本文详细介绍Numpy库的基础知识，包括数组的创建与访问、数据类型、数组计算等关键概念。Numpy是Python科学计算的核心库，掌握其用法对于数据分析、机器学习等领域至关重要。
@Python Numpy
Python Numpy
链接: https://zhuanlan.zhihu.com/p/20878530?refer=intelligentunit
Numpy是Python中用于科学计算的核心库。
1.数组Arrays
维度的数量被称为数组的阶，数组的大小是一个由整型数构成的元组，可以描述数组不同维度上的大小。
我们可以从列表创建数组，然后利用方括号访问其中的元素：
import numpy as np
a = np.array([1, 2, 3]) # Create a rank 1 array
print (type(a)) # Prints "<type 'numpy.ndarray'>"
print (a.shape) # Prints "(3,)"
print (a[0], a[1], a[2]) # Prints "1 2 3"
a[0] = 5 # Change an element of the array
print (a) # Prints "[5, 2, 3]"
b = np.array([[1,2,3],[4,5,6]]) # Create a rank 2 array
print (b) # 显示一下矩阵b
print (b.shape) # Prints "(2, 3)"
print (b[0, 0], b[0, 1], b[1, 0]) # Prints "1 2 4"
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
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]]"
2.访问数组
切片：和Python列表类似，numpy数组可以使用切片语法。因为数组可以是多维的，所以你必须为每个维度指定好切片。
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',a)
# 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:',b)
# 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"
可以同时使用整型和切片语法来访问数组。但是，这样做会产生一个比原数组低阶的新数组。
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]])
# 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)"
整型数组访问：当使用切片语法访问数组时，得到的总是原数组的一个子集。整型数组访问允许我们利用其它数组的数据构建一个新的数组：
import numpy as np
a = np.array([[1,2], [3, 4], [5, 6]])
# 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]"
# 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]"
整型数组访问语法还有个有用的技巧，可以用来选择或者更改矩阵中每行中的一个元素
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]])"
# Create an array of indices
b = np.array([0, 2, 0, 1])
print(b)
print(np.arange(4))
print("===========")
# Select one element from each row of a using the indices in b
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]])
布尔型数组访问：布尔型数组访问可以让你选择数组中任意元素。通常，这种访问方式用于选取数组中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]"
# We can do all of the above in a single concise statement:
print (a[a > 2]) # Prints "[3 4 5 6]"
3.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"
4.数组计算
基本数学计算函数会对数组中元素逐个进行计算，既可以利用操作符重载，也可以使用函数方式：
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))
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))
在Numpy中使用dot来进行矩阵乘法：
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))
Numpy提供了很多计算数组的函数，其中最常用的一个是sum：
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]"
我们还常常改变数组或者操作其中的元素。其中将矩阵转置是常用的一个，在Numpy中，使用T来转置矩阵：
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]]"
# Note that taking the transpose of a rank 1 array does nothing:
v = np.array([1,2,3])
L=v.tolist()
print (v,' v size:',v.shape) # Prints "[1 2 3]"
print (v.T, ' v.T size:',v.T.shape) # Prints "[1 2 3]"
4. 广播Broadcasting
广播是一种强有力的机制，它让Numpy可以让不同大小的矩阵在一起进行数学计算。我们常常会有一个小的矩阵和一个大的矩阵，然后我们会需要用小的矩阵对大的矩阵做一些计算。
把一个向量加到矩阵的每一行
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(y)
# 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)
对两个数组使用广播机制要遵守下列规则：
- 如果数组的秩不同，使用1来将秩较小的数组进行扩展，直到两个数组的尺寸的长度都一样。
- 如果两个数组在某个维度上的长度是一样的，或者其中一个数组在该维度上长度为1，那么我们就说这两个数组在该维度上是相容的。
- 如果两个数组在所有维度上都是相容的，他们就能使用广播。
- 如果两个输入数组的尺寸不同，那么注意其中较大的那个尺寸。因为广播之后，两个数组的尺寸将和那个较大的尺寸一样。
- 在任何一个维度上，如果一个数组的长度为1，另一个数组长度大于1，那么在该维度上，就好像是对第一个数组进行了复制。
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]]"
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 row 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)
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