Case Study (Salary) :

SKLearrn (Simple Linear Regression)

 - Simple Linear Regression: 

      -> aims to find the best-fitting line through the data points

 - Finding the line with the minimum sum of squares

      -> Sum of Squares: = Σ(y - ŷ)2

      -> y: Real Values , ŷ: Predicted Values

 - y = b0 + b1 x

     -> y is the dependent variable (variable on the Y axis)
     -> X is the independent variable (variable on the X axis)
     -> b1 is the slope of the line  (Coefficient)

     -> showing how much (Y) will change given a one-unit shift in (X) 
     -> while holding other variables in the model constant
     -> for 1 more year experience, the person will receive b$ on top of his salary 

 - b0 is the y-intercept (Constant)

     -> the point where the best fitting line cross the y-axis 
     -> when a person has no experience (X = 0), salary = a

Overview

- Importing the Relevant Libraries

- Loading the Data

- Declaring the Dependent and the Independent variables

- Splitting the dataset into the Training set and Test set

- Linear Regression Model

    - Creating a Linear Regression 
    - Fitting The Model
    - Predicting the Test Set Results

- Creating a Summary Table (Test Set Results)

- Making predictions 

    - Making a Single Observation Prediction
    - Making Multiple Observations Prediction

- R-Squared (R²) , Intercept , Coefficient

    - Calculating the R-squared (R²)
    - Finding the intercept
    - Finding the coefficients
    - Final Regression Equation (y = b0 + b1 x)

- Data visualization

    - Visualising the Training Set Results
    - Visualising the Test Set Results
    - Visualising the Train &Test Set Results on the same plot

Importing the Relevant Libraries

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

Loading the data

url = "https://datascienceschools.github.io/Machine_Learning/Sklearn/Case_Study/Salary/Salary_Data.csv"

df = pd.read_csv(url)

df.head()

	YearsExperience	Salary
0	1.1	39343.0
1	1.3	46205.0
2	1.5	37731.0
3	2.0	43525.0
4	2.2	39891.0

Declaring the Dependent and the Independent variables

    - x : (Independent variable)-> Input or Feature
    - y : (dependent variable)-> Output or Target 

X = df.iloc[:, :-1].values

y = df.iloc[:, -1].values

Splitting the dataset into the Training set and Test set

from sklearn.model_selection import train_test_split

X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 1/3, random_state = 0)

Linear Regression Model

Creating a Linear Regression

- LinearRegression Class from linear_model Module of sklearn Library

- model -> Object of LinearRegression Class

from sklearn.linear_model import LinearRegression

model = LinearRegression()

Fitting The Model

- fit method -> training the model

model.fit(X_train, y_train)

LinearRegression()

Predicting the Test Set Results

- y_pred -> the predicted salaries

y_pred = model.predict(X_test)

Creating a Summary Table (Test Set Results)

- Comparing predicted_salary & real_salary

data = pd.DataFrame(X_test).rename(columns={0: "experience_years"})

data['predicted_salary'] = y_pred

data['real_salary'] = y_test

data['difference'] =  y_pred - y_test

data

	experience_years	predicted_salary	real_salary	difference
0	1.5	40835.105909	37731.0	3104.105909
1	10.3	123079.399408	122391.0	688.399408
2	4.1	65134.556261	57081.0	8053.556261
3	3.9	63265.367772	63218.0	47.367772
4	9.5	115602.645454	116969.0	-1366.354546
5	8.7	108125.891499	109431.0	-1305.108501
6	9.6	116537.239698	112635.0	3902.239698
7	4.0	64199.962017	55794.0	8405.962017
8	5.3	76349.687193	83088.0	-6738.312807
9	7.9	100649.137545	101302.0	-652.862455

Making Predictions

Making a Single Observation Prediction

- Predicting the salary of an employee with 12 years experience

model.predict([[12]])

array([138967.5015615])

Making Multiple Observations Prediction

- Predicting salaries of employees with 0, 1, 5 & 10 years experience

new_employees = pd.DataFrame({'years_of_experience': [0,1,5,10]})

new_employees['predicted_salary'] = model.predict(new_employees)

new_employees

	years_of_experience	predicted_salary
0	0	26816.192244
1	1	36162.134687
2	5	73545.904460
3	10	120275.616675

R-Squared (R²) , Intercept , Coefficient

Calculating the R-Squared (R²)

* What is R-squared?

- a statistical measure of how close the data are to the fitted regression line

- also known as the coefficient of determination

- R-squared is always between 0 and 100%

- the higher the R-squared, the better the model fits the data

Rsquared = model.score(X_train,y_train)

print('R-Squared is:', Rsquared)

R-Squared is: 0.9381900012894278

Finding the Intercept (b0)

 -> y = b0 + b1 x

 -> the point where the best fitting line cross the y-axis 

 -> when a person has no experience (X = 0), salary = b0 = 26816.192244

Intercept = model.intercept_

print('Intercept is:', Intercept)

Intercept is: 26816.19224403119

Finding the coefficients (b1)

-> y = b0 + b1 x

-> for 1 more year experience, the person will receive b1 $ on top of his salary 

-> b1 = 9345.94

-> Person with no experience -> Salary: 26816.192244 $

-> Person with 1 year experience -> Salary: 36162.134687 $

-> 36162.134687 - 26816.192244 = 9345.94 $ more for 1 more year experience

Coefficient = model.coef_

print('coefficient is:', Coefficient)

coefficient is: [9345.94244312]

Final Regression Equation (y = b0 + b1 x)

    - b0 = 26816.19
    - b1 = 9345.94

Salary = 26816.19 + 9345.94 × YearsExperience

Data visualization

Visualising the Training Set Results

plt.scatter(X_train, y_train)

plt.plot(X_train, model.predict(X_train), color = 'red')

plt.title('Salary vs Experience (Training set)')
plt.xlabel('Years of Experience')
plt.ylabel('Salary')

plt.show()

Visualising the Test Set Results

- Visualising Regression Line:

    -> plt.plot(X_train, model.predict(X_train), color = 'red')

- The Regression Line is resulting from a unique equation:

    -> Salary = 26816.19 + 9345.94 × YearsExperience

- the predicted salaries of both training & test set

    -> will be on the same regression line

plt.scatter(X_test, y_test)

plt.plot(X_train, model.predict(X_train), color = 'red')

plt.title('Salary vs Experience (Test set)')
plt.xlabel('Years of Experience')
plt.ylabel('Salary')

plt.show()

Visualising the Train &Test Set Results on the same plot

- Training set : Blue points
- Test set: Green points

- Visualising Regression Line:

        -> plt.plot(X_train, model.predict(X_train), color = 'red')

- The Regression Line is resulting from a unique equation:

        -> Salary = 26816.19 + 9345.94 × YearsExperience

- the predicted salaries of both training & test set

        -> will be on the same regression line

plt.scatter(X_train, y_train, color='blue')

plt.scatter(X_test, y_test, color='green')

plt.plot(X_train, model.predict(X_train), color = 'red')

plt.title('Salary vs Experience')
plt.xlabel('Years of Experience')
plt.ylabel('Salary')

plt.show()