From 7e76f58aa54161597663756a0a9d8a4c39744dbb Mon Sep 17 00:00:00 2001 From: xiaowuhu Date: Mon, 1 Apr 2019 17:05:03 +0800 Subject: [PATCH] update (#172) --- .../06.2-线性二分类实现.md | 4 +- ...络的原理.md => 08.1-双层拟合网络的原理.md} | 7 +- ...络的实现.md => 08.2-双层拟合网络的实现.md} | 0 .../09.0-多入多出双层-双变量非线性分类.md | 27 ++- .../09.1-实现逻辑异或门.md | 168 +++++++++++++----- .../09.2-理解工作原理.md | 147 +++++++++++++++ ...2-非线性多分类.md => 09.3-非线性多分类.md} | 4 +- ...设计.md => 09.4-结果可视化与模块化设计.md} | 0 .../Images/9/binary_backward.png | Bin 0 -> 29595 bytes .../Images/9/binary_classifier.png | Bin 0 -> 52786 bytes .../Images/9/binary_forward.png | Bin 0 -> 14244 bytes .../Images/9/multiple_classifier.png | Bin 0 -> 68685 bytes .../Images/9/xor_2_1000_a1.png | Bin 0 -> 11219 bytes .../Images/9/xor_2_1000_loss.png | Bin 0 -> 27487 bytes .../Images/9/xor_2_1000_result.png | Bin 0 -> 19710 bytes .../Images/9/xor_2_1000_z1.png | Bin 0 -> 11057 bytes .../Images/9/xor_2_1000_z2a2.png | Bin 0 -> 18580 bytes .../Images/9/xor_2_500_a1.png 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index ddb24b85..11182b04 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/06.2-线性二分类实现.md +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/06.2-线性二分类实现.md @@ -176,9 +176,9 @@ def ToBool(YData): num_example = YData.shape[1] Y = np.zeros((1, num_example)) for i in range(num_example): - if YData[0,i] == 0: # 第一类的标签设为0 + if YData[0,i] == 1: # 第一类的标签设为0 Y[0,i] = 0 - elif YData[0,i] == 1: # 第二类的标签设为1 + elif YData[0,i] == 2: # 第二类的标签设为1 Y[0,i] = 1 # end if # end for diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/08.1-双层神经网络的原理.md b/B-教学案例与实践/B6-神经网络基本原理简明教程/08.1-双层拟合网络的原理.md similarity index 97% rename from B-教学案例与实践/B6-神经网络基本原理简明教程/08.1-双层神经网络的原理.md rename to B-教学案例与实践/B6-神经网络基本原理简明教程/08.1-双层拟合网络的原理.md index c41e7bad..f1857c1a 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/08.1-双层神经网络的原理.md +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/08.1-双层拟合网络的原理.md @@ -142,20 +142,19 @@ $$ $$\frac{\partial{Z2}}{\partial{A1}} = \frac{\partial{(W2 \cdot A1 + B2)}}{\partial{A1}}=W2^T$$ -$$\frac{\partial{A1}}{\partial{Z1}}=\frac{\partial{(Sigmoid(Z1))}}{\partial{Z1}}=A1 \cdot (1-A1)$$ +$$\frac{\partial{A1}}{\partial{Z1}}=\frac{\partial{(Sigmoid(Z1))}}{\partial{Z1}}=A1 \odot (1-A1)$$ 所以: $$dZ1=\frac{\partial{J}}{\partial{Z1}} = \frac{\partial{J}}{\partial{Z2}} \cdot \frac{\partial{Z2}}{\partial{A1}} \cdot \frac{\partial{A1}}{\partial{Z1}}$$ -$$=W2^T \cdot (Z2-Y) \cdot A1 \cdot (1-A1)$$ -$$=W2^T \cdot dZ2 \cdot A1 \cdot (1-A1) \tag{4}$$ +$$=W2^T \times dZ2 \odot A1 \odot (1-A1) \tag{4}$$ ### 求W1的梯度 $$\frac{\partial{Z1}}{\partial{W1}} = \frac{\partial{(W1 \cdot X+B1)}}{\partial{W1}} = X^T$$ $$ -dW1=\frac{\partial{J}}{\partial{W1}} = \frac{\partial{J}}{\partial{Z1}} \frac{\partial{Z1}}{\partial{W1}}= dZ1 \cdot X^T \tag{5} +dW1=\frac{\partial{J}}{\partial{W1}} = \frac{\partial{J}}{\partial{Z1}} \frac{\partial{Z1}}{\partial{W1}}= dZ1 \times X^T \tag{5} $$ ### 求B1的梯度 diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/08.2-双层神经网络的实现.md b/B-教学案例与实践/B6-神经网络基本原理简明教程/08.2-双层拟合网络的实现.md