Numpy
Numpy是 Python 科学计算的核心库。它提供了一个高性能的多维数组对象，以及用于处理这些数组的工具。
数组
创建数组
numpy 数组所有类型都相同，并由非负整数元组索引。维数是数组的秩；数组的形状是一个整数元组，给出了数组沿每个维度的大小。
我们可以从嵌套的 Python 列表中初始化 numpy 数组，并使用方括号访问元素：
import numpy as np
a = np.array([1,2,3]) # 创建了一个名为a的数组
print(type(a)) # Prints "<class 'numpy.ndarray'>"
print(a.shape) # Prints "(3,)"
print(a[0],a[1],a[2]) # Prints "1 2 3"
a[0] = 5 # 改变了数组中的一个元素
print(a) # Prints "[5, 2, 3]"
print("################")
b = np.array([[1,2,3],[4,5,6]]) # Create a rank 2 array
print(b.shape) # Prints "(2, 3)"
print(b[0, 0], b[0, 1], b[1, 0]) # Prints "1 2 4"输出结果是这样的：
<class 'numpy.ndarray'> (3,) 1 2 3 [5 2 3] ################ (2, 3) 1 2 4
Numpy 还提供了很多创建数组的函数：
| 函数名 | 功能解释 |
| np.zeros((x,y)) | 创建一个x*y的全0数组/矩阵 |
| np.ones((x,y)) | 创建一个x*y的全1数组/矩阵 |
| np.full((x,y),z) | 创建一个x*y的全z数组/矩阵 |
| np.eye(n) | 创建一个n阶单位矩阵 |
import numpy as np
a = np.zeros((2,2)) # 创建一个全零数组/矩阵
print(a) # Prints "[[0. 0.]
# [0. 0.]]"
print("################")
b = np.ones((1,2)) # 创建一个全一数组/矩阵
print(b) # Prints "[[1. 1.]]"
print("################")
c = np.full((2,2),7) # 创建一个常量数组/矩阵
print(c) # Prints "[[7 7]
# [7 7]]"
print("################")
d = np.eye(2) # 创建一个单位矩阵
print(d) # Prints "[[1. 0.]
# [0. 1.]]"
print("################")
e = np.random.random((2,2)) # 创建一个随机数组
print(e) # Might print "[[0.91940167 0.08143941]
# [0.68744134 0.87236687]]"输出结果是这样的：
[[0. 0.] [0. 0.]] ################ [[1. 1.]] ################ [[7 7] [7 7]] ################ [[1. 0.] [0. 1.]] ################ [[0.23857088 0.24130546] [0.98978102 0.51152028]]
数组索引
Numpy 提供了几种索引数组的方法。
切片： 类似于 Python 列表，numpy 数组可以被切片。由于数组可能是多维的，您必须为数组的每个维度指定一个切片：
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]])
print(a)
print("################")
# Use slicing to pull out the subarray consisting of the first 2 rows
# and columns 1 and 2; b is the following array of shape (2, 2):
# [[2 3]
# [6 7]]
b = a[:2, 1:3]
print(b)
print("################")
# A slice of an array is a view into the same data, so modifying it
# will modify the original 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"输出结果是这样的：
[[ 1 2 3 4] [ 5 6 7 8] [ 9 10 11 12]] ################ [[2 3] [6 7]] ################ 2 77
您还可以将整数索引与切片索引混合使用。但是，这样做会产生一个比原始数组更低阶的数组。
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]])
print(a)
print("################")
# Two ways of accessing the data in the middle row of the array.
# 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)"
print("################")
# We can make the same distinction when accessing columns of an array:
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)"输出结果是这样的：
[[ 1 2 3 4] [ 5 6 7 8] [ 9 10 11 12]] ################ [5 6 7 8] (4,) [[5 6 7 8]] (1, 4) ################ [ 2 6 10] (3,) [[ 2] [ 6] [10]] (3, 1)
整数数组索引
当您使用切片对 numpy 数组进行索引时，生成的数组视图将始终是原始数组的子数组。相反，整数数组索引允许您使用来自另一个数组的数据构造任意数组。这是一个例子：
import numpy as np
a = np.array([[1,2], [3, 4], [5, 6]])
print(a)
print("################")
