DataWhale（numpy）：task4 数学函数及逻辑函数
一、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；
二维数组加一维数组
import numpy as np
x = np.arange(4)
y = np.ones((3, 4))
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.]]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.shape) # (4, 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.]]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,) 二、数学函数
2.1 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。
注意 numpy 的广播规则。
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]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]]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]]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.
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]三、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。
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]四、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)Calculate2**pfor allpin the input array.numpy.log2(x, *args, **kwargs)Base-2 logarithm ofx.numpy.log10(x, *args, **kwargs)Return the base 10 logarithm of the input array, element-wise.
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.]五、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。
返回给定轴上的数组元素的总和。
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.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]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.
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]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 diffrecursively.
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]]六、四舍五入
numpy.around 舍入
numpy.around(a, decimals=0, out=None)Evenly round to the given number of decimals.
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.
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.]]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.
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]
# [30 30 30 30 30]]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.
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.
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]七、逻辑函数
7.1 真值测试
numpy.all
numpy.any
numpy.all(a, axis=None, out=None, keepdims=np._NoValue)Test whether all array elements along a given axis evaluate to True.numpy.any(a, axis=None, out=None, keepdims=np._NoValue)Test whether any array element along a given axis evaluates to True.
import numpy as np
a = np.array([0, 4, 5])
b = np.copy(a)
print(np.all(a == b)) # True
print(np.any(a == b)) # True
b[0] = 1
print(np.all(a == b)) # False
print(np.any(a == b)) # True
print(np.all([1.0, np.nan])) # True
print(np.any([1.0, np.nan])) # True
a = np.eye(3)
print(np.all(a, axis=0)) # [False False False]
print(np.any(a, axis=0)) # [ True True True]7.2 数组内容
numpy.isnan
numpy.isnan(x, *args, **kwargs)Test element-wise for NaN and return result as a boolean array.
a=np.array([1,2,np.nan])
print(np.isnan(a))
#[False False True]7.3 逻辑运算
numpy.logical_not
numpy.logical_and
numpy.logical_or
numpy.logical_xor
numpy.logical_not(x, *args, **kwargs)Compute the truth value of NOT x element-wise.numpy.logical_and(x1, x2, *args, **kwargs)Compute the truth value of x1 AND x2 element-wise.numpy.logical_or(x1, x2, *args, **kwargs)Compute the truth value of x1 OR x2 element-wise.numpy.logical_xor(x1, x2, *args, **kwargs)Compute the truth value of x1 XOR x2, element-wise.
计算非x元素的真值。
import numpy as np
print(np.logical_not(3))
# False
print(np.logical_not([True, False, 0, 1]))
# [False True True False]
x = np.arange(5)
print(np.logical_not(x < 3))
# [False False False True True]
计算x1 AND x2元素的真值。
print(np.logical_and(True, False))
# False
print(np.logical_and([True, False], [True, False]))
# [ True False]
print(np.logical_and(x > 1, x < 4))
# [False False True True False]逐元素计算x1 OR x2的真值。
print(np.logical_or(True, False))
# True
print(np.logical_or([True, False], [False, False]))
# [ True False]
print(np.logical_or(x < 1, x > 3))
# [ True False False False True]计算x1 XOR x2的真值，按元素计算。
print(np.logical_xor(True, False))
# True
print(np.logical_xor([True, True, False, False], [True, False, True, False]))
# [False True True False]
print(np.logical_xor(x < 1, x > 3))
# [ True False False False True]
print(np.logical_xor(0, np.eye(2)))
# [[ True False]
# [False True]]7.4 对照
numpy.greater
numpy.greater_equal
numpy.equal
numpy.not_equal
numpy.less
numpy.less_equal
numpy.greater(x1, x2, *args, **kwargs)Return the truth value of (x1 > x2) element-wise.numpy.greater_equal(x1, x2, *args, **kwargs)Return the truth value of (x1 >= x2) element-wise.numpy.equal(x1, x2, *args, **kwargs)Return (x1 == x2) element-wise.numpy.not_equal(x1, x2, *args, **kwargs)Return (x1 != x2) element-wise.numpy.less(x1, x2, *args, **kwargs)Return the truth value of (x1 < x2) element-wise.numpy.less_equal(x1, x2, *args, **kwargs)Return the truth value of (x1 =< x2) element-wise.
