Simple Linear Regression: How It works? (Python Implementation) |

# Linear Regression (Python Implementation)

This article discusses the basics of linear regression and its implementation in Python programming language.

Linear regression is a statistical approach for modelling the relationship between a dependent variable with a given set of independent variables.

**Note:**In this article, we refer dependent variables as

**response**and independent variables as

**features**for simplicity.

In order to provide a basic understanding of linear regression, we start with the most basic version of linear regression, i.e.

**Simple linear regression**.## Simple Linear Regression

Simple linear regression is an approach for predicting a

**response**using a**single feature**.
It is assumed that the two variables are linearly related. Hence, we try to find a linear function that predicts the response value(y) as accurately as possible as a function of the feature or independent variable(x).

Let us consider a dataset where we have a value of response y for every feature x:

For generality, we define:

x as

**feature vector**, i.e x = [x_1, x_2, …., x_n],
y as

**response vector**, i.e y = [y_1, y_2, …., y_n]
for

**n**observations (in above example, n=10).
A scatter plot of above dataset looks like:-

Now, the task is to find a

**line which fits best**in above scatter plot so that we can predict the response for any new feature values. (i.e a value of x not present in the dataset)
This line is called the

**regression line**.
The equation of the regression line is represented as:

Here,

- h(x_i) represents the
**predicted response value**for ith observation. - b_0 and b_1 are regression coefficients and represent
**y-intercept**and**slope**of regression line respectively.

To create our model, we must “learn” or estimate the values of regression coefficients b_0 and b_1. And once we’ve estimated these coefficients, we can use the model to predict responses!

In this article, we are going to use the

**Least Squares technique**.
Now consider:

Here, e_i is a

So, our aim is to minimize the total residual error.

**residual error**in ith observation.So, our aim is to minimize the total residual error.

We define the squared error or cost function, J as:

and our task is to find the value of b_0 and b_1 for which J(b_0,b_1) is minimum!

Without going into the mathematical details, we present the result here:

where SS_xy is the sum of cross-deviations of y and x:

and SS_xx is the sum of squared deviations of x:

Note: The complete derivation for finding least squares estimates in simple linear regression can be found here.

Given below is the python implementation of the above technique on our small dataset:

The output of the above piece of code is:

Estimated coefficients: b_0 = -0.0586206896552 b_1 = 1.45747126437

And the graph obtained looks like this:

Good post keep it up. Keep updating.

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for your estimated coefficients I seem to be getting different figures

ReplyDeleteb_0 = 1.2363636363636363

b_1 = 1.1696969696969697

It will vary with the system configuration

DeleteOk I get you, I further used excel linear regression plotting and got the same result like I got in python

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