A brief introduction to some of the scientific computing used in this course. In particular the NumPy scientific computing package and its use with python.
1 Outline
1.1 Goals
In this lab, you will review the features of NumPy and Python that are used in Course 1.
1.2 Useful References
NumPy Documentation including a basic introduction: NumPy.orgA challenging feature topic: NumPy Broadcasting
2 Python and NumPy
Python是我们本课程使用的编程语言，其有一系列的数据类型和算术运算；NumPy是一个库，扩展了Python的基本功能，以添加更丰富的数据集，包括更多的数据类型、向量、矩阵和许多矩阵函数 二者可以无缝衔接协同工作，Python的算术运算符可以处理NumPy的数据类型，许多NumPy函数可以接受Python的数据类型
import numpy as np # it is an unofficial standard to use np for numpy
import time
3 Vector
3.1 Abstract
vector是有序的数字数组，用小写粗体字母表示
x
\mathbf{x}
xvector中的元素都是相同类型，不能同时包含字符和数字vector中元素的数量通常被称为维度，数学家称其为秩vector中的索引为0至n - 1，可以用索引进行引用，单独引用时会写在下标，如
x
0
x_0
x0 ，此时不加粗
3.2 NumPy Arrays
NumPy的基本数据结构是一个可索引的n维数组（n-demensional array），包含相同类型（dtype）的元素 上面，维度指向量中元素的数量，这里指数组的索引数量 一维数组1-D array有一个索引，在course 1中，将vectors表示为NumPy的1-D arrays
1-D array, shape (n,): n elements indexed [0] through [n-1]
3.3 Vector Creation
NumPy的数据创建通常会由第一个参数，代表对象的shape，this can either be a single value for a 1-D result or a tuple (n,m,…) specifying the shape of the result.
# NumPy routines which allocate memory and fill arrays with value
a = np.zeros(4); print(f"np.zeros(4) : a = {a}, a shape = {a.shape}, a data type = {a.dtype}")
a = np.zeros((4,)); print(f"np.zeros(4,) : a = {a}, a shape = {a.shape}, a data type = {a.dtype}")
a = np.random.random_sample(4); print(f"np.random.random_sample(4): a = {a}, a shape = {a.shape}, a data type = {a.dtype}")
输出如下
np.zeros(4) : a = [0. 0. 0. 0.], a shape = (4,), a data type = float64
np.zeros(4,) : a = [0. 0. 0. 0.], a shape = (4,), a data type = float64
np.random.random_sample(4): a = [0.38919476 0.38019795 0.86953179 0.1653972 ], a shape = (4,), a data type = float64
Some data creation routines do not take a shape tuple.
# NumPy routines which allocate memory and fill arrays with value but do not accept shape as input argument
a = np.arange(4.); print(f"np.arange(4.): a = {a}, a shape = {a.shape}, a data type = {a.dtype}")
a = np.random.rand(4); print(f"np.random.rand(4): a = {a}, a shape = {a.shape}, a data type = {a.dtype}")
输出如下
np.arange(4.): a = [0. 1. 2. 3.], a shape = (4,), a data type = float64
np.random.rand(4): a = [0.56777089 0.44204559 0.45052726 0.41138661], a shape = (4,), a data type = float64
values can be specified manually as well.
# NumPy routines which allocate memory and fill with user specified values
a = np.array([5,4,3,2]); print(f"np.array([5,4,3,2]): a = {a}, a shape = {a.shape}, a data type = {a.dtype}")
a = np.array([5.,4,3,2]); print(f"np.array([5.,4,3,2]): a = {a}, a shape = {a.shape}, a data type = {a.dtype}")
输出如下
np.array([5,4,3,2]): a = [5 4 3 2], a shape = (4,), a data type = int32
np.array([5.,4,3,2]): a = [5. 4. 3. 2.], a shape = (4,), a data type = float64
这些都是创建一个具有四个元素的one-dimensional vector的方法 a.shape返回尺寸，返回数据类型为tuple，对于n行m列的数组，返回值是 (n, m) 此处 a,shape = (4, ) 表示一个具有四个元素的一维数组
3.4 Operations on Vectors
Let’s explore some operations using vectors.
