Non-linear regression is a type of statistical analysis that is used to model relationships between variables that are not linear. In other words, it is used to model relationships where the change in one variable is not directly proportional to the change in another variable. One popular method of non-linear regression is using regression with bagging (also known as bootstrap aggregating) and Classification and Regression Trees (CART) algorithm.

Regression with bagging is a technique that combines multiple decision tree models to improve the stability and accuracy of the predictions. It works by creating multiple subsets of the data using a technique called bootstrapping, and then training a decision tree model on each subset. The final predictions are made by averaging the predictions from all the decision tree models.

Classification and Regression Trees (CART) is a decision tree-based algorithm that can be used for both classification and regression tasks. It works by recursively splitting the data into subsets based on the values of the predictor variables, and then using these subsets to make predictions.

In R, there are several packages available for building regression with bagging and CART models, such as the ‘ipred’ and ‘rpart’ packages. These packages provide functions for creating and training regression with bagging and CART models, as well as functions for evaluating the performance of the model.

The process of building a regression with bagging and CART model in R typically involves the following steps:

1. Prepare the data: The first step is to prepare the data for the model. This may involve cleaning the data, splitting it into training and testing sets, and scaling the variables.
2. Define the model: The next step is to define the structure of the model, including the number of decision trees and the number of features to consider when creating each tree.
3. Train the model: The model is trained using the prepared data. The model will create multiple decision trees using random subsets of the data and features with bagging and using CART algorithm.
4. Evaluate the model: The model’s performance is evaluated using various metrics such as accuracy, precision, recall, and F1 score.
5. Make predictions: Once the model is trained and evaluated, it can be used to make predictions on new data.

By using regression with bagging and CART algorithm in R, you can model non-linear relationship and get accurate predictions. It’s particularly useful when you have multiple variables and complex relationships between them. Regression with bagging and CART algorithm is considered as one of the most accurate and robust algorithm for non-linear regression and it can handle large amount of data and missing values as well.