英文原文链接：http://cs231n.github.io/python-numpy-tutorial/
Numpy
Numpy是Python中科学计算的核心库。它提供了一个高性能的多维数组对象，以及处理这些数组的工具。如果您已经熟悉MATLAB，那么在开始学习Numpy时，您可能会发现本教程非常有用。
Arrays
numpy数组是由所有类型相同的值组成的网格，由非负整数的元组索引。维数为数组的秩;数组的形状是一个整数元组，给出了数组在每个维度上的大小。
我们可以从嵌套的Python列表初始化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还提供了许多函数来创建数组:
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]]"
Array indexing
Numpy提供了几种索引数组的方法。
Slicing:与Python列表类似，numpy数组也可以被切片。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]]) # 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"
您还可以混合使用整数索引和切片索引。但是，这样做将产生一个比原始数组低秩的数组。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)"
Integer array indexing:当您使用切片对numpy数组进行索引时，得到的数组视图将始终是原始数组的子数组。相反，整数数组索引允许使用来自另一个数组的数据构造任意数组。举个例子:
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技巧是从矩阵的每一行中选择或修改一个元素:
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]])
Boolean array indexing:布尔数组索引允许选择数组的任意元素。通常，这种索引类型用于选择满足某种条件的数组元素。举个例子:
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]"
Datatypes
每个numpy数组都是由相同类型的元素组成的网格。Numpy提供了一组可以用来构造数组的大型数字数据类型。在创建数组时，Numpy尝试猜测数据类型，但是构造数组的函数通常还包含一个可选参数来显式指定数据类型。举个例子:
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"
Array math
基本数学函数对数组进行elementwise操作，既可以作为操作符重载，也可以作为numpy模块中的函数:
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))
注意，与MATLAB不同，*是元素乘，而不是矩阵乘。我们用点函数来计算向量的内积，用一个向量乘以一个矩阵，再乘以矩阵。dot既可以作为numpy模块中的函数使用，也可以作为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提供了许多有用的函数来执行数组上的计算;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]"
除了使用数组计算数学函数外，我们还经常需要对数组中的数据进行整形或以其他方式进行操作。这种运算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]"
Broadcasting
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。通常我们有一个更小的数组和一个更大的数组，我们希望多次使用更小的数组来对更大的数组执行一些操作。
例如，假设我们想给矩阵的每一行加上一个常数向量。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)
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)
这可行;然而，当矩阵x非常大时，用Python计算显式循环可能会很慢。注意，将向量v添加到矩阵x的每一行中，等价于通过垂直叠加v的多个副本来形成矩阵vv，然后执行x和vv的元素求和。我们可以这样实现这个方法:
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 broadcast允许我们执行这个计算，而不需要实际创建v的多个副本。
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]]"
行y = x + v工作，即使x有形状(4,3)和v有形状(3,3)，由于广播;这行的工作原理就像v实际上具有形状(4,3)一样，其中每一行都是v的副本，并按elementwise执行求和。
将两个数组一起广播遵循以下规则:
- 如果数组的秩不相同，则将较低秩数组的形状加上1，直到两个形状的长度相同为止。
- 如果两个数组在维度中大小相同，或者其中一个数组在该维度中大小为1，那么这两个数组在维度中是兼容的。
- 如果数组在所有维度上都兼容，则可以将它们一起广播。
- 广播之后，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elementwise最大值。
- 在任何维度中，如果一个数组的大小为1，而另一个数组的大小大于1，那么第一个数组的行为就好像它是沿着该维度复制的
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)
广播通常使您的代码更简洁、更快，所以您应该尽可能地使用它。
SciPy
umpy提供了一个高性能的多维数组和基本工具来计算和操作这些数组。SciPy建立在此基础上，并提供了大量对numpy数组进行操作的函数，适用于不同类型的科学和工程应用程序。
Image operations
SciPy提供了一些处理图像的基本功能。例如，它具有将映像从磁盘读入numpy数组、将numpy数组作为映像写入磁盘以及调整映像大小的功能。下面是一个简单的例子，展示了这些功能:
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 files
scipy.io.loadmat和scipy.io.savemat允许您读取和编写MATLAB文件。
Distance between points
SciPy定义了一些有用的函数来计算点集之间的距离。
scipy.spatial.distance.pdist计算给定集合中所有对点之间的距离:
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)
scipy.spatial.distance.cdist计算两组点之间所有对的距离;
Matplotlib
Matplotlib是一个绘图库。在本节中，将简要介绍matplotlib.pyplot模块，它提供了一个类似于MATLAB的绘图系统。
Plotting
matplotlib中最重要的函数是plot，它允许绘制2D数据。下面是一个简单的例子:
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.
只需做一点额外的工作，我们就可以轻松地同时绘制多条线，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()
Subplots
您可以使用subplot函数在同一图中绘制不同的东西。举个例子:
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()
Images
您可以使用imshow函数来显示图像。举个例子:
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()