3.4.1 Indexing
可以通过索引和切片来访问向量的元素，NumPy提供了一套非常完整的索引和切片功能 在这里只探索课程所需的基础知识有关更多详细信息，请参阅Slicing and Indexing
索引是指通过数组中某个元素的位置来引用该元素，切片意味着根据元素的索引从数组中获取元素的子集
NumPy从零开始索引，因此向量
a
\mathbf{a}
a 的第三个元素是a[2]
#vector indexing operations on 1-D vectors
a = np.arange(10)
print(a)
#access an element
print(f"a[2].shape: {a[2].shape} a[2] = {a[2]}, Accessing an element returns a scalar")
# access the last element, negative indexes count from the end
# -2是8，倒着来
print(f"a[-1] = {a[-1]}")
#indexs must be within the range of the vector or they will produce and error
try:
c = a[10]
except Exception as e:
print("The error message you'll see is:")
print(e)
输出如下
[0 1 2 3 4 5 6 7 8 9]
a[2].shape: () a[2] = 2, Accessing an element returns a scalar
a[-1] = 9
The error message you'll see is:
index 10 is out of bounds for axis 0 with size 10
3.4.2 Slicing
切片使用一组三个值(start: stop: step)创建一个索引数组，只有start / stop也是有效的
#vector slicing operations
a = np.arange(10)
print(f"a = {a}")
#access 5 consecutive elements (start:stop:step)
c = a[2:7:1]; print("a[2:7:1] = ", c)
# access 3 elements separated by two
c = a[2:7:2]; print("a[2:7:2] = ", c)
# access all elements index 3 and above
c = a[3:]; print("a[3:] = ", c)
# access all elements below index 3
c = a[:3]; print("a[:3] = ", c)
# access all elements
c = a[:]; print("a[:] = ", c)
输出如下
a = [0 1 2 3 4 5 6 7 8 9]
a[2:7:1] = [2 3 4 5 6]
a[2:7:2] = [2 4 6]
a[3:] = [3 4 5 6 7 8 9]
a[:3] = [0 1 2]
a[:] = [0 1 2 3 4 5 6 7 8 9]
3.4.3 Single Vector Operations
有许多有用的operations涉及对单个向量的操作
a = np.array([1,2,3,4])
print(f"a : {a}")
# negate elements of a
b = -a
print(f"b = -a : {b}")
# sum all elements of a, returns a scalar
b = np.sum(a)
print(f"b = np.sum(a) : {b}")
b = np.mean(a)
print(f"b = np.mean(a): {b}")
b = a**2
print(f"b = a**2 : {b}")
输出如下
a : [1 2 3 4]
b = -a : [-1 -2 -3 -4]
b = np.sum(a) : 10
b = np.mean(a): 2.5
b = a**2 : [ 1 4 9 16]
3.4.4 Vector Vector Element-wise Operations
大多数NumPy算术、逻辑和比较运算也适用于向量，这些操作符对逐个元素进行操作，如
a
+
b
=
∑
i
=
0
n
−
1
a
i
+
b
i
\mathbf{a} + \mathbf{b} = \sum_{i=0}^{n-1} a_i + b_i
a+b=i=0∑n−1ai+bi
a = np.array([ 1, 2, 3, 4])
b = np.array([-1,-2, 3, 4])
print(f"Binary operators work element wise: {a + b}")
输出如下
Binary operators work element wise: [0 0 6 8]
为了保证运算正确，进行运算的向量必须是相同大小的
#try a mismatched vector operation
c = np.array([1, 2])
try:
d = a + c
except Exception as e:
print("The error message you'll see is:")
print(e)
输出如下
The error message you'll see is:
operands could not be broadcast together with shapes (4,) (2,)
3.4.5 Scalar Vector Operations
vectors可以通过标量值进行缩放，标量值只是一个数字，乘以vectors的所有元素
a = np.array([1, 2, 3, 4])
# multiply a by a scalar
b = 5 * a
print(f"b = 5 * a : {b}")
输出如下
b = 5 * a : [ 5 10 15 20]
3.4.6 Vector Vector Dot Product
点积是线性代数和NumPy的主要内容，是本课程中广泛使用的一个操作
点积将两个vectors中的值逐元素相乘并对结果求和，要求两个vectors的尺寸相同 使用for循环，实现一个返回两个vectors点积的函数，the function to return given inputs
a
a
a and
b
b
b:
x
=
∑
i
=
0
n
−
1
a
i
b
i
x = \sum_{i=0}^{n-1} a_i b_i
x=i=0∑n−1aibi Assume both a and b are the same shape.
