Python Numpy 教程 - 阅读笔记

该教程我看的是范文迪同学的学习笔记，写的很好，并记录我下我看的时候的理解（使用粗体表示）

Python Numpy 教程

感谢Justin Johnson为本教程做出的贡献。

我们将在这门课中全部使用python语言。Python是一门很好的通用编程软件，但在一些流行库（numpy，scipy,matplotlib）的帮助下，使它拥有强大的科学计算环境。

我们期望你们中的很多人有一些Python和numpy的编程经验；而另一部分人，这一部分将提供一个关于python编程语言和使用python进行科学计算的冲刺课程。

你们中的一些人可能过去有matlab经验，那么我们推荐你们浏览 numpy for Matlab users网页。

Python

Python是一种高级的、动态类型的多范型编程语言。Python代码通常被认为是伪代码,它允许你用几行代码就表达出很强大的idea的同时不失可读性。作为例子，下面是用Python表示的快速分类算法：

def quicksort(arr):
if len(arr) <= 1:
return arr
pivot = arr[len(arr) // 2]
left = [x for x in arr if x < pivot]
middle = [x for x in arr if x == pivot]
right = [x for x in arr if x > pivot]
return quicksort(left) + middle + quicksort(right)

print(quicksort([3,6,8,10,1,2,1]))
# Prints "[1, 1, 2, 3, 6, 8, 10]"

以上代码就是二分查找进行排序，在Python里，[括号中可以写一写语句，很像matlab，他会执行，并返回结果，其实就是相当于解释器中的一切皆为函数- -]

Python版本

当前有两个不同的Python支持版本2.7和3.5。Python3.0引入了 ，对语法有所改变。那么在2.7版本下写的程序可能在3.0以后版本不能很好运行或者不能运行。本节课的代码将使用Python3.5（我系统装的是3.6，不过python2和3是有跨度和差异的，具体的话还有大神写的转化器，毕竟的解释型语言，没那么多条条框框，具体的下支版本的代码是通用的，编辑器的话，用的是VS Code，简单易用，只需要在自己电脑安装Python环境以及下载运行插件即可）。

4000

基本数据类型

和很多语言一样，python有很多基本数据类型，包括：整型，浮点型，布尔型和字符串类型（string）。基本数据类型的用法和其他语言一样。

数字：整型和浮点型就像其他语言一样：

x = 3
print(type(x)) # Prints "<class 'int'>"
print(x)       # Prints "3"
print(x + 1)   # Addition; prints "4"
print(x - 1)   # Subtraction; prints "2"
print(x * 2)   # Multiplication; prints "6"
print(x ** 2)  # Exponentiation; prints "9"
x += 1
print(x)  # Prints "4"
x *= 2
print(x)  # Prints "8"
y = 2.5
print(type(y)) # Prints "<class 'float'>"
print(y, y + 1, y * 2, y ** 2) # Prints "2.5 3.5 5.0 6.25"

提示一下，python和其他语言的一点不一样在于没有自加（++）或自减（–）运算符

python对复杂的数字也有内置类型，详情见文档(看到这里，我感觉吧，没有自增自减是为什么呢？弱类型，要什么自增自减。。)

布尔型：python对所有常见运算符使用boolean逻辑表达，但对于英语单词更倾向于使用符号（&&, ||, …）：

Python里没有&&和||这二个我们非常熟悉的符号而是使用and 和 or 来代替，对了 还有!这个符号，这里用的是not 来代替，反正刚才自己试验的时候，就实现了这三种，位运算符还是有的（<<, >> , ~, &, |）

t = True
f = False
print(type(t)) # Prints "<class 'bool'>"
print(t and f) # Logical AND; prints "False"
print(t or f)  # Logical OR; prints "True"
print(not t)   # Logical NOT; prints "False"
print(t != f)  # Logical XOR; prints "True"

字符型String：python对String类型有很好的支持：

hello = 'hello'    # String literals can use single quotes
world = "world"    # or double quotes; it does not matter.
print(hello)       # Prints "hello"
print(len(hello))  # String length; prints "5"
hw = hello + ' ' + world  # String concatenation
print(hw)  # prints "hello world"
hw12 = '%s %s %d' % (hello, world, 12)  # sprintf style string formatting
print(hw12)  # prints "hello world 12"

对String对象有很多有用的方法，例如：

s = "hello"
print(s.capitalize())  # Capitalize a string; prints "Hello"
print(s.upper())       # Convert a string to uppercase; prints "HELLO"
print(s.rjust(7))      # Right-justify a string, padding with spaces; prints "  hello"
print(s.center(7))     # Center a string, padding with spaces; prints " hello "
print(s.replace('l', '(ell)'))  # Replace all instances of one substring with another;
# prints "he(ell)(ell)o"
print('  world '.strip())  # Strip leading and trailing whitespace; prints "world"

