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Decision Trees—
What Are They?
Introduction ...................................................................................... 1 Using Decision Trees with Other Modeling Approaches ...................... 5 Why Are Decision Trees So Useful? .................................................... 8 Level of Measurement ..................................................................... 11
Introduction
Decision trees are a simple, but powerful form of multiple variable analysis. They provide unique capabilities to supplement, complement, and substitute for
- traditional statistical forms of analysis (such as multiple linear regression)
- a variety of data mining tools and techniques (such as neural networks)
- recently developed multidimensional forms of reporting and analysis found in the field of business intelligence
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Decision trees are produced by algorithms that identify various ways of splitting a data set into branch-like segments. These segments form an inverted decision tree that originates with a root node at the top of the tree. The object of analysis is reflected in this root node as a simple, one-dimensional display in the decision tree interface. The name of the field of data that is the object of analysis is usually displayed, along with the spread or distribution of the values that are contained in that field. A sample decision tree is illustrated in Figure 1.1, which shows that the decision tree can reflect both a continuous and categorical object of analysis. The display of this node reflects all the data set records, fields, and field values that are found in the object of analysis. The discovery of the decision rule to form the branches or segments underneath the root node is based on a method that extracts the relationship between the object of analysis (that serves as the target field in the data) and one or more fields that serve as input fields to create the branches or segments. The values in the input field are used to estimate the likely value in the target field. The target field is also called an outcome, response, or dependent field or variable.
The general form of this modeling approach is illustrated in Figure 1.1. Once the relationship is extracted, then one or more decision rules can be derived that describe the relationships between inputs and targets. Rules can be selected and used to display the decision tree, which provides a means to visually examine and describe the tree-like network of relationships that characterize the input and target values. Decision rules can predict the values of new or unseen observations that contain values for the inputs, but might not contain values for the targets.
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decision tree , and each segment or branch is called a node. A node with all its descendent segments forms an additional segment or a branch of that node. The bottom nodes of the decision tree are called leaves (or terminal nodes ). For each leaf, the decision rule provides a unique path for data to enter the class that is defined as the leaf. All nodes, including the bottom leaf nodes, have mutually exclusive assignment rules; as a result, records or observations from the parent data set can be found in one node only. Once the decision rules have been determined, it is possible to use the rules to predict new node values based on new or unseen data. In predictive modeling, the decision rule yields the predicted value.
Figure 1.2: Illustration of Decision Tree Nomenclature
Chapter 1: Decision Trees—What Are They? 5
Although decision trees have been in development and use for over 50 years (one of the earliest uses of decision trees was in the study of television broadcasting by Belson in 1956), many new forms of decision trees are evolving that promise to provide exciting new capabilities in the areas of data mining and machine learning in the years to come. For example, one new form of the decision tree involves the creation of random forests. Random forests are multi-tree committees that use randomly drawn samples of data and inputs and reweighting techniques to develop multiple trees that, when combined, provide for stronger prediction and better diagnostics on the structure of the decision tree.
Besides modeling, decision trees can be used to explore and clarify data for dimensional cubes that can be found in business analytics and business intelligence.
Using Decision Trees with Other Modeling Approaches
Decision trees play well with other modeling approaches, such as regression, and can be used to select inputs or to create dummy variables representing interaction effects for regression equations. For example, Neville (1998) explains how to use decision trees to create stratified regression models by selecting different slices of the data population for in-depth regression modeling.
The essential idea in stratified regression is to recognize that the relationships in the data are not readily fitted for a constant, linear regression equation. As illustrated in Figure 1.3, a boundary in the data could suggest a partitioning so that different regression models of different forms can be more readily fitted in the strata that are formed by establishing this boundary. As Neville (1998) states, decision trees are well suited in identifying regression strata.
Chapter 1: Decision Trees—What Are They? 7
high level in the target. Categories 5, 6, and 7 join together to predict the lowest level of the target. And categories 8, 9, and 10 form the final group.
If a completely unordered grouping of the categorical codes is requested—as would be the case if the input was defined as “nominal”—then the 3 bins as shown in the bottom of Figure 1.4 might be produced. Here the categories 1, 5, 6, 7, 9, and 10 group together as associated with the highest level of the target. The medium target levels produce a grouping of categories 3, 4, and 8. The lone high target level that is associated with category 2 falls out as a category of its own.
Figure 1.4: Illustration of Forming Nodes by Binning Input-Target Relationships
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Since a decision tree allows you to combine categories that have similar values with respect to the level of some target value there is less information loss in collapsing categories together. This leads to improved prediction and classification results. As shown in the figure, it is possible to intuitively appreciate that these collapsed categories can be used as branches in a tree. So, knowing the branch—for example, branch 3 (labeled BIN 3), we are better able to guess or predict the level of the target. In the case of branch 2 we can see that the target level lies in the mid-range, whereas in the last branch—here collapsed categories 1, 5, 6, 7, 9, 10—the target is relatively low.