numpy对以上对照函数进行了运算符的重载。
import numpy as np
x = np.array([1, 2, 3, 4, 5, 6, 7, 8])
y = x > 2
print(y)
print(np.greater(x, 2))
# [False False True True True True True True]
y = x >= 2
print(y)
print(np.greater_equal(x, 2))
# [False True True True True True True True]
y = x == 2
print(y)
print(np.equal(x, 2))
# [False True False False False False False False]
y = x != 2
print(y)
print(np.not_equal(x, 2))
# [ True False True True True True True True]
y = x < 2
print(y)
print(np.less(x, 2))
# [ True False False False False False False False]
y = x <= 2
print(y)
print(np.less_equal(x, 2))
# [ True True False False False False False False]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 > 20
print(y)
print(np.greater(x, 20))
# [[False False False False False]
# [False False False False False]
# [ True True True True True]
# [ True True True True True]
# [ True True True True True]]
y = x >= 20
print(y)
print(np.greater_equal(x, 20))
# [[False False False False False]
# [False False False False True]
# [ True True True True True]
# [ True True True True True]
# [ True True True True True]]
y = x == 20
print(y)
print(np.equal(x, 20))
# [[False False False False False]
# [False False False False True]
# [False False False False False]
# [False False False False False]
# [False False False False False]]
y = x != 20
print(y)
print(np.not_equal(x, 20))
# [[ True True True True True]
# [ True True True True False]
# [ True True True True True]
# [ True True True True True]
# [ True True True True True]]
y = x < 20
print(y)
print(np.less(x, 20))
# [[ True True True True True]
# [ True True True True False]
# [False False False False False]
# [False False False False False]
# [False False False False False]]
y = x <= 20
print(y)
print(np.less_equal(x, 20))
# [[ True True True True True]
# [ True True True True True]
# [False False False False False]
# [False False False False False]
# [False False False False False]]import numpy as np
np.random.seed(20200611)
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.random.randint(10, 40, [5, 5])
print(y)
# [[32 28 31 33 37]
# [23 37 37 30 29]
# [32 24 10 33 15]
# [27 17 10 36 16]
# [25 32 23 39 34]]
z = x > y
print(z)
print(np.greater(x, y))
# [[False False False False False]
# [False False False False False]
# [False False True False True]
# [False True True False True]
# [ True False True False True]]
z = x >= y
print(z)
print(np.greater_equal(x, y))
# [[False False False False False]
# [False False False False False]
# [False False True False True]
# [False True True False True]
# [ True True True False True]]
z = x == y
print(z)
print(np.equal(x, y))
# [[False False False False False]
# [False False False False False]
# [False False False False False]
# [False False False False False]
# [False True False False False]]
z = x != y
print(z)
print(np.not_equal(x, y))
# [[ True True True True True]
# [ True True True True True]
# [ True True True True True]
# [ True True True True True]
# [ True False True True True]]
z = x < y
print(z)
print(np.less(x, y))
# [[ True True True True True]
# [ True True True True True]
# [ True True False True False]
# [ True False False True False]
# [False False False True False]]
z = x <= y
print(z)
print(np.less_equal(x, y))
# [[ True True True True True]
# [ True True True True True]
# [ True True False True False]
# [ True False False True False]
# [False True False True False]]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]])
np.random.seed(20200611)
y = np.random.randint(10, 50, 5)
print(y)
# [32 37 30 24 10]
z = x > y
print(z)
print(np.greater(x, y))
# [[False False False False True]
# [False False False False True]
# [False False False False True]
# [False False False True True]
# [False False True True True]]
z = x >= y
print(z)
print(np.greater_equal(x, y))
# [[False False False False True]
# [False False False False True]
# [False False False True True]
# [False False False True True]
# [False False True True True]]
z = x == y
print(z)
print(np.equal(x, y))
# [[False False False False False]
# [False False False False False]
# [False False False True False]
# [False False False False False]
# [False False False False False]]
z = x != y
print(z)
print(np.not_equal(x, y))
# [[ True True True True True]
# [ True True True True True]
# [ True True True False True]
# [ True True True True True]
# [ True True True True True]]
z = x < y
print(z)
print(np.less(x, y))
# [[ True True True True False]
# [ True True True True False]
# [ True True True False False]
# [ True True True False False]
# [ True True False False False]]
z = x <= y
print(z)
print(np.less_equal(x, y))
# [[ True True True True False]
# [ True True True True False]
# [ True True True True False]
# [ True True True False False]
# [ True True False False False]]numpy.isclose
numpy.allclose
numpy.isclose(a, b, rtol=1.e-5, atol=1.e-8, equal_nan=False)Returns a boolean array where two arrays are element-wise equal within a tolerance.numpy.allclose(a, b, rtol=1.e-5, atol=1.e-8, equal_nan=False)Returns True if two arrays are element-wise equal within a tolerance.
