全网最简单易懂且最综合的Python Numpy新手教程
Numpy是Python中涉及科学计算的核心代码库，使用频次颇高，尤其是在机器学习领域。但是对于初学者（我）而言，找不到一个简单且综合的教程一直使我痛心疾首。最近看到一个英文网页写的颇有大家风范，特意翻译下来与大家共赏，也做自己以后的参考之用，原网页在：http://cs231n.github.io/python-numpy-tutorial/#numpy 。如果你对MATLAB很熟悉，这个教程或许可以帮你快速上手：https://docs.scipy.org/doc/numpy/user/numpy-for-matlab-users.html 。
Arrays
numpy array就是一个元素集合，所有的元素都具有同样的类型（参考C/C++里的数组），元素的索引是一个tuple，tuple里的值为非负整数。array的维度数目就是该array的rank，array的形状（shape）用一个tuple来表示，tuple里的数字一一对应array各个维度中元素的数目。
我们可以用Python list来初始化一个array，用方括号来访问array中的元素：
import numpy as np
a = np.array([1, 2, 3]) # 创建一个1维array
print(type(a)) # Prints ""
print(a.shape) # Prints "(3,)"
print(a[0], a[1], a[2]) # Prints "1 2 3"
a[0] = 5 # 改变array中的一个元素
print(a) # Prints "[5, 2, 3]"
b = np.array([[1,2,3],[4,5,6]]) # Create a rank 2 array，即2维array
print(b.shape) # Prints "(2, 3)"
print(b[0, 0], b[0, 1], b[1, 0]) # Prints "1 2 4"
Numpy也提供了很多创建array的函数：
import numpy as np
a = np.zeros((2,2)) # Create an array of all zeros
print(a) # Prints "[[ 0. 0.]
# [ 0. 0.]]"
b = np.ones((1,2)) # Create an array of all ones
print(b) # Prints "[[ 1. 1.]]"
c = np.full((2,2), 7) # Create a constant array
print(c) # Prints "[[ 7. 7.]
# [ 7. 7.]]"
d = np.eye(2) # Create a 2x2 identity matrix（单位矩阵）
print(d) # Prints "[[ 1. 0.]
# [ 0. 1.]]"
e = np.random.random((2,2)) # Create an array filled with random values
print(e) # Might print "[[ 0.91940167 0.08143941]
# [ 0.68744134 0.87236687]]"
更多array的创建方法可以在这里找到：https://docs.scipy.org/doc/numpy/user/basics.creation.html#arrays-creation 。
Array索引
Numpy提供了很多索引array的方法。
Slicing（切片）
与Python list的切片方法类似。因为array是多维数组，所以你在索引的时候必须给每个维度都切一次片：
import numpy as np
# Create the following rank 2 array with shape (3, 4)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])
# 使用切片来获取包含前两行以及中间两列的子数组 b of shape (2, 2):
# [[2 3]
# [6 7]]
b = a[:2, 1:3]
# A slice of an array is a view into the same data, 修改切片子数组也会同时修改原array
print(a[0, 1]) # Prints "2"
b[0, 0] = 77 # b[0, 0] is the same piece of data as a[0, 1]
print(a[0, 1]) # Prints "77"
你也可以混合使用整数索引和切片索引。需要注意的是，这种索引方式会降低输出array的维度（rank）：
import numpy as np
# Create the following rank 2 array with shape (3, 4)
# [[ 1 2 3 4]
# [ 5 6 7 8]
# [ 9 10 11 12]]
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])
# 提取上述array的中间行有两种方式：
# Mixing integer indexing with slices yields an array of lower rank,
# while using only slices yields an array of the same rank as the
# original array:
row_r1 = a[1, :] # Rank 1 view of the second row of a
row_r2 = a[1:2, :] # Rank 2 view of the second row of a
print(row_r1, row_r1.shape) # Prints "[5 6 7 8] (4,)"
print(row_r2, row_r2.shape) # Prints "[[5 6 7 8]] (1, 4)"
# 提取中间列的效果类似：
col_r1 = a[:, 1]
col_r2 = a[:, 1:2]
print(col_r1, col_r1.shape) # Prints "[ 2 6 10] (3,)"
print(col_r2, col_r2.shape) # Prints "[[ 2]
# [ 6]
# [10]] (3, 1)"
整数数组索引
当你使用切片来索引array时，输出的array是原array的一个子数组。与此相反，整数数组（Integer array）索引可以让你使用一个array的数据来完全重构另一个array。下面是几个例子：
import numpy as np
a = np.array([[1,2], [3, 4], [5, 6]])
# 整数数组索引的一个例子
# The returned array will have shape (3,) and
print(a[[0, 1, 2], [0, 1, 0]]) # Prints "[1 4 5]"
# 上述整数数组索引的效果与以下代码相同
print(np.array([a[0, 0], a[1, 1], a[2, 0]])) # Prints "[1 4 5]"
# When using integer array indexing, you can reuse the same
# element from the source array:
print(a[[0, 0], [1, 1]]) # Prints "[2 2]"
# Equivalent to the previous integer array indexing example
print(np.array([a[0, 1], a[0, 1]])) # Prints "[2 2]"
整数数组索引能做的一个有趣的事情，是在一个矩阵的每一行中选取或修改一个元素：
import numpy as np
# 创建一个新array
a = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
print(a) # prints "array([[ 1, 2, 3],
# [ 4, 5, 6],
# [ 7, 8, 9],
# [10, 11, 12]])"
# Create an array of indices
b = np.array([0, 2, 0, 1])
# 使用b中的每个元素作为索引，在a的每一行选取一个数据
print(a[np.arange(4), b]) # Prints "[ 1 6 7 11]"
# Mutate one element from each row of a using the indices in b
a[np.arange(4), b] += 10
print(a) # prints "array([[11, 2, 3],
# [ 4, 5, 16],
# [17, 8, 9],
# [10, 21, 12]])
Boolean数组索引
Boolean数组索引可以让你从一个array中选取任意个元素。这种索引方式经常被用来选取array里满足某些条件的元素：
import numpy as np
a = np.array([[1,2], [3, 4], [5, 6]])
bool_idx = (a > 2) # Find the elements of a that are bigger than 2;
