向量化和广播
向量化和广播这两个概念是 numpy 内部实现的基础。有了向量化，编写代码时无需使用显式循环。这些循环实际上不能省略，只不过是在内部实现，被代码中的其他结构代替。向量化的应用使得代码更简洁，可读性更强，也可以说使用了向量化方法的代码看上去更“Pythonic”。
广播（Broadcasting）机制描述了 numpy 如何在算术运算期间处理具有不同形状的数组，让较小的数组在较大的数组上“广播”，以便它们具有兼容的形状。并不是所有的维度都要彼此兼容才符合广播机制的要求，但它们必须满足一定的条件。
若两个数组的各维度兼容，也就是两个数组的每一维等长，或其中一个数组为 一维，那么广播机制就适用。如果这两个条件不满足，numpy就会抛出异常，说两个数组不兼容。
总结来说，广播的规则有三个：
- 如果两个数组的维度数dim不相同，那么小维度数组的形状将会在左边补1。
- 如果shape维度不匹配，但是有维度是1，那么可以扩展维度是1的维度匹配另一个数组；
- 如果shape维度不匹配，但是没有任何一个维度是1，则匹配引发错误；
【例】二维数组加一维数组
import numpy as np
x = np.arange(4)
print(x)
#[0 1 2 3]
y = np.ones((3, 4))
print(y)
# [[1. 1. 1. 1.]
# [1. 1. 1. 1.]
# [1. 1. 1. 1.]]
print(x.shape) # (4,)
print(y.shape) # (3, 4)
print((x + y).shape) # (3, 4)
print(x + y)
# [[1. 2. 3. 4.]
# [1. 2. 3. 4.]
# [1. 2. 3. 4.]]
【例】两个数组均需要广播
import numpy as np
x = np.arange(4).reshape(4, 1)
y = np.ones(5)
print(x)
# [[0]
# [1]
# [2]
# [3]]
print(x.shape) # (4, 1)
print(y)
# [1. 1. 1. 1. 1.]
print(y.shape) # (5,)
print((x + y).shape) # (4, 5)
print(x + y)
# [[1. 1. 1. 1. 1.]
# [2. 2. 2. 2. 2.]
# [3. 3. 3. 3. 3.]
# [4. 4. 4. 4. 4.]]
x = np.array([0.0, 10.0, 20.0, 30.0])
y = np.array([1.0, 2.0, 3.0])
z = x[:, np.newaxis] + y
print(z)
# [[ 1. 2. 3.]
# [11. 12. 13.]
# [21. 22. 23.]
# [31. 32. 33.]]
np.newaxis的功能:插入新维度
a=np.array([1,2,3,4,5])
aa=a[:,np.newaxis]
print(aa.shape) # (5, 1)
print (aa)
# [[1]
# [2]
# [3]
# [4]
# [5]]
a=np.array([1,2,3,4,5])
aa=a[np.newaxis,:]
print(aa.shape) # (1, 5)
print (aa)
# [[1 2 3 4 5]]
【例】不匹配报错的例子
import numpy as np
x = np.arange(4)
y = np.ones(5)
print(x.shape) # (4,)
print(y.shape) # (5,)
print(x + y)
# ValueError: operands could not be broadcast together with shapes (4,) (5,)
数学函数
1.算数运算
**numpy.add
numpy.subtract
numpy.multiply
numpy.divide
numpy.floor_divide
numpy.power**
- numpy.add(x1, x2, *args, **kwargs)
Add arguments element-wise. - numpy.subtract(x1, x2, *args, **kwargs)
Subtract arguments element-wise. - numpy.multiply(x1, x2, *args, **kwargs)
Multiply arguments element-wise. - numpy.divide(x1, x2, *args, **kwargs)
Returns a true division of the inputs, element-wise. - numpy.floor_divide(x1, x2, *args, **kwargs)
Return the largest integer smaller or equal to the division of the inputs. - numpy.power(x1, x2, *args, **kwargs)
First array elements raised to powers from second array, element-wise.
