mirror of
https://github.com/PiBrewing/craftbeerpi4.git
synced 2024-11-29 18:24:14 +01:00
456 lines
16 KiB
Python
456 lines
16 KiB
Python
"""
|
|
==============
|
|
Array indexing
|
|
==============
|
|
|
|
Array indexing refers to any use of the square brackets ([]) to index
|
|
array values. There are many options to indexing, which give numpy
|
|
indexing great power, but with power comes some complexity and the
|
|
potential for confusion. This section is just an overview of the
|
|
various options and issues related to indexing. Aside from single
|
|
element indexing, the details on most of these options are to be
|
|
found in related sections.
|
|
|
|
Assignment vs referencing
|
|
=========================
|
|
|
|
Most of the following examples show the use of indexing when
|
|
referencing data in an array. The examples work just as well
|
|
when assigning to an array. See the section at the end for
|
|
specific examples and explanations on how assignments work.
|
|
|
|
Single element indexing
|
|
=======================
|
|
|
|
Single element indexing for a 1-D array is what one expects. It work
|
|
exactly like that for other standard Python sequences. It is 0-based,
|
|
and accepts negative indices for indexing from the end of the array. ::
|
|
|
|
>>> x = np.arange(10)
|
|
>>> x[2]
|
|
2
|
|
>>> x[-2]
|
|
8
|
|
|
|
Unlike lists and tuples, numpy arrays support multidimensional indexing
|
|
for multidimensional arrays. That means that it is not necessary to
|
|
separate each dimension's index into its own set of square brackets. ::
|
|
|
|
>>> x.shape = (2,5) # now x is 2-dimensional
|
|
>>> x[1,3]
|
|
8
|
|
>>> x[1,-1]
|
|
9
|
|
|
|
Note that if one indexes a multidimensional array with fewer indices
|
|
than dimensions, one gets a subdimensional array. For example: ::
|
|
|
|
>>> x[0]
|
|
array([0, 1, 2, 3, 4])
|
|
|
|
That is, each index specified selects the array corresponding to the
|
|
rest of the dimensions selected. In the above example, choosing 0
|
|
means that the remaining dimension of length 5 is being left unspecified,
|
|
and that what is returned is an array of that dimensionality and size.
|
|
It must be noted that the returned array is not a copy of the original,
|
|
but points to the same values in memory as does the original array.
|
|
In this case, the 1-D array at the first position (0) is returned.
|
|
So using a single index on the returned array, results in a single
|
|
element being returned. That is: ::
|
|
|
|
>>> x[0][2]
|
|
2
|
|
|
|
So note that ``x[0,2] = x[0][2]`` though the second case is more
|
|
inefficient as a new temporary array is created after the first index
|
|
that is subsequently indexed by 2.
|
|
|
|
Note to those used to IDL or Fortran memory order as it relates to
|
|
indexing. NumPy uses C-order indexing. That means that the last
|
|
index usually represents the most rapidly changing memory location,
|
|
unlike Fortran or IDL, where the first index represents the most
|
|
rapidly changing location in memory. This difference represents a
|
|
great potential for confusion.
|
|
|
|
Other indexing options
|
|
======================
|
|
|
|
It is possible to slice and stride arrays to extract arrays of the
|
|
same number of dimensions, but of different sizes than the original.
|
|
The slicing and striding works exactly the same way it does for lists
|
|
and tuples except that they can be applied to multiple dimensions as
|
|
well. A few examples illustrates best: ::
|
|
|
|
>>> x = np.arange(10)
|
|
>>> x[2:5]
|
|
array([2, 3, 4])
|
|
>>> x[:-7]
|
|
array([0, 1, 2])
|
|
>>> x[1:7:2]
|
|
array([1, 3, 5])
|
|
>>> y = np.arange(35).reshape(5,7)
|
|
>>> y[1:5:2,::3]
|
|
array([[ 7, 10, 13],
|
|
[21, 24, 27]])
|
|
|
|
Note that slices of arrays do not copy the internal array data but
|
|
only produce new views of the original data. This is different from
|
|
list or tuple slicing and an explicit ``copy()`` is recommended if
|
|
the original data is not required anymore.
