知識點梳理

python工具使用：

sklearn: 數據挖掘，數據分析工具，內置logistic回歸

matplotlib： 做圖工具，可繪制等高線等

繪制散點圖： plt.scatter(X[0, :], X[1, :], c=np.squeeze(Y), s=40, cmap=plt.cm.Spectral); s：繪制點大小 cmap：顏色集

繪制等高線： 先做網格，計算結果，繪圖

x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1h = 0.01xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))Z = model(np.c_[xx.ravel(), yy.ravel()])Z = Z.reshape(xx.shape)#xx是x軸值， yy是y軸值， Z是預測結果值， cmap表示采用什么顏色plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)

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關鍵變量：

• m: 訓練樣本數量
  
• n_x：一個訓練樣本的輸入數量，輸入層大小
  
• n_h：隱藏層大小
• 方括號上標[l]: 第l層
• 圓括號上標(i): 第i個樣本

$$ X = \left[ \begin{matrix} \vdots & \vdots & \vdots & \vdots \\ x^{(1)} & x^{(2)} & \vdots & x^{(m)} \\ \vdots & \vdots & \vdots & \vdots \\ \end{matrix} \right]_{(n\_x, m)} $$

$$ W^{[1]} = \left[ \begin{matrix} \cdots & w^{[1] T}_1 & \cdots \\ \cdots & w^{[1] T}_2 & \cdots \\ \cdots & \cdots & \cdots \\ \cdots & w^{[1] T}_{n\_h} & \cdots \\ \end{matrix} \right]_{(n\_h, n\_x)} $$

$$ b^{[1]} = \left[ \begin{matrix} b^{[1]}_1 \\ b^{[1]}_2 \\ \vdots \\ b^{[1]}_{n\_h} \\ \end{matrix} \right]_{(n\_h, 1)} $$

$$ A^{[1]}= \left[ \begin{matrix} \vdots & \vdots & \vdots & \vdots \\ a^{[1](1)} & a^{[1](2)} & \vdots & a^{[1](m)} \\ \vdots & \vdots & \vdots & \vdots \\ \end{matrix} \right]_{(n\_h, m)} $$

$$ Z^{[1]}= \left[ \begin{matrix} \vdots & \vdots & \vdots & \vdots \\ z^{[1](1)} & z^{[1](2)} & \vdots & z^{[1](m)} \\ \vdots & \vdots & \vdots & \vdots \\ \end{matrix} \right]_{(n\_h, m)} $$

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單隱層神經網絡關鍵公式：

• 前向傳播：

$$Z^{[1]}=W^{[1]}X+b^{[1]}$$ $$A^{[1]}=g^{[1]}(Z^{[1]})$$ $$Z^{[2]}=W^{[2]}A^{[1]}+b^{[2]}$$ $$A^{[2]}=g^{[2]}(Z^{[2]})$$

Z1 = np.dot(W1, X) + b1 A1 = np.tanh(Z1) Z2 = np.dot(W2, A1) + b2 A2 = sigmoid(Z2)

• 反向傳播
  

dZ2 = A2 - Y dW2 = 1 / m * np.dot(dZ2, A1.T) db2 = 1 / m * np.sum(dZ2, axis = 1, keepdims = True) dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2)) dW1 = 1 / m * np.dot(dZ1, X.T) db1 = 1 / m * np.sum(dZ1, axis = 1, keepdims = True)

• cost計算

\[J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large{(} \small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large{)} \small\]

logprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1 - A2), 1 - Y) cost = - 1 / m * np.sum(logprobs)

