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