similarity index 100% rename from B-教学案例与实践/B6-神经网络基本原理简明教程/08.2-双层神经网络的实现.md rename to B-教学案例与实践/B6-神经网络基本原理简明教程/08.2-双层拟合网络的实现.md diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.0-多入多出双层-双变量非线性分类.md b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.0-多入多出双层-双变量非线性分类.md index 331912cc..1b52b62c 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.0-多入多出双层-双变量非线性分类.md +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.0-多入多出双层-双变量非线性分类.md @@ -4,7 +4,8 @@ Copyright © Microsoft Corporation. All rights reserved. # 知识点 - [解决异或问题](09.1-实现逻辑异或门.md) -- [非线性多分类](09.1-非线性多分类.md) +- [理解工作原理](09.2-理解工作原理].md) +- [非线性多分类](09.3-非线性多分类.md) - [结果可视化与模块化设计](09.2-结果可视化与模块化设计.md) @@ -19,6 +20,30 @@ Copyright © Microsoft Corporation. All rights reserved. 单层神经网络只能实现线性分类,虽然可以用一些特征的高阶形式来拟合有限的曲线,但要想实现真正的非线性分类,必须使用两层以上的神经网络。 +# 用于分类的双层神经网络 + +和我们前面学到的线性分类器一样,双层神经网络也有二分类和多分类两种分类器。 + +这是二分类器: + + + +- 输入层可以是任意数量的特征值 +- 隐层的神经元数量要大于等于输入层特征值数量,并且需要Sigmoid或Tanh等激活函数 +- 输出层用一个神经元,后接Sigmoid分类函数 +- 损失函数用二分类交叉熵函数 + +这是多分类器,图中所示为三分类: + + + +- 输入层可以是任意数量的特征值 +- 隐层的神经元数量要大于等于输入层特征值数量,并且需要Sigmoid或Tanh等激活函数 +- 输出层用N个神经元,神经元的数量等于分类的数量,后接Sigmoid分类函数 +- 损失函数用多分类交叉熵函数 + +# 工作原理 + 在两层神经网络的输出层,可以用和单层神经网络一样的结构来完成分类任务,而用隐层来完成非线性到线性的转换工作。我们可以通过以下几张图的比较来理解一下非线性到线性的转换。 以下图片均来自于[Christopher Olah的博客](https://colah.github.io/posts/2014-03-NN-Manifolds-Topology/)。 diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.1-实现逻辑异或门.md b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.1-实现逻辑异或门.md index 6e95ce45..dddb7eb6 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.1-实现逻辑异或门.md +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.1-实现逻辑异或门.md @@ -3,16 +3,60 @@ Copyright © Microsoft Corporation. All rights reserved. # 实现逻辑异或门 +## 简单证明异或的可能性 在1969年,一本著名的书《Perceptrons》(感知器,Minsky、Papert,1969)证明了无法使用单层网络(当时称为感知器)来表示最基本的异或逻辑功能。这本书带来了毁灭性的影响,对于神经网络这一新生领域的资金支持及兴趣都消失了。 +50年前的书,就不要去找了,咱们自己简单地证明一下。先看样本数据: + +|样本|1|2|3|4| +|----|----|----|----|---| +|x1|0|0|1|1| +|x2|0|1|0|1| +|y|0|1|1|0| + +用单个神经元(感知机)的话,就是下图: + +|神经元|激活函数 Sigmoid| +|-|-| +||| + +前向计算公式: + +$$Z = w1 \cdot x1 + w2 \cdot x2 + b \tag{1}$$ +$$A = Sigmoid(Z) \tag{2}$$ + +对于第一个样本数据,x1=0, x2=0, 如果需要A=0的话,从Sigmoid函数曲线看,需要Z<0,于是有: +$$w1 \cdot x1 + w2 \cdot x2 + b < 0$$ +因为x1=0, x2=0, 所以: +$$b < 0 \tag{3}$$ + +同理,第二个样本和第三个样本的y值为1,要求Z值大于0,不等式为: + +$$w2 + b > 0 \tag{4}$$ +$$w1 + b > 0 \tag{5}$$ + +第四个样本要求Z小于0: + +$$w1 + w2 + b < 0 \tag{6}$$ + +把公式6两边都加b,并把公式3接续: + +$$(w1 + b) + (w2 + b) < b < 0 \tag{7}$$ + +再看公式(4)(5),不等式左侧两个因子都大于0,其和必然也大于0,不可能小于b。因此无论如何也不能满足所有的4个样本的条件,所以单个神经元做异或是不可能的。 + +## 脑补理想的分类结果 + 因为当时还没有BP算法,所以深层神经网络无法实现。进入到两层神经网络后,我们将可以轻松地解决这个问题。看下图: 因为单层神经网络只能做线性分割,而上图中所示红蓝两色样本,用一条直线是不能分开的,于是人们脑补了那条曲线的分割方式,相当的fancy。但是神经网络真的会这样工作吗? -我们前边学习过来如何实现与、与非、或、或非,我们看看如何用已有的逻辑搭建异或: +## 利用已有的知识解决问题 + +我们前边学习过如何实现与、与非、或、或非,我们看看如何用已有的逻辑搭建异或门: @@ -20,70 +64,96 @@ Copyright © Microsoft Corporation. All rights reserved. |----|----|----|----|----| |x1|0|0|1|1| |x2|0|1|0|1| -|s1=NAND|1|1|1|0| -|s2=OR|0|1|1|1| -|y=AND|0|1|1|0| +|x1 AND x2|0|0|0|1| +|s1=x1 NAND x2|1|1|1|0| +|s2=x1 OR x1|0|1|1|1| +|y=s1 AND s2|0|1|1|0| -经过以上组合运算后,可以看到y的输出与x1,x2的输入相比,就是异或逻辑了。 +经过以上组合运算后,可以看到y的输出与x1,x2的输入相比,就是异或逻辑了。所以,实践证明两层神经网络可以解决问题。我们可以模拟这个思路,用两层神经网络搭建如下模型: -所以,我们可以模拟这个思路,用两层神经网络搭建如下模型: +## 搭建神经网络 - + -隐层的两个神经元接收x1,x2的输入,输出层有一个神经元负责分类。 +- 输入层两个特征值x1, x2 +- 隐层2x2的权重矩阵和2x1的偏移矩阵 +- 隐层由两个神经元构成 +- 输出层有一个神经元使用Sigmoid函数进行分类 + +## 前向计算 + +根据网络结构,我们有了前向计算图: + + + +$$Z1 = W1 \cdot X + B1 \tag{10}$$ +$$A1 = Sigmoid(Z1) \tag{11}$$ +$$Z2 = W2 \cdot A1 + B2 \tag{12}$$ +$$A2 = Sigmoid(Z2) \tag{13}$$ + +**注意:公式13的Sigmoid在这里不是激活函数,不是激活函数,不是激活函数(重要的概念说三遍),是用作分类函数。** + +## 损失函数 + +我们把异或问题归类成二分类问题,所以使用二分类交叉熵损失函数: + +$$ +J(w,b) = -{1 \over m} \sum^m_{i=1}y_i \ln (a_i)+(1-y_i) \ln (1-a_i) \tag{14} +$$ + +可以简写为: + +$$ +J = -Y \ln A + (1-Y) \ln (1-A) \tag{15} +$$ +## 反向传播 + -||| -|---|---| -||| -||| -||| -||| +### 梯度生成 +对损失函数求导,可以得到损失函数对输出层的梯度值,即上图中的Z2部分。 -以下是隐层为两个神经元时的结果输出: +根据公式15,求A2和Z2的导数: + +$${\partial{J} \over \partial{Z2}}={\partial{J} \over \partial{A2}} \cdot {\partial{A2} \over \partial{Z2}}$$ +$$= {A2-Y \over A2(1-A2)} \cdot A2(1-A2)$$ +$$=A2-Y \tag{15} => dZ2$$ + +### 求W2和B2的梯度 + +$${\partial{J} \over \partial{W2}}={\partial{J} \over \partial{Z2}}{\partial{Z2} \over \partial{W2}}=dZ2 \times A1^T => dW2\tag{16}$$ + +$${\partial{J} \over \partial{B2}}={\partial{J} \over \partial{Z2}}{\partial{Z2} \over \partial{B2}}=dZ2 => dB2 \tag{17}$$ + +### 求损失函数对隐层的梯度 + +$$ +\frac{\partial{J}}{\partial{Z1}} = \frac{\partial{J}}{\partial{Z2}} \cdot \frac{\partial{Z2}}{\partial{A1}} \cdot \frac{\partial{A1}}{\partial{Z1}} +$$ +$$=W2^T \times dZ2 \odot A1 \odot (1-A1)=>dZ1 \tag{18}$$ + +### 求W1和B1的梯度 + +$$ +\frac{\partial{J}}{\partial{W1}} = \frac{\partial{J}}{\partial{Z1}} \frac{\partial{Z1}}{\partial{W1}}= dZ1 \times X^T => dW1\tag{19} +$$ + +$$ +\frac{\partial{J}}{\partial{B1}} = \frac{\partial{J}}{\partial{Z1}} \frac{\partial{Z1}}{\partial{B1}}= dZ1 => dB1\tag{20} +$$ + +## 运行结果 ``` -epoch=6039, loss=0.005000 -epoch=6040, loss=0.004999 -epc=6040,ite=3,los=0.0050 w=[[-7.00777143 -7.01121059] [ 5.51518649 5.51451102]] b=[[ 2.86885647] [-8.53863829]] - -w=[[-12.0633545 -12.23675401]] -b=[[5.95925939]] - -testing... -x=[[0] [0]] y=[[0]] output=[[0.00424183]] -True -x=[[0] [1]] y=[[1]] output=[[0.99453265]] -True -x=[[1] [0]] y=[[1]] output=[[0.99452718]] -True -x=[[1] [1]] y=[[0]] output=[[0.00476486]] -True ``` -后面的testing...为测试结果,精度为1e-2。 + - -||1|2|3|4| -|---|---|---|---|---| -|x1|0|0|1|1| -|x2|0|1|0|1| -|y|0|1|1|0| -|Z1|2.86885647|-4.14235412|-4.13891495|-11.15012554| -||-8.53863829|-3.02412727|-3.0234518|2.49105922| -|A1|9.46285253e-01|1.56370110e-02|1.56900366e-02 |1.43732759e-05| -||1.95718330e-04|4.63477089e-02|4.63775738e-02|9.23512657e-01| -|Z2|-5.45851003|5.20347907|5.20247396|-5.3417112| -|A2|0.00424183|0.99453265|0.99452718|0.00476486| - -||| -|---|---| -||| -||| \ No newline at end of file +代码位置:ch09, Level0, Level1 \ No newline at end of file diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.2-理解工作原理.md b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.2-理解工作原理.md new file mode 100644 index 00000000..3adddec4 --- /dev/null +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.2-理解工作原理.md @@ -0,0 +1,147 @@ +Copyright © Microsoft Corporation. All rights reserved. + 适用于[License](https://github.com/Microsoft/ai-edu/blob/master/LICENSE.md)版权许可 + +# 理解工作原理 + +我们在上一节课的代码基础上再增加些东西,来理解神经网络针对这个异或问题的工作原理。 + +## 隐层神经元数量的选择 + +一般来说,隐层的神经元数量要大于等于输入特征的数量,在本例中是2。我们从下图可以看到,如果隐层只有一个神经元的话,是不能完成分类任务的。 + +||| +|---|---| +||| +|1个神经元|2个神经元,迭代6040次到达精度要求| +||| +|3个神经元,迭代5228次到达精度要求|4个神经元,迭代4331次到达精度要求| +||| +|5个神经元,迭代4171次到达精度要求|6个神经元,迭代3903次到达精度要求| +||| +|7个神经元,迭代5031次到达精度要求|8个神经元,迭代4107次到达精度要求| + +## 隐层有两个神经元的工作原理 + +以下是隐层为两个神经元时的结果输出: + +``` +w=[[-7.00777143 -7.01121059] + [ 5.51518649 5.51451102]] +b=[[ 2.86885647] + [-8.53863829]] +``` + +我们使用上面的权重矩阵结果,把4个样本数据代入到前向计算公式中,依次求得Z1,A1,Z2,A2的值,并列表如下: + +||1|2|3|4| +|---|---|---|---|---| +|x1|0|0|1|1| +|x2|0|1|0|1| +|y|0|1|1|0| +|Z1|2.86885647|-4.14235412|-4.13891495|-11.15012554| +||-8.53863829|-3.02412727|-3.0234518|2.49105922| +|A1|9.46285253e-01|1.56370110e-02|1.56900366e-02 |1.43732759e-05| +||1.95718330e-04|4.63477089e-02|4.63775738e-02|9.23512657e-01| +|Z2|-5.45851003|5.20347907|5.20247396|-5.3417112| +|A2|0.00424183|0.99453265|0.99452718|0.00476486| + +||| +|---|---| +||| +||| + + + +|损失函数值|| +|---|---| +||| +|500次迭代|800次迭代| +||| +|1000次迭代|6000次迭代| + +|Z1|| +|---|---| +||| +|500次迭代|800次迭代| +||| +|1000次迭代|6000次迭代| + +|A1|| +|---|---| +||| +|500次迭代|800次迭代| +||| +|1000次迭代|6000次迭代| + +|Z1:A1|| +|---|---| +||| +|500次迭代|800次迭代| +||| +|1000次迭代|6000次迭代| + +|分类结果|| +|---|---| +||| +|500次迭代|800次迭代| +||| +|1000次迭代|6000次迭代| + + +## 隐层有三个神经元的工作原理 + +最终输出的权重矩阵值: + +``` +[[ 6.70984241 -6.82710114] + [ 6.22781451 -5.99280808] + [-2.74806343 2.4870413 ]] +b=[[-3.51362288] + [ 3.15804745] + [-1.4773994 ]] + +w=[[12.31224597 -9.738143 3.80483026]] +b=[[3.05151459]] +``` + +迭代5000次到达精度要求的各层的网络输出: + +||1|2|3|4| +|---|---|---|---|---| +|x1|0|0|1|1| +|x2|0|1|0|1| +|y|0|1|1|0| +|Z1|3.51362288|-10.34072402|3.19621952|-3.63088162| +||3.15804745|-2.83476063|9.38586196|3.39305388| +||-1.4773994|1.00964189|-4.22546284|-1.73842154| +|A1|2.89270940e-02| 3.22898922e-05 |9.60691763e-01| 2.58090628e-02| +||9.59224645e-01| 5.54744264e-02| 9.99916105e-01| 9.67486745e-01| +||1.85820545e-01| 7.32950061e-01| 1.44079436e-02| 1.49513539e-01| +|Z2|-5.22637905| 5.30044483 | 5.19728163 |-5.48336852| +|A2|0.00534423 |0.9950354 | 0.99449885 |0.00413811| + + +我们可以把Z1,A1看作是一个三维的坐标点,并比较迭代200次与迭代5000次的结果,从而了解神经网络的工作过程: + + +||迭代200次|迭代5000次| +|---|---|---| +|初始|| +|Z1-1||| +||两个红色点距离较远|两个红色点距离很近| +|Z1-2||| +||绿色点从一开始就已经分得很开|绿色点的距离越来越远| +|Z1-3||| +||从某一角度看4个点在一条直线上|| +|A1-1||| +||红点较散|红点缩到了一个角落里| +|A1-2||| +|A1-3||| +|Z2||| +||两种颜色的点没有分开|彻底分开了| +|A2||| +||没有完成|理想分类| + + + +代码位置:ch09, Level2_HowXorGateEorks(两个神经元的情况), Level2_LogicXorGate3D(三个神经元的情况) diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.2-非线性多分类.md b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.3-非线性多分类.md similarity index 96% rename from B-教学案例与实践/B6-神经网络基本原理简明教程/09.2-非线性多分类.md rename to B-教学案例与实践/B6-神经网络基本原理简明教程/09.3-非线性多分类.md index 7bcf5a64..82476581 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.2-非线性多分类.md +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.3-非线性多分类.md @@ -1,12 +1,12 @@ Copyright © Microsoft Corporation. All rights reserved. 