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單隱層神經網絡代碼：

# Package imports import numpy as np import matplotlib.pyplot as plt import sklearn import sklearn.datasets import sklearn.linear_model from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets%matplotlib inlinenp.random.seed(1) # set a seed so that the results are consistentdef initialize_parameters(n_x, n_h, n_y):np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.W1 = np.random.randn(n_h, n_x) * 0.01b1 = np.zeros((n_h, 1))W2 = np.random.randn(n_y, n_h)b2 = np.zeros((n_y, 1))assert (W1.shape == (n_h, n_x))assert (b1.shape == (n_h, 1))assert (W2.shape == (n_y, n_h))assert (b2.shape == (n_y, 1))parameters = {"W1": W1,"b1": b1,"W2": W2,"b2": b2}return parametersdef forward_propagation(X, parameters):# Retrieve each parameter from the dictionary "parameters"W1 = parameters["W1"]b1 = parameters["b1"]W2 = parameters["W2"]b2 = parameters["b2"]# Implement Forward Propagation to calculate A2 (probabilities)Z1 = np.dot(W1, X) + b1A1 = np.tanh(Z1)Z2 = np.dot(W2, A1) + b2A2 = sigmoid(Z2)assert(A2.shape == (1, X.shape[1]))cache = {"Z1": Z1,"A1": A1,"Z2": Z2,"A2": A2}return A2, cachedef compute_cost(A2, Y, parameters):m = Y.shape[1] # number of example# Compute the cross-entropy costlogprobs = np.multiply(np.log(A2), Y) + np.multiply(np.log(1 - A2), 1 - Y)cost = - 1 / m * np.sum(logprobs)cost = np.squeeze(cost) # makes sure cost is the dimension we expect. # E.g., turns [[17]] into 17 assert(isinstance(cost, float))return costdef backward_propagation(parameters, cache, X, Y):m = X.shape[1]# First, retrieve W1 and W2 from the dictionary "parameters".W1 = parameters["W1"]W2 = parameters["W2"]# Retrieve also A1 and A2 from dictionary "cache".A1 = cache["A1"]A2 = cache["A2"]# Backward propagation: calculate dW1, db1, dW2, db2. dZ2 = A2 - Y dW2 = 1 / m * np.dot(dZ2, A1.T)db2 = 1 / m * np.sum(dZ2, axis = 1, keepdims = True)dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2)) dW1 = 1 / m * np.dot(dZ1, X.T)db1 = 1 / m * np.sum(dZ1, axis = 1, keepdims = True)grads = {"dW1": dW1,"db1": db1,"dW2": dW2,"db2": db2}return gradsdef update_parameters(parameters, grads, learning_rate = 0.8):# Retrieve each parameter from the dictionary "parameters"W1 = parameters["W1"]b1 = parameters["b1"]W2 = parameters["W2"]b2 = parameters["b2"]# Retrieve each gradient from the dictionary "grads"dW1 = grads["dW1"]db1 = grads["db1"]dW2 = grads["dW2"]db2 = grads["db2"]# Update rule for each parameterW1 = W1 - learning_rate * dW1b1 = b1 - learning_rate * db1W2 = W2 - learning_rate * dW2b2 = b2 - learning_rate * db2parameters = {"W1": W1,"b1": b1,"W2": W2,"b2": b2}return parametersdef nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):np.random.seed(3)n_x = X.shape[0]n_y = Y.shape[0]# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".parameters = initialize_parameters(n_x, n_h, n_y)W1 = parameters["W1"]b1 = parameters["b1"]W2 = parameters["W2"]b2 = parameters["b2"]# Loop (gradient descent)for i in range(0, num_iterations):# Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".A2, cache = forward_propagation(X, parameters)# Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".cost = compute_cost(A2, Y, parameters)# Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".grads = backward_propagation(parameters, cache, X, Y)# Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".parameters = update_parameters(parameters, grads)# Print the cost every 1000 iterationsif print_cost and i % 1000 == 0:print ("Cost after iteration %i: %f" %(i, cost))return parametersdef predict(parameters, X):# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.A2, cache = forward_propagation(X, parameters)predictions = A2 > 0.5return predictionsX, Y = load_planar_dataset() # Build a model with a n_h-dimensional hidden layer parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)# Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X, np.squeeze(Y)) plt.title("Decision Boundary for hidden layer size " + str(4)) # planar_utils.py import matplotlib.pyplot as plt import numpy as np import sklearn import sklearn.datasets import sklearn.linear_modeldef plot_decision_boundary(model, X, y):# Set min and max values and give it some paddingx_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1h = 0.01# Generate a grid of points with distance h between them# 創造網格，以0.01為間隔劃分整個區間xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))# Predict the function value for the whole grid# 計算每個網格點上的預測結果Z = model(np.c_[xx.ravel(), yy.ravel()])# 將預測結果變形為與網格形式一致Z = Z.reshape(xx.shape)# Plot the contour and training examples# xx是x軸值， yy是y軸值， Z是預測結果值， cmap表示采用什么顏色plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral) #等位線plt.ylabel('x2')plt.xlabel('x1')plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)def sigmoid(x):"""Compute the sigmoid of xArguments:x -- A scalar or numpy array of any size.Return:s -- sigmoid(x)"""s = 1/(1+np.exp(-x))return sdef load_planar_dataset():np.random.seed(1)m = 400 # number of examplesN = int(m/2) # number of points per classD = 2 # dimensionalityX = np.zeros((m,D)) # data matrix where each row is a single exampleY = np.zeros((m,1), dtype='uint8') # labels vector (0 for red, 1 for blue)a = 4 # maximum ray of the flowerfor j in range(2):ix = range(N*j,N*(j+1))t = np.linspace(j*3.12,(j+1)*3.12,N) + np.random.randn(N)*0.2 # thetar = a*np.sin(4*t) + np.random.randn(N)*0.2 # radiusX[ix] = np.c_[r*np.sin(t), r*np.cos(t)]Y[ix] = jX = X.TY = Y.Treturn X, Ydef load_extra_datasets(): N = 200noisy_circles = sklearn.datasets.make_circles(n_samples=N, factor=.5, noise=.3)noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)gaussian_quantiles = sklearn.datasets.make_gaussian_quantiles(mean=None, cov=0.5, n_samples=N, n_features=2, n_classes=2, shuffle=True, random_state=None)no_structure = np.random.rand(N, 2), np.random.rand(N, 2)return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure

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