在 numpy 中对以上函数进行了运算符的重载，且运算符为 元素级。也就是说，它们只用于位置相同的元素之间，所得到的运算结果组成一个新的数组。
【例】注意 numpy 的广播规则。
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import numpy as np
x = np.array([1, 2, 3, 4, 5, 6, 7, 8])
y = x + 1
print(y)
print(np.add(x, 1))
# [2 3 4 5 6 7 8 9]
y = x - 1
print(y)
print(np.subtract(x, 1))
# [0 1 2 3 4 5 6 7]
y = x * 2
print(y)
print(np.multiply(x, 2))
# [ 2 4 6 8 10 12 14 16]
y = x / 2
print(y)
print(np.divide(x, 2))
# [0.5 1. 1.5 2. 2.5 3. 3.5 4. ]
y = x // 2
print(y)
print(np.floor_divide(x, 2))
# [0 1 1 2 2 3 3 4]
y = x ** 2
print(y)
print(np.power(x, 2))
# [ 1 4 9 16 25 36 49 64]
【例】注意 numpy 的广播规则。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = x + 1
print(y)
print(np.add(x, 1))
# [[12 13 14 15 16]
# [17 18 19 20 21]
# [22 23 24 25 26]
# [27 28 29 30 31]
# [32 33 34 35 36]]
y = x - 1
print(y)
print(np.subtract(x, 1))
# [[10 11 12 13 14]
# [15 16 17 18 19]
# [20 21 22 23 24]
# [25 26 27 28 29]
# [30 31 32 33 34]]
y = x * 2
print(y)
print(np.multiply(x, 2))
# [[22 24 26 28 30]
# [32 34 36 38 40]
# [42 44 46 48 50]
# [52 54 56 58 60]
# [62 64 66 68 70]]
y = x / 2
print(y)
print(np.divide(x, 2))
# [[ 5.5 6. 6.5 7. 7.5]
# [ 8. 8.5 9. 9.5 10. ]
# [10.5 11. 11.5 12. 12.5]
# [13. 13.5 14. 14.5 15. ]
# [15.5 16. 16.5 17. 17.5]]
y = x // 2
print(y)
print(np.floor_divide(x, 2))
# [[ 5 6 6 7 7]
# [ 8 8 9 9 10]
# [10 11 11 12 12]
# [13 13 14 14 15]
# [15 16 16 17 17]]
y = x ** 2
print(y)
print(np.power(x, 2))
# [[ 121 144 169 196 225]
# [ 256 289 324 361 400]
# [ 441 484 529 576 625]
# [ 676 729 784 841 900]
# [ 961 1024 1089 1156 1225]]
【例】注意 numpy 的广播规则。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.arange(1, 6)
print(y)
# [1 2 3 4 5]
z = x + y
print(z)
print(np.add(x, y))
# [[12 14 16 18 20]
# [17 19 21 23 25]
# [22 24 26 28 30]
# [27 29 31 33 35]
# [32 34 36 38 40]]
z = x - y
print(z)
print(np.subtract(x, y))
# [[10 10 10 10 10]
# [15 15 15 15 15]
# [20 20 20 20 20]
# [25 25 25 25 25]
# [30 30 30 30 30]]
z = x * y
print(z)
print(np.multiply(x, y))
# [[ 11 24 39 56 75]
# [ 16 34 54 76 100]
# [ 21 44 69 96 125]
# [ 26 54 84 116 150]
# [ 31 64 99 136 175]]
z = x / y
print(z)
print(np.divide(x, y))
# [[11. 6. 4.33333333 3.5 3. ]
# [16. 8.5 6. 4.75 4. ]
# [21. 11. 7.66666667 6. 5. ]
# [26. 13.5 9.33333333 7.25 6. ]
# [31. 16. 11. 8.5 7. ]]
z = x // y
print(z)
print(np.floor_divide(x, y))
# [[11 6 4 3 3]
# [16 8 6 4 4]
# [21 11 7 6 5]
# [26 13 9 7 6]
# [31 16 11 8 7]]
z = x ** np.full([1, 5], 2)
print(z)
print(np.power(x, np.full([5, 5], 2)))
# [[ 121 144 169 196 225]
# [ 256 289 324 361 400]
# [ 441 484 529 576 625]
# [ 676 729 784 841 900]
# [ 961 1024 1089 1156 1225]]
np.full(shape, fill_value)可以生成一个元素为fill_value，形状为shape的array。比如
print(np.full([3, 4], 'a'))
# [['a' 'a' 'a' 'a']
# ['a' 'a' 'a' 'a']
# ['a' 'a' 'a' 'a']]
【例】
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.arange(1, 26).reshape([5, 5])
print(y)
# [[ 1 2 3 4 5]
# [ 6 7 8 9 10]
# [11 12 13 14 15]
# [16 17 18 19 20]
# [21 22 23 24 25]]
z = x + y
print(z)
print(np.add(x, y))
# [[12 14 16 18 20]
# [22 24 26 28 30]
# [32 34 36 38 40]
# [42 44 46 48 50]
# [52 54 56 58 60]]
z = x - y
print(z)
print(np.subtract(x, y))
# [[10 10 10 10 10]
# [10 10 10 10 10]
# [10 10 10 10 10]
# [10 10 10 10 10]
# [10 10 10 10 10]]
z = x * y