# An example of integer array indexing.
# The returned array will have shape (3,) and
print(a[[0, 1, 2], [0, 1, 0]]) # Prints "[1 4 5]"
# The above example of integer array indexing is equivalent to this:
print(np.array([a[0, 0], a[1, 1], a[2, 0]])) # Prints "[1 4 5]"
print("################")
# 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]"输出结果如下所示：
[[1 2] [3 4] [5 6]] ################ [1 4 5] [1 4 5] ################ [2 2] [2 2]
整数数组索引的一个有用技巧是从矩阵的每一行中选择或改变一个元素：
import numpy as np
# Create a new array from which we will select elements
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]])"
print("################")
# Create an array of indices
b = np.array([0, 2, 0, 1])
# Select one element from each row of a using the indices in b
print(a[np.arange(4), b]) # Prints "[ 1 6 7 11]"
print("################")
# 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]])输出结果如下所示：
[[ 1 2 3] [ 4 5 6] [ 7 8 9] [10 11 12]] ################ [ 1 6 7 11] ################ [[11 2 3] [ 4 5 16] [17 8 9] [10 21 12]]
布尔数组索引
布尔数组索引可让您挑选出数组的任意元素。这种类型的索引经常用于选择满足某些条件的数组元素。这是一个例子：
import numpy as np
a = np.array([[1,2], [3, 4], [5, 6]])
print(a)
print("################")
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("################")
print(a[bool_idx]) # Prints "[3 4 5 6]"
# We can do all of the above in a single concise statement:
print(a[a > 2]) # Prints "[3 4 5 6]"输出结果如下所示：
[[1 2] [3 4] [5 6]] ################ [[False False] [ True True] [ True True]] ################ [3 4 5 6] [3 4 5 6]
数据类型
每个 numpy 数组都是相同类型元素的网格。Numpy 提供了大量可用于构造数组的数字数据类型。Numpy 会在您创建数组时尝试猜测数据类型，但构造数组的函数通常还包含一个可选参数以显式指定数据类型。
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"int32 float64 int64
import numpy as np
x=np.array([1,2,3],dtype=np.int64)
xarray([1, 2, 3], dtype=int64)
数组数学
基本数学函数按元素对数组进行操作，并且可以作为运算符重载和 numpy 模块中的函数使用：
import numpy as np
x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64)
print(x)
print("################")
print(y)
print("################")
# Elementwise sum; both produce the array
# [[ 6.0 8.0]
# [10.0 12.0]]
print(x + y)
print(np.add(x, y))
print("################")
# Elementwise difference; both produce the array
# [[-4.0 -4.0]
# [-4.0 -4.0]]
print(x - y)
print(np.subtract(x, y))
print("################")
# Elementwise product; both produce the array
# [[ 5.0 12.0]
# [21.0 32.0]]
print(x * y)
print(np.multiply(x, y))
print("################")
# Elementwise division; both produce the array
# [[ 0.2 0.33333333]
# [ 0.42857143 0.5 ]]
print(x / y)
print(np.divide(x, y))
print("################")
# Elementwise square root; produces the array
# [[ 1. 1.41421356]
# [ 1.73205081 2. ]]
print(np.sqrt(x))[[1. 2.] [3. 4.]] ################ [[5. 6.] [7. 8.]] ################ [[ 6. 8.] [10. 12.]] [[ 6. 8.] [10. 12.]] ################ [[-4. -4.] [-4. -4.]] [[-4. -4.] [-4. -4.]] ################ [[ 5. 12.] [21. 32.]] [[ 5. 12.] [21. 32.]] ################ [[0.2 0.33333333] [0.42857143 0.5 ]] [[0.2 0.33333333] [0.42857143 0.5 ]] ################ [[1. 1.41421356] [1.73205081 2. ]]
我们使用dot函数来计算向量的内积、将向量乘以矩阵以及将矩阵相乘。dot既可用作 numpy 模块中的函数，也可用作数组对象的实例方法：
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))
print("################")
# Matrix / vector product; both produce the rank 1 array [29 67]
print(x.dot(v))
print(np.dot(x, v))
print("################")
# Matrix / matrix product; both produce the rank 2 array
# [[19 22]
# [43 50]]
print(x.dot(y))
print(np.dot(x, y))219 219 ################ [29 67] [29 67] ################ [[19 22] [43 50]] [[19 22] [43 50]]
Numpy 提供了许多有用的函数来对数组进行计算
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]"10 [4 6] [3 7]
除了使用数组计算数学函数外，我们还经常需要重塑或以其他方式操作数组中的数据。此类操作的最简单示例是转置矩阵
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]]"
print("################")
# 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]"[[1 2] [3 4]] [[1 3] [2 4]] ################ [1 2 3] [1 2 3]
广播
广播是一种强大的机制，它允许 numpy 在执行算术运算时处理不同形状的数组。通常我们有一个较小的数组和一个较大的数组，我们想多次使用较小的数组来对较大的数组执行一些操作。
例如，假设我们要向矩阵的每一行添加一个常量向量。我们可以这样做：
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
print(x)
print("################")
# 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)[[ 1 2 3] [ 4 5 6] [ 7 8 9] [10 11 12]] ################ [[ 2 2 4] [ 5 5 7] [ 8 8 10] [11 11 13]]
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]]"
print("################")
y = x + vv # Add x and vv elementwise
print(y) # Prints "[[ 2 2 4
# [ 5 5 7]
# [ 8 8 10]
# [11 11 13]]"[[1 0 1] [1 0 1] [1 0 1] [1 0 1]] ################ [[ 2 2 4] [ 5 5 7] [ 8 8 10] [11 11 13]]
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]]"[[ 2 2 4] [ 5 5 7] [ 8 8 10] [11 11 13]]
一起广播两个数组遵循以下规则：
1. 如果数组的秩不同，则在较低秩数组的形状前面加上 1，直到两个形状具有相同的长度。
2. 如果两个数组在维度中具有相同的大小，或者如果其中一个数组在该维度中的大小为 1，则称这两个数组在维度中兼容。
3. 如果阵列在所有维度上都兼容，则它们可以一起广播。
4. 广播后，每个数组的行为就好像它的形状等于两个输入数组的形状的元素最大值。
5. 在一个数组的大小为 1 而另一个数组的大小大于 1 的任何维度中，第一个数组的行为就好像它是沿该维度复制的一样
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)
print("################")
# Add a vector to each row of a matrix
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)
print("################")
# Add a vector to each column of a matrix
# 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)
print("################")
# 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)))
print("################")
# 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)[[ 4 5] [ 8 10] [12 15]] ################ [[2 4 6] [5 7 9]] ################ [[ 5 6 7] [ 9 10 11]] ################ [[ 5 6 7] [ 9 10 11]] ################ [[ 2 4 6] [ 8 10 12]]
