mirror of
https://github.com/PiBrewing/craftbeerpi4.git
synced 2024-12-27 16:01:44 +01:00
180 lines
5.4 KiB
Python
180 lines
5.4 KiB
Python
"""
|
|
========================
|
|
Broadcasting over arrays
|
|
========================
|
|
|
|
.. note::
|
|
See `this article
|
|
<https://numpy.org/devdocs/user/theory.broadcasting.html>`_
|
|
for illustrations of broadcasting concepts.
|
|
|
|
|
|
The term broadcasting describes how numpy treats arrays with different
|
|
shapes during arithmetic operations. Subject to certain constraints,
|
|
the smaller array is "broadcast" across the larger array so that they
|
|
have compatible shapes. Broadcasting provides a means of vectorizing
|
|
array operations so that looping occurs in C instead of Python. It does
|
|
this without making needless copies of data and usually leads to
|
|
efficient algorithm implementations. There are, however, cases where
|
|
broadcasting is a bad idea because it leads to inefficient use of memory
|
|
that slows computation.
|
|
|
|
NumPy operations are usually done on pairs of arrays on an
|
|
element-by-element basis. In the simplest case, the two arrays must
|
|
have exactly the same shape, as in the following example:
|
|
|
|
>>> a = np.array([1.0, 2.0, 3.0])
|
|
>>> b = np.array([2.0, 2.0, 2.0])
|
|
>>> a * b
|
|
array([ 2., 4., 6.])
|
|
|
|
NumPy's broadcasting rule relaxes this constraint when the arrays'
|
|
shapes meet certain constraints. The simplest broadcasting example occurs
|
|
when an array and a scalar value are combined in an operation:
|
|
|
|
>>> a = np.array([1.0, 2.0, 3.0])
|
|
>>> b = 2.0
|
|
>>> a * b
|
|
array([ 2., 4., 6.])
|
|
|
|
The result is equivalent to the previous example where ``b`` was an array.
|
|
We can think of the scalar ``b`` being *stretched* during the arithmetic
|
|
operation into an array with the same shape as ``a``. The new elements in
|
|
``b`` are simply copies of the original scalar. The stretching analogy is
|
|
only conceptual. NumPy is smart enough to use the original scalar value
|
|
without actually making copies so that broadcasting operations are as
|
|
memory and computationally efficient as possible.
|
|
|
|
The code in the second example is more efficient than that in the first
|
|
because broadcasting moves less memory around during the multiplication
|
|
(``b`` is a scalar rather than an array).
|
|
|
|
General Broadcasting Rules
|
|
==========================
|
|
When operating on two arrays, NumPy compares their shapes element-wise.
|
|
It starts with the trailing dimensions and works its way forward. Two
|
|
dimensions are compatible when
|
|
|
|
1) they are equal, or
|
|
2) one of them is 1
|
|
|
|
If these conditions are not met, a
|
|
``ValueError: operands could not be broadcast together`` exception is
|
|
thrown, indicating that the arrays have incompatible shapes. The size of
|
|
the resulting array is the size that is not 1 along each axis of the inputs.
|
|
|
|
Arrays do not need to have the same *number* of dimensions. For example,
|
|
if you have a ``256x256x3`` array of RGB values, and you want to scale
|
|
each color in the image by a different value, you can multiply the image
|
|
by a one-dimensional array with 3 values. Lining up the sizes of the
|
|
trailing axes of these arrays according to the broadcast rules, shows that
|
|
they are compatible::
|
|
|
|
Image (3d array): 256 x 256 x 3
|
|
Scale (1d array): 3
|
|
Result (3d array): 256 x 256 x 3
|
|
|
|
When either of the dimensions compared is one, the other is
|
|
used. In other words, dimensions with size 1 are stretched or "copied"
|
|
to match the other.
|
|
|
|
In the following example, both the ``A`` and ``B`` arrays have axes with
|
|
length one that are expanded to a larger size during the broadcast
|
|
operation::
|
|
|
|
A (4d array): 8 x 1 x 6 x 1
|
|
B (3d array): 7 x 1 x 5
|
|
Result (4d array): 8 x 7 x 6 x 5
|
|
|
|
Here are some more examples::
|
|
|
|
A (2d array): 5 x 4
|
|
B (1d array): 1
|
|
Result (2d array): 5 x 4
|
|
|
|
A (2d array): 5 x 4
|
|
B (1d array): 4
|
|
Result (2d array): 5 x 4
|
|
|
|
A (3d array): 15 x 3 x 5
|
|
B (3d array): 15 x 1 x 5
|
|
Result (3d array): 15 x 3 x 5
|
|
|
|
A (3d array): 15 x 3 x 5
|
|
B (2d array): 3 x 5
|
|
Result (3d array): 15 x 3 x 5
|
|
|
|
A (3d array): 15 x 3 x 5
|
|
B (2d array): 3 x 1
|
|
Result (3d array): 15 x 3 x 5
|
|
|
|
Here are examples of shapes that do not broadcast::
|
|
|
|
A (1d array): 3
|
|
B (1d array): 4 # trailing dimensions do not match
|
|
|
|
A (2d array): 2 x 1
|
|
B (3d array): 8 x 4 x 3 # second from last dimensions mismatched
|
|
|
|
An example of broadcasting in practice::
|
|
|
|
>>> x = np.arange(4)
|
|
>>> xx = x.reshape(4,1)
|
|
>>> y = np.ones(5)
|
|
>>> z = np.ones((3,4))
|
|
|
|
>>> x.shape
|
|
(4,)
|
|
|
|
>>> y.shape
|
|
(5,)
|
|
|
|
>>> x + y
|
|
ValueError: operands could not be broadcast together with shapes (4,) (5,)
|
|
|
|
>>> xx.shape
|
|
(4, 1)
|
|
|
|
>>> y.shape
|
|
(5,)
|
|
|
|
>>> (xx + y).shape
|
|
(4, 5)
|
|
|
|
>>> xx + y
|
|
array([[ 1., 1., 1., 1., 1.],
|
|
[ 2., 2., 2., 2., 2.],
|
|
[ 3., 3., 3., 3., 3.],
|
|
[ 4., 4., 4., 4., 4.]])
|
|
|
|
>>> x.shape
|
|
(4,)
|
|
|
|
>>> z.shape
|
|
(3, 4)
|
|
|
|
>>> (x + z).shape
|
|
(3, 4)
|
|
|
|
>>> x + z
|
|
array([[ 1., 2., 3., 4.],
|
|
[ 1., 2., 3., 4.],
|
|
[ 1., 2., 3., 4.]])
|
|
|
|
Broadcasting provides a convenient way of taking the outer product (or
|
|
any other outer operation) of two arrays. The following example shows an
|
|
outer addition operation of two 1-d arrays::
|
|
|
|
>>> a = np.array([0.0, 10.0, 20.0, 30.0])
|
|
>>> b = np.array([1.0, 2.0, 3.0])
|
|
>>> a[:, np.newaxis] + b
|
|
array([[ 1., 2., 3.],
|
|
[ 11., 12., 13.],
|
|
[ 21., 22., 23.],
|
|
[ 31., 32., 33.]])
|
|
|
|
Here the ``newaxis`` index operator inserts a new axis into ``a``,
|
|
making it a two-dimensional ``4x1`` array. Combining the ``4x1`` array
|
|
with ``b``, which has shape ``(3,)``, yields a ``4x3`` array.
|
|
|
|
"""
|