.. _arrays.indexing:
Indexing
========
.. sectionauthor:: adapted from "Guide to NumPy" by Travis E. Oliphant
.. currentmodule:: numpy
.. index:: indexing, slicing
:class:`ndarrays <ndarray>` can be indexed using the standard Python
``x[obj]`` syntax, where *x* is the array and *obj* the selection.
There are three kinds of indexing available: field access, basic
slicing, advanced indexing. Which one occurs depends on *obj*.
.. note::
In Python, ``x[(exp1, exp2, ..., expN)]`` is equivalent to
``x[exp1, exp2, ..., expN]``; the latter is just syntactic sugar
for the former.
Basic Slicing and Indexing
--------------------------
Basic slicing extends Python's basic concept of slicing to N
dimensions. Basic slicing occurs when *obj* is a :class:`slice` object
(constructed by ``start:stop:step`` notation inside of brackets), an
integer, or a tuple of slice objects and integers. :const:`Ellipsis`
and :const:`newaxis` objects can be interspersed with these as
well. In order to remain backward compatible with a common usage in
Numeric, basic slicing is also initiated if the selection object is
any non-ndarray sequence (such as a :class:`list`) containing :class:`slice`
objects, the :const:`Ellipsis` object, or the :const:`newaxis` object,
but not for integer arrays or other embedded sequences.
.. index::
triple: ndarray; special methods; getitem
triple: ndarray; special methods; setitem
single: ellipsis
single: newaxis
The simplest case of indexing with *N* integers returns an :ref:`array
scalar <arrays.scalars>` representing the corresponding item. As in
Python, all indices are zero-based: for the *i*-th index :math:`n_i`,
the valid range is :math:`0 \le n_i < d_i` where :math:`d_i` is the
*i*-th element of the shape of the array. Negative indices are
interpreted as counting from the end of the array (*i.e.*, if
:math:`n_i < 0`, it means :math:`n_i + d_i`).
All arrays generated by basic slicing are always :term:`views <view>`
of the original array.
The standard rules of sequence slicing apply to basic slicing on a
per-dimension basis (including using a step index). Some useful
concepts to remember include:
- The basic slice syntax is ``i:j:k`` where *i* is the starting index,
*j* is the stopping index, and *k* is the step (:math:`k\neq0`).
This selects the *m* elements (in the corresponding dimension) with
index values *i*, *i + k*, ..., *i + (m - 1) k* where
:math:`m = q + (r\neq0)` and *q* and *r* are the quotient and remainder
obtained by dividing *j - i* by *k*: *j - i = q k + r*, so that
*i + (m - 1) k < j*.
.. admonition:: Example
>>> x = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> x[1:7:2]
array([1, 3, 5])
- Negative *i* and *j* are interpreted as *n + i* and *n + j* where
*n* is the number of elements in the corresponding dimension.
Negative *k* makes stepping go towards smaller indices.
.. admonition:: Example
>>> x[-2:10]
array([8, 9])
>>> x[-3:3:-1]
array([7, 6, 5, 4])
- Assume *n* is the number of elements in the dimension being
sliced. Then, if *i* is not given it defaults to 0 for *k > 0* and
*n - 1* for *k < 0* . If *j* is not given it defaults to *n* for *k > 0*
and *-n-1* for *k < 0* . If *k* is not given it defaults to 1. Note that
``::`` is the same as ``:`` and means select all indices along this
axis.
.. admonition:: Example
>>> x[5:]
array([5, 6, 7, 8, 9])
- If the number of objects in the selection tuple is less than
*N* , then ``:`` is assumed for any subsequent dimensions.
.. admonition:: Example
>>> x = np.array([[[1],[2],[3]], [[4],[5],[6]]])
>>> x.shape
(2, 3, 1)
>>> x[1:2]
array([[[4],
[5],
[6]]])
- :const:`Ellipsis` expand to the number of ``:`` objects needed to
make a selection tuple of the same length as ``x.ndim``. There may
only be a single ellipsis present.