你可以在文档里找到String方法列表。

容器

python包括几个内置类型：lists, dictionaries, sets, tuples。

Lists

python中list等同于数组，但是可以重定义大小且能包含不同类型元素：

xs = [3, 1, 2]    # Create a list
print(xs, xs[2])  # Prints "[3, 1, 2] 2"
print(xs[-1])     # Negative indices count from the end of the list; prints "2"
xs[2] = 'foo'     # Lists can contain elements of different types
print(xs)         # Prints "[3, 1, 'foo']"
xs.append('bar')  # Add a new element to the end of the list
print(xs)         # Prints "[3, 1, 'foo', 'bar']"
x = xs.pop()      # Remove and return the last element of the list
print(x, xs)      # Prints "bar [3, 1, 'foo']"

详细细节见文档

说到这个list，就非常d了，还记得Java，C家族这些强类型语言吧，对数组的内存操作有着非常强的限制，Python这里的话，在解释器中，将list写成了循环链表（反正我们这里可以这么理解）而且语法跟matlab很像

Slicing：除了一次访问一个列表元素之外，Python还提供了访问子列表的简明语法，这就是所谓的Slicing（切割，切片，Slice）:

nums = list(range(5))     # range is a built-in function that creates a list of integers
print(nums)               # Prints "[0, 1, 2, 3, 4]"
print(nums[2:4])          # Get a slice from index 2 to 4 (exclusive); prints "[2, 3]"
print(nums[2:])           # Get a slice from index 2 to the end; prints "[2, 3, 4]"
print(nums[:2])           # Get a slice from the start to index 2 (exclusive); prints "[0, 1]"
print(nums[:])            # Get a slice of the whole list; prints "[0, 1, 2, 3, 4]"
print(nums[:-1])          # Slice indices can be negative; prints "[0, 1, 2, 3]"
nums[2:4] = [8, 9]        # Assign a new sublist to a slice
print(nums)               # Prints "[0, 1, 8, 9, 4]"

这里的话，下标还是从0开始，支持负数（指上面list的负数），设置步长的时候也支持负数具体可以看这里，分为基础和扩展切片，基础就是默认步长为1，扩展就是设置步长

我们将在numpy数组的环境中再次看到Slicing。

循环：遍历元素：

animals = ['cat', 'dog', 'monkey']
for animal in animals:
print(animal)
# Prints "cat", "dog", "monkey", each on its own line.

如果你想访问循环体中的元素索引，你可以使用内置的enumerate函数:

animals = ['cat', 'dog', 'monkey']
for idx, animal in enumerate(animals):
print('#%d: %s' % (idx + 1, animal))
# Prints "#1: cat", "#2: dog", "#3: monkey", each on its own line

其中enumerate还接受第二个参数，即初始下标，在Python里用缩进来代表块（所以不能随便缩进，也不能没有缩进- -）

列表推导式：

编程时，我们经常会想从一种数据类型转换到另一种。举个例子，计算数的平方：

nums = [0, 1, 2, 3, 4]
squares = []
for x in nums:
squares.append(x ** 2)
print(squares)   # Prints [0, 1, 4, 9, 16]

用列表推导式可以简化：

nums = [0, 1, 2, 3, 4]
squares = [x ** 2 for x in nums]
print(squares)   # Prints [0, 1, 4, 9, 16]

列表推导式同样包含条件语句：

nums = [0, 1, 2, 3, 4]
even_squares = [x ** 2 for x in nums if x % 2 == 0]
print(even_squares)  # Prints "[0, 4, 16]"

Dictionaries：

Dictionary存储键值对（key,value）,就像java里的map和javascript里的object一样。

d = {'cat': 'cute', 'dog': 'furry'}  # Create a new dictionary with some data
print(d['cat'])       # Get an entry from a dictionary; prints "cute"
print('cat' in d)     # Check if a dictionary has a given key; prints "True"
d['fish'] = 'wet'     # Set an entry in a dictionary
print(d['fish'])      # Prints "wet"
# print(d['monkey'])  # KeyError: 'monkey' not a key of d
print(d.get('monkey', 'N/A'))  # Get an element with a default; prints "N/A"
print(d.get('fish', 'N/A'))    # Get an element with a default; prints "wet"
del d['fish']         # Remove an element from a dictionary
print(d.get('fish', 'N/A')) # "fish" is no longer a key; prints "N/A"

