【李宏毅机器学习CP9】(task3下)PM2.5预测+numpy实现神经网络logistic回归

学习心得

（1）小批量梯度下降算法中，初始参数的选择很重要，不同的初始参数，其对应损失函数收敛速度也不一样

（2）learning rate 采用递减的方式选取的，根据经验的选择也很重要，在实验中多积累、多总结。

（3）注意第二部分的《用numpy实现一个简单神经网络logistic回归模型》中， numpy、h5py、matplotlib、PIL、scipy等库的使用逐渐要练习起来。逻辑回归是最基础也是最重要的模型：

通过逻辑回归能演化出很多模型：

• 逻辑回归=线性回归+sigmoid激活函数，从而将回归问题转换为分类问题
• 逻辑回归+矩阵分解，构成了推荐算法中常用的FM模型
• 逻辑回归+softmax，从而将二分类问题转化为多分类问题
• 逻辑回归还可以看做单层神经网络，相当于最简单的深度学习模型

文章目录

第一部分：PM2.5预测

一、作业描述

让机器预测到丰原站在下一个小时会观测到的PM2.5。举例来说，现在是2017-09-29 08：00：00 ，那么要预测2017-09-29 09：00：00丰原站的PM2.5值会是多少。

二、任务要求

• 任务要求：预测PM2.5的值，我们将用梯度下降法(Gradient Descent)预测 PM2.5的值 (Regression回归问题)
• 环境要求：

• 要求python3.5+
• 只能用numpy、scipy、pandas
• 请用梯度下降手写线性回归
• 最好使用Public Simple Baseline
• 对于想加载模型而并不想运行整个训练过程的人：

• 请上传训练代码并命名成​​train.py​​
• 只要用梯度下降的代码就行了

• 最佳要求：

• 要求python3.5+
• 任何库都可以用
• 在Kaggle上获得你选择的更高的分

• 数据介绍：
  本次作业使用豐原站的觀測記錄，分成train set跟test set，train set 是豐原站每個月的前20天所有資料，test set則是從豐原站剩下的資料中取樣出來。
  train.csv:每個月前20天每個小時的氣象資料(每小時有18種測資)。共12個月。
  test.csv:從剩下的資料當中取樣出連續的10小時為一筆，前九小時的所有觀測數據當作feature，第十小時的PM2.5當作answer。一共取出240筆不重複的 test data，請根據feauure預測這240筆的PM2.5。
• 请完成之后参考以下资料：

• Sample_code:https://github.com/datawhalechina/leeml-notes/tree/master/docs/Homework/HW_1

三、大致任务实现

步骤：

（1）定义你的function set（model），可以加上正则化。

（2）基于training data定义loss function

下图训练资料的每一列为该小时对应的数据。

ps：不要将白天的数据当做training data，晚上的数据当做validation data，否则validation data的结果无法反应testing data的结果。（3）找到最好的function

方案1

'''
利用 Linear Regression 线性回归预测 PM2.5 
该方法参考黑桃大哥的优秀作业-|vv|-
'''

# 导入必要的包 numpy、pandas以及scikit-learn归一化预处理
import numpy as np
import pandas as pd
from sklearn.preprocessing import StandardScaler

# 指定相对路径
path = "./Dataset/"

# 利用 pands 进行读取文件操作
train = pd.read_csv(path + 'train.csv', engine='python', encoding='utf-8')
test = pd.read_csv(path + 'test.csv', engine='python', encoding='gbk')
train = train[train['observation'] == 'PM2.5']
# print(train)
test = test[test['AMB_TEMP'] == 'PM2.5']
# 删除无关特征
train = train.drop(['Date', 'stations', 'observation'], axis=1)
test_x = test.iloc[:, 2:]

train_x = []
train_y = []

for i in range(15):
    x = train.iloc[:, i:i + 9]
    # notice if we don't set columns name, it will have different columns name in each iteration
    x.columns = np.array(range(9))
    y = train.iloc[:, i + 9]
    y.columns = np.array(range(1))
    train_x.append(x)
    train_y.append(y)

# review "Python for Data Analysis" concat操作
# train_x and train_y are the type of Dataframe
# 取出 PM2.5 的数据，训练集中一共有 240 天，每天取出 15 组 含有 9 个特征 和 1 个标签的数据，共有 240*15*9个数据
train_x = pd.concat(train_x) # (3600, 9) Dataframe类型
train_y = pd.concat(train_y)

# 将str数据类型转换为 numpy的 ndarray 类型
train_y = np.array(train_y, float)
test_x = np.array(test_x, float)
# print(train_x.shape, train_y.shape)

# 进行标准缩放，即数据归一化
ss = StandardScaler()

# 进行数据拟合
ss.fit(train_x)
# 进行数据转换
train_x = ss.transform(train_x)

ss.fit(test_x)
test_x = ss.transform(test_x)

