Decoding Curves with Polynomial Regression

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Polynomial regression: Everything you need to know! Polynomial regression
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Welcome to our exploration of polynomial regression, a powerful technique in predictive modeling that elegantly captures non-linear relationships between variables. In the pursuit of understanding data and building robust models, the need often arises to move beyond linear approximations. This is where polynomial regression steps in, allowing us to unveil our datasets’ hidden complexities and nuances.

This blog will explore polynomial regression’s principles, applications, implementation, and nuances. Whether you’re a seasoned data scientist seeking deeper insights or a curious enthusiast eager to grasp advanced regression techniques, this blog will equip you with the knowledge to use polynomial regression effectively in your analytical pursuits.

By the end, you’ll comprehend how polynomial regression works and gain the confidence to wield this technique in your own data analysis and machine learning projects.

Join us as we unravel the intricacies of polynomial regression and unlock its potential to transform how we model and interpret data. Let’s dive in!

What is polynomial regression?

Polynomial regression is often considered as a special multiple linear regression

Here’s why- Polynomial regression is a statistical method of determining the relationship between an independent variable (x) and a dependent variable (y) and modeling their relationship as the nth-degree polynomial. 

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

Related read: Regression model: Definition, Types, and Examples

Assumptions in polynomial regression

  • A linear, or curvilinear, additive relationship between a dependent variable and a set of k independent variables (xi, i=1 to k) explains its behavior.
  • The relationship between the dependent variable and any independent variable is linear or curvilinear.
  • The independent variables do not depend on each other either.
  • The errors are independent and normally distributed with a 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:

1. Linear polynomial 

2. Quadratic polynomial

3. Cubic polynomial

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 degrees of the polynomial equations increase, the curve better fits the dataset.

Why do we need polynomial regression?

The problem with linear regression is that it uses the line of best fit. When we have a dataset and 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 line 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 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 (curve) is drawn following that pattern of the points. Another difference is that polynomial regression does not require 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.

Related read: How to Calculate Linear Regression

What are the characteristics of polynomial regression?

  • It is a special case of multiple linear regression that can determine the non-linear relationship between independent and dependent variables. 
  • The 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. Dealing with the outliers beforehand is advised.
  • We can use scatterplots to visualize a curve drawn from the data.

What are the uses of polynomial regression?

  • Many researchers use the polynomial regression equation in their experiments to draw conclusions.
  • It is used to determine the relationship between independent variables and dependent variables. 
  • Polynomial regression is used in the study of sediment 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 also be calculated through programming, like Python and R. 
  • Polynomial regression is a great deal in the machine learning world.

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