NumPy

NEP 21 — Simplified and explicit advanced indexing

Author:Sebastian Berg
Author:Stephan Hoyer <shoyer@google.com>
Status:Draft
Type:Standards Track
Created:2015-08-27

Abstract

NumPy’s “advanced” indexing support for indexing array 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 Examples section in the discussion gives code examples. And the hopefully easier Motivational Example provides 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.legacy_index 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 legacy_index 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, :, index_arr]`` will be deprecated, but arr[:, 0, index_arr] will not for the time being.
  7. 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.

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 legacy_index 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[array_scalar, ...], where array_scalar 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 that arr[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 legacy_index 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 oindex and vindex.

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.legacy_index 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 legacy_index 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:

Some discussion can be found on the original pull request for this NEP:

Python implementations of the indexing operations can be found at:

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.legacy_index, 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 are 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]]
>>> mean_value = 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:

>>> interesting_times = np.array([1, 5, 8, 10], dtype=np.intp)

Now to fetch them, we may try to modify the previous code:

>>> group_at_it = arr[interesting_times, [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):

>>> group_at_it = arr.oindex[interesting_times, [2, 5]]

Now of course one could also have used vindex, but it is much less obvious how to achieve the right thing!:

>>> reshaped_times = interesting_times[:, np.newaxis]
>>> group_at_it = arr.vindex[reshaped_times, [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:

>>> bad_data = arr[:, 0] > 0.5
>>> arr[bad_data, :] = 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[bad_data, [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:

>>> bad_data_indx = np.nonzero(bad_data)[0]
>>> bad_data_indx_reshaped = bad_data_indx[:, np.newaxis]
>>> arr[bad_data_indx_reshaped, [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:

>>> correct_sensors = 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[:, correct_sensors] 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:

>>> new_arr = arr.vindex[:, correct_sensors]

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 correct_sensors
>>> new_arr = arr.vindex[rows, correct_sensors]

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 interesting_times:

>>> interesting_times = np.array([0, 4, 8, 9, 10])
>>> correct_sensors_at_it = correct_sensors[interesting_times, :]
>>> interesting_times_reshaped = interesting_times[:, np.newaxis]
>>> new_arr_it = arr[interesting_times_reshaped, correct_sensors_at_it]

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.

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