# Mastering Polynomial Regression: Beyond Straight Lines in Predictions

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## Understanding Polynomial Regression

Linear regression is an effective method for forecasting continuous outcomes. It works well when the relationship between the independent variable (X) and the dependent variable (Y) is linear. This means that as (X) varies at a consistent rate, (Y) should also change accordingly. To illustrate this, I created a data frame using the following code.

In linear regression, our goal is to identify the optimal line that can predict future values of a target variable. Generally, as (X) increases, (Y) tends to increase as well. Despite some variability, this linear relationship is critical as linear regression relies on this assumption. When we draw a predictive line for unseen data, the distance between the predicted and actual values is what we refer to as 'error'.

### Section 1.1: Evaluating Prediction Error

In the graph below, we can visualize how to calculate this error. By measuring the distance from each data point (represented as blue dots) to the fitting line (the red line), we can derive various metrics. For this discussion, we will employ the root mean square error (RMSE).

This is what the graph looks like with a predictive line drawn:

As indicated, the RMSE allows us to ascertain the accuracy of our predictions. Assuming a linear trend, we can develop a robust model as long as we correctly gauge the slope.

However, if the relationship between (X) and (Y) is non-linear, a straight line is no longer the best fit. Let's examine a scenario where this is the case.

### Section 1.2: Non-Linear Relationships

Here, I generated a new dataset:

In this situation, the graph displays a non-linear relationship that begins to exhibit exponential growth.

As we can see, attempting to fit a straight line fails to capture the underlying relationship, resulting in a significant error due to the curvature of the data.

#### Subsection 1.2.1: Finding Solutions

So, how do we address this challenge?

To resolve this issue, we must transform our independent variables. To expedite the process, I will reveal that the variables demonstrate a cubic relationship. While you may discover this through your own analysis, I will provide this insight for clarity in this article.

By cubing the (X) variable, we established a new foundation for our linear regression.

This transformation resulted in a model that better captures the underlying quadratic relationship in the data.

## Summary

In summary, it is essential to recognize that a linear correlation between the variables and the target is not always present. In this instance, identifying a cubic relationship was relatively straightforward. In most scenarios, employing the PolynomialFeatures() package will be necessary to accurately determine the relationship.

Stay tuned for future articles that delve into using PolynomialFeatures() to streamline these processes.

If you're interested in reviewing the Jupyter Notebook containing this refined code, feel free to reach out.

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