolean indices results are always inserted where the index is:: >>> # Create boolean index with one True value for the last two dimensions: >>> bindx = np.zeros((7, 8), dtype=np.bool) >>> bindx[0, 0] = True >>> arr.oindex[:, 0, bindx].shape (5, 1) >>> arr.oindex[0, :, bindx].shape (6, 1) Nothing changed in the presence of other advanced indices since:: >>> arr.oindex[[0], :, bindx].shape (1, 6, 1) >>> arr.oindex[:, [0, 1], bindx].shape (5, 2, 1) Vectorized/inner indexing ------------------------- Multiple indices are broadcasted and iterated as one like fancy indexing, but the new axes area always inserted at the front:: >>> arr.vindex[:, [0], [0, 1], :].shape (2, 5, 8) >>> arr.vindex[:, [0], :, [0, 1]].shape (2, 5, 7) >>> arr.vindex[:, [0], 0, :].shape (1, 5, 8) >>> arr.vindex[:, [0], :, 0].shape (1, 5, 7) Boolean indices results are always inserted where the index is, exactly as in oindex given how specific they are to the axes they operate on:: >>> # Create boolean index with one True value for the last two dimensions: >>> bindx = np.zeros((7, 8), dtype=np.bool) >>> bindx[0, 0] = True >>> arr.vindex[:, 0, bindx].shape (5, 1) >>> arr.vindex[0, :, bindx].shape (6, 1) But other advanced indices are again transposed to the front:: >>> arr.vindex[[0], :, bindx].shape (1, 6, 1) >>> arr.vindex[:, [0, 1], bindx].shape (2, 5, 1) Motivational Example ~~~~~~~~~~~~~~~~~~~~ Imagine having a data acquisition software storing D channels and N datapoints along the time. She stores this into an (N, D) shaped array. During data analysis, we needs to fetch a pool of channels, for example to calculate a mean over them. This data can be faked using:: >>> arr = np.random.random((100, 10)) Now one may remember indexing with an integer array and find the correct code:: >>> group = arr[:, [2, 5]] >>> meanvalue = arr.mean() However, assume that there were some specific time points (first dimension of the data) that need to be specially considered. These time points are already known and given by:: >>> interestingtimes = np.array([1, 5, 8, 10], dtype=np.intp) Now to fetch them, we may try to modify the previous code:: >>> groupatit = arr[interestingtimes, [2, 5]] IndexError: Ambiguous index, use .oindex or .vindex An error such as this will point to read up the indexing documentation. This should make it clear, that oindex behaves more like slicing. So, out of the different methods it is the obvious choice (for now, this is a shape mismatch, but that could possibly also mention oindex):: >>> groupatit = arr.oindex[interestingtimes, [2, 5]] Now of course one could also have used vindex, but it is much less obvious how to achieve the right thing!:: >>> reshapedtimes = interestingtimes[:, np.newaxis] >>> groupatit = arr.vindex[reshapedtimes, [2, 5]] One may find, that for example our data is corrupt in some places. So, we need to replace these values by zero (or anything else) for these times. The first column may for example give the necessary information, so that changing the values becomes easy remembering boolean indexing:: >>> baddata = arr[:, 0] > 0.5 >>> arr[baddata, :] = 0 # (corrupts further examples) Again, however, the columns may need to be handled more individually (but in groups), and the oindex attribute works well:: >>> arr.oindex[baddata, [2, 5]] = 0 Note that it would be very hard to do this using legacy fancy indexing. The only way would be to create an integer array first:: >>> baddataindx = np.nonzero(baddata)[0] >>> baddataindxreshaped = baddataindx[:, np.newaxis] >>> arr[baddataindxreshaped, [2, 5]] In any case we can use only oindex to do all of this without getting into any trouble or confused by the whole complexity of advanced indexing. But, some new features are added to the data acquisition. Different sensors have to be used depending on the times. Let us assume we already have created an array of indices:: >>> correctsensors = np.random.randint(10, size=(100, 2)) Which lists for each time the two correct sensors in an (N, 2) array. A first try to achieve this may be arr[:, correctsensors] and this does not work. It should be clear quickly that slicing cannot achieve the desired thing. But hopefully users will remember that there is vindex as a more powerful and flexible approach to advanced indexing. One may, if trying vindex randomly, be confused about:: >>> newarr = arr.vindex[:, correctsensors] which is neither the same, nor the correct result (see transposing rules)! This is because slicing works still the same in vindex. However, reading the documentation and examples, one can hopefully quickly find the desired solution:: >>> rows = np.arange(len(arr)) >>> rows = rows[:, np.newaxis] # make shape fit with correctsensors >>> newarr = arr.vindex[rows, correctsensors] At this point we have left the straight forward world of oindex but can do random picking of any element from the array. Note that in the last example a method such as mentioned in the Related Questions section could be more straight forward. But this approach is even more flexible, since rows does not have to be a simple arange, but could be interstingtimes:: >>> interestingtimes = np.array([0, 4, 8, 9, 10]) >>> correctsensorsatit = correctsensors[interestingtimes, :] >>> interestingtimesreshaped = interestingtimes[:, np.newaxis] >>> newarrit = arr[interestingtimesreshaped, correctsensorsatit] Truly complex situation would arise now if you would for example pool L experiments into an array shaped (L, N, D). But for oindex this should not result into surprises. vindex, being more powerful, will quite certainly create some confusion in this case but also cover pretty much all eventualities. Copyright --------- This document is placed under the CC0 1.0 Universell (CC0 1.0) Public Domain Dedication [1]. References and Footnotes ------------------------ .. [1] To the extent possible under law, the person who associated CC0 with this work has waived all copyright and related or neighboring rights to this work. The CC0 license may be found at https://creativecommons.org/publicdomain/zero/1.0/ .. [2] e.g., see NEP 18, http://www.numpy.org/neps/nep-0018-array-function-protocol.html

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