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) = - [ylog 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 o 6
]
[v = a + u o 11
]
[J = 3v o 33
]
7、反向传播
求出(da) = 3，表示(J)对(a)的偏导数
[J = 3v o frac {dJ} {da} = frac {dJ} {dv} frac {dv} {da} = 3
]
求出(db) = 6
[J = 3v o frac {dJ} {dc} = frac {dJ} {dv} frac {dv} {du} frac {du} {dc} = 3c = 6
]
求出(dc) = 6
[J = 3v o 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 ('
' + "-------------------------------------------------------" + '
')
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.