Data Science and Machine Learning Algorithms from Scratch-Linear Regression-02

Simple linear regression

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

Question 1. The relationship between working experience and salary? The relationship between working experience and salary?

# 数据
experience=[1,2,3,4,5,6,7,8,9,10,2,2,3,3,3,2,4,5,5,2]
salary    =[5000,7000,8000,10000,15000,18000,19000,20000,16000,30000,12000,6000,8000,5500,9000,7800,9200,14000,16000,8000]
# 数据合并
list_of_tuples=list(zip(experience,salary))
# 转换成dataframe类型
df=pd.DataFrame(list_of_tuples,columns=['experience','salary'])
df

df.describe()#生成描述性统计

df.corr()

plt.scatter(df.experience,df.salary) # 绘制散点图
plt.xlabel('experience')
plt.ylabel('salary')
plt.show

df.plot(kind="hist",y="experience",bins=10 ,range=(0,50),density= True)

# 绘制热力图
f, ax =plt.subplots(figsize=(20,10))
sns.heatmap(df,annot=True,linewidths=0.5,linecolor="red",fmt=".1f",ax=ax)
plt.show()

#定义平均值计算函数
# First , Let's calculate the mean!
def calculate_mean(a_list_of_values): # 传入一组数据
    mean=sum(a_list_of_values)/float(len(a_list_of_values)) # 数据的总和除以数据的个数
    return mean # 返回平均值

# 计算方差函数
# Then , Let's calculate the variance!
def calculate_variance(a_list_of_values,mean):# 传入一组数据和平均值
    variance_sum=sum((x-mean)**2 for x in a_list_of_values)#  通过for循环遍历数据中的每个值  每个值减去数据的平方值的平方的总和
    variance=variance_sum/(len(a_list_of_values)-1) # 总和除以（数据个数-1）得出方差值
    return variance

# 测试
experience=[1,2,3,4,5,6,7,8,9,10,2,2,3,3,3,2,4,5,5,2]
salary    =[5000,7000,8000,10000,15000,18000,19000,20000,16000,30000,12000,6000,8000,5500,9000,7800,9200,14000,16000,8000]

mean_experience=calculate_mean(experience)
print(mean_experience)
variance_experience=calculate_variance(experience,mean_experience)
print(variance_experience)

mean_salary=calculate_mean(salary)
print(mean_salary)
variance_salary=calculate_variance(salary,mean_salary)
print(variance_salary)

# And then we calculate the Covariance
#计算协方差
def calculate_covariance(a_list_of_Xs,the_mean_of_Xs,a_list_of_Ys,the_mean_of_Ys):# 传入x数据，x的平均值，y数据，y平均值
    cov_sum=0 # 首先定义结果为0通过下面循环+进去
    for i in range(len(a_list_of_Xs)):# 循环数据的长度
        cov_sum+=(a_list_of_Xs[i]-the_mean_of_Xs)*(a_list_of_Ys[i]-the_mean_of_Ys)#总和等于(x[i]-x_mean) 乘以 (y[i]-y_mean)
    the_covariance=cov_sum/(len(a_list_of_Xs)-1)#协方差等于 总和 除以（数据个数-1）
    return the_covariance

df.cov()# 调用pandas.DataFrame.cov协方差函数方法计算数据和我们自定义函数做验证

# 测试
covariance_of_experience_and_salary=calculate_covariance(experience,mean_experience,salary,mean_salary)
print(covariance_of_experience_and_salary)

# Let's calculate standard deviation!
# 计算标准差
def calculate_the_standard_deviation(a_list_values):
    the_mean_of_the_list_values=sum(a_list_values)/float(len(a_list_values))# 计算平均值
    variance=sum([(a_list_values[i]-the_mean_of_the_list_values)**2 for i in range(len(a_list_values)) ]) / float(len(a_list_values)-1)# sum（data[i]-mean）**2/(len(data)-1)
    return variance**0.5# 开方

# 测试标准差
a=[1,2,3,4,5,6,7,8,9,10]
calculate_the_standard_deviation(a) # 因为我们是计算样本的标准差 样本个数-1

import pandas as pd
b=np.array(a)
b.std() # 是计算整体的标准差
# 默认标准偏差类型在 numpy 的 .std() 和 pandas 的 .std() 函数之间是不同的。
# 默认情况下，numpy 计算的是总体标准偏差，ddof = 0。另一方面，pandas 计算的是样本标准偏差，ddof = 1。如果我们知道所有的分数，那么我们就有了总体——因此，要使用 pandas 进行归一化处理，我们需要将“ddof”设置为 0。

