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Polynomial regression: Everything you need to know!

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Polynomial regression: Everything you need to know! Polynomial regression
Table of Contents

What is polynomial regression?

Polynomial regression is often considered as a special multiple linear regression. Why? Let us understand – polynomial regression is a statistical method of determining the relationship between an independent variable (x) and a dependent variable (y) and model their relationship as the nth degree polynomial. 

The relationship of the independent and dependent variable on a graph turns out as a curvilinear relationship with the help of a polynomial equation. Polynomial regression is used when there is no linear correlation between the variables. Hence, it explains why it looks more like a non-linear function.

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Assumptions in polynomial regression

  • The behaviour of a dependent variable is explained by a linear, or curvilinear, additive relationship between the dependent variable and a set of k independent variables (xi, i=1 to k).
  • The relationship between the dependent variable and any independent variable is linear or curvilinear.
  • The independent variables do no depend on each other too.
  • The errors are independent, normally distributed with mean zero and a constant variance.

Polynomial regression equation

Polynomial regression equation of nth degree can be written as:

Y= b0+a1x+a2x^2+a3x^3+…. anx^n

There are three types of polynomials:

 

Polynomial regression: Everything you need to know! Polynomial regression

As you can see, the linear polynomial has a degree of 1, the quadratic polynomial has a degree of 2 and the cubic polynomial has a degree of 3. As the degree of the polynomial equations goes up, the curve better fits the dataset.

 

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Why do we need polynomial regression?

The problem with linear regression was, it uses the line of best fit. Meaning, when we have a dataset and we plot it on a graph, there has to be a straight line where the scatterplots lie. But what if we have a dataset that gives us no straight but a curve? This is when polynomial regression comes in.

Polynomial regression: Everything you need to know! Polynomial regression

The difference between linear regression and a polynomial regression is that the line of best fit is a curve in polynomial regression. The scatterplots are scanned for a pattern and the line is drawn (curve) following that pattern of the points. Another difference is, polynomial regression does not make it compulsory for the data to have a linear relationship between them. 

So when linear regression fails to determine a linear relationship between variables, polynomial regression does it for us.

What are the characteristics of polynomial regression?

  • It is a special case of multiple linear regression and can determine the relationship between independent and dependent variables having a non-linear relationship. 
  • Polynomial regression equation has a degree that decides the line of best fit for the data.
  • The polynomial regression model is prone to outliers and can change the results. It is advised to deal with the outliers beforehand.
  • To draw out a curve through the data, we can make use of scatterplots which helps us to visualize the curve.

What are the uses of polynomial regression?

  • The polynomial regression equation is used by many of the researchers in their experiments to draw out conclusions.
  • It is used to determine the relationship between independent variables and dependent variables. 
  • Polynomial regression is used in the study of sediments isotopes. 
  • It is also used to study the spreading of a disease in the population. 
  • Organizations also use polynomial regression when they meet non-linear data. 
  • Polynomial regression can be calculated through programming too, like python and R. 
  • Polynomial regression is a great deal in the machine learning world.

 

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