适用于[License](https://github.com/Microsoft/ai-edu/blob/master/LICENSE.md)版权许可 -# 非线性多分类实现 +# 非线性多分类 # 提出问题 -前面我们学习了线性二分类与多分类。升级为两层的神经网络之后,我们看看如何用它来做非线性的二分类与多分类。 +前面用异或问题学习了二分类,现在我们看看如何用它来做非线性多分类。 我们有如下1000个样本和标签: diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/09.3-结果可视化与模块化设计.md b/B-教学案例与实践/B6-神经网络基本原理简明教程/09.4-结果可视化与模块化设计.md similarity index 100% rename from B-教学案例与实践/B6-神经网络基本原理简明教程/09.3-结果可视化与模块化设计.md rename to B-教学案例与实践/B6-神经网络基本原理简明教程/09.4-结果可视化与模块化设计.md diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/Images/9/binary_backward.png b/B-教学案例与实践/B6-神经网络基本原理简明教程/Images/9/binary_backward.png new file mode 100644 index 0000000000000000000000000000000000000000..b1d96418d50cf3033905fd53188829aee210087c GIT binary patch literal 29595 zcmagFbx>Q~7dDEN^0q)JP%PL>aff2XLR*TvyF+nz2v7>1_lQ90tN== zKmR-ieyK9MBLF@yJv3zBVN{P(?*LyO*}PSLi-A#>fOlt(1AKqtDyQ#(f$^;O?}<6+ zQfh^P;lV5a{;ig;$-yFShLQFh?y=9vMcNWoh$oi(zhTcX3)$Y;zI>#m7Coz`mIyDc z`!e~WFzVyGH%~g{Qca}fvFO7Y!_G{EkjPQUW$-3!u|2u0UGuOLg_eMedu)yG?;q~Z 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z02etFkNQvC3UQ}%)Ij>5A?MQt2Bz-wJL4hrvyXs*VN5V7phJSt9W(KdBE-^B5H)M^ zAC8z5VPPdcVwluV0fYF6afDc8QlLuCg6~_H8+b^H$jf{45Bl{z8nM*;?{<*A1p^89 zg#x6Aj5ElPzg=U26wzeA5Q= 0.5: + plt.plot(x[0,0], x[1,0], 's', c='m') + else: + plt.plot(x[0,0], x[1,0], 's', c='y') + # end if + # end for + # end for + plt.title(title) +#end def + +def ShowData(X, Y): + for i in range(X.shape[1]): + if Y[0,i] == 0: + plt.plot(X[0,i], X[1,i], '^', c='r') + elif Y[0,i] == 1: + plt.plot(X[0,i], X[1,i], '.', c='g') + # end if + # end for + plt.xlabel("x1") + plt.ylabel("x2") + plt.show() + +def Test(dataReader, net, wb1, wb2): + print("testing...") + for i in range(dataReader.num_example): + x,y = dataReader.GetBatchSamples(1, i) + dict_output = net.ForwardCalculationBatch2(x, wb1, wb2) + output = dict_output["Output"] + print(str.format("x={0} y={1} output={2}", x, y, output)) + if np.abs(output - y) < 1e-2: + print("True") + else: + print("False") + #end if + #end for + +def SaveWeights(wb1, wb2): + np.save("xor_w1_2_2.npy", wb1.W) + np.save("xor_w1_2_1.npy", wb1.B) + np.save("xor_w2_1_2.npy", wb2.W) + np.save("xor_w2_1_1.npy", wb2.B) + + +def train(): + + dataReader = XOR_DataReader() + dataReader.ReadData() + + n_input, n_output = dataReader.num_feature, dataReader.num_category + n_hidden = 2 + eta, batch_size, max_epoch = 0.1, 1, 10000 + eps = 0.005 + + params = CParameters(n_input, n_hidden, n_output, eta, max_epoch, batch_size, eps, LossFunctionName.CrossEntropy2) + + loss_history = CLossHistory() + net = TwoLayerClassificationNet() + + #ShowData(XData, YData) + + wb1, wb2 = net.train(dataReader, params, loss_history) + + trace = loss_history.GetMinimalLossData() + print(trace.toString()) + title = loss_history.ShowLossHistory(params) + + print(wb1.toString()) + print(wb2.toString()) + + print("wait for 10 