# Polynomial

Polynomial regression modeling explained in detail.

**Polynomial Regression Modeling**

Polynomial regression is used to model the relationship between a dependent variable and an independent variable using a polynomial function. It is an extension of simple linear regression and multiple regression, where the relationship between the variables is modeled as a polynomial equation instead of a linear equation.

In polynomial regression, the goal is to estimate the parameters of a polynomial function that best fits the observed data. The polynomial function Y = β0 + β1X + β2X^2 + … + βn*X^n where:

- Y is the dependent variable, or response that we want to model or predict.
- X is the independent variable, or predictor that we believe may have an effect on the dependent variable.
- β0, β1, β2, …, βn are the coefficients, or parameters), of the polynomial function that represent the strength and direction of the relationship between the variables.
- X^2, X^3, …, X^n are the higher-order terms of the independent variable, raised to different powers (2, 3, …, n), allowing for non-linear relationships between the variables.
- n is the degree of the polynomial, which determines the complexity of the model and the number of higher-order terms included.

The goal of polynomial regression is to estimate the values of the coefficients (β0, β1, β2, …, βn) that best fit the observed data. This is often accomplished using ordinary least squares (OLS) estimation or other regression estimation methods.

When the polynomial regression model is estimated, it can be used for purposes such as prediction, estimation, hypothesis testing, and inference. Polynomial regression can be used to predict the value of the dependent variable for new observations, assess the significance and direction of the relationship between variables, identify influential variables, and evaluate the overall fit of the model.

We use polynomial regression modeling when there is evidence of non-linear relationships between variables, and a linear model is not adequate to capture the underlying patterns in the data. Polynomial regression allows us greater flexibility in regression modeling non-linear relationships between variables and can provide more accurate predictions in situations where linear regression may not be sufficient.