[Python知识库]Python数据可视化(Numpy_5_算术运算和广播)

算术运算和广播

我们已经学到“ NumPy ”课程的最后一节课了。在最后一节课，我们将了解 NumPy 如何对 ndarray 进行算术运算。NumPy 允许对 ndarray 执行元素级运算以及矩阵运算。在这节课，我们将仅了解如何对 ndarray 进行元素级运算。为了进行元素级运算，NumPy 有时候会用到广播功能。广播一词用于描述 NumPy 如何对具有不同形状的 ndarray 进行元素级算术运算。例如，在标量和 ndarray 之间进行算术运算时，会隐式地用到广播。

我们先在 ndarray 之间进行元素级加减乘除运算。为此，我们可以在 NumPy 中使用 np.add() 等函数，或者使用 + 等算术符号，后者与数学方程式的写法更像。这两种形式都执行相同的运算，唯一的区别是如果采用函数方式，函数通常都具有各种选项，可以通过关键字调整这些选项。请注意，在进行元素级运算时，对其执行运算的 ndarray 必须具有相同的形状或者可以广播。我们将在这节课的稍后阶段详细讲解这方面的知识。我们先对秩为 1 的 ndarray 执行元素级算术运算：

# We create two rank 1 ndarrays
x = np.array([1,2,3,4])
y = np.array([5.5,6.5,7.5,8.5])

# We print x
print()
print('x = ', x)

# We print y
print()
print('y = ', y)
print()

# We perfrom basic element-wise operations using arithmetic symbols and functions
print('x + y = ', x + y)
print('add(x,y) = ', np.add(x,y))
print()
print('x - y = ', x - y)
print('subtract(x,y) = ', np.subtract(x,y))
print()
print('x * y = ', x * y)
print('multiply(x,y) = ', np.multiply(x,y))
print()
print('x / y = ', x / y)
print('divide(x,y) = ', np.divide(x,y))

x = [1 2 3 4]

y = [ 5.5 6.5 7.5 8.5]

x + y = [ 6.5 8.5 10.5 12.5]
add(x,y) = [ 6.5 8.5 10.5 12.5]

x - y = [-4.5 -4.5 -4.5 -4.5]
subtract(x,y) = [-4.5 -4.5 -4.5 -4.5]

x * y = [ 5.5 13. 22.5 34. ]
multiply(x,y) = [ 5.5 13. 22.5 34. ]

x / y = [ 0.18181818 0.30769231 0.4 0.47058824]
divide(x,y) = [ 0.18181818 0.30769231 0.4 0.47058824]

我们还可以对秩为 2 的 ndarray 执行相同的元素级算术运算。同样，为了执行这些运算，ndarray 的形状必须一样或者可广播。

# We create two rank 2 ndarrays
X = np.array([1,2,3,4]).reshape(2,2)
Y = np.array([5.5,6.5,7.5,8.5]).reshape(2,2)

# We print X
print()
print('X = \n', X)

# We print Y
print()
print('Y = \n', Y)
print()

# We perform basic element-wise operations using arithmetic symbols and functions
print('X + Y = \n', X + Y)
print()
print('add(X,Y) = \n', np.add(X,Y))
print()
print('X - Y = \n', X - Y)
print()
print('subtract(X,Y) = \n', np.subtract(X,Y))
print()
print('X * Y = \n', X * Y)
print()
print('multiply(X,Y) = \n', np.multiply(X,Y))
print()
print('X / Y = \n', X / Y)
print()
print('divide(X,Y) = \n', np.divide(X,Y))

X =
[[1 2]
 [3 4]]

Y =
[[ 5.5 6.5]
 [ 7.5 8.5]]

X + Y =
[[ 6.5 8.5]
 [ 10.5 12.5]]

add(X,Y) =
[[ 6.5 8.5]
 [ 10.5 12.5]]

X - Y =
[[-4.5 -4.5]
 [-4.5 -4.5]]

subtract(X,Y) =
[[-4.5 -4.5]
 [-4.5 -4.5]]

X * Y =
[[ 5.5 13. ]
 [ 22.5 34. ]]

multiply(X,Y) =
[[ 5.5 13. ]
 [ 22.5 34. ]]

X / Y =
[[ 0.18181818 0.30769231]
 [ 0.4 0.47058824]]

divide(X,Y) =
[[ 0.18181818 0.30769231]
 [ 0.4 0.47058824]]

我们还可以同时对 ndarray 的所有元素应用数学函数，例如 sqrt(x)。

# We create a rank 1 ndarray
x = np.array([1,2,3,4])

# We print x
print()
print('x = ', x)

# We apply different mathematical functions to all elements of x
print()
print('EXP(x) =', np.exp(x))
print()
print('SQRT(x) =',np.sqrt(x))
print()
print('POW(x,2) =',np.power(x,2)) # We raise all elements to the power of 2

x = [1 2 3 4]

EXP(x) = [ 2.71828183 7.3890561 20.08553692 54.59815003]

