碎碎念：达叔说 逻辑回归相当于小型的神经网络 就是没有隐含层嘛

先简单介绍一下需要做的事情  共分为7步，接下来将达叔作业里的所有程序 粘贴出来

1.,导入数据集，查看数据格式

操作：train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

           train_set_x_orig.shape  

一般会得到四个值（n,num_px,num_px,3） n 是训练集的个数，如 100张猫咪的训练照片， num_px 是图片的长宽？比如 64,64  代表一张64×64的图片 3 代表 RGB三个值

2.reshape

操作：train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T

X_flatten=X.shape(X.shape[0],-1).T  此操作可以将（a,b,c,d）的矩阵转化为（b*c*d,a）的二维矩阵

将（n,num_px,num_px,3)维度的数据reshape成（num_px×num_px×3，n）样子的数据，如（12288，n）12288行，n列。每一列是一张图片,每张图片有12288个特征。n代表 共有n张图片。这样所有训练集的维度就是一样的了

同时不要忘记对测试集也进行相同的操作

3.预处理

操作:train_set_x = train_set_x_flatten/255.

一般来说 预处理包含center 去中心化和standardize标准化两步，但是因为这里每个12288个值，其实每一值代表图片上64×64个点中的RGB中的一个值，所以每个值得范围是（0,255）。所以每个值除以255就ok了

4.初始化参数

操作：w = np.zeros((dim, 1))

           b = 0

一般来说有多少个输入特征，行数 这里是12288 一般就有多少个w

5.代价函数最小化（利用梯度下降法）

参考公式

操作： m = X.shape[1]

            A = sigmoid(np.dot(w.T, X) + b) # compute activation    因为w是一个与X的行数一样，一列的向量 ，所以求点积时，需要转置再求

            cost = -1 / m * np.sum(Y * np.log(A) + (1 - Y) * np.log(1 - A))        # compute cost

            dw = 1 / m * np.dot(X, (A - Y).T)

            db = 1 / m * np.sum(A - Y)

            w = w - learning_rate * dw

            b = b - learning_rate * db

6.更换学习率 α 画图观察不同学习率下的 cost变化情况 得到算法的正确率

操作：learning_rates = [0.01, 0.001, 0.0001]

           plt.plot(np.squeeze(models[str(i)]["costs"]), label= str(models[str(i)]["learning_rate"])

7.得出结论

一步一步定义函数，然后定义模型，在模型中调用函数，画图观察cost  确定合适的学习率α

程序部分

import numpy as np

import matplotlib.pyplot as plt

import h5py

import scipy                               

from PIL import Image

from scipy import ndimage

from lr_utils import load_dataset

train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

m_train = train_set_x_orig.shape[0]

m_test = test_set_x_orig.shape[0]

num_px = train_set_x_orig.shape[1]

train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T

test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T

train_set_x = train_set_x_flatten/255.

test_set_x = test_set_x_flatten/255.

def sigmoid(z):

    s = 1 / (1 + np.exp(-z))

    return s

def sigmoid_derivative(x):

    s = sigmoid(x)

    ds = s * (1 - s)

    return ds     

def initialize_with_zeros(dim):

    w = np.zeros((dim, 1))

    b = 0

    return w, b

def propagate(w, b, X, Y):     

    m = X.shape[1]

    A = sigmoid(np.dot(w.T, X) + b)            # compute activation

    cost = -1 / m * np.sum(Y * np.log(A) + (1 - Y) * np.log(1 - A))

    dw = 1 / m * np.dot(X, (A - Y).T)

    db = 1 / m * np.sum(A - Y)

    cost = np.squeeze(cost)

    grads = {"dw": dw,

            "db": db}

    return grads, cost   

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):

#通过梯度下降法实现w,b的更新

    costs = []

    for i in range(num_iterations):

        # Cost and gradient calculation (≈ 1-4 lines of code)

        grads, cost = propagate(w, b, X, Y) 

        # Retrieve derivatives from grads

        dw = grads["dw"]

        db = grads["db"]

        w = w - learning_rate * dw

        b = b - learning_rate * db

        if i % 100 == 0:

            costs.append(cost)

        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

def predict(w, b, X):   

    m = X.shape[1]

    Y_prediction = np.zeros((1,m))

    w = w.reshape(X.shape[0], 1)       

    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

#将以上所有函数结合在一起形成一个model

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):

#print_cost -- Set to true to print the cost every 100 iterations

    D={}

    w, b = initialize_with_zeros(X_train.shape[0])   

    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)   

    w = parameters["w"]

    b = parameters["b"]   

    Y_prediction_test = predict(w, b, X_test)

    Y_prediction_train = predict(w, b, X_train)

    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))))

    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

d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)

#把学习率为0.05 情况下的cost 图画出来

costs = np.squeeze(d['costs'])

plt.plot(costs)

plt.ylabel('cost')

plt.xlabel('iterations (per hundreds)')

plt.title("Learning rate =" + str(d["learning_rate"]))

plt.show()

#更换不同学习率

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')

legend = plt.legend(loc='upper center', shadow=True)

frame = legend.get_frame()

frame.set_facecolor('0.90')

plt.show()