## Introduction to Polynomial Regression

In the world of data analysis, regression analysis is a powerful tool used to establish relationships between variables. Linear regression, in particular, has been extensively used for many years.

But, what do you do when your data does not fit a straight line? Polynomial regression comes to the rescue! In this article, we will take a deep dive into polynomial regression, its uses, and how it is different from linear regression.

## Definition and Explanation of Polynomial Regression

Polynomial regression is a form of regression analysis used to model complex non-linear relationships between the independent variable (X) and the dependent variable (Y). It is a type of multiple regression where the relationship between the independent variable and the dependent variable is a polynomial of degree n.

Essentially, it is a curve-fitting technique that finds the best-fit curve through the given data points.

In simple terms, polynomial regression is used when the relationship between the independent and dependent variables is non-linear and cannot be explained through a straight line.

A simple linear regression model would not be an effective fit for such complex relationships. The equation for polynomial regression is a bit more complicated than the simple linear regression equation.

## It is represented as:

Y = 0 + 1X + 2X^2 + + nX^n

The variables in the above equation are as follows:

- Y represents the dependent variable (the outcome)
- X represents the independent variable or the predictor variable
- The values (beta coefficients) represent the model coefficients that need to be estimated from the data.

The degree of the polynomial (n) is determined based on the amount of curvature present in the relationship between the variables.

The higher the degree, the more curved the line is.

## Quick Revision to Linear Regression

To better understand polynomial regression, let’s take a quick revision of linear regression.

### Simple Linear Regression

Simple linear regression is a statistical tool used to find the relationship between two variables. One variable is the independent variable (X), and the other is the dependent variable (Y).

A single linear equation is used to represent the relationship between X and Y. The equation is represented as:

Y = 0 + 1X

Here, 0 and 1 are the coefficients of the equation that need to be estimated from the given data.

0 represents the y-intercept, i.e., the value of Y when the value of X is zero. 1 represents the slope of the line, i.e., how much Y changes when X increases by 1.

### Multiple Linear Regression

In multiple linear regression, we consider more than one independent variable and find the relationship with the dependent variable. The multiple linear regression equation is represented as:

Y = 0 + 1X1 + 2X2 + 3X3 + + nxn

Here, Y represents the dependent variable, and X1, X2, X3, Xn represent the independent variables.

The coefficients of the line 0, 1, 2, 3, n need to be estimated from the given data. The matrix of features is used to represent the multiple regression model.

## Difference between Polynomial and Linear Regression

Now that we have understood polynomial and linear regression let’s discuss the difference between the two.

The most significant difference between the two is that polynomial regression allows for more complex relationships between variables than linear regression.

Linear regression only allows for a straight line relationship between the variables.

Polynomial regression, on the other hand, allows for a curve relationship between dependent and independent variables.

This enables you to capture the complexities of the data, allowing for more accurate predictions. Another major difference lies in the model equation.

In linear regression, the equation is a straight line, while in polynomial regression, it is a higher-degree polynomial function. This means that the degree of the polynomial is determined based on the number of turning points in the data.

## Conclusion

In conclusion, polynomial regression is a valuable tool for data scientists to model complex non-linear relationships. It is an extension of linear regression that allows for more accurate predictions and curve-fitting.

Linear regression, on the other hand, is a well-established technique that is limited to only linear relationships between variables. By understanding the difference between the two, data analysts can choose the right regression model for their data.

## 3) Understanding Polynomial Regression

Polynomial regression is a statistical technique that helps to establish a relationship between the independent variable and the dependent variable by fitting a polynomial equation to the data points. This method is often used in many fields, including economics, physics, biology, and engineering, where the relationship between variables is not linear.

## Advantages of Using Polynomial Regression

One of the main advantages of using polynomial regression is its ability to capture the nonlinear relationship between variables. While linear regression assumes a linear relationship between the dependent and independent variables, polynomial regression does not have this limitation and can better fit the data.