|
|
|
|
It is possible to index arrays with other arrays for the purposes of
|
|
selecting lists of values out of arrays into new arrays. There are
|
|
two different ways of accomplishing this. One uses one or more arrays
|
|
of index values. The other involves giving a boolean array of the proper
|
|
shape to indicate the values to be selected. Index arrays are a very
|
|
powerful tool that allow one to avoid looping over individual elements in
|
|
arrays and thus greatly improve performance.
|
|
|
|
It is possible to use special features to effectively increase the
|
|
number of dimensions in an array through indexing so the resulting
|
|
array acquires the shape needed for use in an expression or with a
|
|
specific function.
|
|
|
|
Index arrays
|
|
============
|
|
|
|
NumPy arrays may be indexed with other arrays (or any other sequence-
|
|
like object that can be converted to an array, such as lists, with the
|
|
exception of tuples; see the end of this document for why this is). The
|
|
use of index arrays ranges from simple, straightforward cases to
|
|
complex, hard-to-understand cases. For all cases of index arrays, what
|
|
is returned is a copy of the original data, not a view as one gets for
|
|
slices.
|
|
|
|
Index arrays must be of integer type. Each value in the array indicates
|
|
which value in the array to use in place of the index. To illustrate: ::
|
|
|
|
>>> x = np.arange(10,1,-1)
|
|
>>> x
|
|
array([10, 9, 8, 7, 6, 5, 4, 3, 2])
|
|
>>> x[np.array([3, 3, 1, 8])]
|
|
array([7, 7, 9, 2])
|
|
|
|
|
|
The index array consisting of the values 3, 3, 1 and 8 correspondingly
|
|
create an array of length 4 (same as the index array) where each index
|
|
is replaced by the value the index array has in the array being indexed.
|
|
|
|
Negative values are permitted and work as they do with single indices
|
|
or slices: ::
|
|
|
|
>>> x[np.array([3,3,-3,8])]
|
|
array([7, 7, 4, 2])
|
|
|
|
It is an error to have index values out of bounds: ::
|
|
|
|
>>> x[np.array([3, 3, 20, 8])]
|
|
<type 'exceptions.IndexError'>: index 20 out of bounds 0<=index<9
|
|
|
|
Generally speaking, what is returned when index arrays are used is
|
|
an array with the same shape as the index array, but with the type
|
|
and values of the array being indexed. As an example, we can use a
|
|
multidimensional index array instead: ::
|
|
|
|
>>> x[np.array([[1,1],[2,3]])]
|
|
array([[9, 9],
|
|
[8, 7]])
|
|
|
|
Indexing Multi-dimensional arrays
|
|
=================================
|
|
|
|
Things become more complex when multidimensional arrays are indexed,
|
|
particularly with multidimensional index arrays. These tend to be
|
|
more unusual uses, but they are permitted, and they are useful for some
|
|
problems. We'll start with the simplest multidimensional case (using
|
|
the array y from the previous examples): ::
|
|
|
|
>>> y[np.array([0,2,4]), np.array([0,1,2])]
|
|
array([ 0, 15, 30])
|
|
|
|
In this case, if the index arrays have a matching shape, and there is
|
|
an index array for each dimension of the array being indexed, the
|
|
resultant array has the same shape as the index arrays, and the values
|
|
correspond to the index set for each position in the index arrays. In
|
|
this example, the first index value is 0 for both index arrays, and
|
|
thus the first value of the resultant array is y[0,0]. The next value
|
|
is y[2,1], and the last is y[4,2].