SQRT(x) = [ 1. 1.41421356 1.73205081 2. ]

POW(x,2) = [ 1 4 9 16]

NumPy 的另一个重要特性是具有大量不同的统计学函数。统计学函数为我们提供了关于 ndarray 中元素的统计学信息。我们来看一些示例：

# We create a 2 x 2 ndarray
X = np.array([[1,2], [3,4]])

# We print x
print()
print('X = \n', X)
print()

print('Average of all elements in X:', X.mean())
print('Average of all elements in the columns of X:', X.mean(axis=0))
print('Average of all elements in the rows of X:', X.mean(axis=1))
print()
print('Sum of all elements in X:', X.sum())
print('Sum of all elements in the columns of X:', X.sum(axis=0))
print('Sum of all elements in the rows of X:', X.sum(axis=1))
print()
print('Standard Deviation of all elements in X:', X.std())
print('Standard Deviation of all elements in the columns of X:', X.std(axis=0))
print('Standard Deviation of all elements in the rows of X:', X.std(axis=1))
print()
print('Median of all elements in X:', np.median(X))
print('Median of all elements in the columns of X:', np.median(X,axis=0))
print('Median of all elements in the rows of X:', np.median(X,axis=1))
print()
print('Maximum value of all elements in X:', X.max())
print('Maximum value of all elements in the columns of X:', X.max(axis=0))
print('Maximum value of all elements in the rows of X:', X.max(axis=1))
print()
print('Minimum value of all elements in X:', X.min())
print('Minimum value of all elements in the columns of X:', X.min(axis=0))
print('Minimum value of all elements in the rows of X:', X.min(axis=1))

X =
[[1 2]
 [3 4]]

Average of all elements in X: 2.5
Average of all elements in the columns of X: [ 2. 3.]
Average of all elements in the rows of X: [ 1.5 3.5]

Sum of all elements in X: 10
Sum of all elements in the columns of X: [4 6]
Sum of all elements in the rows of X: [3 7]

Standard Deviation of all elements in X: 1.11803398875
Standard Deviation of all elements in the columns of X: [ 1. 1.]
Standard Deviation of all elements in the rows of X: [ 0.5 0.5]

Median of all elements in X: 2.5
Median of all elements in the columns of X: [ 2. 3.]
Median of all elements in the rows of X: [ 1.5 3.5]

Maximum value of all elements in X: 4
Maximum value of all elements in the columns of X: [3 4]
Maximum value of all elements in the rows of X: [2 4]

Minimum value of all elements in X: 1
Minimum value of all elements in the columns of X: [1 2]
Minimum value of all elements in the rows of X: [1 3]

最后，我们来看看 NumPy 如何使 ndarray 中的所有元素与单个数字相加，而不使用复杂的循环。

# We create a 2 x 2 ndarray
X = np.array([[1,2], [3,4]])

# We print x
print()
print('X = \n', X)
print()

print('3 * X = \n', 3 * X)
print()
print('3 + X = \n', 3 + X)
print()
print('X - 3 = \n', X - 3)
print()
print('X / 3 = \n', X / 3)

X =
[[1 2]
 [3 4]]

3 * X =
[[ 3 6]
 [ 9 12]]

3 + X =
[[4 5]
 [6 7]]

X - 3 =
[[-2 -1]
 [ 0 1]]

X / 3 =
[[ 0.33333333 0.66666667]
 [ 1. 1.33333333]]

在上述示例中，NumPy 在后台对 ndarray 广播 3，使它们具有相同的形状。这样我们仅使用一行代码，就可以使 X 的每个元素加 3。

Numpy 可以对两个形状不同的 ndarray 执行相同的操作，但是存在一些限制，如下所示。

# We create a rank 1 ndarray
x = np.array([1,2,3])

# We create a 3 x 3 ndarray
Y = np.array([[1,2,3],[4,5,6],[7,8,9]])

# We create a 3 x 1 ndarray
Z = np.array([1,2,3]).reshape(3,1)

# We print x
print()
print('x = ', x)
print()

# We print Y
print()
print('Y = \n', Y)
print()

# We print Z
print()
print('Z = \n', Z)
print()

print('x + Y = \n', x + Y)
print()
print('Z + Y = \n',Z + Y)

x = [1 2 3]

Y =
[[1 2 3]
 [4 5 6]
 [7 8 9]]

Z =
[[1]
 [2]
 [3]]

x + Y =
[[ 2 4 6]
 [ 5 7 9]
 [ 8 10 12]]

Z + Y =
[[ 2 3 4]
 [ 6 7 8]
 [10 11 12]]

和之前一样，NumPy 能够通过沿着大的 ndarray 对更小的 ndarray 进行广播，将 1 x 3 和 3 x 1 ndarray 加到 3 x 3 ndarray 上。通常，NumPy 能够这么操作的前提是，更小的 ndarray（例如我们的示例中的 1 x 3 ndarray）可以扩展成更大的 ndarray 的形状，并且生成的广播很清晰明确。