This is because by fitting a polynomial equation to the data points, instead of a straight line, polynomial regression can capture more complex patterns. As a result, polynomial regression models can provide a more accurate prediction of the dependent variable.

Another advantage of polynomial regression is its ability to model interaction effects. Interaction effects occur when the combined effect of two or more independent variables on the dependent variable is different from their individual effects.

Polynomial regression can capture these interaction effects by including cross-product terms, such as X1*X2 and X1*X3, in the regression equation. This allows for a more precise modeling of the relationship between variables.

## Comparison with Linear Regression

One of the primary differences between linear and polynomial regression is that linear regression models a straight line relationship between the dependent variable and the independent variable, while polynomial regression models a curved relationship. In linear regression, the regression line is represented by a straight line, while in polynomial regression, the regression line is represented by a curve.

In linear regression, the relationship between the independent variable and the dependent variable is modeled as follows:

Y = 0 + 1X +

Where Y is the dependent variable, X is the independent variable, 0 is the y-intercept, 1 is the slope, and is the error term. In polynomial regression, the relationship between the independent variable and the dependent variable is modeled by a polynomial function of degree n, as follows:

Y = 0 + 1X + 2X^2 + 3X^3 + + nX^n +

Where Y is the dependent variable, X is the independent variable, 0 is the y-intercept, 1, 2, 3, n are the coefficients of the polynomial equation, and is the error term.

If we plot the regression line for linear regression, it will be a straight line, while for polynomial regression, it will be a curved line.

## 4) Polynomial Regression – Linear or Non-Linear

Despite polynomial regression modeling a curve relationship between the dependent variable and the independent variable, it is still referred to as a linear model. This is because the polynomial function can be expressed as a linear combination of coefficients.

## Explanation of Why Polynomial Regression is Called Linear

The term “linear” in polynomial regression refers to the relationship between the coefficients, not the relationship between the dependent and independent variables. Polynomial regression is still considered linear because the coefficients themselves form a linear function.

## We can express the polynomial function as a linear combination of the coefficients as follows:

Y = 0 + 1X + 2X^2 + 3X^3 + + nX^n +

= 0 + 1X1 + 2X2 + 3X3 + + nXn +

Where X1 = X, X2 = X^2, X3 = X^3, , Xn = X^n. By expressing the polynomial function as a linear combination of the coefficients, we can calculate the coefficients using matrix algebra.

This makes it easier to compute the coefficients for higher degree polynomials.

## How Function can be Expressed as a Linear Combination of Coefficients

In matrix notation, the polynomial function can be expressed as:

Y = X +

Where Y is an n x 1 vector of the dependent variable, X is an n x (p+1) matrix of the independent variable, is a (p+1) x 1 vector of coefficients, and is an n x 1 vector of the error term. We can use the method of least squares to estimate the coefficients , which minimizes the sum of squared errors between the predicted and actual values of the dependent variable.

In conclusion, polynomial regression is a powerful tool that can capture complex relationships between the dependent and independent variables. While it may appear to be non-linear due to the curve relationship between the variables, polynomial regression is still considered a linear model because it can be expressed as a linear combination of the coefficients.

Understanding the difference between linear and polynomial regression can help data analysts choose the right model for their data and obtain more accurate predictions.

## 5) Example of Polynomial Regression in Python

In this section, we will provide an example of performing polynomial regression in Python. We will use a simple dataset of salaries and years of experience to demonstrate the steps involved in performing polynomial regression.

We will use the Pandas library to import and preprocess the data, scikit-learn library to perform linear and polynomial regression, and matplotlib library to visualize the results.

### Importing the Dataset

The first step is to import the dataset into our script. We will use the read_csv method of the Pandas library to read the dataset from a .csv file.

The following code will read the dataset and store it in a DataFrame:

```
import pandas as pd
dataset = pd.read_csv('salary_data.csv')
```

### Data Preprocessing

Once we have imported the dataset, we need to preprocess the data. We need to create a matrix of features and a dependent variable vector.