# Let's calculate the correlation!
# 计算相关性
def calculate_the_correlation(a_list_of_Xs,the_mean_of_Xs,a_list_of_Ys,the_mean_of_Ys):# 传入x数据，x的平均值，y数据，y平均值
    X_std=calculate_the_standard_deviation(a_list_of_Xs)# 使用标准差函数计算x数据标准差
    Y_std=calculate_the_standard_deviation(a_list_of_Ys) # 计算y数据标准差
    X_Y_Cov=calculate_covariance(a_list_of_Xs,the_mean_of_Xs,a_list_of_Ys,the_mean_of_Ys)# 使用协方差函数计算x，y的协方差
    
    Corr=(X_Y_Cov)/(X_std*Y_std) # 相关性等于 相关性除以（x标准差乘y标准差）
    return Corr

    

calculate_the_correlation(experience,mean_experience,salary,mean_salary)

df.corr()# 使用pandas—相关系数函数corr() 进行验证

Make Prediction: Use one variable to make prediction about another

dependent variable: Y dependent variable y

independent variable: X Independent variable x
• Regression analysis is used to establish an equation to simulate how two or more variables are related • The variable to be predicted is called: dependent variable, output • Used The variables used for prediction are called: independent variable, input • One-variable linear regression contains one independent variable and one dependent variable • The relationship between the above two variables is simulated by a straight line • If it contains more than two independent variables Variables are called multiple regression analysis (multiple regression)

# 计算系数
def calculate_the_coefficients(dataset):# 传入数据集
    x=[row[0] for row in dataset]# 循环数据集放入x和y中
    y=[row[1] for row in dataset]
    x_mean,y_mean=calculate_mean(x),calculate_mean(y) # 调用自定义平均函数算出x,y的平均值
    b1=calculate_covariance(x,x_mean,y,y_mean)/calculate_variance(x,x_mean) # b1=（计算协方差（x数据，x平均值，y数据，y平均值） ）除以（方差（x数据，x平均值））
    b0=y_mean-b1*x_mean # b0 = y的平均值减去b1乘以x的平均值
    
    return [b0,b1]

# 数据
experience=[1,2,3,4,5,6,7,8,9,10,2,2,3,3,3,2,4,5,5,2]
salary    =[5000,7000,8000,10000,15000,18000,19000,20000,16000,30000,12000,6000,8000,5500,9000,7800,9200,14000,16000,8000]
# 数据合并
list_of_tuples=list(zip(experience,salary))
list_of_tuples

list_of_lists=[list(elem) for elem in list_of_tuples] # 将每个元素变成列表
list_of_lists

#测试系数
b0,b1=calculate_the_coefficients(list_of_lists) 
b0,b1

y=2555+2237X

# 简单线性回归函数
def simple_linear_regression(training_data,testing_data):# 传入参数
    predictions=[]# 定义个空列表存储预测结果
    b0,b1=calculate_the_coefficients(training_data)# 在相关系数函数中传入测试数据算出b0和b1
    for row in testing_data: # 循环测试数据
        y=b0+b1*row[0]# y等于b0+b1乘以测试数据的每个值
        predictions.append(y)
    return predictions

# 使用均方根误差来查看预测如何
from math import sqrt #  从math即数学库中导入用于开根运算的方法sqrt
def calculate_the_RMSE(predicted_data,actual_data):
    the_sum_of_error=0
    for i in range(len(actual_data)):
        prediction_error=predicted_data[i]-actual_data[i] # 循环遍历预测值和真实值相减的结果
        the_sum_of_error += (prediction_error**2) # 预测值和真实值相减的结果的平方
    RMSE=sqrt(the_sum_of_error/float(len(actual_data))) # 预测值和真实值相减的结果的平方除以数据个数的开根运算
    return RMSE

# 将数据分离只要工作年限最后在模型中预测出收入结果
data_to_be_put_into_the_model=[]
for row in list_of_lists:
    row_copy=list(row)
    row_copy[-1]=None
    data_to_be_put_into_the_model.append(row_copy)
data_to_be_put_into_the_model

# 利用模型预测收入结果
predictions=simple_linear_regression(list_of_lists,data_to_be_put_into_the_model) # 传入数据 传入分离出来的工作时间进行预测
predictions # 模型预测收入结果

actual_data=[row[-1] for row in list_of_lists]# 提出真实收入做对比
actual_data

# 查看我们的模型有多好
def how_good_is_our_model(dataset,some_model_to_be_evaluated): # 真实值，模型评估的值
    test_data=[]
    for row in dataset:
        row_copy=list(row)
        row_copy[-1]=None
        test_data.append(row_copy)
    predict_data=some_model_to_be_evaluated(dataset,test_data)
    print(predict_data)# 预测值
    actual_data=[row[-1] for row in dataset]
    print(actual_data) # 真实值
    RMSE=calculate_the_RMSE(predict_data,actual_data)
    return RMSE
    

result=how_good_is_our_model(list_of_lists,simple_linear_regression)
print(result)

Coefficient of determination


TSS = true value minus the square of the average
RSS = predicted value minus the square of the average
SSE = true value minus the square of the predicted value