seconds...") + + ShowAreaResult(net, trace.wb1, trace.wb2, title) + ShowData(dataReader.X, dataReader.Y) + + Test(dataReader, net, wb1, wb2) + + SaveWeights(wb1, wb2) + +if __name__ == '__main__': + train() \ No newline at end of file diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_HowXorGateWorks.py b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_HowXorGateWorks.py new file mode 100644 index 00000000..31d217f0 --- /dev/null +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_HowXorGateWorks.py @@ -0,0 +1,145 @@ +# Copyright (c) Microsoft. All rights reserved. +# Licensed under the MIT license. See LICENSE file in the project root for full license information. + +import numpy as np +from pathlib import Path +import matplotlib.pyplot as plt +import math + +from LossFunction import * +from Activators import * +from Level0_TwoLayerClassificationNet import * +from DataReader import * +from WeightsBias import * + + +# x1=0,x2=0,y=0 +# x1=0,x2=1,y=1 +# x1=1,x2=0,y=1 +# x1=1,x2=1,y=0 +class XOR_DataReader(): + def __init__(self): + pass + + def ReadData(self): + self.X = np.array([0,0,1,1,0,1,0,1]).reshape(2,4) + self.Y = np.array([0,1,1,0]).reshape(1,4) + self.num_example = self.X.shape[1] + self.num_feature = self.X.shape[0] + self.num_category = 1 + + def GetBatchSamples(self, batch_size, iteration): + start = iteration * batch_size + end = start + batch_size + batch_X = self.X[0:self.num_feature, start:end].reshape(self.num_feature, batch_size) + batch_Y = self.Y[:, start:end].reshape(-1, batch_size) + return batch_X, batch_Y + + +def ShowAreaResult(net, wb1, wb2, title): + count = 50 + x1 = np.linspace(0,1,count) + x2 = np.linspace(0,1,count) + for i in range(count): + for j in range(count): + x = np.array([x1[i],x2[j]]).reshape(2,1) + dict_cache = net.ForwardCalculationBatch2(x, wb1, wb2) + output = dict_cache["Output"] + if output[0,0] >= 0.5: + plt.plot(x[0,0], x[1,0], 's', c='m') + else: + plt.plot(x[0,0], x[1,0], 's', c='y') + # end if + # end for + # end for + plt.title(title) +#end def + +def ShowData(X, Y): + for i in range(X.shape[1]): + if Y[0,i] == 0: + plt.plot(X[0,i], X[1,i], '^', c='r') + elif Y[0,i] == 1: + plt.plot(X[0,i], X[1,i], '.', c='g') + # end if + # end for + plt.xlabel("x1") + plt.ylabel("x2") + plt.show() + + +def LoadWeights(wb1, wb2): + wb1.W = np.load("xor_w1_2_2.npy") + wb1.B = np.load("xor_w1_2_1.npy") + wb2.W = np.load("xor_w2_1_2.npy") + wb2.B = np.load("xor_w2_1_1.npy") + + +def ShowZ1A1Z2A2(): + wb1 = WeightsBias(2,2,0.1,InitialMethod.Xavier) + wb2 = WeightsBias(1,2,0.1,InitialMethod.Xavier) + LoadWeights(wb1, wb2) + print(wb1.toString()) + print(wb2.toString()) + + dataReader = XOR_DataReader() + dataReader.ReadData() + + + Z1 = np.dot(wb1.W, dataReader.X) + wb1.B + A1 = Sigmoid().forward(Z1) + # layer 2 + Z2 = np.dot(wb2.W, A1) + wb2.B + A2 = Sigmoid().forward(Z2) + print(Z1) + print(A1) + print(Z2) + print(A2) + + for i in range(dataReader.num_example): + if dataReader.Y[0,i] == 0: + plt.plot(dataReader.X[0,i],dataReader.X[1,i],'^',c='r') + else: + plt.plot(dataReader.X[0,i],dataReader.X[1,i],'o',c='g') + plt.grid() + plt.title("X1:X2") + plt.show() + + + for i in range(dataReader.num_example): + if dataReader.Y[0,i] == 0: + plt.plot(Z1[0,i],Z1[1,i],'^',c='r') + else: + plt.plot(Z1[0,i],Z1[1,i],'o',c='g') + plt.grid() + plt.title("Z1") + plt.show() + + + for i in range(dataReader.num_example): + if dataReader.Y[0,i] == 0: + plt.plot(A1[0,i],A1[1,i],'^',c='r') + else: + plt.plot(A1[0,i],A1[1,i],'o',c='g') + plt.grid() + plt.title("A1") + plt.show() + + x = np.linspace(-6,6) + a = Sigmoid().forward(x) + plt.plot(x,a) + + for i in range(dataReader.num_example): + if dataReader.Y[0,i] == 0: + plt.plot(Z2[0,i],A2[0,i],'^',c='r') + else: + plt.plot(Z2[0,i],A2[0,i],'o',c='g') + plt.grid() + plt.title("Z2:A2") + plt.show() + + +if __name__ == '__main__': + # 每次运行之前,需要把level1中的Line 98, max_epoch的数字从小改到大,比如200,400,600,800,1000,..., + # 每改一次max_epoch,运行一次level1,生成权重值并自动保存,再运行一次本程序绘图 + ShowZ1A1Z2A2() \ No newline at end of file diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_LogicXorGate.py b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_LogicXorGate3D.py similarity index 85% rename from B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_LogicXorGate.py rename to B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_LogicXorGate3D.py index bf66b501..0eefe0c7 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_LogicXorGate.py +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_LogicXorGate3D.py @@ -4,7 +4,7 @@ import numpy as np from pathlib import Path import matplotlib.pyplot as plt -import math +from mpl_toolkits.mplot3d import Axes3D from LossFunction import * from Activators import * @@ -94,7 +94,7 @@ def train(): dataReader.ReadData() n_input, n_output = dataReader.num_feature, dataReader.num_category - n_hidden = 2 + n_hidden = 3 eta, batch_size, max_epoch = 0.1, 1, 10000 eps = 0.005 @@ -122,7 +122,7 @@ def train(): Test(dataReader, net, wb1, wb2) SaveWeights(wb1, wb2) - + def LoadWeights(wb1, wb2): wb1.W = np.load("xor_w1_2_2.npy") wb1.B = np.load("xor_w1_2_1.npy") @@ -160,22 +160,28 @@ def ShowZ1A1Z2A2(): plt.title("X1:X2") plt.show() - + fig = plt.figure() + ax = Axes3D(fig) for i in range(dataReader.num_example): if dataReader.Y[0,i] == 0: - plt.plot(Z1[0,i],Z1[1,i],'^',c='r') + ax.scatter(Z1[0,i],Z1[1,i],Z1[2,i],c='r') + #ax.plot(Z1[0,i],Z1[1,i],Z1[2,i],c='r') else: - plt.plot(Z1[0,i],Z1[1,i],'o',c='g') + ax.scatter(Z1[0,i],Z1[1,i],Z1[2,i],c='g') + #ax.plot(Z1[0,i],Z1[1,i],Z1[2,i],c='g') plt.grid() plt.title("Z1") plt.show() - + fig = plt.figure() + ax = Axes3D(fig) for i in range(dataReader.num_example): if dataReader.Y[0,i] == 0: - plt.plot(A1[0,i],A1[1,i],'^',c='r') + ax.scatter(A1[0,i],A1[1,i],A1[2,i],'^',c='r') + #plt.plot(A1[0,i],A1[1,i],A1[2,i],'^',c='r') else: - plt.plot(A1[0,i],A1[1,i],'o',c='g') + ax.scatter(A1[0,i],A1[1,i],A1[2,i],'o',c='g') + #plt.plot(A1[0,i],A1[1,i],A1[2,i],'o',c='g') plt.grid() plt.title("A1") plt.show() @@ -195,4 +201,7 @@ def ShowZ1A1Z2A2(): if __name__ == '__main__': + train() + # 每次运行之前,需要把level1中的Line 98, max_epoch的数字从小改到大,比如200,400,600,800,1000,..., + # 