print(z)
print(np.multiply(x, y))
# [[ 11 24 39 56 75]
# [ 96 119 144 171 200]
# [231 264 299 336 375]
# [416 459 504 551 600]
# [651 704 759 816 875]]
z = x / y
print(z)
print(np.divide(x, y))
# [[11. 6. 4.33333333 3.5 3. ]
# [ 2.66666667 2.42857143 2.25 2.11111111 2. ]
# [ 1.90909091 1.83333333 1.76923077 1.71428571 1.66666667]
# [ 1.625 1.58823529 1.55555556 1.52631579 1.5 ]
# [ 1.47619048 1.45454545 1.43478261 1.41666667 1.4 ]]
z = x // y
print(z)
print(np.floor_divide(x, y))
# [[11 6 4 3 3]
# [ 2 2 2 2 2]
# [ 1 1 1 1 1]
# [ 1 1 1 1 1]
# [ 1 1 1 1 1]]
z = x ** np.full([5, 5], 2)
print(z)
print(np.power(x, np.full([5, 5], 2)))
# [[ 121 144 169 196 225]
# [ 256 289 324 361 400]
# [ 441 484 529 576 625]
# [ 676 729 784 841 900]
# [ 961 1024 1089 1156 1225]]
**numpy.sqrt
numpy.square**
- numpy.sqrt(x, *args, **kwargs)
Return the non-negative square-root of an array, element-wise. - numpy.square(x, *args, **kwargs)
Return the element-wise square of the input.
【例】
import numpy as np
x = np.arange(1, 5)
print(x) # [1 2 3 4]
y = np.sqrt(x)
print(y)
# [1. 1.41421356 1.73205081 2. ]
print(np.power(x, 0.5))
# [1. 1.41421356 1.73205081 2. ]
y = np.square(x)
print(y)
# [ 1 4 9 16]
print(np.power(x, 2))
# [ 1 4 9 16]
2.三角函数
**numpy.sin
numpy.cos
numpy.tan
numpy.arcsin
numpy.arccos
numpy.arctan**
- numpy.sin(x, *args, **kwargs)
Trigonometric sine, element-wise. - numpy.cos(x, *args, **kwargs)
Cosine element-wise. - numpy.tan(x, *args, **kwargs)
Compute tangent element-wise.
numpy.arcsin(x, *args, **kwargs) Inverse sine, element-wise. - numpy.arccos(x, *args, **kwargs)
Trigonometric inverse cosine, element-wise. - numpy.arctan(x, *args, **kwargs)
Trigonometric inverse tangent, element-wise.
通用函数（universal function）通常叫作ufunc，它对数组中的各个元素逐一进行操作。这表明，通用函数分别处理输入数组的每个元素，生成的结果组成一个新的输出数组。输出数组的大小跟输入数组相同。
三角函数等很多数学运算符合通用函数的定义，例如，计算平方根的sqrt()函数、用来取对数的log()函数和求正弦值的sin()函数。
【例】
import numpy as np
x = np.linspace(start=0, stop=np.pi / 2, num=10)
print(x)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]
y = np.sin(x)
print(y)
# [0. 0.17364818 0.34202014 0.5 0.64278761 0.76604444
# 0.8660254 0.93969262 0.98480775 1. ]
z = np.arcsin(y)
print(z)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]
y = np.cos(x)
print(y)
# [1.00000000e+00 9.84807753e-01 9.39692621e-01 8.66025404e-01
# 7.66044443e-01 6.42787610e-01 5.00000000e-01 3.42020143e-01
# 1.73648178e-01 6.12323400e-17]
z = np.arccos(y)
print(z)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]
y = np.tan(x)
print(y)
# [0.00000000e+00 1.76326981e-01 3.63970234e-01 5.77350269e-01
# 8.39099631e-01 1.19175359e+00 1.73205081e+00 2.74747742e+00
# 5.67128182e+00 1.63312394e+16]
z = np.arctan(y)
print(z)
# [0. 0.17453293 0.34906585 0.52359878 0.6981317 0.87266463
# 1.04719755 1.22173048 1.3962634 1.57079633]
3.指数和对数
**numpy.exp
numpy.log
numpy.exp2
numpy.log2
numpy.log10**
- numpy.exp(x, *args, **kwargs)
Calculate the exponential of all elements in the input array. - numpy.log(x, *args, **kwargs)
Natural logarithm, element-wise. - numpy.exp2(x, *args, **kwargs)
Calculate 2**p for all p in the input array. - numpy.log2(x, *args, **kwargs)
Base-2 logarithm of x. - numpy.log10(x, *args, **kwargs)
Return the base 10 logarithm of the input array, element-wise.