numpy.allclose() 等价于 numpy.all(isclose(a, b, rtol=rtol, atol=atol, equal_nan=equal_nan))。
The tolerance values are positive, typically very small numbers. The relative difference (rtol * abs(b)) and the absolute difference atol are added together to compare against the absolute difference between a and b.
判断是否为True的计算依据：
np.absolute(a - b) <= (atol + rtol * absolute(b))
- atol：float，绝对公差。
- rtol：float，相对公差。NaNs are treated as equal if they are in the same place and if equal_nan=True. Infs are treated as equal if they are in the same place and of the same sign in both arrays.
比较两个数组是否可以认为相等。
import numpy as np
x = np.isclose([1e10, 1e-7], [1.00001e10, 1e-8])
print(x) # [ True False]
x = np.allclose([1e10, 1e-7], [1.00001e10, 1e-8])
print(x) # False
x = np.isclose([1e10, 1e-8], [1.00001e10, 1e-9])
print(x) # [ True True]
x = np.allclose([1e10, 1e-8], [1.00001e10, 1e-9])
print(x) # True
x = np.isclose([1e10, 1e-8], [1.0001e10, 1e-9])
print(x) # [False True]
x = np.allclose([1e10, 1e-8], [1.0001e10, 1e-9])
print(x) # False
x = np.isclose([1.0, np.nan], [1.0, np.nan])
print(x) # [ True False]
x = np.allclose([1.0, np.nan], [1.0, np.nan])
print(x) # False
x = np.isclose([1.0, np.nan], [1.0, np.nan], equal_nan=True)
print(x) # [ True True]
x = np.allclose([1.0, np.nan], [1.0, np.nan], equal_nan=True)
print(x) # True八、练习
数学函数
8.1 将数组a中大于30的值替换为30，小于10的值替换为10。
a = np.random.uniform(1, 50, 20)
【知识点：数学函数、搜索】
- 如何将大于给定值的所有值替换为给定的截止值？
import numpy as np
np.set_printoptions(precision=2)
np.random.seed(100)
a = np.random.uniform(1, 50, 20)
print(a)
# [27.63 14.64 21.8 42.39 1.23 6.96 33.87 41.47 7.7 29.18 44.67 11.25
# 10.08 6.31 11.77 48.95 40.77 9.43 41. 14.43]
# 方法1
b = np.clip(a, a_min=10, a_max=30)
print(b)
# [27.63 14.64 21.8 30. 10. 10. 30. 30. 10. 29.18 30. 11.25
# 10.08 10. 11.77 30. 30. 10. 30. 14.43]
# 方法2
b = np.where(a < 10, 10, a)
b = np.where(b > 30, 30, b)
print(b)
# [27.63 14.64 21.8 30. 10. 10. 30. 30. 10. 29.18 30. 11.25
# 10.08 10. 11.77 30. 30. 10. 30. 14.43]8.2 找到一个一维数字数组a中的所有峰值。峰顶是两边被较小数值包围的点。
a = np.array([1, 3, 7, 1, 2, 6, 0, 1])
【知识点：数学函数、搜索】
- 如何在一维数组中找到所有的局部极大值（或峰值）？
import numpy as np
a = np.array([1, 3, 7, 1, 2, 6, 0, 1])
b1 = np.diff(a)
b2 = np.sign(b1)
b3 = np.diff(b2)
print(b1) # [ 2 4 -6 1 4 -6 1]
print(b2) # [ 1 1 -1 1 1 -1 1]
print(b3) # [ 0 -2 2 0 -2 2]
index = np.where(np.equal(b3, -2))[0] + 1
print(index) # [2 5]8.3 对于给定的一维数组，计算窗口大小为3的移动平均值。
z = np.random.randint(10, size=10)
【知识点：数学函数】
- 如何计算numpy数组的移动平均值？
import numpy as np
np.random.seed(100)
z = np.random.randint(10, size=10)
print(z)