Why Are Decision Trees So Useful?
Decision trees are a form of multiple variable (or multiple effect) analyses. All forms of multiple variable analyses allow us to predict, explain, describe, or classify an outcome (or target). An example of a multiple variable analysis is a probability of sale or the likelihood to respond to a marketing campaign as a result of the combined effects of multiple input variables, factors, or dimensions. This multiple variable analysis capability of decision trees enables you to go beyond simple one-cause, one-effect relationships and to discover and describe things in the context of multiple influences. Multiple variable analysis is particularly important in current problem-solving because almost all critical outcomes that determine success are based on multiple factors. Further, it is becoming increasingly clear that while it is easy to set up one-cause, one-effect relationships in the form of tables or graphs, this approach can lead to costly and misleading outcomes.
According to research in cognitive psychology (Miller 1956; Kahneman, Slovic, and Tversky 1982) the ability to conceptually grasp and manipulate multiple chunks of knowledge is limited by the physical and cognitive processing limitations of the short- term memory portion of the brain. This places a premium on the utilization of dimensional manipulation and presentation techniques that are capable of preserving and reflecting high-dimensionality relationships in a readily comprehensible form so that the relationships can be more easily consumed and applied by humans.
There are many multiple variable techniques available. The appeal of decision trees lies in their relative power, ease of use, robustness with a variety of data and levels of measurement, and ease of interpretability. Decision trees are developed and presented incrementally; thus, the combined set of multiple influences (which are necessary to fully explain the relationship of interest) is a collection of one-cause, one-effect relationships
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Gender Weight Height Ht_Cent. BMIndex BodyType Female 179 4’10 147 162 slim Female 160 5’ 4 163 161 slim Male 191 5’ 8 173 182 average Male 132 5’1 155 143 slim Female 167 5’1 180 174 average Female 128 5’2 157 142 slim Female 150 5’2 157 154 slim Male 150 5’2 157 154 slim Female 215 5’2 157 184 heavy Female 89 5’3 160 119 slim Female 167 5’3 160 163 slim Male 180 5’4 163 171 average Male 206 5’4 163 183 average Male 239 5’5 165 199 heavy Male 161 5’6 168 164 average Male 188 5’6 168 178 average Male 284 5’6 168 218 heavy Female 117 5’7 170 141 slim Male 163 5’7 170 167 average Male 194 5’7 170 182 average Male 201 5’7 170 185 heavy Male 254 5’8 173 209 heavy Male 201 5’9 175 188 heavy Male 206 5’9 175 190 heavy Male 216 5’9 175 195 heavy Male 206 6’0 183 194 heavy Male 220 6’1 185 202 heavy Female 182 6’2 188 185 heavy
In this display, the overall average height is 5’6 and the overall average weight is 183. Among males, the average height is 5’7, while among females, the average height is 5’ (males weigh 200 on average, versus 155 for females).
Knowing the gender puts us in a better position to predict the height and weight of the individuals, and knowing the relationship between gender and height and weight puts us in a better position to understand the characteristics of the target. Based on the relationship between height and weight and gender, you can infer that females are both smaller and lighter than males. As a result, you can see how this sort of knowledge that is based on gender can be used to determine the height and weight of unseen humans.
From the display, you can construct a branch with three leaves to illustrate how decision trees are formed by grouping input values based on their relationship to the target.
Chapter 1: Decision Trees—What Are They? 11
Figure 1.5: Illustration of Decision Tree Partitioning of Physical Measurements
Level of Measurement
The example as shown here illustrates an important characteristic of decision trees: both quantitative and qualitative data can be accommodated in decision tree construction. Quantitative data, like height and weight, refers to quantities that can be manipulated with arithmetic operations such as addition, subtraction, and multiplication. Qualitative data, such as gender, cannot be used in arithmetic operations, but can be presented in tables or decision trees. In the previous example, the target field is weight and is presented as an average. Height, BMIndex, or BodyType could have been used as inputs to form the decision tree.
Some data, such as shoe size, behaves like both qualitative and quantitative data. For example, you might not be able to do meaningful arithmetic with shoe size, even though the sequence of numbers in shoe sizes is in an observable order. For example, with shoe size, size 10 is larger than size 9, but it is not twice as large as size 5.
Figure 1.6 displays a decision tree developed with a categorical target variable. This figure shows the general, tree-like characteristics of a decision tree and illustrates how decision trees display multiple relationships—one branch at a time. In subsequent figures, decision trees are shown with continuous or numeric fields as targets. This shows how decision trees are easily developed using targets and inputs that are both qualitative (categorical data) and quantitative (continuous, numeric data).
