文章目錄

• Sigmoid函數由下列公式定義：
• 其對x的導數可以用自身表示：
• Sigmoid函數的圖形如S曲線：
• Sigmoid函數的級數表示：
• 用python打印Sigmoid函數曲線：
• • 結果：
• 若想讓sigmoid函數計算numpy數組，可用以下編碼方式表示

Sigmoid函數由下列公式定義：

其對x的導數可以用自身表示：

Sigmoid函數的圖形如S曲線：

Sigmoid函數的級數表示：

Sigmoid常用數值：

用python打印Sigmoid函數曲線：

import numpy as np import matplotlib.pyplot as plt def sigmoid(x):return 1.0/(1+np.exp(-x))sigmoid_inputs = np.arange(-10,10,0.1) sigmoid_outputs = sigmoid(sigmoid_inputs) print("Sigmoid Function Input :: {}".format(sigmoid_inputs)) print("Sigmoid Function Output :: {}".format(sigmoid_outputs))plt.plot(sigmoid_inputs,sigmoid_outputs) plt.xlabel("Sigmoid Inputs") plt.ylabel("Sigmoid Outputs") plt.show()

結果：

Sigmoid Function Input :: [-1.00000000e+01 -9.90000000e+00 -9.80000000e+00 -9.70000000e+00-9.60000000e+00 -9.50000000e+00 -9.40000000e+00 -9.30000000e+00-9.20000000e+00 -9.10000000e+00 -9.00000000e+00 -8.90000000e+00-8.80000000e+00 -8.70000000e+00 -8.60000000e+00 -8.50000000e+00-8.40000000e+00 -8.30000000e+00 -8.20000000e+00 -8.10000000e+00-8.00000000e+00 -7.90000000e+00 -7.80000000e+00 -7.70000000e+00-7.60000000e+00 -7.50000000e+00 -7.40000000e+00 -7.30000000e+00-7.20000000e+00 -7.10000000e+00 -7.00000000e+00 -6.90000000e+00-6.80000000e+00 -6.70000000e+00 -6.60000000e+00 -6.50000000e+00-6.40000000e+00 -6.30000000e+00 -6.20000000e+00 -6.10000000e+00-6.00000000e+00 -5.90000000e+00 -5.80000000e+00 -5.70000000e+00-5.60000000e+00 -5.50000000e+00 -5.40000000e+00 -5.30000000e+00-5.20000000e+00 -5.10000000e+00 -5.00000000e+00 -4.90000000e+00-4.80000000e+00 -4.70000000e+00 -4.60000000e+00 -4.50000000e+00-4.40000000e+00 -4.30000000e+00 -4.20000000e+00 -4.10000000e+00-4.00000000e+00 -3.90000000e+00 -3.80000000e+00 -3.70000000e+00-3.60000000e+00 -3.50000000e+00 -3.40000000e+00 -3.30000000e+00-3.20000000e+00 -3.10000000e+00 -3.00000000e+00 -2.90000000e+00-2.80000000e+00 -2.70000000e+00 -2.60000000e+00 -2.50000000e+00-2.40000000e+00 -2.30000000e+00 -2.20000000e+00 -2.10000000e+00-2.00000000e+00 -1.90000000e+00 -1.80000000e+00 -1.70000000e+00-1.60000000e+00 -1.50000000e+00 -1.40000000e+00 -1.30000000e+00-1.20000000e+00 -1.10000000e+00 -1.00000000e+00 -9.00000000e-01-8.00000000e-01 -7.00000000e-01 -6.00000000e-01 -5.00000000e-01-4.00000000e-01 -3.00000000e-01 -2.00000000e-01 -1.00000000e-01-3.55271368e-14 1.00000000e-01 2.00000000e-01 3.00000000e-014.00000000e-01 5.00000000e-01 6.00000000e-01 7.00000000e-018.00000000e-01 9.00000000e-01 1.00000000e+00 1.10000000e+001.20000000e+00 1.30000000e+00 1.40000000e+00 1.50000000e+001.60000000e+00 1.70000000e+00 1.80000000e+00 1.90000000e+002.00000000e+00 2.10000000e+00 2.20000000e+00 2.30000000e+002.40000000e+00 2.50000000e+00 2.60000000e+00 2.70000000e+002.80000000e+00 2.90000000e+00 3.00000000e+00 3.10000000e+003.20000000e+00 3.30000000e+00 3.40000000e+00 3.50000000e+003.60000000e+00 3.70000000e+00 3.80000000e+00 3.90000000e+004.00000000e+00 4.10000000e+00 4.20000000e+00 4.30000000e+004.40000000e+00 4.50000000e+00 4.60000000e+00 4.70000000e+004.80000000e+00 4.90000000e+00 5.00000000e+00 5.10000000e+005.20000000e+00 5.30000000e+00 5.40000000e+00 5.50000000e+005.60000000e+00 5.70000000e+00 5.80000000e+00 5.90000000e+006.00000000e+00 6.10000000e+00 6.20000000e+00 6.30000000e+006.40000000e+00 6.50000000e+00 6.60000000e+00 6.70000000e+006.80000000e+00 6.90000000e+00 7.00000000e+00 7.10000000e+007.20000000e+00 7.30000000e+00 7.40000000e+00 7.50000000e+007.60000000e+00 7.70000000e+00 7.80000000e+00 7.90000000e+008.00000000e+00 8.10000000e+00 8.20000000e+00 8.30000000e+008.40000000e+00 8.50000000e+00 8.60000000e+00 8.70000000e+008.80000000e+00 