.. admonition:: Example
>>> x[...,0]
array([[1, 2, 3],
[4, 5, 6]])
- Each :const:`newaxis` object in the selection tuple serves to expand
the dimensions of the resulting selection by one unit-length
dimension. The added dimension is the position of the :const:`newaxis`
object in the selection tuple.
.. admonition:: Example
>>> x[:,np.newaxis,:,:].shape
(2, 1, 3, 1)
- An integer, *i*, returns the same values as ``i:i+1``
**except** the dimensionality of the returned object is reduced by
1. In particular, a selection tuple with the *p*-th
element an integer (and all other entries ``:``) returns the
corresponding sub-array with dimension *N - 1*. If *N = 1*
then the returned object is an array scalar. These objects are
explained in :ref:`arrays.scalars`.
- If the selection tuple has all entries ``:`` except the
*p*-th entry which is a slice object ``i:j:k``,
then the returned array has dimension *N* formed by
concatenating the sub-arrays returned by integer indexing of
elements *i*, *i+k*, ..., *i + (m - 1) k < j*,
- Basic slicing with more than one non-``:`` entry in the slicing
tuple, acts like repeated application of slicing using a single
non-``:`` entry, where the non-``:`` entries are successively taken
(with all other non-``:`` entries replaced by ``:``). Thus,
``x[ind1,...,ind2,:]`` acts like ``x[ind1][...,ind2,:]`` under basic
slicing.
.. warning:: The above is **not** true for advanced indexing.
- You may use slicing to set values in the array, but (unlike lists) you
can never grow the array. The size of the value to be set in
``x[obj] = value`` must be (broadcastable) to the same shape as
``x[obj]``.
.. index::
pair: ndarray; view
.. note::
Remember that a slicing tuple can always be constructed as *obj*
and used in the ``x[obj]`` notation. Slice objects can be used in
the construction in place of the ``[start:stop:step]``
notation. For example, ``x[1:10:5,::-1]`` can also be implemented
as ``obj = (slice(1,10,5), slice(None,None,-1)); x[obj]`` . This
can be useful for constructing generic code that works on arrays
of arbitrary dimension.
.. data:: newaxis
The :const:`newaxis` object can be used in all slicing operations to
create an axis of length one. :const:`newaxis` is an alias for
'None', and 'None' can be used in place of this with the same result.
Advanced Indexing
-----------------
Advanced indexing is triggered when the selection object, *obj*, is a
non-tuple sequence object, an :class:`ndarray` (of data type integer or bool),
or a tuple with at least one sequence object or ndarray (of data type
integer or bool). There are two types of advanced indexing: integer
and Boolean.
Advanced indexing always returns a *copy* of the data (contrast with
basic slicing that returns a :term:`view`).
.. warning::
The definition of advanced indexing means that ``x[(1,2,3),]`` is
fundamentally different than ``x[(1,2,3)]``. The latter is
equivalent to ``x[1,2,3]`` which will trigger basic selection while
the former will trigger advanced indexing. Be sure to understand
why this occurs.
Also recognize that ``x[[1,2,3]]`` will trigger advanced indexing,
whereas ``x[[1,2,slice(None)]]`` will trigger basic slicing.
Integer array indexing
^^^^^^^^^^^^^^^^^^^^^^
Integer array indexing allows selection of arbitrary items in the array
based on their *N*-dimensional index. Each integer array represents a number
of indexes into that dimension.
Purely integer array indexing
"""""""""""""""""""""""""""""
When the index consists of as many integer arrays as the array being indexed
has dimensions, the indexing is straight forward, but different from slicing.
Advanced indexes always are :ref:`broadcast<ufuncs.broadcasting>` and
iterated as *one*::
result[i_1, ..., i_M] == x[ind_1[i_1, ..., i_M], ind_2[i_1, ..., i_M],
..., ind_N[i_1, ..., i_M]]
Note that the result shape is identical to the (broadcast) indexing array
shapes ``ind_1, ..., ind_N``.
.. admonition:: Example
From each row, a specific element should be selected. The row index is just
``[0, 1, 2]`` and the column index specifies the element to choose for the
corresponding row, here ``[0, 1, 0]``. Using both together the task
can be solved using advanced indexing:
>>> x = np.array([[1, 2], [3, 4], [5, 6]])
>>> x[[0, 1, 2], [0, 1, 0]]
array([1, 4, 5])
To achieve a behaviour similar to the basic slicing above, broadcasting can be
used. The function :func:`ix_` can help with this broadcasting. This is best
understood with an example.
