Merge pull request #19369 from Mukulikaa/broadcast-docs-merge

DOC: Moved VQ example & target from duplicate array broadcasting doc
author: Matti Picus <matti.picus@gmail.com> 2021-07-18 17:31:51 +0300
committer: GitHub <noreply@github.com> 2021-07-18 17:31:51 +0300
commit: 131b6bf29e20e801368907f71e5d8594a484f742 (patch)
tree: 5366ce4d39f65eb46f193b932d1f11ee8c648dd3
parent: 8eed39d6562045c650f4c8aeec5319f4bf34402b (diff)
parent: fcee8a8178d133514d6b3d41dbf881a4e4fc9c15 (diff)
download: numpy-131b6bf29e20e801368907f71e5d8594a484f742.tar.gz
12 files changed, 4498 insertions, 263 deletions
diff --git a/doc/source/user/basics.broadcasting.rst b/doc/source/user/basics.broadcasting.rst
index 5eae3eb32..c31fc98bd 100644
--- a/doc/source/user/basics.broadcasting.rst
+++ b/doc/source/user/basics.broadcasting.rst
@@ -1,22 +1,15 @@
 .. _basics.broadcasting:
+.. _array-broadcasting-in-numpy:
 
 ************
 Broadcasting
 ************
 
 .. seealso::
-    :class:`numpy.broadcast`
+    :class:`numpy.broadcast`   
 
-    :ref:`array-broadcasting-in-numpy`
-        An introduction to the concepts discussed here
 
-.. 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
+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
@@ -47,11 +40,23 @@ 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
+``b``, as shown in :ref:`broadcasting.figure-1`, 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.
 
+.. figure:: broadcasting_1.svg
+    :alt: A scalar is broadcast to match the shape of the 1-d array it
+          is being multiplied to.
+    :name: broadcasting.figure-1
+
+    *Figure 1*
+
+    *In the simplest example of broadcasting, the scalar* ``b`` *is
+    stretched to become an array of same shape as* ``a`` *so the shapes
+    are compatible for element-by-element multiplication.*
+
 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).
@@ -123,50 +128,49 @@ Here are examples of shapes that do not broadcast::
   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,)
+An example of broadcasting when a 1-d array is added to a 2-d array::
 
- >>> y.shape
- (5,)
-
- >>> x + y
- ValueError: operands could not be broadcast together with shapes (4,) (5,)
-
- >>> xx.shape
- (4, 1)
+  >>> a = array([[ 0.0,  0.0,  0.0],
+  ...            [10.0, 10.0, 10.0],
+  ...            [20.0, 20.0, 20.0],
+  ...            [30.0, 30.0, 30.0]])
+  >>> b = array([1.0, 2.0, 3.0])
+  >>> a + b
+  array([[  1.,   2.,   3.],
+          [ 11.,  12.,  13.],
+          [ 21.,  22.,  23.],
+          [ 31.,  32.,  33.]])
+  >>> b = array([1.0, 2.0, 3.0, 4.0])
+  >>> a + b 
+  ValueError: operands could not be broadcast together with shapes (4,3) (4,)
 
- >>> y.shape
- (5,)
+As shown in :ref:`broadcasting.figure-2`, ``b`` is added to each row of ``a``.
+In :ref:`broadcasting.figure-3`, an exception is raised because of the
+incompatible shapes.
 
- >>> (xx + y).shape
- (4, 5)
+.. figure:: broadcasting_2.svg
+    :alt: A 1-d array with shape (3) is strectched to match the 2-d array of
+          shape (4, 3) it is being added to, and the result is a 2-d array of shape
+          (4, 3).
+    :name: broadcasting.figure-2
 
- >>> xx + y
- array([[ 1.,  1.,  1.,  1.,  1.],
-        [ 2.,  2.,  2.,  2.,  2.],
-        [ 3.,  3.,  3.,  3.,  3.],
-        [ 4.,  4.,  4.,  4.,  4.]])
+    *Figure 2*
 
- >>> x.shape
- (4,)
+    *A one dimensional array added to a two dimensional array results in
+    broadcasting if number of 1-d array elements matches the number of 2-d
+    array columns.*
 
- >>> z.shape
- (3, 4)
+.. figure:: broadcasting_3.svg
+    :alt: A huge cross over the 2-d array of shape (4, 3) and the 1-d array
+          of shape (4) shows that they can not be broadcast due to mismatch
+          of shapes and thus produce no result.
+    :name: broadcasting.figure-3
 
- >>> (x + z).shape
- (3, 4)
+    *Figure 3*
 
- >>> x + z
- array([[ 1.,  2.,  3.,  4.],
-        [ 1.,  2.,  3.,  4.],
-        [ 1.,  2.,  3.,  4.]])
+    *When the trailing dimensions of the arrays are unequal, broadcasting fails
+    because it is impossible to align the values in the rows of the 1st array
+    with the elements of the 2nd arrays for element-by-element addition.*
 
 Broadcasting provides a convenient way of taking the outer product (or
 any other outer operation) of two arrays. The following example shows an
@@ -180,8 +184,94 @@ outer addition operation of two 1-d arrays::
          [ 21.,  22.,  23.],
          [ 31.,  32.,  33.]])
 
