實現邏輯迴歸-神經網路

神經網路 · 發表 2019-02-17 21:52:15

摘要： 1、邏輯迴歸與線性迴歸的區別? 線性迴歸預測得到的是一個數值，而邏輯迴歸預測到的數值只有0、1兩個值。邏輯迴歸是線上性迴歸的基礎上，加上一個 sigmoid函式，讓其值位於 0-1 之間，最後獲得的值大於 0.5 判斷為 1 ，小於等於 0.5 判斷為 0 二、邏輯迴歸...

1、邏輯迴歸與線性迴歸的區別?

線性迴歸 預測得到的是一個數值，而 邏輯迴歸 預測到的數值只有0、1兩個值。 邏輯迴歸 是線上性迴歸的基礎上，加上一個 sigmoid函式 ，讓其值位於 0-1 之間，最後獲得的值大於 0.5 判斷為 1 ，小於等於 0.5 判斷為 0

二、邏輯迴歸的推導

1、一般公式

y 為標籤值，另一個 y hat 為預測值。

2、向量化

3、啟用函式

引入sigmoid函式（用

$\sigma$ 表示)，使 $\hat y$

值位於0-1

4、損失函式

損失函式用

$L$

表示

因 梯度下降 效果不好，換用 交叉熵損失函式

5、代價函式

代價函式用

$J$

表示

展開

6、正向傳播

7、反向傳播

求出

$da$ = 3，表示 $J$ 對 $a$ 的 偏導數

求出

$db$

= 6

求出

$dc$

= 6

對 Sigmod函式 求導

8、反向傳播的意義

修正引數，使 代價函式值 減少， 預測值 接近 實際值 。

舉個例子：

(1) 玩一個猜數遊戲，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) 設定初始權重: 設定一個權重值，比如權重w設為0.5

(4) 正向計算： 進行計算，獲得值wx

(5) 求出代價函式： 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) 重複(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、神經網路實現

1、實現啟用函式Sigmoid

def sigmoid(z):
s = 1.0 / (1.0 + np.exp(-z))
return s

2、引數初始化

def initialize_with_zeros(dim):
w = np.zeros([dim,1])
b = 0
return w, b

3、前後向傳播

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、預測函式

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、程式碼模組整合

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、執行程式

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.