【例】The natural logarithm log is the inverse of the exponential function, so that log(exp(x)) = x. The natural logarithm is logarithm in base e.
import numpy as np
x = np.arange(1, 5)
print(x)
# [1 2 3 4]
y = np.exp(x)
print(y)
# [ 2.71828183 7.3890561 20.08553692 54.59815003]
z = np.log(y)
print(z)
# [1. 2. 3. 4.]
4.加法函数、乘法函数
numpy.sum
- numpy.sum(a[, axis=None, dtype=None, out=None, …])
Sum of array elements over a given axis.
通过不同的 axis，numpy 会沿着不同的方向进行操作：如果不设置，那么对所有的元素操作；如果axis=0，则沿着纵轴进行操作；axis=1，则沿着横轴进行操作。但这只是简单的二位数组，如果是多维的呢？可以总结为一句话：设axis=i，则 numpy 沿着第i个下标变化的方向进行操作。
numpy.cumsum
- numpy.cumsum(a, axis=None, dtype=None, out=None)
Return the cumulative sum of the elements along a given axis.
聚合函数 是指对一组值（比如一个数组）进行操作，返回一个单一值作为结果的函数。因而，求数组所有元素之和的函数就是聚合函数。ndarray类实现了多个这样的函数。
【例】返回给定轴上的数组元素的总和。
mport numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.sum(x)
print(y) # 575
y = np.sum(x, axis=0)
print(y) # [105 110 115 120 125]
y = np.sum(x, axis=1)
print(y) # [ 65 90 115 140 165]
【例】返回给定轴上的数组元素的累加和。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.cumsum(x)
print(y)
# [ 11 23 36 50 65 81 98 116 135 155 176 198 221 245 270 296 323 351
# 380 410 441 473 506 540 575]
y = np.cumsum(x, axis=0)
print(y)
# [[ 11 12 13 14 15]
# [ 27 29 31 33 35]
# [ 48 51 54 57 60]
# [ 74 78 82 86 90]
# [105 110 115 120 125]]
y = np.cumsum(x, axis=1)
print(y)
# [[ 11 23 36 50 65]
# [ 16 33 51 70 90]
# [ 21 43 66 90 115]
# [ 26 53 81 110 140]
# [ 31 63 96 130 165]]
numpy.prod 乘积
- numpy.prod(a[, axis=None, dtype=None, out=None, …])
Return the product of array elements over a given axis.
numpy.cumprod 累乘
- numpy.cumprod(a, axis=None, dtype=None, out=None)
Return the cumulative product of elements along a given axis.
【例】返回给定轴上数组元素的乘积。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.prod(x)
print(y) # 788529152
y = np.prod(x, axis=0)
print(y)
# [2978976 3877632 4972968 6294624 7875000]
y = np.prod(x, axis=1)
print(y)
# [ 360360 1860480 6375600 17100720 38955840]
【例】返回给定轴上数组元素的累乘。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.cumprod(x)
print(y)
# [ 11 132 1716 24024 360360 5765760
# 98017920 1764322560 -837609728 427674624 391232512 17180672
# 395155456 893796352 870072320 1147043840 905412608 -418250752
# 755630080 1194065920 -1638662144 -897581056 444596224 -2063597568
# 788529152]
y = np.cumprod(x, axis=0)
print(y)
# [[ 11 12 13 14 15]
# [ 176 204 234 266 300]
# [ 3696 4488 5382 6384 7500]
# [ 96096 121176 150696 185136 225000]
# [2978976 3877632 4972968 6294624 7875000]]
y = np.cumprod(x, axis=1)
print(y)
# [[ 11 132 1716 24024 360360]
# [ 16 272 4896 93024 1860480]
# [ 21 462 10626 255024 6375600]
# [ 26 702 19656 570024 17100720]
# [ 31 992 32736 1113024 38955840]]
numpy.diff 差值
- numpy.diff(a, n=1, axis=-1, prepend=np._NoValue, append=np._NoValue)
Calculate the n-th discrete difference along the given axis. - a：输入矩阵
- n：可选，代表要执行几次差值
- axis：默认是最后一个
The first difference is given by out[i] = a[i+1] - a[i] along the given axis, higher differences are calculated by using diff recursively.
