一, 基本概念
1, 逻辑回归与线性回归的区别?
线性回归预测得到的是一个数值, 而逻辑回归预测到的数值只有 0,1 两个值. 逻辑回归是在线性回归的基础上, 加上一个 sigmoid 函数, 让其值位于 0-1 之间, 最后获得的值大于 0.5 判断为 1, 小于等于 0.5 判断为 0
二, 逻辑回归的推导
\(\hat y\)表示预测值,\(y\)表示训练标签值
1, 一般公式
\[ \hat y = wx + b \]
2, 向量化
\[ \hat y = w^Tx+b \]
3, 激活函数
引入 sigmoid 函数 (用 \(\sigma\) 表示), 使 \(\hat y\)值位于 0-1
\[ \hat y = \sigma (w^Tx+b) \]
4, 损失函数
损失函数用 \(L\)表示
\[ L(\hat y ,y) = \frac 1 2 (\hat y - y) ^ 2 \]
因梯度下降效果不好, 换用交叉熵损失函数
\[ L(\hat y,y) = - [y\log \hat y + (1-y)\log(1-\hat y)] \]
5, 代价函数
代价函数用 \(J\)表示
\[ J(w,b) = \frac 1 m \sum_{i=1}^m L(\hat y^{(i)},y^{(i)}) \]
展开
\[ J(w,b) = - \frac 1 m \sum_{i=1}^m [y^{(i)}\log \hat y^{(i)} + (1-y^{(i)})\log(1-\hat y ^ {(i)})] \]
6, 正向传播
- \[ a = 5,b=3,c=2 \]
- \[ u = bc \to 6 \]
- \[ v = a + u \to 11 \]
- \[ J = 3v \to 33 \]
7, 反向传播
求出 \(da\) = 3, 表示 \(J\)对 \(a\)的偏导数
\[ J = 3v \to \frac {dJ} {da} = \frac {dJ} {dv} \frac {dv} {da} = 3 \]
求出 \(db\) = 6
\[ J = 3v \to \frac {dJ} {dc} = \frac {dJ} {dv} \frac {dv} {du} \frac {du} {dc} = 3c = 6 \]
求出 \(dc\) = 6
\[ J = 3v \to \frac {dJ} {db} = \frac {dJ} {dv} \frac {dv} {du} \frac {du} {db} = 3b = 9 \]
对 Sigmod 函数求导
- \[ \sigma (z) = \frac 1 {
- 1 + e ^ {
- -z
- }
- } = s \]
- \[ dz = \frac {
- e^{
- -z
- }
- }{
- (1 + e^{
- -z
- }) ^ 2
- } \]
- \[ dz = \frac 1 {
- 1 + e^{
- -z
- }
- } (1 - \frac 1 {
- 1+e^{
- -z
- }
- }) \]
- \[ =\sigma(z)(1-\sigma(z)) \]
- \[ =s(1-s) \]
8, 反向传播的意义
修正参数, 使代价函数值减少, 预测值接近实际值.
举个例子:
(1) 玩一个猜数游戏, 目标数字为 150.
(2) 输入训练样本值: 你第一次猜出一个数字为 x = 10
(3) 设置初始权重: 设置一个权重值, 比如权重 w 设为 0.5
(4) 正向计算: 进行计算, 获得值 wx
(5) 求出代价函数: 出题人说差了多少(说的不是具体数字, 而是用 0-10 表示, 10 表示差的离谱, 1 表示非常接近, 0 表示正确)
(6) 反向传播或求导: 你通过出题人的结论, 去一点点修正权重(增加 w 或减少 w).
(7) 重复 (4) 操作, 直到无限接近或等于目标数字.
机器学习, 就是在训练中改进, 优化, 找到最有泛化能力的规则.