# 定义评估函数
# 计算均方误差（Mean Squared Error，MSE）
# r^2 用于度量因变量的变异中 可以由自变量解释部分所占的比例 取值一般为 0~1
def r2_score(y_true, y_predict):
    # 计算y_true和y_predict之间的MSE
    MSE = np.sum((y_true - y_predict) ** 2) / len(y_true)
    # 计算y_true和y_predict之间的R Square
    return 1 - MSE / np.var(y_true)

# 线性回归
class LinearRegression:

    def __init__(self):
        # 初始化 Linear Regression 模型
        self.coef_ = None
        self.intercept_ = None
        self._theta = None

    def fit_normal(self, X_train, y_train):
        # 根据训练数据集X_train, y_train训练Linear Regression模型
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        # 对训练数据集添加 bias
        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        self._theta = np.linalg.inv(X_b.T.dot(X_b)).dot(X_b.T).dot(y_train)

        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]

        return self

    def fit_gd(self, X_train, y_train, eta=0.01, n_iters=1e4):
        '''
        :param X_train: 训练集
        :param y_train: label
        :param eta: 学习率
        :param n_iters: 迭代次数
        :return: theta 模型参数
        '''
        # 根据训练数据集X_train, y_train, 使用梯度下降法训练Linear Regression模型
        assert X_train.shape[0] == y_train.shape[0], \
            "the size of X_train must be equal to the size of y_train"

        # 定义损失函数
        def J(theta, X_b, y):
            try:
                return np.sum((y - X_b.dot(theta)) ** 2) / len(y)
            except:
                return float('inf')
        # 对损失函数求导
        def dJ(theta, X_b, y):
            return X_b.T.dot(X_b.dot(theta) - y) * 2. / len(y)

        def gradient_descent(X_b, y, initial_theta, eta, n_iters=1e4, epsilon=1e-8):
            '''
            :param X_b: 输入特征向量
            :param y: lebel
            :param initial_theta: 初始参数
            :param eta: 步长
            :param n_iters: 迭代次数
            :param epsilon: 容忍度
            :return:theta：模型参数
            '''
            theta = initial_theta
            cur_iter = 0

            while cur_iter < n_iters:
                gradient = dJ(theta, X_b, y)
                last_theta = theta
                theta = theta - eta * gradient
                if (abs(J(theta, X_b, y) - J(last_theta, X_b, y)) < epsilon):
                    break
                cur_iter += 1
            return theta

        X_b = np.hstack([np.ones((len(X_train), 1)), X_train])
        initial_theta = np.zeros(X_b.shape[1]) # 初始化theta
        self._theta = gradient_descent(X_b, y_train, initial_theta, eta, n_iters)
        self.intercept_ = self._theta[0]
        self.coef_ = self._theta[1:]
        return self

    def predict(self, X_predict):
        # 给定待预测数据集X_predict，返回表示X_predict的结果向量
        assert self.intercept_ is not None and self.coef_ is not None, \
            "must fit before predict!"
        assert X_predict.shape[1] == len(self.coef_), \
            "the feature number of X_predict must be equal to X_train"
        X_b = np.hstack([np.ones((len(X_predict), 1)), X_predict])
        return X_b.dot(self._theta)

    def score(self, X_test, y_test):
        # 根据测试数据集 X_test 和 y_test 确定当前模型的准确度
        y_predict = self.predict(X_test)
        return r2_score(y_test, y_predict)

    def __repr__(self):
        return

# 模型训练
LR = LinearRegression().fit_gd(train_x, train_y)
# 评分
LR.score(train_x, train_y)
# 预测
result = LR.predict(test_x)

# 结果保存
sampleSubmission = pd.read_csv(path + 'sampleSubmission.csv', engine='python', encoding='gbk')
sampleSubmission['value'] = result
sampleSubmission.to_csv(path + 'result.csv')

方案2

#-*- coding:utf-8 -*-
# @File    : Predict_PM2dot5.py
# @Author  : 追风者
# @Software: PyCharm
# @Python Version: python 3.6

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

# 数据读取与预处理
train_data = pd.read_csv("./Dataset/train.csv")
train_data.drop(['Date', 'stations'], axis=1, inplace=True)
column = train_data['observation'].unique()
# print(column)
new_train_data = pd.DataFrame(np.zeros([24*240, 18]), columns=column)

for i in column:
    train_data1 = train_data[train_data['observation'] == i]
    # Be careful with the inplace, as it destroys any data that is dropped!
    train_data1.drop(['observation'], axis=1, inplace=True)
    train_data1 = np.array(train_data1)
    train_data1[train_data1 == 'NR'] = '0'
    train_data1 = train_data1.astype('float')
    train_data1 = train_data1.reshape(1, 5760)
    train_data1 = train_data1.T
    new_train_data[i] = train_data1

label = np.array(new_train_data['PM2.5'][9:], dtype='float32')

# 探索性数据分析 EDA
# 最简单粗暴的方式就是根据 HeatMap 热力图分析各个指标之间的关联性
f, ax = plt.subplots(figsize=(9, 6))
sns.heatmap(new_train_data.corr(), fmt="d", linewidths=0.5, ax=ax)
plt.show()