+.. figure:: broadcasting_4.svg
+    :alt: A 2-d array of shape (4, 1) and a 1-d array of shape (3) are
+          stretched to match their shapes and produce a resultant array
+          of shape (4, 3).
+    :name: broadcasting.figure-4
+
+    *Figure 4*
+
+    *In some cases, broadcasting stretches both arrays to form an output array
+    larger than either of the initial arrays.*
+
 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.
 
-
+A Practical Example: Vector Quantization
+========================================
+
+Broadcasting comes up quite often in real world problems. A typical example
+occurs in the vector quantization (VQ) algorithm used in information theory,
+classification, and other related areas. The basic operation in VQ finds
+the closest point in a set of points, called ``codes`` in VQ jargon, to a given
+point, called the ``observation``. In the very simple, two-dimensional case
+shown below, the values in ``observation`` describe the weight and height of an
+athlete to be classified. The ``codes`` represent different classes of
+athletes. [#f1]_ Finding the closest point requires calculating the distance
+between observation and each of the codes. The shortest distance provides the
+best match. In this example, ``codes[0]`` is the closest class indicating that
+the athlete is likely a basketball player.
+
+  >>> from numpy import array, argmin, sqrt, sum
+  >>> observation = array([111.0, 188.0])
+  >>> codes = array([[102.0, 203.0],
+  ...                [132.0, 193.0],
+  ...                [45.0, 155.0],
+  ...                [57.0, 173.0]])
+  >>> diff = codes - observation    # the broadcast happens here
+  >>> dist = sqrt(sum(diff**2,axis=-1))
+  >>> argmin(dist)
+  0
+
+In this example, the ``observation`` array is stretched to match
+the shape of the ``codes`` array::
+
+  Observation      (1d array):      2
+  Codes            (2d array):  4 x 2
+  Diff             (2d array):  4 x 2
+
+.. figure:: broadcasting_5.svg
+    :alt: A height versus weight graph that shows data of a female
+          gymnast, marathon runner, basketball player, football
+          lineman and the athlete to be classified. Shortest distance
+          is found between the basketball player and the athlete
+          to be classified. 
+    :name: broadcasting.figure-5
+
+    *Figure 5*
+
+    *The basic operation of vector quantization calculates the distance between
+    an object to be classified, the dark square, and multiple known codes, the
+    gray circles. In this simple case, the codes represent individual classes.
+    More complex cases use multiple codes per class.*
+
+Typically, a large number of ``observations``, perhaps read from a database,
+are compared to a set of ``codes``. Consider this scenario::
+
+  Observation      (2d array):      10 x 3
+  Codes            (2d array):       5 x 3
+  Diff             (3d array):  5 x 10 x 3 
+
+The three-dimensional array, ``diff``, is a consequence of broadcasting, not a
+necessity for the calculation. Large data sets will generate a large
+intermediate array that is computationally inefficient. Instead, if each
+observation is calculated individually using a Python loop around the code
+in the two-dimensional example above, a much smaller array is used.
+
+Broadcasting is a powerful tool for writing short and usually intuitive code
+that does its computations very efficiently in C. However, there are cases
+when broadcasting uses unnecessarily large amounts of memory for a particular
+algorithm. In these cases, it is better to write the algorithm's outer loop in
+Python. This may also produce more readable code, as algorithms that use
+broadcasting tend to become more difficult to interpret as the number of
+dimensions in the broadcast increases.
+
+.. rubric:: Footnotes
+
+.. [#f1]
+    In this example, weight has more impact on the distance calculation
+    than height because of the larger values. In practice, it is important to
+    normalize the height and weight, often by their standard deviation across the
+    data set, so that both have equal influence on the distance calculation.
diff --git a/doc/source/user/broadcasting_1.svg b/doc/source/user/broadcasting_1.svg
new file mode 100644
index 000000000..4031c3af4
--- /dev/null
+++ b/doc/source/user/broadcasting_1.svg
@@ -0,0 +1,669 @@
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diff --git a/doc/source/user/theory.broadcast_1.gif b/doc/source/user/theory.broadcast_1.gif
deleted file mode 100644
index 541ec734b..000000000
--- a/doc/source/user/theory.broadcast_1.gif
+++ /dev/null
diff --git a/doc/source/user/theory.broadcast_2.gif b/doc/source/user/theory.broadcast_2.gif
deleted file mode 100644
index 163a8473f..000000000
--- a/doc/source/user/theory.broadcast_2.gif
+++ /dev/null
diff --git a/doc/source/user/theory.broadcast_3.gif b/doc/source/user/theory.broadcast_3.gif
deleted file mode 100644
index 83f61f2df..000000000
--- a/doc/source/user/theory.broadcast_3.gif
+++ /dev/null
diff --git a/doc/source/user/theory.broadcast_4.gif b/doc/source/user/theory.broadcast_4.gif
deleted file mode 100644
index 9b21ff582..000000000
--- a/doc/source/user/theory.broadcast_4.gif
+++ /dev/null
diff --git a/doc/source/user/theory.broadcast_5.png b/doc/source/user/theory.broadcast_5.png
deleted file mode 100644
index 3aa2f0536..000000000
--- a/doc/source/user/theory.broadcast_5.png
+++ /dev/null
diff --git a/doc/source/user/theory.broadcasting.rst b/doc/source/user/theory.broadcasting.rst
index a82d78e6c..a4973e4e6 100644
--- a/doc/source/user/theory.broadcasting.rst
+++ b/doc/source/user/theory.broadcasting.rst
@@ -1,7 +1,5 @@
 :orphan:
 