三, 神经网络实现
1, 实现激活函数 Sigmoid
- def sigmoid(z):
- s = 1.0 / (1.0 + np.exp(-z))
- return s
2, 参数初始化
- def initialize_with_zeros(dim):
- w = np.zeros([dim,1])
- b = 0
- return w, b
3, 前后向传播
- def propagate(w, b, X, Y):
- m = X.shape[1]
- A = sigmoid(np.dot(w.T,X) + b)
- cost = (- 1.0 / m ) * np.sum(Y*np.log(A) + (1-Y)*np.log(1-A))
- dw = (1.0 / m) * np.dot(X,(A - Y).T)
- db = (1.0 / m) * np.sum(A - Y)
- cost = np.squeeze(cost)
- grads = {"dw": dw,"db": db}
- return grads, cost
4, 优化器实现
- def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
- costs = []
- for i in range(num_iterations):
- # Cost and gradient calculation
- grads, cost = propagate(w,b,X,Y)
- # Retrieve derivatives from grads
- dw = grads["dw"]
- db = grads["db"]
- # update rule
- w = w - learning_rate * dw
- b = b - learning_rate * db
- # Record the costs
- if i % 100 == 0:
- costs.append(cost)
- # Print the cost every 100 training iterations
- if print_cost and i % 100 == 0:
- print ("Cost after iteration %i: %f" %(i, cost))
- params = {"w": w,
- "b": b}
- grads = {"dw": dw,
- "db": db}
- return params, grads, costs
5, 预测函数
- def predict(w, b, X):
- m = X.shape[1]
- Y_prediction = np.zeros((1,m))
- w = w.reshape(X.shape[0], 1)
- # Compute vector "A" predicting the probabilities of a cat being present in the picture
- A = sigmoid(np.dot(w.T,X)+b)
- for i in range(A.shape[1]):
- # Convert probabilities A[0,i] to actual predictions p[0,i]
- if A[0][i] <= 0.5:
- Y_prediction[0][i] = 0
- else:
- Y_prediction[0][i] = 1
- return Y_prediction
6, 代码模块整合
- def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
- # initialize parameters with zeros (≈ 1 line of code)
- w, b = initialize_with_zeros(train_set_x.shape[0])
- # Gradient descent (≈ 1 line of code)
- parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)
- # Retrieve parameters w and b from dictionary "parameters"
- w = parameters["w"]
- b = parameters["b"]
- Y_prediction_test = predict(w, b, X_test)
- Y_prediction_train = predict(w, b, X_train)
- # Print train/test Errors
- print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
- print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))
- d = {"costs": costs,
- "Y_prediction_test": Y_prediction_test,
- "Y_prediction_train" : Y_prediction_train,
- "w" : w,
- "b" : b,
- "learning_rate" : learning_rate,
- "num_iterations": num_iterations}
- return d
7, 运行程序
d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)
结果
- Cost after iteration 0: 0.693147
- Cost after iteration 100: 0.584508
- Cost after iteration 200: 0.466949
- Cost after iteration 300: 0.376007
- Cost after iteration 400: 0.331463
- Cost after iteration 500: 0.303273
- Cost after iteration 600: 0.279880
- Cost after iteration 700: 0.260042
- Cost after iteration 800: 0.242941
- Cost after iteration 900: 0.228004
- Cost after iteration 1000: 0.214820
- Cost after iteration 1100: 0.203078
- Cost after iteration 1200: 0.192544
- Cost after iteration 1300: 0.183033
- Cost after iteration 1400: 0.174399
- Cost after iteration 1500: 0.166521
- Cost after iteration 1600: 0.159305
- Cost after iteration 1700: 0.152667
- Cost after iteration 1800: 0.146542
- Cost after iteration 1900: 0.140872
- train accuracy: 99.04306220095694 %
- test accuracy: 70.0 %
8, 更多的分析
- learning_rates = [0.01, 0.001, 0.0001]
- models = {}
- for i in learning_rates:
- print ("learning rate is:" + str(i))
- models[str(i)] = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 1500, learning_rate = i, print_cost = False)
- print ('\n' + "-------------------------------------------------------" + '\n')
- for i in learning_rates:
- plt.plot(np.squeeze(models[str(i)]["costs"]), label= str(models[str(i)]["learning_rate"]))
- plt.ylabel('cost')
- plt.xlabel('iterations (hundreds)')
- legend = plt.legend(loc='upper center', shadow=True)
- frame = legend.get_frame()
- frame.set_facecolor('0.90')
- plt.show()
9, 测试图片
- ## START CODE HERE ## (PUT YOUR IMAGE NAME)
- my_image = "my_image.jpg" # change this to the name of your image file
- ## END CODE HERE ##
- # We preprocess the image to fit your algorithm.
- fname = "images/" + my_image
- image = np.array(ndimage.imread(fname, flatten=False))
- my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((1, num_px*num_px*3)).T
- my_predicted_image = predict(d["w"], d["b"], my_image)
- plt.imshow(image)
- print("y =" + str(np.squeeze(my_predicted_image)) + ", your algorithm predicts a \"" + classes[int(np.squeeze(my_predicted_image)),].decode("utf-8") + "\" picture.")
结果
y = 0.0, your algorithm predicts a "non-cat" picture.
参考文档
神经网络和深度学习 - 吴恩达 --Course1 Week2
来源: https://www.cnblogs.com/fonxian/p/10459762.html