Electron microscopy
 
Regression Tree/Decision Tree for Regression
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A regression tree, also known as a decision tree for regression, is a machine learning algorithm used for solving regression problems. It is a type of supervised learning technique that is primarily employed for predicting a continuous numeric output variable, as opposed to a classification tree, which is used for predicting categorical labels.

Here's how a regression tree works:

  1. Tree Structure: A regression tree consists of a tree-like structure where each node represents a decision or a splitting point based on one of the input features. The tree starts with a root node and branches into multiple child nodes based on specific criteria.

  2. Splitting Criteria: At each internal node (non-leaf node) of the tree, a decision is made about which feature and which threshold value should be used to split the data into two subsets. The goal is to find the feature and threshold that minimizes the variance or some other measure of error in the target variable within each subset. Common measures of error for regression trees include mean squared error (MSE) or mean absolute error (MAE).

  3. Leaf Nodes: The splitting process continues recursively until a stopping criterion is met. This criterion could be a predefined depth limit, a minimum number of data points in a node, or a minimum error threshold. When the stopping criterion is met, the node becomes a leaf node, and it contains the predicted value for the target variable. This prediction is typically the mean (or another statistic) of the target variable values in that leaf node's subset of data.

  4. Prediction: To make predictions for new data points, you traverse the tree from the root node down to a leaf node, following the decision rules at each node. When you reach a leaf node, you use the predicted value stored in that leaf node as the final prediction for the input data.

Regression trees are easy to interpret and visualize, making them useful for gaining insights into the relationships between input features and the target variable. However, they are prone to overfitting the training data, especially when the tree becomes too deep and complex. To mitigate this issue, techniques like pruning (removing or simplifying branches of the tree) and using ensemble methods like Random Forests are often employed.

JMP (pronounced "jump") is a powerful statistical software package primarily used for data analysis, visualization, and exploration. It offers a wide range of statistical and data mining tools, including decision trees, which can be used for both classification and regression tasks.

In JMP, you can create regression trees using the "Fit Model" platform or the "Partition" platform, depending on the version and edition of JMP you are using. Here's how you can create a regression tree in JMP:

  1. Using the Fit Model Platform:

    • Open your dataset in JMP.
    • Go to the "Analyse" menu, and then select "Fit Model."
    • In the "Fit Model" dialog box, specify your response (dependent) variable and predictor (independent) variables.
    • Click the "Run" button to fit the model.
    • In the results, you can explore various statistics and visualizations, including a decision tree plot, which displays the tree structure and the variable splits.
  2. Using the Partition Platform:
    • Open your dataset in JMP.
    • Go to the "Analyse" menu, and then select "Partition."
    • In the "Partition" dialog box, specify your response variable and predictor variables.
    • You can also set options related to tree complexity, such as the maximum depth of the tree or the minimum number of observations per leaf node.
    • Click the "OK" button to create the regression tree.

Once you've created the regression tree in JMP, you can visualize and interpret the results. JMP provides various graphical and tabular representations of the tree, making it easy to understand the relationships between the predictor variables and the response variable.

Keep in mind that the specific steps and features available in JMP may vary slightly depending on the version of the software you are using, so it's a good idea to consult the documentation or help resources provided with your version of JMP for more detailed instructions.

The key difference between classification trees and regression trees lies in the nature of the response variable they are designed to predict:

  1. Classification Trees:

    • Response Variable: Classification trees are used for predicting categorical or discrete class labels. The response variable in classification trees represents categories or classes. Examples of classification tasks include spam detection (classifying emails as spam or not spam), disease diagnosis (classifying patients into disease categories), and sentiment analysis (classifying text as positive, negative, or neutral).
    • Node Outputs: In a classification tree, the leaf nodes (end nodes) contain the predicted class label for the input data point. The prediction is typically the majority class label among the data points that reach that leaf node during the tree traversal.
  2. Regression Trees:
    • Response Variable: Regression trees are used for predicting a continuous numeric output variable. The response variable in regression trees represents a real-valued quantity. Examples of regression tasks include predicting house prices, estimating a person's age based on certain features, or forecasting stock prices.
    • Node Outputs: In a regression tree, the leaf nodes contain the predicted numeric value for the input data point. The prediction is typically the mean, median, or some other statistical measure of the target variable values among the data points that reach that leaf node.

The predicted value for a regression tree node can be given by,

          prediction of regression tree --------------------------------------- [4003a]

where,

  • represents the predicted value for the regression tree node .
  • is the sum of the actual target values (yi) for all data points in the region .
  • The region is the set of data points that fall into the particular node of the regression tree.
  • |Rm| is the number of data points in the region , i.e., the cardinality or count of the set .

Therefore, is calculated as the average of the target values for all the data points in the region Rm. Then, we have the mean squared error (MSE) for the regression tree node , given by,

          prediction of regression tree --------------------------------------- [4003b]

where,

          The numerator on the right side is the sum of the squared differences between the actual target values ) and the predicted values (y^m) for all data points in the region .

The mean squared error is a commonly used metric to assess the performance of regression models.

Figure 4003a plots month and latitude which are suitable for ski around the world.

Month and latitude which are suitable for ski around the world

Figure 4003a. Month and latitude which are suitable for ski around the world. (Code)

Figure 4003b plots the regression tree with the month and latitude, which are suitable for ski around the world, described in Figure 4003a.

Month and latitude which are suitable for ski around the world

Figure 4003b. Regression tree with the month and latitude, which are suitable for ski around the world, described in Figure 4003a. (Code)

Note that decision trees can be used for both classification and regression tasks, and they are capable of handling both categorical and numerical variables.

Furthermore, decision trees are fairly high variance models. (see page4313)

Table 4003. Applications and related concepts of decision tree.

Applications Page
Categorical variables Introduction

 

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