# 模型选择
# a.数据归一化
# 使用前九个小时的 PM2.5 来预测第十个小时的 PM2.5，使用线性回归模型
PM = new_train_data['PM2.5']
PM_mean = int(PM.mean())
PM_theta = int(PM.var()**0.5)
PM = (PM - PM_mean) / PM_theta
w = np.random.rand(1, 10)
theta = 0.1
m = len(label)
for i in range(100):
    loss = 0
    i += 1
    gradient = 0
    for j in range(m):
        x = np.array(PM[j : j + 9])
        x = np.insert(x, 0, 1)
        error = label[j] - np.matmul(w, x)
        loss += error**2
        gradient += error * x

    loss = loss/(2*m)
    print(loss)
    w = w+theta*gradient/m

热力图分析 由热力图可直接看出，与PM2.5相关性较高的指标有PM10、NO2、SO2、NOX、O3、THC.

# 打印损失函数
[313.93192714]
[234.85253287]
[194.58981571]
[163.68386696]
[138.78909523]
[118.62599157]
[102.26572091]
[88.9697716]
[78.14562861]
[69.3172079]
[62.10166483]
[56.19090086]
.........
[22.15055994]
[22.11594415]
[22.08201703]
[22.04875752]
[22.0161454]
[21.98416128]
[21.95278654]
[21.92200329]
[21.89179437]
[21.86214327]
[21.83303413]
[21.80445167]
[21.77638122]

方案3

模型设计

• 数据预处理
  处理训练样本为 (18*9) 的矩阵

#-*- coding:utf-8 -*-
# @File    : PM2.5Prediction.py
# @Author  : 追风者
# @Software: PyCharm
# @Python Version: python 3.6

'''
利用线性回归Linear Regression模型预测 PM2.5

特征工程中的特征选择与数据可视化的直观分析
通过选择的特征进一步建立回归模型
'''
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import seaborn as sns

'''数据读取与预处理'''
# DataFrame类型
train_data = pd.read_csv("./Dataset/train.csv")
train_data.drop(['Date', 'stations', 'observation'], axis=1, inplace=True)

ItemNum=18
#训练样本features集合
X_Train=[]
#训练样本目标PM2.5集合
Y_Train=[]

for i in range(int(len(train_data)/ItemNum)):
    observation_data = train_data[i*ItemNum:(i+1)*ItemNum] #一天的观测数据
    for j in range(15):
        x = observation_data.iloc[:, j:j + 9]
        y = int(observation_data.iloc[9,j+9])
        # 将样本分别存入X_Train、Y_Train中
        X_Train.append(x)
        Y_Train.append(y)
# print(X_Train)
# print(Y_Train)

• 数据可视化
  绘制散点图，预测各特征与 PM2.5 的关系

下面这份代码先mark下（有点问题）

'''绘制散点图'''
x_AMB=[]
x_CH4=[]
x_CO=[]
x_NMHC=[]