# [8 8 3 7 7 0 4 2 5 2]
def MovingAverage(arr, n=3):
a = np.cumsum(arr)
a[n:] = a[n:] - a[:-n]
return a[n - 1:] / n
r = MovingAverage(z, 3)
print(np.around(r, 2))
# [6.33 6. 5.67 4.67 3.67 2. 3.67 3. ]8.4 对一个5x5的随机矩阵做归一化
【知识点：数学函数】
- (提示: (x - min) / (max - min))
Z = np.random.random((5,5))
Zmax, Zmin = Z.max(), Z.min()
Z = (Z - Zmin)/(Zmax - Zmin)
print(Z)
[[0.65852707 0.23160203 0.9700425 0.43393441 0.36950354]
[0.58585632 0.01877358 0. 0.27782335 0.89654906]
[0.71307774 0.32030221 0.55187887 0.01411576 0.38903753]
[0.05702349 0.40824762 0.23277188 0.74335934 0.72125366]
[0.59816442 0.98727663 1. 0.86466287 0.42760849]]8.6 用五种不同的方法去提取一个随机数组的整数部分
【知识点：数学函数】
- (提示: %, np.floor, np.ceil, astype, np.trunc)
Z = np.random.uniform(0,10,10)
print (Z - Z%1)
print (np.floor(Z))
print (np.ceil(Z)-1)
print (Z.astype(int))
print (np.trunc(Z))
#[8. 5. 5. 1. 2. 7. 0. 1. 5. 9.]
#[8. 5. 5. 1. 2. 7. 0. 1. 5. 9.]
#[8. 5. 5. 1. 2. 7. 0. 1. 5. 9.]
#[8 5 5 1 2 7 0 1 5 9]
#[8. 5. 5. 1. 2. 7. 0. 1. 5. 9.]8.7 考虑一维数组Z，构建一个二维数组，其第一行为（Z [0]，Z [1]，Z [2]），随后的每一行都移位1（最后一行应为（Z [ -3]，Z [-2]，Z [-1]）
【知识点：数学函数】
- （提示np.lib.stride_tricks）
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]-------->
np.arange(10).itemsize
from numpy.lib import stride_tricks
def rolling(a, window):
shape = (a.size - window + 1, window)
strides = (a.itemsize, a.itemsize)
return stride_tricks.as_strided(a, shape=shape, strides=strides)
Z = rolling(np.arange(10), 3)
print (Z)
#
[[0 1 2]
[1 2 3]
[2 3 4]
[3 4 5]
[4 5 6]
[5 6 7]
[6 7 8]
[7 8 9]]8.8 考虑两组点集P0和P1去描述一组线(二维)和一个点p,如何计算点p到每一条线 i (P0[i],P1[i])的距离？
【知识点：数学函数】
- 提示 设P(x0,y0)，直线方程为：Ax+By+C=0 则P到直线的距离为:d=|Ax0+By0+C|/√(A²+B²)
import numpy as np
def distance(P0,P1,p):
A=-1/(P1[:,0]-P0[:,0])
B=1/(P1[:,1]-P0[:,1])
C=P0[:,0]/(P1[:,0]-P0[:,0])-P0[:,1]/(P1[:,1]-P0[:,1])
return np.abs(A*p[:,0]+B*p[:,1]+C)/np.sqrt(A**2+B**2)
P0 = np.random.uniform(-10,10,(10,2))
P1 = np.random.uniform(-10,10,(10,2))
p = np.random.uniform(-10,10,( 1,2))
print (distance(P0, P1, p))
#[ 2.41120585 7.05924485 5.94906338 3.89096895 7.68364539 14.01851392
# 4.28148124 2.29582076 6.19155339 15.55087594]8.9 画正弦函数和余弦函数， x = np.arange(0, 3 * np.pi, 0.1)？
import numpy as np
from matplotlib import pyplot as plt
x = np.arange(0, 3*np.pi, 0.1)
y1 = np.sin(x)
y2 = np.cos(x)
plt.plot(x, y1)
plt.plot(x, y2)[<matplotlib.lines.Line2D at 0x236a41aac88>]
8.10 减去矩阵每一行的平均值 ？
X = np.random.rand(5, 10)
# 新
Y = X - X.mean(axis=1, keepdims=True)
# 旧
Y = X - X.mean(axis=1).reshape(-1, 1)
print(Y)
#[[-0.22611265 0.18681928 0.13113673 -0.42713648 0.13273158 0.01160406