8.90000000e+00 9.00000000e+00 9.10000000e+009.20000000e+00 9.30000000e+00 9.40000000e+00 9.50000000e+009.60000000e+00 9.70000000e+00 9.80000000e+00 9.90000000e+00] Sigmoid Function Output :: [4.53978687e-05 5.01721647e-05 5.54485247e-05 6.12797396e-056.77241496e-05 7.48462275e-05 8.27172229e-05 9.14158739e-051.01029194e-04 1.11653341e-04 1.23394576e-04 1.36370327e-041.50710358e-04 1.66558065e-04 1.84071905e-04 2.03426978e-042.24816770e-04 2.48455082e-04 2.74578156e-04 3.03447030e-043.35350130e-04 3.70606141e-04 4.09567165e-04 4.52622223e-045.00201107e-04 5.52778637e-04 6.10879359e-04 6.75082731e-047.46028834e-04 8.24424686e-04 9.11051194e-04 1.00677082e-031.11253603e-03 1.22939862e-03 1.35851995e-03 1.50118226e-031.65880108e-03 1.83293894e-03 2.02532039e-03 2.23784852e-032.47262316e-03 2.73196076e-03 3.01841632e-03 3.33480731e-033.68423990e-03 4.07013772e-03 4.49627316e-03 4.96680165e-035.48629890e-03 6.05980149e-03 6.69285092e-03 7.39154134e-038.16257115e-03 9.01329865e-03 9.95180187e-03 1.09869426e-021.21284350e-02 1.33869178e-02 1.47740317e-02 1.63024994e-021.79862100e-02 1.98403057e-02 2.18812709e-02 2.41270214e-022.65969936e-02 2.93122308e-02 3.22954647e-02 3.55711893e-023.91657228e-02 4.31072549e-02 4.74258732e-02 5.21535631e-025.73241759e-02 6.29733561e-02 6.91384203e-02 7.58581800e-028.31726965e-02 9.11229610e-02 9.97504891e-02 1.09096821e-011.19202922e-01 1.30108474e-01 1.41851065e-01 1.54465265e-011.67981615e-01 1.82425524e-01 1.97816111e-01 2.14165017e-012.31475217e-01 2.49739894e-01 2.68941421e-01 2.89050497e-013.10025519e-01 3.31812228e-01 3.54343694e-01 3.77540669e-014.01312340e-01 4.25557483e-01 4.50166003e-01 4.75020813e-015.00000000e-01 5.24979187e-01 5.49833997e-01 5.74442517e-015.98687660e-01 6.22459331e-01 6.45656306e-01 6.68187772e-016.89974481e-01 7.10949503e-01 7.31058579e-01 7.50260106e-017.68524783e-01 7.85834983e-01 8.02183889e-01 8.17574476e-018.32018385e-01 8.45534735e-01 8.58148935e-01 8.69891526e-018.80797078e-01 8.90903179e-01 9.00249511e-01 9.08877039e-019.16827304e-01 9.24141820e-01 9.30861580e-01 9.37026644e-019.42675824e-01 9.47846437e-01 9.52574127e-01 9.56892745e-019.60834277e-01 9.64428811e-01 9.67704535e-01 9.70687769e-019.73403006e-01 9.75872979e-01 9.78118729e-01 9.80159694e-019.82013790e-01 9.83697501e-01 9.85225968e-01 9.86613082e-019.87871565e-01 9.89013057e-01 9.90048198e-01 9.90986701e-019.91837429e-01 9.92608459e-01 9.93307149e-01 9.93940199e-019.94513701e-01 9.95033198e-01 9.95503727e-01 9.95929862e-019.96315760e-01 9.96665193e-01 9.96981584e-01 9.97268039e-019.97527377e-01 9.97762151e-01 9.97974680e-01 9.98167061e-019.98341199e-01 9.98498818e-01 9.98641480e-01 9.98770601e-019.98887464e-01 9.98993229e-01 9.99088949e-01 9.99175575e-019.99253971e-01 9.99324917e-01 9.99389121e-01 9.99447221e-019.99499799e-01 9.99547378e-01 9.99590433e-01 9.99629394e-019.99664650e-01 9.99696553e-01 9.99725422e-01 9.99751545e-019.99775183e-01 9.99796573e-01 9.99815928e-01 9.99833442e-019.99849290e-01 9.99863630e-01 9.99876605e-01 9.99888347e-019.99898971e-01 9.99908584e-01 9.99917283e-01 9.99925154e-019.99932276e-01 9.99938720e-01 9.99944551e-01 9.99949828e-01]

若想讓sigmoid函數計算numpy數組，可用以下編碼方式表示

def sigmoid(x):x_ravel = x.ravel() # 將numpy數組展平length = len(x_ravel)y = []for index in range(length):if x_ravel[index] >= 0:y.append(1.0 / (1 + np.exp(-x_ravel[index])))else:y.append(np.exp(x_ravel[index]) / (np.exp(x_ravel[index]) + 1))return np.array(y).reshape(x.shape)

參考文章1：Sigmoid函數

參考文章2：python計算警告：overflow encountered in exp（指數函數溢出）（sigmoid函數的numpy數組計算方式）

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