I haven’t been exposed to mathematics for a long time. Machine learning requires some mathematical knowledge. Here I will review the relevant basic mathematical knowledge.
Vector
Vectors are ordered arrays of numbers. In notation, vectors are represented by lowercase bold letters. The elements of a vector are all of the same type. For example, vectors do not contain characters and numbers. The number of elements in an array is often called a dimension. The number of elements in an array is often called a dimension. The elements of a vector can be referenced using indices. In mathematical settings, the index usually goes from 1 to n. In computer science and these labs, indexes usually run from 0 to n-1. The following is a comparison of the two. In computers, we use the code on the left, which is 0 to n-1.
Arrays in Numpy
The basic data structure of NumPy is an indexable n-dimensional array containing elements of the same type (dtype).
Operations on one-dimensional vectors:
vector creation
Create a one-dimensional vector of the specified shape. The parameters can be integers, primitives, etc., and the parameters represent the shape of the sequence to be created.
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}")Create a one-dimensional vector without specifying a shape
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}")Creates a one-dimensional vector of 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}")Vector operations
Get vector elements: It can be obtained through index and slicing. This operation is similar to the list operation.
-
Obtained through index:
a = np.arange(10)
print(a)
# 获取Index = 2的元素
print(f"a[2].shape: {a[2].shape} a[2] = {a[2]}, Accessing an element returns a scalar")
# 获取最后一个元素
print(f"a[-1]={a[-1]}")
# index 必须在向量有效范围以内，否则会报错
try:
c = a[10]
except Exception as e:
print(e)- Slice to get elements
# 切片操作通过(start:stop:step)这个三个参数来控制要获取的元素，
# 切片操作是左闭右开，也就是包括index=start的值，但是不包括index=stop的值
# 向量切片操作
a = np.arange(10)
print(f"a = {a}")
#获取向量中的从index=2开始到Index=7结束的5个元素, 第三个参数1表示step=1，代表连续取值 (start:stop:step)
c = a[2:7:1]; print("a[2:7:1] = ", c)
# 获取向量中的从index=2开始到Index=7结束的元素, 第三个参数1表示step=2，代表隔一个index取一个值 (start:stop:step)
c = a[2:7:2]; print("a[2:7:2] = ", c)
# 取index 大于3的所有值
c = a[3:]; print("a[3:] = ", c)
# 取index小于3的所有值
c = a[:3]; print("a[:3] = ", c)
# 取所有的值
c = a[:]; print("a[:] = ", c)- Operations on a single vector
a = np.array([1, 2, 3, 4])
print(f"a: {a}")
# 将向量中的元素全部变为相反数
b = -a
print(f"b: {b}")
# 计算向量中所有元素的和并返回一个和的标量
b = np.sum(a)
print(f"b = np.sum(a) : {b}")
# 求向量的平均值
b = np.mean(a)
print(f"b = np.mean(a): {b}")
# 对向量中每个元素求平法
b = a**2
print(f"b = a**2 : {b}")- For operations on vector elements, many numpy operations on numbers are also used on vectors.
# 向量a+向量b, 两个向量长度必须相同，不然会报error
a = np.array([1, 2, 3, 4])
b= np.array([-1, -2, 3, 4])
print(f"Binary operators work element wise: {a + b}")
# 标量和向量的操作
a = np.array([1, 2, 3, 4])
b = 5 * a
print(f"b = 5 * a : {b}")- Dot product of vectors and vectors
Customize a method to implement dot product:
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 my_dot()
a = np.array([1,2,3,4])
b = np.array([-1, 4, 3, 2])
print(f"my_dot(a, b) = {my_dot(a, b)}")Using the dot product method in Numpy:
# 使用numpy中的dot来计算点积，返回一个标量
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} ")Then make an efficiency comparison between the above two methods of calculating dot products.
# 对比一下numpy 的dot和自己写的my_dot的效率如何，可以看出numpy中的效率要高很多
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 memoryThe running results are: You can see that numpy takes much less time
my_dot(a, b) = 24
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 = ()
np.dot(a, b) = 2501072.5817
Vectorized version duration: 6.5184 ms
my_dot(a, b) = 2501072.5817
loop version duration: 2430.3420 ms matrix
A matrix is a two-dimensional array whose elements are all of the same type. Generally represented by capital bold letters. It is represented by two subscripts m and n, where m represents the number of rows and n represents the number of columns. Specified elements can be accessed through two subscripts
Matrix operations
Create matrix
The same method as creating a vector, except that the parameters here need to be replaced by the ancestor
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}")
# 2.创建指定元素的矩阵
a= np.array([[5],
[4],
[3]])
print(f" a shape = {a.shape}, np.array: a = {a}")
Matrix operations
# 3.矩阵的操作
# 3.1 下标访问
# reshape 是一种比较方便的方法创建矩阵,
a = np.arange(6).reshape(-1, 2) #reshape(-1,2) 表示生成一个6/2行，2列的矩阵，也就是3行两列的矩阵
print(f"a.shape:{a.shape},\na={a}")
# 访问一个元素
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")
# 访问一行
print(f"a[2].shape:{a[2].shape},a[2] = {a[2]},type(a[2]) = {type(a[2])}")
# 3.2切片访问
a = np.arange(20).reshape(-1, 10)
print(f"a=\n{a}")
# 访问一行中5个连续的元素(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-1D array")
# 访问两行中5个连续的元素(start:stop:step)
print("a[:, 2:7:1] = \n", a[:, 2:7:1], ", a[:, 2:7:1].shape =", a[:, 2:7:1].shape, "a 2-D array")
# 访问矩阵所有元素
print("a[:,:] = \n", a[:,:], ", a[:,:].shape =", a[:,:].shape)
# 访问一行中的所有元素，方法1
print("a[1,:] = ", a[1,:], ", a[1,:].shape =", a[1,:].shape, "a 1-D array")
# 访问一行中的所有元素，方法2
print("a[1] = ", a[1], ", a[1].shape =", a[1].shape, "a 1-D array")