.. admonition:: Example
From a 4x3 array the corner elements should be selected using advanced
indexing. Thus all elements for which the column is one of ``[0, 2]`` and
the row is one of ``[0, 3]`` need to be selected. To use advanced indexing
one needs to select all elements *explicitly*. Using the method explained
previously one could write:
>>> x = array([[ 0, 1, 2],
... [ 3, 4, 5],
... [ 6, 7, 8],
... [ 9, 10, 11]])
>>> rows = np.array([[0, 0],
... [3, 3]], dtype=np.intp)
>>> columns = np.array([[0, 2],
... [0, 2]], dtype=np.intp)
>>> x[rows, columns]
array([[ 0, 2],
[ 9, 11]])
However, since the indexing arrays above just repeat themselves,
broadcasting can be used (compare operations such as
``rows[:, np.newaxis] + columns``) to simplify this:
>>> rows = np.array([0, 3], dtype=np.intp)
>>> columns = np.array([0, 2], dtype=np.intp)
>>> rows[:, np.newaxis]
array([[0],
[3]])
>>> x[rows[:, np.newaxis], columns]
array([[ 0, 2],
[ 9, 11]])
This broadcasting can also be achieved using the function :func:`ix_`:
>>> x[np.ix_(rows, columns)]
array([[ 0, 2],
[ 9, 11]])
Note that without the ``np.ix_`` call, only the diagonal elements would
be selected, as was used in the previous example. This difference is the
most important thing to remember about indexing with multiple advanced
indexes.
Combining advanced and basic indexing
"""""""""""""""""""""""""""""""""""""
When there is at least one slice (``:``), ellipsis (``...``) or ``np.newaxis``
in the index (or the array has more dimensions than there are advanced indexes),
then the behaviour can be more complicated. It is like concatenating the
indexing result for each advanced index element
In the simplest case, there is only a *single* advanced index. A single
advanced index can for example replace a slice and the result array will be
the same, however, it is a copy and may have a different memory layout.
A slice is preferable when it is possible.
.. admonition:: Example
>>> x[1:2, 1:3]
array([[4, 5]])
>>> x[1:2, [1, 2]]
array([[4, 5]])
The easiest way to understand the situation may be to think in
terms of the result shape. There are two parts to the indexing operation,
the subspace defined by the basic indexing (excluding integers) and the
subspace from the advanced indexing part. Two cases of index combination
need to be distinguished:
* The advanced indexes are separated by a slice, ellipsis or newaxis.
For example ``x[arr1, :, arr2]``.
* The advanced indexes are all next to each other.
For example ``x[..., arr1, arr2, :]`` but *not* ``x[arr1, :, 1]``
since ``1`` is an advanced index in this regard.
In the first case, the dimensions resulting from the advanced indexing
operation come first in the result array, and the subspace dimensions after
that.
In the second case, the dimensions from the advanced indexing operations
are inserted into the result array at the same spot as they were in the
initial array (the latter logic is what makes simple advanced indexing
behave just like slicing).
.. admonition:: Example
Suppose ``x.shape`` is (10,20,30) and ``ind`` is a (2,3,4)-shaped
indexing :class:`intp` array, then ``result = x[...,ind,:]`` has
shape (10,2,3,4,30) because the (20,)-shaped subspace has been
replaced with a (2,3,4)-shaped broadcasted indexing subspace. If
we let *i, j, k* loop over the (2,3,4)-shaped subspace then
``result[...,i,j,k,:] = x[...,ind[i,j,k],:]``. This example
produces the same result as :meth:`x.take(ind, axis=-2) <ndarray.take>`.
