一、基本概念
1、邏輯回歸與線性回歸的區別?
線性回歸預測得到的是一個數值，而邏輯回歸預測到的數值只有0、1兩個值。邏輯回歸是在線性回歸的基礎上，加上一個sigmoid函數，讓其值位於0-1之間，最后獲得的值大於0.5判斷為1，小於等於0.5判斷為0
二、邏輯回歸的推導
\(\hat y\)表示預測值，\(y\)表示訓練標簽值
1、一般公式
\[\hat y = wx + b \]
2、向量化
\[\hat y = w^Tx+b \]
3、激活函數
引入sigmoid函數（用\(\sigma\)表示)，使\(\hat y\)值位於0-1
\[\hat y = \sigma (w^Tx+b) \]
4、損失函數
損失函數用\(L\)表示
\[L(\hat y ,y) = \frac 1 2 (\hat y - y) ^ 2 \]
因梯度下降效果不好，換用交叉熵損失函數
\[L(\hat y,y) = - [y\log \hat y + (1-y)\log(1-\hat y)] \]
5、代價函數
代價函數用\(J\)表示
\[J(w,b) = \frac 1 m \sum_{i=1}^m L(\hat y^{(i)},y^{(i)}) \]
展開
\[J(w,b) = - \frac 1 m \sum_{i=1}^m [y^{(i)}\log \hat y^{(i)} + (1-y^{(i)})\log(1-\hat y ^ {(i)})] \]
6、正向傳播
\[a = 5,b=3,c=2 \]
\[u = bc \to 6 \]
\[v = a + u \to 11 \]
\[J = 3v \to 33 \]
7、反向傳播
求出\(da\) = 3，表示\(J\)對\(a\)的偏導數
\[J = 3v \to \frac {dJ} {da} = \frac {dJ} {dv} \frac {dv} {da} = 3 \]
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
\[J = 3v \to \frac {dJ} {db} = \frac {dJ} {dv} \frac {dv} {du} \frac {du} {db} = 3b = 9 \]
對Sigmod函數求導
\[\sigma (z) = \frac 1 {1 + e ^ {-z}} = s \]
\[dz = \frac {e^{-z}}{(1 + e^{-z}) ^ 2} \]
\[dz = \frac 1 {1 + e^{-z} } (1 - \frac 1 {1+e^{-z}}) \]
\[=\sigma(z)(1-\sigma(z)) \]
\[=s(1-s) \]
8、反向傳播的意義
修正參數，使代價函數值減少，預測值接近實際值。
舉個例子：
(1) 玩一個猜數游戲，目標數字為150。
(2) 輸入訓練樣本值: 你第一次猜出一個數字為x = 10
(3) 設置初始權重: 設置一個權重值，比如權重w設為0.5
(4) 正向計算： 進行計算，獲得值wx
(5) 求出代價函數： 出題人說差了多少（說的不是具體數字，而是用0-10表示，10表示差的離譜，1表示非常接近，0表示正確）
(6) 反向傳播或求導： 你通過出題人的結論，去一點點修正權重（增加w或減少w）。
(7) 重復(4)操作，直到無限接近或等於目標數字。
機器學習，就是在訓練中改進、優化，找到最有泛化能力的規則。
三、神經網絡實現
1、實現激活函數Sigmoid
def sigmoid(z):
s = 1.0 / (1.0 + np.exp(-z))
return s
2、參數初始化
def initialize_with_zeros(dim):
w = np.zeros([dim,1])
b = 0
return w, b
3、前后向傳播
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、優化器實現
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、預測函數
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、代碼模塊整合
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、運行程序
d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)
結果
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、測試圖片
## 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.")
結果
y = 0.0, your algorithm predicts a "non-cat" picture.