见文档

d = {'1':'2', 1:'3'};
print(d[1] + d['1'], end='');
# Print 32

循环：在字典中对键进行迭代是很容易的：

d = {'person': 2, 'cat': 4, 'spider': 8}
for animal in d:
legs = d[animal]
print('A %s has %d legs' % (animal, legs))
# Prints "A person has 2 legs", "A cat has 4 legs", "A spider has 8 legs"

如果你想访问键及其对应的键值，使用items函数。

字典推导式：同列表推导式类似，但允许你更简单的创建字典：

nums = [0, 1, 2, 3, 4]
even_num_to_square = {x: x ** 2 for x in nums if x % 2 == 0}
print(even_num_to_square)  # Prints "{0: 0, 2: 4, 4: 16}"

sets

set是离散元的无序集，一个简单的例子：

animals = {'cat', 'dog'}
print('cat' in animals)   # Check if an element is in a set; prints "True"
print('fish' in animals)  # prints "False"
animals.add('fish')       # Add an element to a set
print('fish' in animals)  # Prints "True"
print(len(animals))       # Number of elements in a set; prints "3"
animals.add('cat')        # Adding an element that is already in the set does nothing
print(len(animals))       # Prints "3"
animals.remove('cat')     # Remove an element from a set
print(len(animals))       # Prints "2"

见文档

循环：遍历set和遍历list的语法相同；然而，由于集合是无序的，所以您不能对您访问集合元素的顺序作出假设:

animals = {'cat', 'dog', 'fish'}
for idx, animal in enumerate(animals):
print('#%d: %s' % (idx + 1, animal))
# Prints "#1: fish", "#2: dog", "#3: cat"

集合推导式：和list，dictionary一样，我们可以很容易地用集合推导式来构造集合：

from math import sqrt
nums = {int(sqrt(x)) for x in range(30)}
print(nums)  # Prints "{0, 1, 2, 3, 4, 5}"

Tuples

tuple是一个(不可变的)有序的值列表。元组在许多方面与列表相似;最重要的区别之一是，元组可以用作字典中的键，也可以作为集合的元素，而列表则不能。这里有一个简单的例子：

d = {(x, x + 1): x for x in range(10)}  # Create a dictionary with tuple keys
t = (5, 6)        # Create a tuple
print(type(t))    # Prints "<class 'tuple'>"
print(d[t])       # Prints "5"
print(d[(1, 2)])  # Prints "1"

见文档

函数

Python函数是使用def关键字定义的。例如:

def sign(x):
if x > 0:
return 'positive'
elif x < 0:
return 'negative'
else:
return 'zero'

for x in [-1, 0, 1]:
print(sign(x))
# Prints "negative", "zero", "positive"

我们通常会定义一些函数来选择可选的关键字参数（有默认值），比如：

def hello(name, loud=False):
if loud:
print('HELLO, %s!' % name.upper())
else:
print('Hello, %s' % name)

hello('Bob') # Prints "Hello, Bob"
hello('Fred', loud=True)  # Prints "HELLO, FRED!"

见文档

类

在Python中定义类的语法很简单:

class Greeter(object):

# Constructor
def __init__(self, name):
self.name = name  # Create an instance variable

# Instance method
def greet(self, loud=False):
if loud:
print('HELLO, %s!' % self.name.upper())
else:
print('Hello, %s' % self.name)

g = Greeter('Fred')  # Construct an instance of the Greeter class
g.greet()            # Call an instance method; prints "Hello, Fred"
g.greet(loud=True)   # Call an instance method; prints "HELLO, FRED!"

见文档

NumPy安装，直接用pip install NumPy即可

Scipy安装，直接用pip install Scipy即可

如果不能用pip下载，则需要使用github下载，接着用cmd安装python setup.py install

再不行就上这个网站进行下载(根据自己的系统，注意，这里还要安装numpy+MKL)

Numpy

Numpy是Python中科学计算的核心库。它提供了高性能多维数组对象和用于处理这些数组的工具。如果你已经熟悉了MATLAB，你可能会发现本教程对于开始学习Numpy很有用。

数组

numpy数组是一个值网格，所有类型都是相同类型的，并且被一个非负整数的元组索引。维数是数组的秩；形状是每个维度给定尺寸的的整元组。

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])   # Create a rank 1 array
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                  # 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.shape)                     # Prints "(2, 3)"
print(b[0, 0], b[0, 1], b[1, 0])   # Prints "1 2 4"

Numpy也提供了许多创建数组的函数：

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
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]]"

详情见文档

跟matlab很像

数组索引

Numpy提供了几种索引数组的方法。

切片（Slicing）:类似于Python列表，可以将numpy数组切片。由于数组可能是多维的，所以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]])