.. admonition:: Example
Let ``x.shape`` be (10,20,30,40,50) and suppose ``ind_1``
and ``ind_2`` can be broadcast to the shape (2,3,4). Then
``x[:,ind_1,ind_2]`` has shape (10,2,3,4,40,50) because the
(20,30)-shaped subspace from X has been replaced with the
(2,3,4) subspace from the indices. However,
``x[:,ind_1,:,ind_2]`` has shape (2,3,4,10,30,50) because there
is no unambiguous place to drop in the indexing subspace, thus
it is tacked-on to the beginning. It is always possible to use
:meth:`.transpose() <ndarray.transpose>` to move the subspace
anywhere desired. Note that this example cannot be replicated
using :func:`take`.
Boolean array indexing
^^^^^^^^^^^^^^^^^^^^^^
This advanced indexing occurs when obj is an array object of Boolean
type, such as may be returned from comparison operators. A single
boolean index array is practically identical to ``x[obj.nonzero()]`` where,
as described above, :meth:`obj.nonzero() <ndarray.nonzero>` returns a
tuple (of length :attr:`obj.ndim <ndarray.ndim>`) of integer index
arrays showing the :const:`True` elements of *obj*. However, it is
faster when ``obj.shape == x.shape``.
If ``obj.ndim == x.ndim``, ``x[obj]`` returns a 1-dimensional array
filled with the elements of *x* corresponding to the :const:`True`
values of *obj*. The search order will be :term:`row-major`,
C-style. If *obj* has :const:`True` values at entries that are outside
of the bounds of *x*, then an index error will be raised. If *obj* is
smaller than *x* it is identical to filling it with :const:`False`.
.. admonition:: Example
A common use case for this is filtering for desired element values.
For example one may wish to select all entries from an array which
are not NaN:
>>> x = np.array([[1., 2.], [np.nan, 3.], [np.nan, np.nan]])
>>> x[~np.isnan(x)]
array([ 1., 2., 3.])
Or wish to add a constant to all negative elements:
>>> x = np.array([1., -1., -2., 3])
>>> x[x < 0] += 20
>>> x
array([ 1., 19., 18., 3.])
In general if an index includes a Boolean array, the result will be
identical to inserting ``obj.nonzero()`` into the same position
and using the integer array indexing mechanism described above.
``x[ind_1, boolean_array, ind_2]`` is equivalent to
``x[(ind_1,) + boolean_array.nonzero() + (ind_2,)]``.
If there is only one Boolean array and no integer indexing array present,
this is straight forward. Care must only be taken to make sure that the
boolean index has *exactly* as many dimensions as it is supposed to work
with.
.. admonition:: Example
From an array, select all rows which sum up to less or equal two:
>>> x = np.array([[0, 1], [1, 1], [2, 2]])
>>> rowsum = x.sum(-1)
>>> x[rowsum <= 2, :]
array([[0, 1],
[1, 1]])
But if ``rowsum`` would have two dimensions as well:
>>> rowsum = x.sum(-1, keepdims=True)
>>> rowsum.shape
(3, 1)
>>> x[rowsum <= 2, :] # fails
IndexError: too many indices
>>> x[rowsum <= 2]
array([0, 1])
The last one giving only the first elements because of the extra dimension.
Compare ``rowsum.nonzero()`` to understand this example.
Combining multiple Boolean indexing arrays or a Boolean with an integer
indexing array can best be understood with the
:meth:`obj.nonzero() <ndarray.nonzero>` analogy. The function :func:`ix_`
also supports boolean arrays and will work without any surprises.
.. admonition:: Example
Use boolean indexing to select all rows adding up to an even
number. At the same time columns 0 and 2 should be selected with an
advanced integer index. Using the :func:`ix_` function this can be done
with:
>>> x = array([[ 0, 1, 2],
... [ 3, 4, 5],
... [ 6, 7, 8],
... [ 9, 10, 11]])
>>> rows = (x.sum(-1) % 2) == 0
>>> rows
array([False, True, False, True])
>>> columns = [0, 2]
>>> x[np.ix_(rows, columns)]
array([[ 3, 5],
[ 9, 11]])
Without the ``np.ix_`` call or only the diagonal elements would be
selected.