-.. _array-broadcasting-in-numpy:
-
 ===========================
 Array Broadcasting in Numpy
 ===========================
@@ -12,218 +10,6 @@ Array Broadcasting in Numpy
    `github repo
    <https://github.com/scipy/old-wiki/blob/gh-pages/pages/EricsBroadcastingDoc.html>`_
 
-Let's explore a more advanced concept in numpy called broadcasting. 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 also
-cases where broadcasting is a bad idea because it leads to inefficient use of
-memory that slows computation. This article provides a gentle introduction to
-broadcasting with numerous examples ranging from simple to involved. It also
-provides hints on when and when not to use broadcasting.
-
-numpy operations are usually done element-by-element which requires two arrays
-to have exactly the same shape:
-
-.. code-block:: python
-    :caption: Example 1
-    :name: example-1
-
-    >>> from numpy import array
-    >>> a = array([1.0, 2.0, 3.0])
-    >>> b = 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:
-
-.. code-block:: python
-    :caption: Example 2
-    :name: example-2
-
-    >>> from numpy import array
-    >>> a = 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``, as shown
-in :ref:`figure-1`, 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. Because :ref:`example-2`
-moves less memory, (``b`` is a scalar, not an array) around during the
-multiplication, it is about 10% faster than :ref:`example-1` using the standard
-numpy on Windows 2000 with one million element arrays.
-
-.. figure:: theory.broadcast_1.gif
-    :alt: Vector-Scalar multiplication
-    :name: figure-1
-
-    *Figure 1*
-
-    *In the simplest example of broadcasting, the scalar ``b`` is
-    stretched to become an array of same shape as ``a`` so the shapes
-    are compatible for element-by-element multiplication.*
-
-
-The rule governing whether two arrays have compatible shapes for broadcasting
-can be expressed in a single sentence.
-
-.. admonition:: The Broadcasting Rule
-
-    **In order to broadcast, the size of the trailing axes for both arrays in
-    an operation must either be the same size or one of them must be one.**
-
-If this condition is not met, a ``ValueError('frames are not aligned')``
-exception is thrown indicating that the arrays have incompatible shapes. The
-size of the result array created by broadcast operations is the maximum size
-along each dimension from the input arrays. Note that the rule does not say
-anything about the two arrays needing to have the same number of dimensions.
-So, for example, if you have a 256 x 256 x 3 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 rule 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 |
-+-------+------------+-------+-------+---+
-
-In the following example, both the ``A`` and ``B`` arrays have axes with length
-one that are expanded to a larger size in a 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 |
-+-------+------------+-----+-----+-----+---+
-
-Below, are several code examples and graphical representations that help make
-the broadcast rule visually obvious. :ref:`example-3` adds a one-dimensional array
-to a two-dimensional array:
-
-.. code-block:: python
-    :caption: Example 3
-    :name: example-3
-
-    >>> from numpy import array
-    >>> a = array([[ 0.0,  0.0,  0.0],
-    ...            [10.0, 10.0, 10.0],
-    ...            [20.0, 20.0, 20.0],
-    ...            [30.0, 30.0, 30.0]])
-    >>> b = array([1.0, 2.0, 3.0])
-    >>> a + b
-    array([[  1.,   2.,   3.],
-           [ 11.,  12.,  13.],
-           [ 21.,  22.,  23.],
-           [ 31.,  32.,  33.]])
-
-As shown in :ref:`figure-2`, ``b`` is added to each row of ``a``. When ``b`` is
-longer than the rows of ``a``, as in :ref:`figure-3`, an exception is raised
-because of the incompatible shapes.