# this returns a numpy array of Booleans of the same
# shape as a, where each slot of bool_idx tells
# whether that element of a is > 2.
print(bool_idx) # Prints "[[False False]
# [ True True]
# [ True True]]"
# We use boolean array indexing to construct a rank 1 array
# consisting of the elements of a corresponding to the True values
# of bool_idx
print(a[bool_idx]) # Prints "[3 4 5 6]"
# 上述操作可以在一句代码里完成:
print(a[a > 2]) # Prints "[3 4 5 6]"
本部分省略了array索引的很多细节，感兴趣的同学可以阅读一下官方教程：https://docs.scipy.org/doc/numpy/reference/arrays.indexing.html 。
数据类型（Datatypes)
前面说过，array中所有元素都有同样的数据类型。Numpy提供了很多数据类型，当你构建一个array时，Numpy会为你定义的元素选择最匹配的类型。当然你也可以在构造array的函数中手动指定数据类型：
import numpy as np
x = np.array([1, 2]) # Let numpy choose the datatype
print(x.dtype) # Prints "int64"
x = np.array([1.0, 2.0]) # Let numpy choose the datatype
print(x.dtype) # Prints "float64"
x = np.array([1, 2], dtype=np.int64) # Force a particular datatype
print(x.dtype) # Prints "int64"
最全面的数据类型永远在官方文档中：https://docs.scipy.org/doc/numpy/reference/arrays.dtypes.html 。
Array运算
Array中，基础数学运算都是元素级（elementwise ）的操作，你可以使用运算符，也可以使用函数来进行这些运算：
import numpy as np
x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64)
# Elementwise sum; both produce the array
# [[ 6.0 8.0]
# [10.0 12.0]]
print(x + y)
print(np.add(x, y))
# Elementwise difference; both produce the array
# [[-4.0 -4.0]
# [-4.0 -4.0]]
print(x - y)
print(np.subtract(x, y))
# Elementwise product; both produce the array
# [[ 5.0 12.0]
# [21.0 32.0]]
print(x * y)
print(np.multiply(x, y))
# Elementwise division; both produce the array
# [[ 0.2 0.33333333]
# [ 0.42857143 0.5 ]]
print(x / y)
print(np.divide(x, y))
# Elementwise square root; produces the array
# [[ 1. 1.41421356]
# [ 1.73205081 2. ]]
print(np.sqrt(x))
注意，Numpy的*运算符是元素级（elementwise）操作。vector的内积运算，vector与矩阵乘法，以及矩阵乘法都使用dot函数来做。dot函数可以是Numpy的模块内定函数，也可以是array对象的私有函数，两者效果相同：
import numpy as np
x = np.array([[1,2],[3,4]])
y = np.array([[5,6],[7,8]])
v = np.array([9,10])
w = np.array([11, 12])
# Inner product of vectors; both produce 219
print(v.dot(w))
print(np.dot(v, w))
# Matrix / vector product; both produce the rank 1 array [29 67]
print(x.dot(v))
print(np.dot(x, v))
# Matrix / matrix product; both produce the rank 2 array
# [[19 22]
# [43 50]]
print(x.dot(y))
print(np.dot(x, y))
sum运算非常有用：
import numpy as np
x = np.array([[1,2],[3,4]])
print(np.sum(x)) # Compute sum of all elements; prints "10"
print(np.sum(x, axis=0)) # Compute sum of each column; prints "[4 6]"
print(np.sum(x, axis=1)) # Compute sum of each row; prints "[3 7]"
你可以在这个页面中找到所有的数学运算函数：https://docs.scipy.org/doc/numpy/reference/routines.math.html 。
除了对array进行数学运算之外，你可能还需要改变array的形状（reshape），或者操作array中的元素。例如，求矩阵的转置，可以使用array对象的T属性：
import numpy as np
x = np.array([[1,2], [3,4]])
print(x) # Prints "[[1 2]
# [3 4]]"
print(x.T) # Prints "[[1 3]
# [2 4]]"
# Note that taking the transpose of a rank 1 array does nothing:
v = np.array([1,2,3])
print(v) # Prints "[1 2 3]"
print(v.T) # Prints "[1 2 3]"
Numpy提供的所有操纵array的函数，都可以在这里找到：https://docs.scipy.org/doc/numpy/reference/routines.array-manipulation.html 。
广播（broadcasting）
广播是Numpy提供的一个非常重要的运算机制，它可以让你在两个形状不同的array之间做数学运算。例如我们有一个vector和一个矩阵，我们想把这个vector加到矩阵的每一行上：
import numpy as np