def my_dot(a, b):
"""
Compute the dot product of two vectors
Args:
a (ndarray (n,)): input vector
b (ndarray (n,)): input vector with same dimension as a
Returns:
x (scalar):
"""
x=0
for i in range(a.shape[0]):
x = x + a[i] * b[i]
return x
# test 1-D
a = np.array([1, 2, 3, 4])
b = np.array([-1, 4, 3, 2])
print(f"my_dot(a, b) = {my_dot(a, b)}")
输出如下
my_dot(a, b) = 24
注意，点积应返回标量值 尝试使用np,dot来完成点积操作
# test 1-D
a = np.array([1, 2, 3, 4])
b = np.array([-1, 4, 3, 2])
c = np.dot(a, b)
print(f"NumPy 1-D np.dot(a, b) = {c}, np.dot(a, b).shape = {c.shape} ")
c = np.dot(b, a)
print(f"NumPy 1-D np.dot(b, a) = {c}, np.dot(a, b).shape = {c.shape} ")
输出如下
NumPy 1-D np.dot(a, b) = 24, np.dot(a, b).shape = ()
NumPy 1-D np.dot(b, a) = 24, np.dot(a, b).shape = ()
结果相同
3.4.7 The Need for Speed: Vector vs For-loop
使用NumPy库是因为其提高了速度和内存效率，演示如下
np.random.seed(1)
a = np.random.rand(10000000) # very large arrays
b = np.random.rand(10000000)
tic = time.time() # capture start time
c = np.dot(a, b)
toc = time.time() # capture end time
print(f"np.dot(a, b) = {c:.4f}")
print(f"Vectorized version duration: {1000*(toc-tic):.4f} ms ")
tic = time.time() # capture start time
c = my_dot(a,b)
toc = time.time() # capture end time
print(f"my_dot(a, b) = {c:.4f}")
print(f"loop version duration: {1000*(toc-tic):.4f} ms ")
del(a);del(b) #remove these big arrays from memory
输出如下
np.dot(a, b) = 2501072.5817
Vectorized version duration: 1107.5144 ms
my_dot(a, b) = 2501072.5817
loop version duration: 4505.1224 ms
因此在本例中，矢量化提供了很大的速度提升，这是因为NumPy在底层硬件对可用的数据并行性进行了更好地利用 GPU和现代CPU实现单指令多数据（SIMD）管道，允许并行发布多个操作，这在数据集通常非常大的机器学习中至关重要
3.4.8 Vector Vector Operations in Course 1
Vector Vector operations will appear frequently in course 1. 下面是原因：
接下来，我们的例子将存储在一个数组中，X_train of dimension (m,n). 需要注意的是这是一个二维数组或矩阵w will be a 1-dimensional vector of shape (n,).我们将通过循环遍历示例来执行操作，通过索引X来提取每个示例以单独处理，例如X[i]X[i]返回 a value of shape (n,), a 1-dimensional vector. 因此涉及X[i]的运算通常是vector-vector.
# show common Course 1 example
X = np.array([[1],[2],[3],[4]])
w = np.array([2])
c = np.dot(X[1], w)
print(f"X[1] has shape {X[1].shape}")
print(f"w has shape {w.shape}")
print(f"c has shape {c.shape}")
输出如下
X[1] has shape (1,)
w has shape (1,)
c has shape ()
Matrices
4.1 Abstract
矩阵是二维数组，用大写粗体字母表示
X
\mathbf{X}
X，元素均为同一类型 在Lab中，m通常是行数，n通常是列数，矩阵中的元素可以用二维索引来引用
4.2 NumPy Arrays
Matrices have a two-dimensional (2-D) index [m,n]. 在Course 1中，2-D matrices用来保存训练数据 Training data is m examples by n features creating an (m,n) array. Course 1不直接对矩阵进行运算，但通常提取一个例子作为向量并对其进行运算
4.3 Matrix Creation
The same functions that created 1-D vectors will create 2-D or n-D arrays. Below, the shape tuple is provided to achieve a 2-D result. 请注意NumPy是如何使用括号来表示每个维度的，在打印时，NumPy将每行分别打印一行
a = np.zeros((1, 5))
print(f"a shape = {a.shape}, a = {a}")
a = np.zeros((2, 1))
print(f"a shape = {a.shape}, a = {a}")
a = np.random.random_sample((1, 1))
print(f"a shape = {a.shape}, a = {a}")
输出如下
a shape = (1, 5), a = [[0. 0. 0. 0. 0.]]
a shape = (2, 1), a = [[0.]