# x_NO=[]
# x_NO2=[]
# x_NOX=[]
# x_O3=[]

# x_PM10=[]
# x_PM2Dot5=[]
# x_RAINFALL=[]
# x_RH=[]

# x_SO2=[]
# x_THC=[]
# x_WD_HR=[]
# x_WIND_DIREC=[]

# x_WIND_SPEED=[]
# x_WS_HR=[]
#
y=[]
#
# for i in range(len(Y_Train)):
#     y.append(Y_Train[i])
#     x=X_Train[i]
#     # print(type(x.iloc[0,0]))
#     # 求各测项的平均值
#     x_WIND_SPEED_sum = 0
#     x_WS_HR_sum = 0
#     for j in range(9):
#         x_WIND_SPEED_sum = x_WIND_SPEED_sum + float(x.iloc[0, j])
#         x_WS_HR_sum = x_WS_HR_sum + float(x.iloc[1, j])
#     x_WIND_SPEED.append(x_WIND_SPEED_sum / 9)
#     x_WS_HR.append(x_WS_HR_sum / 9)
# plt.figure(figsize=(10, 6))
# plt.subplot(1, 2, 1)
# plt.title('WIND_SPEED')
# plt.scatter(x_WIND_SPEED, y)
# plt.subplot(1, 2, 2)
# plt.title('WS_HR')
# plt.scatter(x_WS_HR, y)
# plt.show()
#     x_SO2_sum = 0
#     x_THC_sum = 0
#     x_WD_HR_sum = 0
#     x_WIND_DIREC_sum = 0
#     for j in range(9):
#         x_SO2_sum = x_SO2_sum + float(x.iloc[0, j])
#         x_THC_sum = x_THC_sum + float(x.iloc[1, j])
#         x_WD_HR_sum = x_WD_HR_sum + float(x.iloc[2, j])
#         x_WIND_DIREC_sum = x_WIND_DIREC_sum + float(x.iloc[3, j])
#     x_SO2.append(x_SO2_sum / 9)
#     x_THC.append(x_THC_sum / 9)
#     x_WD_HR.append(x_WD_HR_sum / 9)
#     x_WIND_DIREC.append(x_WIND_DIREC_sum / 9)
# plt.figure(figsize=(10, 6))
# plt.subplot(2, 2, 1)
# plt.title('SO2')
# plt.scatter(x_SO2, y)
# plt.subplot(2, 2, 2)
# plt.title('THC')
# plt.scatter(x_THC, y)
# plt.subplot(2, 2, 3)
# plt.title('WD_HR')
# plt.scatter(x_WD_HR, y)
# plt.subplot(2, 2, 4)
# plt.title('WIND_DIREC')
# plt.scatter(x_WIND_DIREC, y)
# plt.show()
#     x_PM10_sum = 0
#     x_PM2Dot5_sum = 0
#     x_RAINFALL_sum = 0
#     x_RH_sum = 0
#     for j in range(9):
#         x_PM10_sum = x_PM10_sum + float(x.iloc[0, j])
#         x_PM2Dot5_sum = x_PM2Dot5_sum + float(x.iloc[1, j])
#         x_RAINFALL_sum = x_RAINFALL_sum + float(x.iloc[2, j])
#         x_RH_sum = x_RH_sum + float(x.iloc[3, j])
#     x_PM10.append(x_PM10_sum / 9)
#     x_PM2Dot5.append(x_PM2Dot5_sum / 9)
#     x_RAINFALL.append(x_RAINFALL_sum / 9)
#     x_RH.append(x_RH_sum / 9)
# plt.figure(figsize=(10, 6))
# plt.subplot(2, 2, 1)
# plt.title('PM10')
# plt.scatter(x_PM10, y)
# plt.subplot(2, 2, 2)
# plt.title('PM2.5')
# plt.scatter(x_PM2Dot5, y)
# plt.subplot(2, 2, 3)
# plt.title('RAINFALL')
# plt.scatter(x_RAINFALL, y)
# plt.subplot(2, 2, 4)
# plt.title('RH')
# plt.scatter(x_RH, y)
# plt.show()
    x_AMB_sum=0
    x_CH4_sum=0
    x_CO_sum=0
    x_NMHC_sum=0
    for j in range(9):
        x_AMB_sum = x_AMB_sum + float(x.iloc[0,j])
        x_CH4_sum = x_CH4_sum + float(x.iloc[1, j])
        x_CO_sum = x_CO_sum + float(x.iloc[2, j])
        x_NMHC_sum = x_NMHC_sum + float(x.iloc[3, j])
    x_AMB.append(x_AMB_sum / 9)
    x_CH4.append(x_CH4_sum / 9)
    x_CO.append(x_CO_sum / 9)
    x_NMHC.append(x_NMHC_sum / 9)
plt.figure(figsize=(10, 6))
plt.subplot(2, 2, 1)
plt.title('AMB')
plt.scatter(x_AMB, y)
plt.subplot(2, 2, 2)
plt.title('CH4')
plt.scatter(x_CH4, y)
plt.subplot(2, 2, 3)
plt.title('CO')
plt.scatter(x_CO, y)
plt.subplot(2, 2, 4)
plt.title('NMHC')
plt.scatter(x_NMHC, y)
plt.show()

• 特征选择
  选择最具代表性的特征：PM10、PM2.5、SO2
• 模型建立
  建立线性回归模型
  $y=b+\sum_{i=1}^{27} \mathcal{W}_{i} \times \mathcal{X}_{i} \\ 等价于 \\ \mathrm{y}=b+w_{1} \times x_{1}+w_{2} \times x_{2}+\cdots+w_{27} \times x_{27}$
  其中x1到x9是前九个时间点的PM10值，x10到x18是前9个时间点的PM2.5值，x19到x27是前9个时间点的SO2值，w为对应参数，b为偏移量
• 定义损失函数 (Gradient Descent)
  $L \mathrm{oss}=\frac{1}{2} \sum_{i=1}^{m}\left(y_{i}-y_{\text {ireal}}\right)^{2}$
  其中m为每次更新参数时使用的样本数,yi为预测值，yireal为真实值
  采用小批量梯度下降算法，并且设定批量样本大小为50，即每次随机在训练样本中选取50个用来更新参数
  设定学习率learning rate分别为0.000000001、0.0000001、0.000001时，比较不同的learning rate对损失函数收敛速度的影响
• 模型评估
  $\mathrm{Model_-Evaluation}=\frac{1}{n} \sum_{i=1}^{n}\left(y_{i}-y_{\text {ireal}}\right)^{2}$

第二部分：用numpy实现一个简单神经网络logistic回归模型

此部分为吴恩达第一课第二周的task作业（已翻译）。
$logistic$ $regression$经常被直译为逻辑回归，然鹅逻辑的英文是logic而不是logit或logistic，而logistic是对率的意思，因此也有人译为对率回归。

一、包（库）的介绍

• numpy是py科学计算中最重要的库
• h5py是py与H5数据文件交互的库
• matplotlib是py画图神器
• PIL and scipy是最后test your model with your own picture.