0.54266804 -0.15637426 0.16445073 -0.35978702]
[-0.34359842 0.26798252 0.35328871 0.40386463 -0.45020099 0.40738698
0.10061077 -0.32664784 0.1162996 -0.52898595]
[ 0.08892378 -0.10264121 0.27575287 -0.00792712 0.01868823 0.18782665
0.00795426 -0.12014542 -0.35177329 0.00334125]
[-0.46156881 -0.40474198 0.15298977 0.17155644 0.17332436 -0.06791347
-0.12123457 0.00349192 0.28866751 0.26542884]
[-0.39891856 -0.01008639 0.50523343 -0.28914192 0.01382436 -0.15406104
-0.22350635 0.45768516 -0.17832199 0.2772933 ]]8.11 进行概率统计分析 ？
arr1 = np.random.randint(1,10,10)arr2 = np.random.randint(1,10,10))
f
arr1 = np.random.randint(1,10,10)
arr2 = np.random.randint(1,10,10)
print("arr1的平均数为:%s" %np.mean(arr1))
print("arr1的中位数为:%s" %np.median(arr1))
print("arr1的方差为:%s" %np.var(arr1))
print("arr1的标准差为:%s" %np.std(arr1))
print("arr1,arr的相关性矩阵为:%s" %np.cov(arr1,arr2))
print("arr1,arr的协方差矩阵为:%s" %np.corrcoef(arr1,arr2))逻辑函数
8.12 获取a和b元素匹配的位置。
a = np.array([1, 2, 3, 2, 3, 4, 3, 4, 5, 6])b = np.array([7, 2, 10, 2, 7, 4, 9, 4, 9, 8])
【知识点：逻辑函数】
- 如何得到两个数组元素匹配的位置？
import numpy as np
a = np.array([1, 2, 3, 2, 3, 4, 3, 4, 5, 6])
b = np.array([7, 2, 10, 2, 7, 4, 9, 4, 9, 8])
mask = np.equal(a, b)
# 方法1
x = np.where(mask)
print(x) # (array([1, 3, 5, 7], dtype=int64),)
# 方法2
x = np.nonzero(mask)
print(x) # (array([1, 3, 5, 7], dtype=int64),)
#(array([1, 3, 5, 7], dtype=int64),)
#(array([1, 3, 5, 7], dtype=int64),)8.13 获取5到10 之间的所有元素。
a = np.array([2, 6, 1, 9, 10, 3, 27])
【知识点：逻辑函数】
- 如何从numpy数组中提取给定范围内的所有元素？
import numpy as np
a = np.array([2, 6, 1, 9, 10, 3, 27])
mask = np.logical_and(np.greater_equal(a, 5), np.less_equal(a, 10))
# 方法1
x = np.where(mask)
print(a[x]) # [ 6 9 10]
# 方法2
x = np.nonzero(mask)
print(a[x]) # [ 6 9 10]
# 方法3
x = a[np.logical_and(a >= 5, a <= 10)]
print(x) # [ 6 9 10]
#[ 6 9 10]
#[ 6 9 10]
#[ 6 9 10]8.14 对于两个随机数组A和B,检查他们是否相等
【知识点：逻辑函数】
- (提示: np.allclose, np.array_equal)
A = np.random.randint(0,2,5)
B = np.random.randint(0,2,5)
# Assuming identical shape of the arrays and a tolerance for the comparison of values
equal = np.allclose(A,B)
print(equal)
#False8.15 何对布尔值取反，或者原位(in-place)改变浮点数的符号(sign)？
【知识点：逻辑函数】
- (提示: np.logical_not, np.negative)
Z = np.array([0,1])
print(Z)
np.logical_not(Z, out=Z)
# Z = np.random.uniform(-1.0,1.0,100)
# np.negative(Z, out=Z)
#[0 1]
#array([1, 0])
Z = np.array([0.2,1.15])
print(Z)
np.negative(Z, out=Z)
#[0.2 1.15]
#array([-0.2 , -1.15])8.16 找出数组中与给定值最接近的数
Z=np.array([[0,1,2,3],[4,5,6,7]])
print(Z)
z=5.1
np.abs(Z - z).argmin()
print(Z.flat[np.abs(Z - z).argmin()])
#
[[0 1 2 3]
[4 5 6 7]]
5