Numpy入門教程：05. 邏輯函數

背景

什麼是 NumPy 呢？

NumPy 這個詞來源於兩個單詞 – Numerical和Python。其是一個功能強大的 Python 庫，可以幫助程序員輕鬆地進行數值計算，通常應用於以下場景：

• 執行各種數學任務，如：數值積分、微分、內插、外推等。因此，當涉及到數學任務時，它形成了一種基於 Python 的 MATLAB 的快速替代。
• 計算機中的圖像表示爲多維數字數組。NumPy 提供了一些優秀的庫函數來快速處理圖像。例如，鏡像圖像、按特定角度旋轉圖像等。
• 在編寫機器學習算法時，需要對矩陣進行各種數值計算。如：矩陣乘法、求逆、換位、加法等。NumPy 數組用於存儲訓練數據和機器學習模型的參數。

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邏輯函數

真值測試

• 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]

邏輯運算

• 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.

【例】

import numpy as np

x = np.arange(5)
print(np.logical_not(3))  
# False
print(np.logical_not([True, False, 0, 1]))
# [False  True  True False]
print(np.logical_not(x < 3))
# [False False False  True  True]

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]

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]

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]]

對照

• 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.

【例】numpy對以上對照函數進行了運算符的重載。

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]

【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 = 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]]

【例】

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]]

【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]])

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(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.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.

判斷是否爲True的計算依據：

np.absolute(a - b) <= (atol + rtol * absolute(b))

- atol：float，絕對公差。
- rtol：float，相對公差。

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.

【例】比較兩個數組是否可以認爲相等。

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

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