|
|
|
|
If the index arrays do not have the same shape, there is an attempt to
|
|
broadcast them to the same shape. If they cannot be broadcast to the
|
|
same shape, an exception is raised: ::
|
|
|
|
>>> y[np.array([0,2,4]), np.array([0,1])]
|
|
<type 'exceptions.ValueError'>: shape mismatch: objects cannot be
|
|
broadcast to a single shape
|
|
|
|
The broadcasting mechanism permits index arrays to be combined with
|
|
scalars for other indices. The effect is that the scalar value is used
|
|
for all the corresponding values of the index arrays: ::
|
|
|
|
>>> y[np.array([0,2,4]), 1]
|
|
array([ 1, 15, 29])
|
|
|
|
Jumping to the next level of complexity, it is possible to only
|
|
partially index an array with index arrays. It takes a bit of thought
|
|
to understand what happens in such cases. For example if we just use
|
|
one index array with y: ::
|
|
|
|
>>> y[np.array([0,2,4])]
|
|
array([[ 0, 1, 2, 3, 4, 5, 6],
|
|
[14, 15, 16, 17, 18, 19, 20],
|
|
[28, 29, 30, 31, 32, 33, 34]])
|
|
|
|
What results is the construction of a new array where each value of
|
|
the index array selects one row from the array being indexed and the
|
|
resultant array has the resulting shape (number of index elements,
|
|
size of row).
|
|
|
|
An example of where this may be useful is for a color lookup table
|
|
where we want to map the values of an image into RGB triples for
|
|
display. The lookup table could have a shape (nlookup, 3). Indexing
|
|
such an array with an image with shape (ny, nx) with dtype=np.uint8
|
|
(or any integer type so long as values are with the bounds of the
|
|
lookup table) will result in an array of shape (ny, nx, 3) where a
|
|
triple of RGB values is associated with each pixel location.
|
|
|
|
In general, the shape of the resultant array will be the concatenation
|
|
of the shape of the index array (or the shape that all the index arrays
|
|
were broadcast to) with the shape of any unused dimensions (those not
|
|
indexed) in the array being indexed.
|
|
|
|
Boolean or "mask" index arrays
|
|
==============================
|
|
|
|
Boolean arrays used as indices are treated in a different manner
|
|
entirely than index arrays. Boolean arrays must be of the same shape
|
|
as the initial dimensions of the array being indexed. In the
|
|
most straightforward case, the boolean array has the same shape: ::
|
|
|
|
>>> b = y>20
|
|
>>> y[b]
|
|
array([21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34])
|
|
|
|
Unlike in the case of integer index arrays, in the boolean case, the
|
|
result is a 1-D array containing all the elements in the indexed array
|
|
corresponding to all the true elements in the boolean array. The
|
|
elements in the indexed array are always iterated and returned in
|
|
:term:`row-major` (C-style) order. The result is also identical to
|
|
``y[np.nonzero(b)]``. As with index arrays, what is returned is a copy
|
|
of the data, not a view as one gets with slices.
|
|
|
|
The result will be multidimensional if y has more dimensions than b.
|
|
For example: ::
|
|
|
|
>>> b[:,5] # use a 1-D boolean whose first dim agrees with the first dim of y
|
|
array([False, False, False, True, True])
|
|
>>> y[b[:,5]]
|
|
array([[21, 22, 23, 24, 25, 26, 27],
|
|
[28, 29, 30, 31, 32, 33, 34]])
|
|
|
|
Here the 4th and 5th rows are selected from the indexed array and
|
|
combined to make a 2-D array.
|
|
|
|
In general, when the boolean array has fewer dimensions than the array
|
|
being indexed, this is equivalent to y[b, ...], which means
|
|
y is indexed by b followed by as many : as are needed to fill
|
|
out the rank of y.
|
|
Thus the shape of the result is one dimension containing the number
|
|
of True elements of the boolean array, followed by the remaining
|
|
dimensions of the array being indexed.