# 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]

# 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"

您还可以将整型索引与片索引相混合。但是，这样做会产生比原始数组更低级的数组。注意，这与MATLAB处理数组切片的方式有很大的不同:

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)"

# 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)"

整数数组索引： 当使用切片索引到numpy数组时，生成的数组视图将始终是原始数组的子阵列。相反，整数数组索引允许您使用另一个数组的数据来构造任意数组。这是一个例子：

import numpy as np

a = np.array([[1,2], [3, 4], [5, 6]])

# 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]"

整数数组索引的一个有用的技巧就是从一个矩阵的每一行中选择一个元素：

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])

# 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]])

布尔数组索引： 布尔数组索引允许您选择数组的任意元素。通常，这种类型的索引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]])

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]"

为了简洁起见，我们省略了很多关于numpy数组索引的细节; 如果您想了解更多，您应该 阅读文档。

数据类型

每个numpy数组都是相同类型元素的网格。Numpy提供了一组可用于构造数组的数字数据类型。Numpy尝试在创建数组时猜测数据类型，但构造数组的函数通常还包含可选参数以显式指定数据类型。这是一个例子：

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"

您可以阅读文档中的所有关于numpy数据类型 。

数组数学

基本的数学函数在数组上以元素方式运算，并且可用作numpy模块中的操作符重载和函数：

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))

请注意，与MATLAB不同，*是元素乘法，而不是矩阵乘法。我们使用该dot函数来计算向量的内积，乘以一个向量乘以一个矩阵，并乘以矩阵。dot可用作numpy模块中的函数和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))

# 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))

Numpy提供了许多有用的功能，用于对数组执行计算; 其中最有用的是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]"

您可以在文档中找到由numpy提供的数学函数的完整列表 。

除了使用数组来计算数学函数，我们经常需要重新整形或以其他方式处理数组中的数据。这种类型的操作的最简单的例子是转置矩阵; 要转置矩阵，只需使用数组对象的属性T：

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提供了更多的功能来操作数组; 您可以在文档中看到完整列表 。

Broadcasting（广播？）

Broadcasting是一种强大的机制，允许numpy在执行算术运算时使用不同形状的数组。通常我们有一个较小的数组和一个更大的数组，我们想要使用较小的数组多次来对较大的数组执行一些操作。

例如，假设我们要为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

# 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)

然而，当矩阵x非常大时，在Python中计算显式循环可能很慢。请注意，将向量v加到矩阵x的每一行中，就等于形成一个矩阵vv，通过对v垂直地叠加多个拷贝，然后对x和vv进行元素的求和。我们可以这样来实现这个方法:

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广播允许我们在不实际创建v的多个拷贝的情况下进行计算，考虑这个版本，使用广播:

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]]"

直线y=x+v，尽管因为广播,x为4×3矩阵，v为3×1矩阵;对这条线来说v实际上 被当作4×3矩阵，每一行都是v的拷贝，并且求和是在元素上进行的。

广播两个数组遵循以下规则:(shape为矩阵大小，如4×3)

1. 如果数组没有相同的秩，让所有输入数组都向其中shape最长的数组看齐，shape中不足的部分都通过在前面加1补齐。

2. 如果两个数组在维度中具有相同的大小，或者在维度中有一个数组的大小为1，那么这两个数组在维度上是兼容的。

3. 如果在所有维度上兼容，阵列就可以进行广播。

4. 在广播之后，每一个数组的shape等于两个输入数组中元素最大值的shape。

5. 在任何一个数组中，一个数组的大小为1，而另一个数组的大小大于1，第一个数组在第二个数组的维度上复制。

如果这个解释没有意义，请尝试从文档 或解释中阅读 。

支持广播的功能称为通用功能。您可以在文档中找到所有通用功能的列表 。

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)

# 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)

# 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)
# 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)

广播通常会使您的代码更简洁快捷，因此您应尽可能地努力使用它。

Numpy文档

这个简短的概述已经涉及到您需要了解的许多重要的事情，但是远远不够。查看 numpy参考 ，了解更多关于numpy的信息。

SciPy

Numpy提供了一个高性能的多维数组和基本的工具来计算和操纵这些数组。 SciPy 建立在此基础之上，并提供了大量功能，可用于numpy数组，并且可用于不同类型的科学和工程应用程序。