每改一次max_epoch,运行一次level1,生成权重值并自动保存,再运行一次本程序绘图 ShowZ1A1Z2A2() \ No newline at end of file diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_Classifier.py b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level3_Classifier.py similarity index 95% rename from B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_Classifier.py rename to B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level3_Classifier.py index d5ebab58..90a197e2 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level2_Classifier.py +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/Level3_Classifier.py @@ -8,7 +8,7 @@ import math from LossFunction import * from Activators import * -from Level1_TwoLayerClassificationNet import * +from Level0_TwoLayerClassificationNet import * from DataReader import * from WeightsBias import * @@ -22,7 +22,7 @@ def ShowAreaResult(net, wb1, wb2, title): for i in range(count): for j in range(count): x = np.array([x1[i],x2[j]]).reshape(2,1) - dict_cache = net.ForwardCalculationBatch(x, wb1, wb2) + dict_cache = net.ForwardCalculationBatch3(x, wb1, wb2) output = dict_cache["Output"] r = np.argmax(output, axis=0) if r == 0: diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/ch09.pyproj b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/ch09.pyproj index 13858903..74276c58 100644 --- a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/ch09.pyproj +++ b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/ch09.pyproj @@ -4,7 +4,7 @@ 2.0 3de29cfe-16a6-493d-97b3-e617074af796 . - Level2_LogicXorGate.py + Level3_Classifier.py . @@ -24,8 +24,10 @@ - - + + + + Code diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w1_2_1.npy b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w1_2_1.npy index 43de23b2daa9e890c5353427ead9d30ed79bd4c1..b649bcbd9f047509e75f086b087e66042c8e223e 100644 GIT binary patch delta 36 ucmV+<0NekN0hj@hJpnV3K0+Atx_?mS8w|ii&!J(iLkU2<$Lo%5qxZiD%n+{t delta 28 kcmbQiIDv73Eu+yyJ0}5U-D|AbpV=JNm|jgiA*^@+0D(IR@&Et; diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w1_2_2.npy b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w1_2_2.npy index ee2ae023c970fbb7f5f0e2873ea3a25aa0517e5c..fc859cddba4383db3e17c279d231cb8f5b45a17f 100644 GIT binary patch delta 60 zcmV-C0K@;F0k8p(JpnV3K0+`kly&Xk)*3*vO3s<{P8+~_Xbv?<=@>wRF1fCv_!q#+ SYJRQ=`~|?eS&Zm)*V14;Db;u>ktp4i-Ra CS`b+P diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w2_1_1.npy b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w2_1_1.npy index c883784cef61561deeabfb854233a3f3a419c83b..0a45a11f3017017c5943d9c8a2b07e0ed4b77d20 100644 GIT binary patch delta 15 WcmeBR>|mVG!13_=&#Hz@4hH}<4F;zG delta 15 WcmeBR>|mVGz%eJATf*a-xB~zu7zD}y diff --git a/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w2_1_2.npy b/B-教学案例与实践/B6-神经网络基本原理简明教程/微软-方案1/NeuralNetwork/ch09/xor_w2_1_2.npy index 7b85500a2c2679cb3cdff5942dc90a3643d0abf3..8862cfb5d38b6051990157fb191fd272e5d32f28 100644 GIT binary patch delta 36 scmbQhID>J51EcXoM|+8c7@IBk=4&|Y@Hc(_kA|vBXuL3>?0PtQ7Qvd(} delta 28 kcmbQiIDv731EbMIM|**{+mjdy6f_QOd~bBgyi(%;0FNjO2><{9