H5文件是层次数据格式第5代的版本（Hierarchical Data Format，HDF5），它是用于存储科学数据的一种文件格式和库文件。它是由美国超级计算与应用中心研发的文件格式，用以存储和组织大规模数据。目前由非营利组织HDF小组提供支持。
很多商业和非商业组织都支持这种文件格式，如Java，MATLAB，Python，R等.
H5将文件结构简化成两个主要的对象类型：
1、数据集，就是同一类型数据的多维数组。
2、组，是一种容器结构，可以包含数据集和其他组。
这导致了H5文件是一种真正的层次结构、文件系统式的数据类型。

import numpy as np
import copy
import matplotlib.pyplot as plt
import h5py
import scipy
from PIL import Image
from scipy import ndimage
from lr_utils import load_dataset
from public_tests import *

%matplotlib inline
%load_ext autoreload
%autoreload 2

注意上面的​​lr_utils​​不是python附带的什么库，而是吴恩达课程coursera上附带的一个文件，需要将下面的代码文件放在同一目录中：

import numpy as np
import h5py
    
    
def load_dataset():
    train_dataset = h5py.File('datasets/train_catvnoncat.h5', "r")
    train_set_x_orig = np.array(train_dataset["train_set_x"][:]) # your train set features
    train_set_y_orig = np.array(train_dataset["train_set_y"][:]) # your train set labels

    test_dataset = h5py.File('datasets/test_catvnoncat.h5', "r")
    test_set_x_orig = np.array(test_dataset["test_set_x"][:]) # your test set features
    test_set_y_orig = np.array(test_dataset["test_set_y"][:]) # your test set labels

    classes = np.array(test_dataset["list_classes"][:]) # the list of classes
    
    train_set_y_orig = train_set_y_orig.reshape((1, train_set_y_orig.shape[0]))
    test_set_y_orig = test_set_y_orig.reshape((1, test_set_y_orig.shape[0]))
    
    return train_set_x_orig, train_set_y_orig, test_set_x_orig, test_set_y_orig,

二、问题（数据集）概述

本模块3个点预告：

• Figure out the dimensions and shapes of the problem (m_train, m_test, num_px, …)
• Reshape the datasets such that each example is now a vector of size (num_px * num_px * 3, 1)
• “Standardize” the data

吴恩达作业给出的数据集data.h5包括一个训练集（是猫就标签1，不是就标签0），一个测试机（也是标记是猫or不是猫）。每张图片的shape是三维张量（num_px,num_px,3），其中3是3 channels（RGB）.

为了表示彩色图像，必须为每个像素指定红色、绿色和蓝色通道(RGB)，因此像素值实际上是一个从0到255的三个数字的向量。

导入数据集

# Loading the data (cat/non-cat)
train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

我们在图像数据集(训练和测试)的末尾添加了“_orig”，因为我们要对它们进行预处理。预处理之后，我们将得到train_set_x和test_set_x(标签train_set_y和test_set_y不需要任何预处理)。

train_set_x_orig和test_set_x_orig的每一行都是一个表示图像的数组，如下代码，也可以随意更改索引值并重新运行以查看其他图像。

# Example of a picture
index = 25
plt.imshow(train_set_x_orig[index])
print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")
# 打印出y = [1], it's a 'cat' picture.

ps：DL中很多bug来自矩阵or向量的维度$dimensions$不匹配，注意。

练习1：找值

• m_train(训练例子个数)
• m_test(测试例子个数)
• num_px(=身高=训练图像的宽度)

注意train_set_x_orig numpy-array的形状(m_train, num_px num_px 3)。例如,你可以用​​train_set_x_orig.shape[0]​​​访问​​m_train​​。

### START CODE HERE ### (≈ 3 lines of code)
m_train = train_set_x_orig.shape[0]
m_test = test_set_x_orig.shape[0]
num_px = train_set_x_orig.shape[1]
### END CODE HERE ###

print ("Number of training examples: m_train = " + str(m_train))
print ("Number of testing examples: m_test = " + str(m_test))
print ("Height/Width of each image: num_px = " + str(num_px))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_set_x shape: " + str(train_set_x_orig.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x shape: " + str(test_set_x_orig.shape))
print ("test_set_y shape: " + str(test_set_y.shape))

Number of training examples: m_train = 209
Number of testing examples: m_test = 50
Height/Width of each image: num_px = 64
Each image is of size: (64, 64, 3)
train_set_x shape: (209, 64, 64, 3)
train_set_y shape: (1, 209)
test_set_x shape: (50, 64, 64, 3)
test_set_y shape: (1, 50)

为了方便起见，现在应该在一个numpy- shape数组(num_px∗num_px∗3,1)中重塑形状(num_px, num_px, 3)的图像。在此之后，我们的训练(和测试)数据集是一个numpy-数组，其中每一列代表一个扁平的图像。应该有m_train(分别m_test个)列。

