Fast-and-frugal trees
In the study of decision-making, including the disciplines of psychology, artificial intelligence, and management science, a fast-and-frugal tree is a type of classification tree or decision tree. As shown in Figure 1--which will be explained in detail later--fast-and-frugal trees are simple graphical structures that ask one question at a time. The goal is to classify an object into a category for the purpose of making a decision. Unlike other classification and decision trees, such as Leo Breiman's CART, fast-and-frugal trees have been defined to be intentionally simple, both in their construction as well as their execution, and operate speedily with little information. For example, the tree of Figure 1 only asks from one to maximum three questions.
Fast-and-frugal trees were introduced and conceptualized in 2003 by Laura Martignon, Vitouch, Takezawa and Forster and constitute a family of simple heuristics in the tradition of Gerd Gigerenzer and Herbert A. Simon's view of formal models of heuristics. Before the term fast-and-frugal trees was coined in 2003, these models of heuristics had been used in several contexts without having been explicitly conceptualized or defined as such .
In tasks where a binary decision or classification needs to be made and there are m cues, available for making such a decision, an FFT is defined as follows:
A fast-and-frugal tree is a decision tree that has m+1 exits, with one exit for each of the first m -1 cues and two exits for the last cue.
Mathematically, fast-and-frugal trees can be viewed as lexicographic heuristics or as linear models with non-compensatory weights as proven by Martignon, Katsikopoulos and Woike in 2008. Their formal properties and construction have also been analyzed using signal detection theory by Luan, Schooler and Gigerenzer in 2011 .
How a fast-and-frugal tree works
This section describes how to construct and use a fast-and-frugal tree.Construction
Recall that the basic elements for making a binary classification are the cues, which are here assumed to be binary. In a fast-and-frugal tree, the cues are ranked, with one cue at each level of the tree and an exit node at each level. Whenever a cue is used, a question is asked about the value of the cue. The answers to the questions might immediately lead to an exit, or they might lead to a further question. A characteristic property of fast-and-frugal trees is that, for each question, there is at least one possible answer that leads to an exit.In the literature on fast-and-frugal trees, many different algorithms have been proposed for ordering cues and deciding which possible answer to a question about a cue leads immediately to an exit. Note that if and are done, a fast-and-frugal tree is fully defined. Often, in order to keep construction simple and intuitive, the algorithms use simple measures of cue "goodness" and make simple choices about exits, but more complex algorithms have been proposed as well.
Execution
To use a fast-and-frugal tree, begin at the root and check one cue at a time. At each step, one of the possible outcomes is an exit node which allows for a decision -- if an exit is reached, stop; otherwise, continue until an exit is reached. you take an exit, stop; otherwise, continue and ask more questions until an exit is reached.Figure 1 illustrates a fast-and-frugal tree for classifying a patient as “high risk” of having a heart stroke and thus having to be sent to the “coronary care unit” or as "low risk” and thus having to be sent to a “regular nursing bed" .
Consider three patients, John, Mary, and Jack:
- John has ST segment changes thus is classified as "high risk" and sent to the coronary care unit-- without considering other cues.
- Mary has no ST segment changes, does have chest pain as her chief complaint, but does not have any of the remaining five factors, thus is classified as "low risk" and sent to a regular nursing bed, after all three cues are checked.
- Jack has no ST segment change and does not have chest pain as his chief complaint, thus is classified as "low risk" and sent to a regular nursing bed, by considering these two cues.
Performance
A signal detection analysis of fast-and-frugal trees
Fast-and-frugal trees are used to perform binary classifications or decisions. In psychology, medicine, and other fields, signal detection theory has been the classic theory under which such tasks are analyzed.The theory assumes that there are two categories of events or people, of which the category more relevant to us is referred as “signal” while the other is referred to as “noise.” The two differ in their distribution on an observation scale that we may call “evidence,” with the signal distribution having a higher mean. One can make two possible classifications, namely “signal” or “noise,” upon gathering the evidence. This leads to four possible outcomes: hit, correct rejection, miss, and false alarm. To maximize overall accuracy or the expected value of a classification, the theory posits that we need to carefully select the classification criterion on the evidence scale, above which we make a “signal” decision and below which “noise.” Specially, when the cost of a miss is very high, a lower, more “liberal” criterion needs to be selected, whereas when the cost of a false alarm is very high, a higher, more “conservative” criterion will be better. This implies that a good decision-maker needs to be properly biased in most real-world situations; this is the most critical and relevant insight from signal detection theory on classification and decision making.
In 2011, Luan, Schooler, and Gigerenzer analyzed characteristics of fast-and-frugal trees from the perspective of signal detection theory. There are several key findings from this analysis. First, the choice of the exit structure of a fast-and-frugal tree corresponds to the setting of the decision criterion in signal detection. In a nutshell, the earlier a "signal exit" appears in a fast-and-frugal tree, the more liberally biased is the tree. The relative biases of two fast-and-frugal trees are determined by the first exit in which the two differ, with the one having the “signal exit” - denoted by "s" - always being more liberal as the one having the "noise exit" - denoted by "n". For example, an FFTsnnn is more liberally biased than an FFTnsss. This principle is referred to as the “lexicographic decision bias” of fast-and-frugal trees.
Second, a series of simulations show that fast-and-frugal trees with different exit structures will lead to different—sometimes drastically different—expected value of a decision when the consequences of a miss and a false alarm differ. Therefore, when constructing and applying a fast-and-frugal tree, one needs to choose an exit structure that matches well the decision payoff structure of a task.
Third, the overall sensitivity of a fast-and-frugal tree—that is, how well the tree can discriminate a signal from a noise and which can be measured by d’ or A’ from signal detection theory—is affected by properties of the cues that make up the tree, such as the mean and variance of the cues’ sensitivities and the inter-cue correlations among the cues, but not much by the exit structure of the tree.
And finally, the performance of fast-and-frugal trees is robust and comparable to much more sophisticated decision algorithms developed in signal detection theory, including the ideal observer analysis model and the optimal sequential sampling model. In the context of out-of-sample predictions, fast-and-frugal trees perform the best relative to other models when the learning sample size is relatively small.