【例】沿着指定轴计算第N维的离散差值。
import numpy as np
A = np.arange(2, 14).reshape((3, 4))
A[1, 1] = 8
print(A)
# [[ 2 3 4 5]
# [ 6 8 8 9]
# [10 11 12 13]]
print(np.diff(A))
# [[1 1 1]
# [2 0 1]
# [1 1 1]]
print(np.diff(A, axis=0))
# [[4 5 4 4]
# [4 3 4 4]]
5. 四舍五入
numpy.around 舍入
- numpy.around(a, decimals=0, out=None)
Evenly round to the given number of decimals.
【例】将数组舍入到给定的小数位数。
import numpy as np
x = np.random.rand(3, 3) * 10
print(x)
# [[6.59144457 3.78566113 8.15321227]
# [1.68241475 3.78753332 7.68886328]
# [2.84255822 9.58106727 7.86678037]]
y = np.around(x)
print(y)
# [[ 7. 4. 8.]
# [ 2. 4. 8.]
# [ 3. 10. 8.]]
y = np.around(x, decimals=2)
print(y)
# [[6.59 3.79 8.15]
# [1.68 3.79 7.69]
# [2.84 9.58 7.87]]
**numpy.ceil 上限
numpy.floor 下限**
- numpy.ceil(x, *args, **kwargs)
Return the ceiling of the input, element-wise. - numpy.floor(x, *args, **kwargs)
Return the floor of the input, element-wise.
【例】
import numpy as np
x = np.random.rand(3, 3) * 10
print(x)
# [[0.67847795 1.33073923 4.53920122]
# [7.55724676 5.88854047 2.65502046]
# [8.67640444 8.80110812 5.97528726]]
y = np.ceil(x)
print(y)
# [[1. 2. 5.]
# [8. 6. 3.]
# [9. 9. 6.]]
y = np.floor(x)
print(y)
# [[0. 1. 4.]
# [7. 5. 2.]
# [8. 8. 5.]]
6.杂项
numpy.clip 裁剪
- numpy.clip(a, a_min, a_max, out=None, **kwargs):
Clip (limit) the values in an array.
Given an interval, values outside the interval are clipped to the interval edges. For example, if an interval of [0, 1] is specified, values smaller than 0 become 0, and values larger than 1 become 1.
【例】裁剪（限制）数组中的值。
import numpy as np
x = np.array([[11, 12, 13, 14, 15],
[16, 17, 18, 19, 20],
[21, 22, 23, 24, 25],
[26, 27, 28, 29, 30],
[31, 32, 33, 34, 35]])
y = np.clip(x, a_min=20, a_max=30)
print(y)
# [[20 20 20 20 20]
# [20 20 20 20 20]
# [21 22 23 24 25]
# [26 27 28 29 30]
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numpy.absolute 绝对值
numpy.abs
- numpy.absolute(x, *args, **kwargs)
Calculate the absolute value element-wise. - numpy.abs(x, *args, **kwargs)
is a shorthand for this function.
【例】
import numpy as np
x = np.arange(-5, 5)
print(x)
# [-5 -4 -3 -2 -1 0 1 2 3 4]
y = np.abs(x)
print(y)
# [5 4 3 2 1 0 1 2 3 4]
y = np.absolute(x)
print(y)
# [5 4 3 2 1 0 1 2 3 4]
numpy.sign 返回数字符号的逐元素指示
- numpy.sign(x, *args, **kwargs)
Returns an element-wise indication of the sign of a number.
【例】
x = np.arange(-5, 5)
print(x)
#[-5 -4 -3 -2 -1 0 1 2 3 4]
print(np.sign(x))
#[-1 -1 -1 -1 -1 0 1 1 1 1]