练习2：reshape图片张量

训练和测试数据集，使大小(num_px, num_px, 3)的图像被平化为单个形状的向量(num_px∗num_px∗3,1)。

当你想把形状为(A,b,c,d)的矩阵X压平到形状为X_flatten (b∗c∗d, A)的矩阵X_flatten时，有一个窍门:

X_flatten = X.reshape(X.shape[0], -1).T      # X.T is the transpose of X

# Reshape the training and test examples

### START CODE HERE ### (≈ 2 lines of code)
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
### END CODE HERE ###

print ("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x_flatten shape: " + str(test_set_x_flatten.shape))
print ("test_set_y shape: " + str(test_set_y.shape))
print ("sanity check after reshaping: " + str(train_set_x_flatten[0:5,0]))

结果为：

train_set_x_flatten shape: (12288, 209)
train_set_y shape: (1, 209)
test_set_x_flatten shape: (12288, 50)
test_set_y shape: (1, 50)
sanity check after reshaping: [17 31 56 22 33]

预处理数据集：
机器学习中一个常见的预处理步骤是集中和标准化你的数据集，这意味着你从每个例子中减去整个numpy数组的平均值，然后用整个numpy数组的标准差除以每个例子。但是对于图片数据集，将数据集的每一行都除以255(像素通道的最大值)会更简单、更方便。

train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.

三、学习算法的大致框架

是时候用算法识别猪/狗/猫了，下面我们用逻辑回归和神经网络的思想，下图解释了为啥逻辑回归是一个简单的神经网络：

数学表达上述网络：

一个栗子，对于$x^{(i)}$：

$\begin{gathered} z^{(i)}=w^{T} x^{(i)}+b \\ \hat{y}^{(i)}=a^{(i)}=\operatorname{sigmoid}\left(z^{(i)}\right) \\ \mathcal{L}\left(a^{(i)}, y^{(i)}\right)=-y^{(i)} \log \left(a^{(i)}\right)-\left(1-y^{(i)}\right) \log \left(1-a^{(i)}\right) \end{gathered}$

损失函数（根据所有的训练样本）：

$J=\frac{1}{m} \sum_{i=1}^{m} \mathcal{L}\left(a^{(i)}, y^{(i)}\right)$

在后面的练习中，我们也是按照如下步骤：

• Initialize the parameters of the model
• Learn the parameters for the model by minimizing the cost
• Use the learned parameters to make predictions (on the test set)
• Analyse the results and conclude

四、构建自己的算法

搭建神经网络的如下主要步骤，并且我们一般将如下步骤组成​​model​​​函数：
（1）定义模型结构，如输入的特征个数
（2）初始化模型变量值
（3）循环：
——计算当前的损失函数（forward propagation）
——计算当前的梯度（backward propagation）
——更新parameters（梯度下降）

1.复习Helper函数

正如刚才上面的图所示，需要计算：
$\operatorname{sigmoid}\left(w^{T} x+b\right)=\frac{1}{1+e^{-\left(u^{T} x+b\right)}}$

练习3——sigmoid

# GRADED FUNCTION: sigmoid
def sigmoid(z):
    s = 1 / (1 + np.exp(-z))
    return s

print ("sigmoid([0, 2]) = " + str(sigmoid(np.array([0,2]))))
# 输出sigmoid([0, 2]) = [0.5        0.88079708]

2.Initializing parameters

练习4——初始化参数为0向量

# GRADED FUNCTION: initialize_with_zeros

def initialize_with_zeros(dim):
    """
    This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.
    
    Argument:
    dim -- size of the w vector we want (or number of parameters in this case)
    
    Returns:
    w -- initialized vector of shape (dim, 1)
    b -- initialized scalar (corresponds to the bias)
    """
    
    ### START CODE HERE ### (≈ 1 line of code)
    w = np.zeros((dim, 1))
    b = .0
    ### END CODE HERE ###

    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))
    
    return w, b

dim = 2
w, b = initialize_with_zeros(dim)
print ("w = " + str(w))
print ("b = " + str(b))

结果为：

w = [[0.]
 [0.]]
b = 0.0

3.向前传播和向后传播

这步需要学习出参数值

练习5——传播函数propagate计算损失值和梯度

（1）获得X
（2）计算：
$A=\sigma\left(w^{T} X+b\right)=\left(a^{(0)}, a^{(1)}, \ldots, a^{(m-1)}, a^{(m)}\right)$
（3）计算损失函数：
$J=-\frac{1}{m} \sum_{i=1}^{m} y^{(i)} \log \left(a^{(i)}\right)+\left(1-y^{(i)}\right) \log \left(1-a^{(i)}\right)$
ps：有2个式子后面会用到：
$\begin{aligned} &\frac{\partial J}{\partial w}=\frac{1}{m} X(A-Y)^{T} \\ &\frac{\partial J}{\partial b}=\frac{1}{m} \sum_{i=1}^{m}\left(a^{(i)}-y^{(i)}\right) \end{aligned}$