|
|
|
|
For example, using a 2-D boolean array of shape (2,3)
|
|
with four True elements to select rows from a 3-D array of shape
|
|
(2,3,5) results in a 2-D result of shape (4,5): ::
|
|
|
|
>>> x = np.arange(30).reshape(2,3,5)
|
|
>>> x
|
|
array([[[ 0, 1, 2, 3, 4],
|
|
[ 5, 6, 7, 8, 9],
|
|
[10, 11, 12, 13, 14]],
|
|
[[15, 16, 17, 18, 19],
|
|
[20, 21, 22, 23, 24],
|
|
[25, 26, 27, 28, 29]]])
|
|
>>> b = np.array([[True, True, False], [False, True, True]])
|
|
>>> x[b]
|
|
array([[ 0, 1, 2, 3, 4],
|
|
[ 5, 6, 7, 8, 9],
|
|
[20, 21, 22, 23, 24],
|
|
[25, 26, 27, 28, 29]])
|
|
|
|
For further details, consult the numpy reference documentation on array indexing.
|
|
|
|
Combining index arrays with slices
|
|
==================================
|
|
|
|
Index arrays may be combined with slices. For example: ::
|
|
|
|
>>> y[np.array([0, 2, 4]), 1:3]
|
|
array([[ 1, 2],
|
|
[15, 16],
|
|
[29, 30]])
|
|
|
|
In effect, the slice and index array operation are independent.
|
|
The slice operation extracts columns with index 1 and 2,
|
|
(i.e. the 2nd and 3rd columns),
|
|
followed by the index array operation which extracts rows with
|
|
index 0, 2 and 4 (i.e the first, third and fifth rows).
|
|
|
|
This is equivalent to::
|
|
|
|
>>> y[:, 1:3][np.array([0, 2, 4]), :]
|
|
array([[ 1, 2],
|
|
[15, 16],
|
|
[29, 30]])
|
|
|
|
Likewise, slicing can be combined with broadcasted boolean indices: ::
|
|
|
|
>>> b = y > 20
|
|
>>> b
|
|
array([[False, False, False, False, False, False, False],
|
|
[False, False, False, False, False, False, False],
|
|
[False, False, False, False, False, False, False],
|
|
[ True, True, True, True, True, True, True],
|
|
[ True, True, True, True, True, True, True]])
|
|
>>> y[b[:,5],1:3]
|
|
array([[22, 23],
|
|
[29, 30]])
|
|
|
|
Structural indexing tools
|
|
=========================
|
|
|
|
To facilitate easy matching of array shapes with expressions and in
|
|
assignments, the np.newaxis object can be used within array indices
|
|
to add new dimensions with a size of 1. For example: ::
|
|
|
|
>>> y.shape
|
|
(5, 7)
|
|
>>> y[:,np.newaxis,:].shape
|
|
(5, 1, 7)
|
|
|
|
Note that there are no new elements in the array, just that the
|
|
dimensionality is increased. This can be handy to combine two
|
|
arrays in a way that otherwise would require explicitly reshaping
|
|
operations. For example: ::
|
|
|
|
>>> x = np.arange(5)
|
|
>>> x[:,np.newaxis] + x[np.newaxis,:]
|
|
array([[0, 1, 2, 3, 4],
|
|
[1, 2, 3, 4, 5],
|
|
[2, 3, 4, 5, 6],
|
|
[3, 4, 5, 6, 7],
|
|
[4, 5, 6, 7, 8]])
|
|
|
|
The ellipsis syntax maybe used to indicate selecting in full any
|
|
remaining unspecified dimensions. For example: ::
|
|
|
|
>>> z = np.arange(81).reshape(3,3,3,3)
|
|
>>> z[1,...,2]
|
|
array([[29, 32, 35],
|
|
[38, 41, 44],
|
|
[47, 50, 53]])
|
|
|
|
This is equivalent to: ::
|
|
|
|
>>> z[1,:,:,2]
|
|
array([[29, 32, 35],
|
|
[38, 41, 44],
|
|
[47, 50, 53]])
|
|
|
|
Assigning values to indexed arrays
|
|
==================================
|
|
|
|
As mentioned, one can select a subset of an array to assign to using
|
|
a single index, slices, and index and mask arrays. The value being
|
|
assigned to the indexed array must be shape consistent (the same shape
