Broadcasting

Broadcasting
See also
numpy.broadcast

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.

广播用以描述numpy中对两个形状不同的阵列进行数学计算的处理机制。较小的阵列“广播”到较大阵列相同的形状尺度上，使它们对等以可以进行数学计算。广播提供了一种向量化阵列的操作方式，因此Python不需要像C一样循环。广播操作不需要数据复制，通常执行效率非常高。然而，有时广播是个坏主意，可能会导致内存浪费以致计算减慢。

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:

Numpy操作通常由成对的阵列完成，阵列间逐个元素对元素地执行。最简单的情形是两个阵列有一样的形状，例如：

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:

Numpy的广播机制放宽了对阵列形状的限制。最简单的情形是一个阵列和一个尺度值相乘：

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 stretchedduring 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.

上面两种结果是一样的，我们可以认为尺度值b在计算时被延展得和a一样的形状。延展后的b的每一个元素都是原来尺度值的复制。延展的类比只是一种概念性的。实际上，Numpy并不需要真的复制这些尺度值，所以广播运算在内存和计算效率上尽量高效。

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).

上面的第二个例子比第一个更高效，因为广播在乘法计算时动用更少的内存。

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

they are equal, or
one of them is 1
对两个阵进行操作时，NumPy逐元素地比较他们的形状，从后面的维度向前执行。当以下情形出现时，两个维度是兼容的：

1，它们相等
2，其中一个是1

If these conditions are not met, a ValueError: frames are not aligned exception is thrown, indicating that the arrays have incompatible shapes. The size of the resulting array is the maximum size along each dimension of the input arrays.

如果这些条件都没有达到，将会抛出错误：frames are not aligned exception，表示两个阵列形状不兼容。结果阵列的尺寸与输入阵列的各维度最大尺寸相同。

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:

阵列不需要有相同的维度。例如，如果你有一个256x256x3的RGB阵列，你想要对每一种颜色加一个权重，你就可以乘以一个拥有3个元素的一维阵列。将两个阵列的各维度尺寸展开，从后往前匹配，如果满足了上面的两个条件，则这两个阵列是兼容的。

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.

当任何一个维度是1，那么另一个不为1的维度将被用作最终结果的维度。也就是说，尺寸为1的维度将延展或“复制”到与另一个维度匹配。

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:

下面的例子，A和B两个阵列中尺寸为1的维度在广播过程中都被拓展了。

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:

更多例子：

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:

下面这些例子不能广播

A (1d array): 3
B (1d array): 4 # trailing dimensions do not match #维度尺寸不兼容

A (2d array): 2 x 1
B (3d array): 8 x 4 x 3 # second from last dimensions mismatched #倒数第二个维度不兼容
An example of broadcasting in practice:

一个实际使用中用到广播的例子：

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
: shape mismatch: objects cannot be broadcast to a single shape

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:

广播提供了一种计算外积（或者任何外部运算）的便捷的方式。下面的例子展示了两个一维阵列外加运算。

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 4x1array with b, which has shape (3,), yields a 4x3 array.

这里的newaxis表示加入一个新的坐标轴到a中，是它成为一个二维4x1的阵列。b的形状为一维（3,）结合4x1d，产生一个4x3的阵列。

See this article for illustrations of broadcasting concepts.

作者：starxhong
来源：CSDN
原文：https://blog.csdn.net/hongxingabc/article/details/53149655
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