# GRADED FUNCTION: propagate

def propagate(w, b, X, Y):
    """
    Implement the cost function and its gradient for the propagation explained above

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

    Return:
    cost -- negative log-likelihood cost for logistic regression
    dw -- gradient of the loss with respect to w, thus same shape as w
    db -- gradient of the loss with respect to b, thus same shape as b
    
    Tips:
    - Write your code step by step for the propagation. np.log(), np.dot()
    """
    
    m = X.shape[1]
    
    # FORWARD PROPAGATION (FROM X TO COST)
    ### START CODE HERE ### (≈ 2 lines of code)
    A = sigmoid(np.dot(w.T, X) + b)  # compute activation
    cost = -1 / m * np.sum(np.dot(Y, np.log(A).T) + np.dot((1 - Y), np.log(1 - A).T))  # compute cost
    ### END CODE HERE ###
    
    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 lines of code)
    dw = 1 / m * np.dot(X, (A - Y).T)
    db = 1 / m * np.sum((A - Y))
    ### END CODE HERE ###

    assert(dw.shape == w.shape)
    assert(db.dtype == float)
    cost = np.squeeze(cost)
    assert(cost.shape == ())
    
    grads = {"dw": dw,
             "db": db}
    
    return grads, cost

w, b, X, Y = np.array([[1],[2]]), 2, np.array([[1,2],[3,4]]), np.array([[1,0]])
grads, cost = propagate(w, b, X, Y)
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
print ("cost = " + str(cost))

结果为：

dw = [[0.99993216]
 [1.99980262]]
db = 0.49993523062470574
cost = 6.000064773192205

4.优化器

你已经初始化了参数，也能够用刚才的propagate计算损失函数及其梯度，现在需要更新参数值（用梯度下降）。

练习6——优化optimize

目标是通过使损失函数J最小时学习到w和b参数，更新的表达式为：$\theta=\theta-\alpha d \theta, \text { where } \alpha \text { is the learning rate. }$

# GRADED FUNCTION: optimize

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
    """
    This function optimizes w and b by running a gradient descent algorithm
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of shape (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)
    num_iterations -- number of iterations of the optimization loop
    learning_rate -- learning rate of the gradient descent update rule
    print_cost -- True to print the loss every 100 steps
    
    Returns:
    params -- dictionary containing the weights w and bias b
    grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
    costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.
    
    Tips:
    You basically need to write down two steps and iterate through them:
        1) Calculate the cost and the gradient for the current parameters. Use propagate().
        2) Update the parameters using gradient descent rule for w and b.
    """
    
    costs = []
    
    for i in range(num_iterations):
        
        
        # Cost and gradient calculation (≈ 1-4 lines of code)
        ### START CODE HERE ###
        grads, cost = propagate(w, b, X, Y)
        ### END CODE HERE ###

        # Retrieve derivatives from grads
        dw = grads["dw"]
        db = grads["db"]

        # update rule (≈ 2 lines of code)
        ### START CODE HERE ###
        w = w - learning_rate * dw
        b = b - learning_rate * db
        ### END CODE HERE ###
        
        # Record the costs
        if i % 100 == 0:
            costs.append(cost)
        
        # Print the cost every 100 training examples
        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

params, grads, costs = optimize(w, b, X, Y, num_iterations= 100, learning_rate = 0.009, print_cost = False)

print ("w = " + str(params["w"]))
print ("b = " + str(params["b"]))
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))

输出为：

w = [[0.1124579 ]
 [0.23106775]]
b = 1.5593049248448891
dw = [[0.90158428]
 [1.76250842]]
db = 0.4304620716786828

练习7——预测predict

前面的函数将输出学习到的w和b。我们能够使用w和b预测数据集x的标签。实现predict()函数。计算预测有两个步骤:
$\hat{Y}=A=\sigma\left(w^{T} X+b\right)$然后将a转换为0(如果$activation$激活<= 0.5)或1(如果激活> 0.5)，将预测存储在向量Y_prediction中。你也可以在for循环中使用If /else语句(尽管也有一种方法可以将其向量化)。

# GRADED FUNCTION: predict

def predict(w, b, X):
    '''
    Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    
    Returns:
    Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in 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
    ### START CODE HERE ### (≈ 1 line of code)
    A = sigmoid(np.dot(w.T, X) + b)
    ### END CODE HERE ###

    for i in range(A.shape[1]):
        # Convert probabilities A[0,i] to actual predictions p[0,i]
        ### START CODE HERE ### (≈ 4 lines of code)
        if A[0, i] <= 0.5:
            Y_prediction[0, i] = 0
        elif A[0, i] > 0.5:
            Y_prediction[0, i] = 1
        ### END CODE HERE ###
    
    assert(Y_prediction.shape == (1, m))
    
    return Y_prediction

print ("predictions = " + str(predict(w, b, X)))
# 打印出predictions = [[1. 1.]]