-
-.. figure:: theory.broadcast_2.gif
-    :alt: Matrix-Vector
-    :name: figure-2
-
-    *Figure 2*
-
-    *A two dimensional array multiplied by a one dimensional array results in
-    broadcasting if number of 1-d array elements matches the number of 2-d
-    array columns.*
-
-.. figure:: theory.broadcast_3.gif
-    :alt: Matrix-Vector-with-error
-    :name: figure-3
-
-    *Figure 3*
-
-    *When the trailing dimensions of the arrays are unequal, broadcasting fails
-    because it is impossible to align the values in the rows of the 1st array
-    with the elements of the 2nd arrays for element-by-element addition.*
-
-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 that produces the same result as
-:ref:`example-3`
-
-.. code-block:: python
-    :caption: Example 4
-    :name: example-4
-
-    >>> from numpy import array, newaxis
-    >>> a = array([0.0, 10.0, 20.0, 30.0])
-    >>> b = array([1.0, 2.0, 3.0])
-    >>> a[:,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. :ref:`figure-4` illustrates the stretching of both
-arrays to produce the desired 4x3 output array.
-
-.. figure:: theory.broadcast_4.gif
-    :alt: vector-vector with newaxis
-    :name: figure-4
-
-    *Figure 4*
-
-    In some cases, broadcasting stretches both arrays to form an output array
-    larger than either of the initial arrays.*
-
-A Practical Example: Vector Quantization.
-=========================================
-
-Broadcasting comes up quite often in real world problems. A typical example
-occurs in the vector quantization (VQ) algorithm used in information theory,
-classification, and other related areas. The basic operation in VQ [#f0] finds
-the closest point in a set of points, called codes in VQ jargon, to a given
-point, called the observation. In the very simple, two-dimensional case shown
-in :ref:`figure-5`, the values in observation describe the weight and height of an
-athlete to be classified. The codes represent different classes of
-athletes. [#f1]_ Finding the closest point requires calculating the distance
-between observation and each of the codes. The shortest distance provides the
-best match. In this example, ``codes[0]`` is the closest class indicating that
-the athlete is likely a basketball player.
-
-.. figure:: theory.broadcast_5.png
-    :alt: vector quantitization example
-    :name: figure-5
-
-    *Figure 5*
-
-    *The basic operation of vector quantization calculates the distance between
-    an object to be classified, the dark square, and multiple known codes, the
-    gray circles. In this simple case, the codes represent individual classes.
-    More complex cases use multiple codes per class.*
-
-
-.. rubric:: Footnotes
-
-.. [#f0] Vector Quantization J. Makhoul, S. Roucos, and H. Gish, "Vector Quantization in Speech Coding," Proc. IEEE, vol. 73, pp. 1551-1587, Nov. 1985.
-.. [#f1]
-    In this example, weight has more impact on the distance calculation
-    than height because of the larger values. In practice, it is important to
-    normalize the height and weight, often by their standard deviation across the
-    data set, so that both have equal influence on the distance calculation.
-
 .. note::
-
-    The code to produce the figures is part of the `AstroML book
-    <http://www.astroml.org/book_figures/appendix/fig_broadcast_visual.html>`_
+    Please refer to the updated :doc:`basics.broadcasting` document.
author	Matti Picus <matti.picus@gmail.com>	2021-07-18 17:31:51 +0300
committer	GitHub <noreply@github.com>	2021-07-18 17:31:51 +0300
commit	131b6bf29e20e801368907f71e5d8594a484f742 (patch)
tree	5366ce4d39f65eb46f193b932d1f11ee8c648dd3
parent	8eed39d6562045c650f4c8aeec5319f4bf34402b (diff)
parent	fcee8a8178d133514d6b3d41dbf881a4e4fc9c15 (diff)
download	numpy-131b6bf29e20e801368907f71e5d8594a484f742.tar.gz