# We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = np.empty_like(x) # Create an empty matrix with the same shape as x
# Add the vector v to each row of the matrix x with an explicit loop
for i in range(4):
y[i, :] = x[i, :] + v
# Now y is the following
# [[ 2 2 4]
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]
print(y)
上面的代码可行，但是如果矩阵x非常大，那么在Python中的循环会非常费时，尤其是在进行机器学习训练时。我们可以换种思路：垂直堆叠多个vector v，以获取一个与矩阵x相同形状的新矩阵vv，然后把x与vv相加即可：
import numpy as np
# We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
vv = np.tile(v, (4, 1)) # Stack 4 copies of v on top of each other
print(vv) # Prints "[[1 0 1]
# [1 0 1]
# [1 0 1]
# [1 0 1]]"
y = x + vv # Add x and vv elementwise
print(y) # Prints "[[ 2 2 4
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]"
Numpy的广播机制可以极大地缩减上述代码：
import numpy as np
# We will add the vector v to each row of the matrix x,
# storing the result in the matrix y
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
v = np.array([1, 0, 1])
y = x + v # Add v to each row of x using broadcasting
print(y) # Prints "[[ 2 2 4]
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]"
因为广播机制，即使x的形状是(4, 3)而v的形状是(3,)，y = x + v仍然可以正常运算。广播机制的运算效果就是把v堆叠成与x形状相同的矩阵，然后再进行元素级数学运算。
广播机制遵循如下规则：
- 如果两个array的rank（维度）不同，堆叠低维array，直至两个array形状相同。
- 如果两个array在某个维度上的数据量（size）相同，或者其中一个array在某个维度上只有1个元素，那么这两个array是兼容的（compatible）。
- 如果两个array在所有维度上都是兼容的，那么它们会被同时广播（即会被同时堆叠出更大的矩阵）。
- 如果在任一维度上，一个array只有1个元素，另一个array的元素数目大于1，则在这个维度上堆叠第一个array。
关于广播的讲解如果您还不满意，那么请移步官方文档：https://docs.scipy.org/doc/numpy/user/basics.broadcasting.html 。
支持广播的函数叫做universal functions。所有的universal functions都可以在这个页面找到：https://docs.scipy.org/doc/numpy/reference/ufuncs.html#available-ufuncs 。
以下是广播的一些应用：
import numpy as np
# Compute outer product of vectors
v = np.array([1,2,3]) # v has shape (3,)
w = np.array([4,5]) # w has shape (2,)
# To compute an outer product, we first reshape v to be a column
# vector of shape (3, 1); we can then broadcast it against w to yield
# an output of shape (3, 2), which is the outer product of v and w:
# [[ 4 5]
# [ 8 10]
# [12 15]]
print(np.reshape(v, (3, 1)) * w)
# 矩阵每一行都与一个vector相加
x = np.array([[1,2,3], [4,5,6]])
# x has shape (2, 3) and v has shape (3,) so they broadcast to (2, 3),
# giving the following matrix:
# [[2 4 6]
# [5 7 9]]
print(x + v)
# 矩阵的每一列都与一个vector相加
# x has shape (2, 3) and w has shape (2,).
# If we transpose x then it has shape (3, 2) and can be broadcast
# against w to yield a result of shape (3, 2); transposing this result
# yields the final result of shape (2, 3) which is the matrix x with
# the vector w added to each column. Gives the following matrix:
# [[ 5 6 7]
# [ 9 10 11]]
print((x.T + w).T)
# Another solution is to reshape w to be a column vector of shape (2, 1);
# we can then broadcast it directly against x to produce the same
# output.
print(x + np.reshape(w, (2, 1)))
# Multiply a matrix by a constant:
# x has shape (2, 3). Numpy treats scalars as arrays of shape ();
# these can be broadcast together to shape (2, 3), producing the
# following array:
# [[ 2 4 6]
# [ 8 10 12]]
print(x * 2)
广播可以让你的代码更简洁高效，能用就用才是王道。
Numpy Documentation
本文涉及到很多重要的知识点，但是忽略掉了很多细节。请查看官方文档，以了解更多用法：https://docs.scipy.org/doc/numpy/reference/ 。