原文博客:Doi技术团队 链接地址:https://blog.doiduoyi.com/authors/1584446358138 初心:记录优秀的Doi技术团队学习经历
μ=1N∑i=1Nxi(x:x1,x2,...,xN)\mu=\frac{1}{N}\sum_{i=1}^Nx_i\left(x:x_1,x_2,...,x_N\right)μ=N1i=1∑Nxi(x:x1,x2,...,xN)
import numpy as np
# 一维数组
x = np.array([-0.02964322, -0.11363636, 0.39417967, -0.06916996, 0.14260276])
print('数据:', x)
# 求均值
avg = np.mean(x)
print('均值:', avg)
输出:
数据: [-0.02964322 -0.11363636 0.39417967 -0.06916996 0.14260276]
均值: 0.064866578
σ=1N∑i=1N(xi−μ)2(x:x1,x2,...,xN)\sigma=\sqrt{\frac{1}{N}\sum_{i=1}^N\left(x_i-\mu\right)^2} \left(x:x_1,x_2,...,x_N\right)σ=N1i=1∑N(xi−μ)2(x:x1,x2,...,xN)
import numpy as np
# 一维数组
x = np.array([-0.02964322, -0.11363636, 0.39417967, -0.06916996, 0.14260276])
print('数据:', x)
# 求标准差
std = np.std(x)
print('标准差:', std)
输出:
数据: [-0.02964322 -0.11363636 0.39417967 -0.06916996 0.14260276]
标准差: 0.18614576055671836
正态分布的概率密度函数: f(x)=12πσe−(x−μ)22σ2f(x)=\frac{1}{\sqrt {2\pi\sigma}}e^{-\frac{\left(x-\mu\right)^2}{2\sigma^2}}f(x)=2πσ1e−2σ2(x−μ)2
import numpy as np
from matplotlib import pyplot as plt
def nd(x, u=-0, d=1):
return 1/np.sqrt(2*np.pi*d)*np.exp(-(x-u)**2/2/d**2)
x = np.linspace(-3, 3, 50)
y = nd(x)
plt.plot(x, y)
# 调整坐标
ax = plt.gca()
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['bottom'].set_position(('data', 0))
ax.spines['left'].set_position(('data', 0))
plt.show()
y=x−μσy=\frac{x-\mu}{\sigma}y=σx−μ
import numpy as np
from matplotlib import pyplot as plt
def nd(x, u=0, d=1):
return 1/np.sqrt(2*np.pi*d)*np.exp(-(x-u)**2/2/d**2)
x = np.linspace(-5, 5, 50)
y1 = nd(x)
y2 = nd(x, 0.5, 2)
plt.plot(x, y1)
plt.plot(x, y2)
# 调整坐标
ax = plt.gca()
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['bottom'].set_position(('data', 0))
ax.spines['left'].set_position(('data', 0))
plt.show()
y1=2xy_1=2^xy1=2x y2=exy_2=e^xy2=ex
import numpy as np
from matplotlib import pyplot as plt
x = np.linspace(-2, 3, 100)
y1 = 2**x
y2 = np.exp(x)
plt.plot(x,y1, color='red')
plt.plot(x, y2, color='blue')
# 调整坐标
ax = plt.gca()
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['bottom'].set_position(('data', 0))
ax.spines['left'].set_position(('data', 0))
plt.show()
y1=log2xy_1=log_2xy1=log2x y2=lnxy_2=lnxy2=lnx y3=lgxy_3=lgxy3=lgx
import numpy as np
from matplotlib import pyplot as plt
x = np.linspace(0.01, 10, 100)
y1 = np.log2(x)
y2 = np.log(x)
y3 = np.log10(x)
plt.plot(x, y1, color='red')
plt.plot(x, y2, color='blue')
plt.plot(x, y3, color='green')
# 调整坐标
ax = plt.gca()
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['bottom'].set_position(('data', 0))
ax.spines['left'].set_position(('data', 0))
plt.show()
Pi=exi∑i=1Nexi(x:x1,x2,…,xN)P_i=\frac{e^{x_i}}{\sum_{i=1}^Ne^{x_i}}\left(x:x_1,x_2,\ldots,x_N\right)Pi=∑i=1Nexiexi(x:x1,x2,…,xN)
import numpy as np
