Simple Linear Regression is a type of Regression algorithms that models the relationship between a dependent variable and a single independent variable. The relationship shown by a Simple Linear Regression model is linear or a sloped straight line, hence it is called Simple Linear Regression.

The key point in Simple Linear Regression is that the ** dependent variable must be a continuous/real value**. However, the independent variable can be measured on continuous or categorical values.

Simple Linear regression algorithm has mainly two objectives:

**Model the relationship between the two variables.**Such as the relationship between Income and expenditure, experience and Salary, etc.**Forecasting new observations.**Such as Weather forecasting according to temperature, Revenue of a company according to the investments in a year, etc.

## Simple Linear Regression Model:

The Simple Linear Regression model can be represented using the below equation:

y= a_{0}+a_{1}x+ Îµ

Where,

**a0= It is the intercept of the Regression line (can be obtained putting x=0)**

**a1= It is the slope of the regression line, which tells whether the line is increasing or decreasing. **

**Îµ = The error term. (For a good model it will be negligible)**

**Problem Statement example for Simple Linear Regression:**

Here we are taking a dataset that has two variables: salary (dependent variable) and experience (Independent variable). The goals of this problem is:

**We want to find out if there is any correlation between these two variables****We will find the best fit line for the dataset.****How the dependent variable is changing by changing the independent variable.**

1. Important the necessary libraries

2. Important the necessary data set and extract two variables

Check for conformation of two variables

3. Now Splitting the dataset into the Training set and Test set

Display the all splits

4. Now Train the Algorithm on your data set. If Training is successful, it will generate a model.

As you can see on the 8th line output, the model was generated. In the above code, we have used a **fit()** method to fit
our Simple Linear Regression object to the training set. In the fit()
function, we have passed the x_train and y_train, which is our training
dataset for the dependent and an independent variable. We have fitted
our regressor object to the training set so that the model can easily
learn the correlations between the predictor and target variables.

5. ** Prediction of test set result:**

dependent (salary) and an independent variable (Experience). So, now, our model is ready to predict the output for the new observations. In this step, we will provide the test dataset (new observations) to the model to check whether it can predict the correct output or not.

We will create a prediction vector **y_pred**, and **x_pred**, which will contain predictions of test dataset, and prediction of training set respectively.

6.

**visualizing the Training set results:**

Now in this step, we will visualize the training set result. To do
so, we will use the scatter() function of the pyplot library, which we
have already imported in the pre-processing step. The **scatter () function** will create a scatter plot of observations.

In the x-axis, we will plot the Years of Experience of employees and on the y-axis, salary of employees. In the function, we will pass the real values of training set, which means a year of experience x_train, training set of Salaries y_train, and color of the observations. Here we are taking a green color for the observation, but it can be any color as per the choice.

Now, we need to plot the regression line, so for this, we will use the **plot() function**
of the pyplot library. In this function, we will pass the years of
experience for training set, predicted salary for training set x_pred,
and color of the line.

Next, we will give the title for the plot. So here, we will use the **title()** function of the **pyplot** library and pass the name ("Salary vs Experience (Training Dataset)".

In the above plot, we can see the real values observations in red dots and predicted values are covered by the blue regression line. The regression line shows a correlation between the dependent and independent variable.

The good fit of the line can be observed by calculating the difference between actual values and predicted values. But as we can see in the above**plot, most of the observations are close to the regression line, hence our model is good for the training set**.

7. **visualizing the Test set results:**

In the previous step, we have visualized the performance of our model on the training set. Now, we will do the same for the Test set. The complete code will remain the same as the above code, except in this, we will use x_test, and y_test instead of x_train and y_train.

In the above plot, there are observations given by the red color, and prediction is given by the blue regression line. As we can see, most of the observations are close to the regression line, hence we can say our Simple Linear Regression is a good model and able to make good predictions.

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