ndarrays 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.
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
start:stop:step notation inside of brackets), an
integer, or a tuple of slice objects and integers.
newaxis objects can be interspersed with these as
Deprecated since version 1.15.0: 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 and non-tuple sequence (such as a
slice objects, the
Ellipsis object, or the
object, but not for integer arrays or other embedded sequences.
The simplest case of indexing with N integers returns an array scalar representing the corresponding item. As in Python, all indices are zero-based: for the i-th index \(n_i\), the valid range is \(0 \le n_i < d_i\) where \(d_i\) 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 \(n_i < 0\), it means \(n_i + d_i\)).
All arrays generated by basic slicing are always views of the original array.
NumPy slicing creates a view instead of a copy as in the case of
builtin Python sequences such as string, tuple and list.
Care must be taken when extracting
a small portion from a large array which becomes useless after the
extraction, because the small portion extracted contains a reference
to the large original array whose memory will not be released until
all arrays derived from it are garbage-collected. In such cases an
copy() is recommended.
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:kwhere i is the starting index, j is the stopping index, and k is the step (\(k\neq0\)). This selects the m elements (in the corresponding dimension) with index values i, i + k, …, i + (m - 1) k where \(m = q + (r\neq0)\) 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.
>>> 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.
>>> 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.
>>> 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.
>>> x = np.array([[,,], [,,]]) >>> x.shape (2, 3, 1) >>> x[1:2] array([[, , ]])
Ellipsisexpands to the number of
:objects needed for the selection tuple to index all dimensions. In most cases, this means that length of the expanded selection tuple is
x.ndim. There may only be a single ellipsis present.
>>> x[...,0] array([[1, 2, 3], [4, 5, 6]])
newaxisobject 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
newaxisobject in the selection tuple.
>>> x[:,np.newaxis,:,:].shape (2, 1, 3, 1)
An integer, i, returns the same values as
i:i+1except 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 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
x[ind1][...,ind2,:]under basic slicing.
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] = valuemust be (broadcastable) to the same shape as
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
notation. For example,
x[1:10:5,::-1] can also be implemented
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.
Advanced indexing is triggered when the selection object, obj, is a
non-tuple sequence object, an
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
Advanced indexing always returns a copy of the data (contrast with basic slicing that returns a view).
The definition of advanced indexing means that
fundamentally different than
x[(1,2,3)]. The latter is
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 due to the deprecated Numeric compatibility mentioned above,
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 broadcast 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
ind_1, ..., ind_N.
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
ix_ can help with this broadcasting. This is best
understood with an 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 = np.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([, ]) >>> x[rows[:, np.newaxis], columns] array([[ 0, 2], [ 9, 11]])
This broadcasting can also be achieved using the function
>>> 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
Combining advanced and basic indexing¶
When there is at least one slice (
:), ellipsis (
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.
>>> 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,
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
1is 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).
x.shape is (10,20,30) and
ind is a (2,3,4)-shaped
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
x.shape be (10,20,30,40,50) and suppose
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
.transpose() to move the subspace
anywhere desired. Note that this example cannot be replicated
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
as described above,
obj.nonzero() returns a
tuple (of length
obj.ndim) of integer index
arrays showing the
True elements of obj. However, it is
obj.shape == x.shape.
obj.ndim == x.ndim,
x[obj] returns a 1-dimensional array
filled with the elements of x corresponding to the
values of obj. The search order will be row-major,
C-style. If obj has
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
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.
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]])
Combining multiple Boolean indexing arrays or a Boolean with an integer
indexing array can best be understood with the
obj.nonzero() analogy. The function
also supports boolean arrays and will work without any surprises.
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
ix_ function this can be done
>>> x = np.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]])
np.ix_ call, only the diagonal elements would be
np.ix_ (compare the integer array examples):
>>> rows = rows.nonzero() >>> x[rows[:, np.newaxis], columns] array([[ 3, 5], [ 9, 11]])
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
intpand may differ from the default integer array type.
intpis 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
xis 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.
nonzeroequivalence 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
IndexErroris raised is undefined (e.g.
123being 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.
ndarray object is a structured array the fields
of the array can be accessed by indexing the array with strings,
x['field-name'] returns a new 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
record array scalars can be “indexed” this way.
Indexing into a structured array can also be done with a list of field names,
x[['field-name1','field-name2']]. As of NumPy 1.16 this returns a
view containing only those fields. In older versions of numpy it returned a
copy. See the user guide section on Structured arrays for more
information on multifield indexing.
If the accessed field is a sub-array, the dimensions of the sub-array are appended to the shape of the result.
>>> 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¶
x.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
x.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.