x = np.array([-0.02964322, -0.11363636, 0.39417967, -0.06916996, 0.14260276])
print('原始输出:', x)
prob = np.exp(x)/np.sum(np.exp(x))
print('概率输出:', prob)
输出:
原始输出: [-0.02964322 -0.11363636 0.39417967 -0.06916996 0.14260276]
概率输出: [0.17868493 0.16428964 0.27299323 0.17175986 0.21227234]
示例: 假如有5类,则编码为: 第一类:[1, 0, 0, 0, 0] 第二类:[0, 1, 0, 0, 0] 第三类:[0, 0, 1, 0, 0] 第四类:[0, 0, 0, 1, 0] 第五类:[0, 0, 0, 0, 1]
Label:[l1,l2,…,lN]Label:[l_1, l_2, \ldots, l_N]Label:[l1,l2,…,lN] ——经过One-hot编码
predict:[P1,P2,…,PN]predict:[P_1, P_2, \ldots, P_N]predict:[P1,P2,…,PN] ——经过Softmax函数
ce=−∑i=1Nli⋅logPice=-\sum_{i=1}^Nl_i\cdot\log^{P_i}ce=−i=1∑Nli⋅logPi
import numpy as np
# one-hot 编码的标签
label = np.array([0,0,1,0,0])
print('分类标签:', label)
# 网络实际输出
x1 = np.array([-0.02964322, -0.11363636, 3.39417967, -0.06916996, 0.14260276])
x2 = np.array([-0.02964322, -0.11363636, 1.39417967, -0.06916996, 5.14260276])
print('网络输出1:', x1)
print('网络输出2:', x2)
# softmax 之后的模拟概率
p1 = np.exp(x1) / np.sum(np.exp(x1))
p2 = np.exp(x2) / np.sum(np.exp(x2))
print('概率输出1:', p1)
print('概率输出2:', p2)
# 交叉熵
ce1 = -np.sum(label * np.log(p1))
ce2 = -np.sum(label * np.log(p2))
print('交叉熵1:', ce1)
print('交叉熵2:', ce2)
输出:
分类标签: [0 0 1 0 0]
网络输出1: [-0.02964322 -0.11363636 3.39417967 -0.06916996 0.14260276]
网络输出2: [-0.02964322 -0.11363636 1.39417967 -0.06916996 5.14260276]
概率输出1: [0.02877271 0.02645471 0.88293386 0.0276576 0.03418112]
概率输出2: [0.00545423 0.00501482 0.02265122 0.00524284 0.96163688]
交叉熵1: 0.12450498821197674
交叉熵2: 3.787541448750617
δ(x)=11+e−x\delta(x)=\frac{1}{1+e^{-x}}δ(x)=1+e−x1 Tanh(x)=ex−e−xex+e−xTanh(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}Tanh(x)=ex+e−xex−e−x ReLU(x)=max(x,0)ReLU(x)=max(x, 0)ReLU(x)=max(x,0)
import numpy as np
from matplotlib import pyplot as plt
x = np.linspace(-10, 10, 100)
# plt.figure(31)
plt.figure(figsize=(10, 20))
# Sigmoid
sigmoid = 1 / (1 + np.exp(-x))
top = np.ones(100)
plt.subplot(311)
plt.plot(x, sigmoid, color='blue')
plt.plot(x, top, color='red', linestyle='-.', linewidth=0.5)
plt.title(s='Sigmoid')
# Tanh
tanh = (np.exp(x) - np.exp(-x)) / (np.exp(x) + np.exp(-x))
top = np.ones(100)
bottom = -np.ones(100)
plt.subplot(312)
plt.plot(x, tanh, color='blue')
plt.plot(x, top, color='red', linestyle='-.', linewidth=0.5)
plt.plot(x, bottom, color='red', linestyle='-.', linewidth=0.5)
plt.title('Tanh')
# ReLU
relu = np.maximum(x, 0)
plt.subplot(313)
plt.plot(x, relu, color='blue')
plt.title('ReLU')
# 调整坐标
ax = plt.gca()
ax.spines['right'].set_color('none')
ax.spines['top'].set_color('none')
ax.xaxis.set_ticks_position('bottom')
ax.yaxis.set_ticks_position('left')
ax.spines['bottom'].set_position(('data', 0))
ax.spines['left'].set_position(('data', 0))
plt.show()
源代码地址:https://aistudio.baidu.com/aistudio/projectdetail/176057