The matrix of features will contain the independent variable ‘YearsExperience,’ and the dependent variable vector will contain the ‘Salary’. We will use the following code to create the matrix of features and the dependent variable vector:

```
X = dataset.iloc[:, :-1].values
y = dataset.iloc[:, 1].values
```

### Fitting a Linear Regression Model

Before we jump into polynomial regression, let’s first fit a simple linear regression model to the data. We can use the scikit-learn library to fit a linear regression model.

The following code will fit the model and print the coefficients:

```
from sklearn.linear_model import LinearRegression
regressor = LinearRegression()
regressor.fit(X, y)
print("Coefficients:", regressor.coef_)
print("Intercept:", regressor.intercept_)
```

### Visualizing Results of the Linear Regression Model

After fitting the linear regression model, we can visualize the results using a scatter plot and a regression line. The scatter plot will show the relationship between the ‘YearsExperience’ and ‘Salary,’ while the regression line will show the best-fit line.

The following code will plot the scatter plot and the regression line:

```
import matplotlib.pyplot as plt
plt.scatter(X, y, color='red')
plt.plot(X, regressor.predict(X), color='blue')
plt.title('Salary vs Years of Experience (Linear Regression)')
plt.xlabel('Years of Experience')
plt.ylabel('Salary')
plt.show()
```

### Fitting a Polynomial Regression Model

Now, let’s fit a polynomial regression model to the data. To perform polynomial regression, we need to add polynomial features to the matrix of features.

We will use the PolynomialFeatures class of the scikit-learn library to add polynomial features. The following code will add polynomial features and fit a linear regression object to the data:

```
from sklearn.preprocessing import PolynomialFeatures
poly_reg = PolynomialFeatures(degree = 4)
X_poly = poly_reg.fit_transform(X)
lin_reg = LinearRegression()
lin_reg.fit(X_poly, y)
```

### Visualizing the Polynomial Regression Model

After fitting the polynomial regression model, we can visualize the results by plotting a higher resolution X-axis and predicting the corresponding Y-axis. The following code will plot the polynomial regression curve:

```
import numpy as np
X_grid = np.arange(min(X), max(X), 0.1)
X_grid = X_grid.reshape((len(X_grid), 1))
plt.scatter(X, y, color='red')
plt.plot(X_grid, lin_reg.predict(poly_reg.fit_transform(X_grid)), color='blue')
plt.title('Salary vs Years of Experience (Polynomial Regression)')
plt.xlabel('Years of Experience')
plt.ylabel('Salary')
plt.show()
```

### Predicting the Result

Finally, we can use the predict method of the linear regression object to predict the salary corresponding to the years of experience. The following code will predict the salary corresponding to 7.5 years of experience:

```
y_pred = lin_reg.predict(poly_reg.fit_transform([[7.5]]))
print("Predicted Salary:", y_pred)
```

## Conclusion

In conclusion, polynomial regression is a powerful tool for modeling complex relationships between variables. By adding polynomial features to the matrix of features, polynomial regression can capture nonlinear relationships.

In this article, we went through a step-by-step guide on how to perform polynomial regression in Python using scikit-learn and matplotlib libraries. We hope this article has given you a better understanding of how polynomial regression works and how to apply it to real-world datasets.

Polynomial regression is a powerful tool used to model complex non-linear relationships between variables. While linear regression assumes a linear relationship between the independent and dependent variables, polynomial regression can capture more complex patterns.

Polynomial regression is advantageous as it models interaction effects between independent variables and captures non-linear relationships more accurately, yielding better predictions. Despite the curve relationship between the variables, polynomial regression is still considered a linear model because it can be expressed as a linear combination of the coefficients.

Performing polynomial regression in Python is a useful skill for data analysts, using libraries such as Pandas, scikit-learn, and matplotlib to import, preprocess, fit, visualize, and predict datasets. Understanding the difference between linear and polynomial regression can help data analysts choose the right model for their data and obtain more accurate predictions.