已经实现了几个函数:

• 初始化(w,b)
• 迭代地优化损失以学习参数(w,b):
• 计算代价及其梯度
• 使用梯度下降更新参数
• 使用学习到的(w,b)来预测给定一组示例的标签

五、将所有的函数合并成一个模型

• Y_prediction用于测试集上的预测
• Y_prediction_train用于训练集上的预测
• w, cost, gradient用于optimize()函数的输出

练习8——合成model

# GRADED FUNCTION: model

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
    """
    Builds the logistic regression model by calling the function you've implemented previously
    
    Arguments:
    X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
    Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
    X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
    Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
    num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
    learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
    print_cost -- Set to true to print the cost every 100 iterations
    
    Returns:
    d -- dictionary containing information about the model.
    """
    
    ### START CODE HERE ###

    # initialize parameters with zeros (≈ 1 line of code)
    w, b = initialize_with_zeros(X_train.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"]

    # Predict test/train set examples (≈ 2 lines of code)
    Y_prediction_test = predict(w, b, X_test)
    Y_prediction_train = predict(w, b, X_train)

    ### END CODE HERE ###

    # 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

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

可以看到如下的迭代中的损失函数值：

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 %

由上面结果：
训练准确率接近100%。这是一个很好的完整性检查：模型有足够的容量来匹配训练数据。测试错误为68%。考虑到我们使用的数据集很小，而且逻辑回归是一个线性分类器，对于这个简单的模型来说，它实际上是不错的。后面建立一个更好的分类器。

上面的模型显然是过度拟合训练数据。如何减少过拟合——例如使用正则化。使用下面的代码(并更改索引变量)，可以查看测试集图片上的预测。

# Example of a picture that was wrongly classified.
index = 1
plt.imshow(test_set_x[:,index].reshape((num_px, num_px, 3)))
print ("y = " + str(test_set_y[0,index]) + ", you predicted that it is a \"" + classes[int(d["Y_prediction_test"][0,index])].decode("utf-8") +  "\" picture.")

可视化损失函数和梯度：

# Plot learning curve (with costs)
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()

由上图可以看出损失函数在下降，说明参数正在被学习。可以在训练集中训练更多的模型，可以试试增大迭代次数，将会看到训练集的准确率提升，但是测试集的准确率下降——过拟合。

让我们将模型的学习曲线与几种学习速率的选择进行比较。也可以尝试不同的值，而不是我们初始化learning_rates变量要包含的三个值。

六、拓展分析

学习速率𝛼决定了我们更新参数的速度。如果学习率太大，我们可能会“超过”最优值。类似地，如果它太小，我们将需要太多的迭代来收敛到最佳值。

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()

数据为：

learning rate is: 0.01
train accuracy: 99.52153110047847 %
test accuracy: 68.0 %

-------------------------------------------------------

learning rate is: 0.001
train accuracy: 88.99521531100478 %
test accuracy: 64.0 %

-------------------------------------------------------

learning rate is: 0.0001
train accuracy: 68.42105263157895 %
test accuracy: 36.0 %

-------------------------------------------------------

解读：

（1）如果学习速率太大（0.01），损失函数可能会上下波动（$oscilate$），还可能会产生分叉（虽然在上面的栗子中用0.01不会这样）

（2）低损失函数值并不意味着更好。你必须检查是否存在过拟合（当训练的准确度远远高于测试的准确度时）。

七、用自己的图片测试

如果直接这样会报错​​module 'scipy.ndimage' has no attribute 'imread'​​：

## START CODE HERE ## (PUT YOUR IMAGE NAME) 
my_image = "cat_in_iran.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.")

另外有三种方法

## START CODE HERE ##
import matplotlib
my_image = "dog1.jpg" # change this to the name of your image file 
my_label_y = [1] # the true class of your image (1 -> cat, 0 -> non-cat)
## END CODE HERE ##

fname = "images/" + my_image
image = np.array(matplotlib.pyplot.imread(fname))
my_image = np.array(Image.fromarray(image).resize(size=(num_px,num_px))).reshape((1, num_px*num_px*3)).T
# image = np.array(ndimage.imread(fname, flatten=False))
# my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((num_px*num_px*3,1))
my_predicted_image = predict(my_image, my_label_y, parameters)

plt.imshow(image)
print ("y = " + str(np.squeeze(my_predicted_image)) + ", your L-layer model predicts a \"" + classes[int(np.squeeze(my_predicted_image)),].decode("utf-8") +  "\" picture.")

识别出不是猫，输出结果为：

y = 0.0, your algorithm predicts a "non-cat" picture.

我们再来个猪八戒上场：

同样是输出：

y = 0.0, your algorithm predicts a "non-cat" picture.

而用猫照片则能够识别出是猫猫：

Reference

• ​​https://github.com/datawhalechina/leeml-notes/tree/master/docs/Homework/HW_1​​