[0.]]
a shape = (1, 1), a = [[0.44236513]]
也可以手动指定数据，尺寸是用额外的括号指定的，与上面打印的格式相匹配
# NumPy routines which allocate memory and fill with user specified values
a = np.array([[5], [4], [3]]); print(f" a shape = {a.shape}, np.array: a = {a}")
a = np.array([[5], # One can also
[4], # separate values
[3]]); #into separate rows
print(f" a shape = {a.shape}, np.array: a = {a}")
输出如下
a shape = (3, 1), np.array: a = [[5]
[4]
[3]]
a shape = (3, 1), np.array: a = [[5]
[4]
[3]]
4.4 Operations on Matrices
Let’s explore some operations using matrices.
4.4.1 Indexing
矩阵包括第二个索引,这两个索引描述[row, column]，访问可以返回一个元素，也可以返回一行/列
#vector indexing operations on matrices
a = np.arange(6).reshape(-1, 2) #reshape is a convenient way to create matrices
print(f"a.shape: {a.shape}, \na= {a}")
#access an element
print(f"\na[2,0].shape: {a[2, 0].shape}, a[2,0] = {a[2, 0]}, type(a[2,0]) = {type(a[2, 0])} Accessing an element returns a scalar\n")
#access a row
print(f"a[2].shape: {a[2].shape}, a[2] = {a[2]}, type(a[2]) = {type(a[2])}")
输出如下
a.shape: (3, 2),
a= [[0 1]
[2 3]
[4 5]]
a[2,0].shape: (), a[2,0] = 4, type(a[2,0]) =
a[2].shape: (2,), a[2] = [4 5], type(a[2]) =
最后一个例子，仅通过指定行访问将返回一个1-D vector
Reshape
The previous example used reshape to shape the array. a = np.arange(6).reshape(-1, 2) This line of code first created a 1-D Vector of six elements. It then reshaped that vector into a 2-D array using the reshape command. This could have been written: a = np.arange(6).reshape(3, 2) To arrive at the same 3 row, 2 column array. The -1 argument tells the routine to compute the number of rows given the size of the array and the number of columns.
4.4.2 Slicing
切片使用一组三个值(start: stop: step)创建一个索引数组，只有start / stop也是有效的
#vector 2-D slicing operations
a = np.arange(20).reshape(-1, 10)
print(f"a = \n{a}")
#access 5 consecutive elements (start:stop:step)
print("a[0, 2:7:1] = ", a[0, 2:7:1], ", a[0, 2:7:1].shape =", a[0, 2:7:1].shape, "a 1-D array")
#access 5 consecutive elements (start:stop:step) in two rows
print("a[:, 2:7:1] = \n", a[:, 2:7:1], ", a[:, 2:7:1].shape =", a[:, 2:7:1].shape, "a 2-D array")
# access all elements
print("a[:,:] = \n", a[:,:], ", a[:,:].shape =", a[:,:].shape)
# access all elements in one row (very common usage)
print("a[1,:] = ", a[1,:], ", a[1,:].shape =", a[1,:].shape, "a 1-D array")
# same as
print("a[1] = ", a[1], ", a[1].shape =", a[1].shape, "a 1-D array")
输出如下
a =
[[ 0 1 2 3 4 5 6 7 8 9]
[10 11 12 13 14 15 16 17 18 19]]
a[0, 2:7:1] = [2 3 4 5 6] , a[0, 2:7:1].shape = (5,) a 1-D array
a[:, 2:7:1] =
[[ 2 3 4 5 6]
[12 13 14 15 16]] , a[:, 2:7:1].shape = (2, 5) a 2-D array
a[:,:] =
[[ 0 1 2 3 4 5 6 7 8 9]
[10 11 12 13 14 15 16 17 18 19]] , a[:,:].shape = (2, 10)
a[1,:] = [10 11 12 13 14 15 16 17 18 19] , a[1,:].shape = (10,) a 1-D array
a[1] = [10 11 12 13 14 15 16 17 18 19] , a[1].shape = (10,) a 1-D array
Congratulations!
In this lab you mastered the features of Python and NumPy that are needed for Course 1.
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