Low weight Average: 138 lb
Medium weight Average: 183 lb
Heavy weight Average: 227 lb
Root Node Average Weight: 183 lb
Chapter 1: Decision Trees—What Are They? 13
The splitting techniques that are used to split the 1–0 responses in the data set are used to identify alternative inputs (for example, income or purchase history) for gender. These techniques are based on numerical and statistical techniques that show an improvement over a simple, uninformed guess at the value of a target (in this example, best–other), as well as the reproducibility of this improvement with a new set of data.
Knowing the gender enables us to guess that females are 5% more likely to be a best customer than males. You could set up a separate, independent hold out or validation data set and (having determined that the gender effect is useful or interesting) you might see whether the strength and direction of the effect is reflected in the hold out or validation data set. The separate, independent data set will show the results if the decision tree is applied to a new data set, which indicates the generality of the results. Another way to assess the generality of the results is to look at data distributions that have been studied and developed by statisticians who know the properties of the data and who have developed guidelines based on the properties of the data and data distributions. The results could be compared to these data distributions and, based on the comparisons, you could determine the strength and reproducibility of the results. These approaches are discussed at greater length in Chapter 3, “The Mechanics of Decision Tree Construction.”
Under the female node in the decision tree in Figure 1.6, female customers can be further categorized into best–other categories based on the total lifetime visits that they have made to HomeStuff stores: those who have made fewer than 3.5 visits are less likely to be best customers compared to those who have made more than 4.5 visits: 29% versus 100%. (In the survey, a shopping visit of less than 20 minutes was characterized as a half visit.)
On the right side of the figure, the decision tree is asymmetric; a new field— Net sales — has entered the analysis. This suggests that Net sales is a stronger or more relevant predictor of customer status than Total lifetime visits , which was used to analyze females. It was this kind of asymmetry that spurred the initial development of decision trees in the statistical community: these kinds of results demonstrate the importance of the combined (or interactive) effect of two indicators in displaying the drivers of an outcome. In the case of males, when Net sales exceed $281.50, then the likelihood of being a best customer increases from 45% to 77%.
As shown in the asymmetry of the decision tree, female behavior and male behavior have different nuances. To explain or predict female behavior, you have to look at the interaction of gender (in this case, female) with Total lifetime visits. For males, Net sales is an important characteristic to look at.
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In Figure 1.6, of all the k-way or n-way branches that could have been formed in this decision tree, the 2-way branch is identified as best. This indicates that a 2-way branch produces the strongest effect. The strength of the effect is measured through a criterion that is based on strength of separation, statistical significance, or reproducibility, with respect to a validation process. These measures, as applied to the determination of branch formation and splitting criterion identification, are discussed further in Chapter 3.
Decision trees can accommodate categorical (gender), ordinal (number of visits), and continuous (net sales) types of fields as inputs or classifiers for the purpose of forming the decision tree. Input classifiers can be created by binning quantitative data types (ordinal and continuous) into categories that might be used in the creation of branches— or splits—in the decision tree. The bins that form total lifetime visits have been placed into three branches:
- < 3.5 … less than 3.
- [3.5 – 4.5) … between 3.5 to strictly less than 4.
= 4.5 … greater than or equal to 4.
Various nomenclatures are used to indicate which values fall in a given range. Meyers (2000) proposes an alternative, which is shown below:
- < 3.5 … less than 3.
- [3.5 – 4.5[ … between 3.5 to strictly less than 4.
= 4.5 … greater than or equal to 4.
The key difference from the convention used in the SAS decision tree is in the second range of values, where the designator “[” is used to indicate the interval that includes the lower number and includes up to any number that is strictly less than the upper number in the range.
A variety of techniques exist to cast bins into branches: 2-way (binary branches), n-way (where n equals the number of bins or categories), or k-way (where k represents an attempt to create an optimal number of branches and is some number greater than or equal to 2 and less than or equal to n).
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Decision trees can contain both categorical and numeric (continuous) information in the nodes of the tree. Similarly, the characteristics that define the branches of the decision tree can be both categorical or numeric (in this latter case, the numeric values are collapsed into bins—sometimes called buckets or collapsed groupings of categories—to enable them to form the branches of the decision tree).
Figure 1.8 shows how the Fisher-Anderson iris data can yield three different types of branches when classifying the target SETOSA versus OTHER (Fisher 1936); in this case, 2-, 3-, and 5-leaf branches. There are 50 SETOSA records in the data set. With the binary partition, these records are classified perfectly by the rule petal width <= 6 mm. The 3- way and 5-way branch partitions are not as effective as the 2-way partition and are shown only for illustration. More examples are provided in Chapter 2, “Descriptive, Predictive, and Explanatory Analyses,” including examples that show how 3-way and n-way partitions are better than 2-way partitions.
Figure 1.8: Illustration of Fisher-Anderson Iris Data and Decision Tree Options
(a) Two Branch Solution
(b) Three Branch Solution
(c) Five Branch Solution