Or without ``np.ix_`` (compare the integer array examples):
>>> rows = rows.nonzero()[0]
>>> x[rows[:, np.newaxis], columns]
array([[ 3, 5],
[ 9, 11]])
Detailed notes
--------------
These are some detailed notes, which are not of importance for day to day
indexing (in no particular order):
* The native NumPy indexing type is ``intp`` and may differ from the
default integer array type. ``intp`` is the smallest data type
sufficient to safely index any array; for advanced indexing it may be
faster than other types.
* For advanced assignments, there is in general no guarantee for the
iteration order. This means that if an element is set more than once,
it is not possible to predict the final result.
* An empty (tuple) index is a full scalar index into a zero dimensional array.
``x[()]`` returns a *scalar* if ``x`` is zero dimensional and a view
otherwise. On the other hand ``x[...]`` always returns a view.
* If a zero dimensional array is present in the index *and* it is a full
integer index the result will be a *scalar* and not a zero dimensional array.
(Advanced indexing is not triggered.)
* When an ellipsis (``...``) is present but has no size (i.e. replaces zero
``:``) the result will still always be an array. A view if no advanced index
is present, otherwise a copy.
* the ``nonzero`` equivalence for Boolean arrays does not hold for zero
dimensional boolean arrays.
* When the result of an advanced indexing operation has no elements but an
individual index is out of bounds, whether or not an ``IndexError`` is
raised is undefined (e.g. ``x[[], [123]]`` with ``123`` being out of bounds).
* When a *casting* error occurs during assignment (for example updating a
numerical array using a sequence of strings), the array being assigned
to may end up in an unpredictable partially updated state.
However, if any other error (such as an out of bounds index) occurs, the
array will remain unchanged.
* The memory layout of an advanced indexing result is optimized for each
indexing operation and no particular memory order can be assumed.
* When using a subclass (especially one which manipulates its shape), the
default ``ndarray.__setitem__`` behaviour will call ``__getitem__`` for
*basic* indexing but not for *advanced* indexing. For such a subclass it may
be preferable to call ``ndarray.__setitem__`` with a *base class* ndarray
view on the data. This *must* be done if the subclasses ``__getitem__`` does
not return views.
.. _arrays.indexing.fields:
Field Access
-------------
.. seealso:: :ref:`arrays.dtypes`, :ref:`arrays.scalars`
If the :class:`ndarray` object is a structured array the :term:`fields <field>`
of the array can be accessed by indexing the array with strings,
dictionary-like.
Indexing ``x['field-name']`` returns a new :term:`view` to the array,
which is of the same shape as *x* (except when the field is a
sub-array) but of data type ``x.dtype['field-name']`` and contains
only the part of the data in the specified field. Also
:ref:`record array <arrays.classes.rec>` scalars can be "indexed" this way.
Indexing into a structured array can also be done with a list of field names,
*e.g.* ``x[['field-name1','field-name2']]``. Currently this returns a new
array containing a copy of the values in the fields specified in the list.
As of NumPy 1.7, returning a copy is being deprecated in favor of returning
a view. A copy will continue to be returned for now, but a FutureWarning
will be issued when writing to the copy. If you depend on the current
behavior, then we suggest copying the returned array explicitly, i.e. use
x[['field-name1','field-name2']].copy(). This will work with both past and
future versions of NumPy.
If the accessed field is a sub-array, the dimensions of the sub-array
are appended to the shape of the result.
.. admonition:: Example
>>> x = np.zeros((2,2), dtype=[('a', np.int32), ('b', np.float64, (3,3))])
>>> x['a'].shape
(2, 2)
>>> x['a'].dtype
dtype('int32')
>>> x['b'].shape
(2, 2, 3, 3)
>>> x['b'].dtype
dtype('float64')
Flat Iterator indexing
----------------------
:attr:`x.flat <ndarray.flat>` returns an iterator that will iterate
over the entire array (in C-contiguous style with the last index
varying the fastest). This iterator object can also be indexed using
basic slicing or advanced indexing as long as the selection object is
not a tuple. This should be clear from the fact that :attr:`x.flat
<ndarray.flat>` is a 1-dimensional view. It can be used for integer
indexing with 1-dimensional C-style-flat indices. The shape of any
returned array is therefore the shape of the integer indexing object.
.. index::
single: indexing
single: ndarray