熟悉SciPy的最佳方法是 浏览文档。我们将重点介绍SciPy的一些可能对此类有用的部分。

图像操作

SciPy提供了一些基本功能来处理图像。例如，它具有从磁盘读取图像到numpy数组，将numpy数组写入磁盘作为图像并调整图像大小的功能。这是一个简单的例子来展示这些功能：

from scipy.misc import imread, imsave, imresize

# Read an JPEG image into a numpy array
img = imread('assets/cat.jpg')
print(img.dtype, img.shape)  # Prints "uint8 (400, 248, 3)"

# We can tint the image by scaling each of the color channels
# by a different scalar constant. The image has shape (400, 248, 3);
# we multiply it by the array [1, 0.95, 0.9] of shape (3,);
# numpy broadcasting means that this leaves the red channel unchanged,
# and multiplies the green and blue channels by 0.95 and 0.9
# respectively.
img_tinted = img * [1, 0.95, 0.9]

# Resize the tinted image to be 300 by 300 pixels.
img_tinted = imresize(img_tinted, (300, 300))

# Write the tinted image back to disk
imsave('assets/cat_tinted.jpg', img_tinted)

MATLAB文件

函数scipy.io.loadmat和scipy.io.savemat允许您读取和写入MATLAB文件。您可以在文档中阅读它们 。

点间距

SciPy定义了一些有用的功能，用于计算点集合之间的距离。

该函数scipy.spatial.distance.pdist计算给定集合中所有点对之间的距离：

import numpy as np
from scipy.spatial.distance import pdist, squareform

# Create the following array where each row is a point in 2D space:
# [[0 1]
#  [1 0]
#  [2 0]]
x = np.array([[0, 1], [1, 0], [2, 0]])
print(x)

# Compute the Euclidean distance between all rows of x.
# d[i, j] is the Euclidean distance between x[i, :] and x[j, :],
# and d is the following array:
# [[ 0.          1.41421356  2.23606798]
#  [ 1.41421356  0.          1.        ]
#  [ 2.23606798  1.          0.        ]]
d = squareform(pdist(x, 'euclidean'))
print(d)

您可以在文档中阅读有关此功能的所有详细信息 。

类似的函数（scipy.spatial.distance.cdist）计算两组点之间的所有对之间的距离; 您可以在文档中阅读 。

Matplotlib

Matplotlib是一个绘图库。本节简要介绍了该matplotlib.pyplot模块，该模块提供了与MATLAB类似的绘图系统。

绘制

matplotlib中最重要的功能是plot允许您绘制2D数据。这是一个简单的例子：

import numpy as np
import matplotlib.pyplot as plt

# Compute the x and y coordinates for points on a sine curve
x = np.arange(0, 3 * np.pi, 0.1)
y = np.sin(x)

# Plot the points using matplotlib
plt.plot(x, y)
plt.show()  # You must call plt.show() to make graphics appear.

只需一点额外的工作，我们可以轻松地绘制多行，并添加标题，图例和轴标签：

import numpy as np
import matplotlib.pyplot as plt

# Compute the x and y coordinates for points on sine and cosine curves
x = np.arange(0, 3 * np.pi, 0.1)
y_sin = np.sin(x)
y_cos = np.cos(x)

# Plot the points using matplotlib
plt.plot(x, y_sin)
plt.plot(x, y_cos)
plt.xlabel('x axis label')
plt.ylabel('y axis label')
plt.title('Sine and Cosine')
plt.legend(['Sine', 'Cosine'])
plt.show()

您可以使用该subplot功能在同一个图中绘制不同的东西。这是一个例子：

import numpy as np
import matplotlib.pyplot as plt

# Compute the x and y coordinates for points on sine and cosine curves
x = np.arange(0, 3 * np.pi, 0.1)
y_sin = np.sin(x)
y_cos = np.cos(x)

# Set up a subplot grid that has height 2 and width 1,
# and set the first such subplot as active.
plt.subplot(2, 1, 1)

# Make the first plot
plt.plot(x, y_sin)
plt.title('Sine')

# Set the second subplot as active, and make the second plot.
plt.subplot(2, 1, 2)
plt.plot(x, y_cos)
plt.title('Cosine')

# Show the figure.
plt.show()

您可以使用该imshow功能显示图像。这是一个例子：

import numpy as np
from scipy.misc import imread, imresize
import matplotlib.pyplot as plt

img = imread('assets/cat.jpg')
img_tinted = img * [1, 0.95, 0.9]

# Show the original image
plt.subplot(1, 2, 1)
plt.imshow(img)

# Show the tinted image
plt.subplot(1, 2, 2)

# A slight gotcha with imshow is that it might give strange results
# if presented with data that is not uint8. To work around this, we
# explicitly cast the image to uint8 before displaying it.
plt.imshow(np.uint8(img_tinted))
plt.show()