【Interaction Terms and Nonlinear Relationships】: Handling Interaction Terms and Nonlinear Relationships in Linear Regression Models

# Interactions and Nonlinear Relationships in Linear Regression Models

In the fields of data analysis and machine learning, we often encounter issues involving interactions and nonlinear relationships. An interaction term refers to the product of two or more variables, used to capture the mutual influence between them; nonlinear relationships, on the other hand, indicate that the relationship between the target variable and features is not a simple linear one, but might be curvilinear or of some other form. Understanding interactions and nonlinear relationships is crucial for building more accurate models and improving predictive accuracy.

By studying this chapter, we will delve into the concepts and significance of interactions and nonlinear relationships, as well as how to consider them when building models, laying the groundwork for the content of subsequent chapters.

# 2. Basics of Linear Regression Models

### 2.1 Principle of Linear Regression
Linear regression is a linear method used to model the relationship between a target variable and one or more independent variables. Its principle involves finding the best fit line by minimizing the difference between actual observed values and model predictions, thus describing the relationship between variables.

The linear regression model can be represented as: $y = b_0 + b_1 * x$, where $y$ is the target variable, $x$ is the independent variable, $b_0$ is the intercept, and $b_1$ is the slope. By fitting data points, we obtain the optimal values of $b_0$ and $b_1$.

### 2.2 Ordinary Least Squares
Ordinary least squares is a commonly used method for estimating parameters in linear regression, aiming to minimize the sum of squared residuals between actual observed values and model predictions. By minimizing the sum of squared residuals, the optimal regression coefficients are determined, resulting in the best-fit line.

In ordinary least squares, we attempt to find a line such that the sum of distances from all data points to this line is minimized. This can be achieved by minimizing a loss function, which is typically defined as the sum of squared residuals.

### 2.3 Evaluation Metrics for Regression Models

In practical applications, ***monly used regression model evaluation metrics include Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Coefficient of Determination ($R^2$), etc.

- **Mean Squared Error (MSE)**: Calculates the mean of squared differences between predicted values and actual values, reflecting the model's predictive accuracy.
- **Root Mean Squared Error (RMSE)**: The square root of MSE, offering a better representation of the differences between predicted values and actual values.
- **Coefficient of Determination ($R^2$)**: Describes how much of the variance in the dependent variable can be explained by changes in the independent variables, with values ranging from 0 to 1, where a value closer to 1 indicates a better model fit.

In practical applications, choosing the right evaluation metrics can effectively determine the strengths and weaknesses of a model and guide model selection and tuning.

# 3.1 What Are Interaction Terms

In linear regression, interaction terms are new variables obtained by multiplying two or more independent variables, used to capture the relationship between different independent variables. They are typically represented as $X_1 \times X_2$. In actual modeling, introducing interaction terms can help describe nonlinear relationships more accurately, enhancing the model's fit.

### 3.2 Why Introduce Interaction Terms

Introducing interaction terms helps explore the relationship between different independent variables, bringing the model closer to real-world scenarios. In the real world, the impact of many variables is not independent; interactions can lead to changes in the final outcome. Therefore, by introducing interaction terms, we can better understand the complex relationships between these variables.

### 3.3 How to Construct Interaction Terms

The methods for constructing interaction terms mainly include the following:

- **Direct Multiplication**: Simply multiply two independent variables to form an interaction term.
- **Centering**: First, center the original variables, then multiply to obtain the interaction term.
- **Standardization**: Standardize the variables before multiplying to form the interaction term.
- **Higher-order Interaction Terms**: Consider introducing higher-order interaction terms, such as $X_1 \times X_2 \times X_3$.

By employing suitable interaction term construction methods, we can better uncover the relationships between variables and enhance the model's performance.

In this section, we have delved into the application of interaction terms in linear regression. We introduced the concept of interaction terms, explained why they are needed, and described methods for constructing them. In the next section, we will see the application of interaction terms in actual modeling and their impact on the model.

# 4. Methods for Handling Nonlinear Relationships

### 4.1 Polynomial Regression
Polynomial regression is a regression analysis method in which the relationship between independent variables and dependent variables can be approximated using a polynomial function. In the following, we will explore the concept of polynomial regression, its application scenarios, and deepen our understanding