|
|
or broadcastable to the shape the index produces). For example, it is
|
|
permitted to assign a constant to a slice: ::
|
|
|
|
>>> x = np.arange(10)
|
|
>>> x[2:7] = 1
|
|
|
|
or an array of the right size: ::
|
|
|
|
>>> x[2:7] = np.arange(5)
|
|
|
|
Note that assignments may result in changes if assigning
|
|
higher types to lower types (like floats to ints) or even
|
|
exceptions (assigning complex to floats or ints): ::
|
|
|
|
>>> x[1] = 1.2
|
|
>>> x[1]
|
|
1
|
|
>>> x[1] = 1.2j
|
|
TypeError: can't convert complex to int
|
|
|
|
|
|
Unlike some of the references (such as array and mask indices)
|
|
assignments are always made to the original data in the array
|
|
(indeed, nothing else would make sense!). Note though, that some
|
|
actions may not work as one may naively expect. This particular
|
|
example is often surprising to people: ::
|
|
|
|
>>> x = np.arange(0, 50, 10)
|
|
>>> x
|
|
array([ 0, 10, 20, 30, 40])
|
|
>>> x[np.array([1, 1, 3, 1])] += 1
|
|
>>> x
|
|
array([ 0, 11, 20, 31, 40])
|
|
|
|
Where people expect that the 1st location will be incremented by 3.
|
|
In fact, it will only be incremented by 1. The reason is because
|
|
a new array is extracted from the original (as a temporary) containing
|
|
the values at 1, 1, 3, 1, then the value 1 is added to the temporary,
|
|
and then the temporary is assigned back to the original array. Thus
|
|
the value of the array at x[1]+1 is assigned to x[1] three times,
|
|
rather than being incremented 3 times.
|
|
|
|
Dealing with variable numbers of indices within programs
|
|
========================================================
|
|
|
|
The index syntax is very powerful but limiting when dealing with
|
|
a variable number of indices. For example, if you want to write
|
|
a function that can handle arguments with various numbers of
|
|
dimensions without having to write special case code for each
|
|
number of possible dimensions, how can that be done? If one
|
|
supplies to the index a tuple, the tuple will be interpreted
|
|
as a list of indices. For example (using the previous definition
|
|
for the array z): ::
|
|
|
|
>>> indices = (1,1,1,1)
|
|
>>> z[indices]
|
|
40
|
|
|
|
So one can use code to construct tuples of any number of indices
|
|
and then use these within an index.
|
|
|
|
Slices can be specified within programs by using the slice() function
|
|
in Python. For example: ::
|
|
|
|
>>> indices = (1,1,1,slice(0,2)) # same as [1,1,1,0:2]
|
|
>>> z[indices]
|
|
array([39, 40])
|
|
|
|
Likewise, ellipsis can be specified by code by using the Ellipsis
|
|
object: ::
|
|
|
|
>>> indices = (1, Ellipsis, 1) # same as [1,...,1]
|
|
>>> z[indices]
|
|
array([[28, 31, 34],
|
|
[37, 40, 43],
|
|
[46, 49, 52]])
|
|
|
|
For this reason it is possible to use the output from the np.nonzero()
|
|
function directly as an index since it always returns a tuple of index
|
|
arrays.
|
|
|
|
Because the special treatment of tuples, they are not automatically
|
|
converted to an array as a list would be. As an example: ::
|
|
|
|
>>> z[[1,1,1,1]] # produces a large array
|
|
array([[[[27, 28, 29],
|
|
[30, 31, 32], ...
|
|
>>> z[(1,1,1,1)] # returns a single value
|
|
40
|
|
|
|
"""
|