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or if-then
rules are constructed. Heuristics are often used to arrange the sequence of decision
attributes. For example, if we find salary to be more powerful than personal assets
for credit analysis, then salary will be evaluated in front of personal assets in the
resulting decision tree.
ID3 and other similar methods deal with problems with a single-dimensional class
only, which can be called single-decision-tree-induction (SDTI) (Liu et al. 2000).
In the real world, there are problems with a multi-dimensional class, so multidecision-
tree-induction (MDTI) methods are needed (Liu et al. 2000). For instance,
in the car-buying decision problem, people may not consider which car but many
properties of a car to buy, e.g. a car with moderate luxury, low price and high
controllability. Methods can deal with this kind of problem may: (1) view the
problem as many independent and separated problems with a single-dimensional
class (Babicˇ et al. 1999), (2) transform a multi-dimensional class into another singledimensional
class (Babicˇ et al. 1999, Liu et al. 2000) or (3) integrate information
measurements for each class dimension into a single measurement (Suzuki et al.
2001).
Although existing methods can deal with multi-dimensional decision problems,
they also have a couple of known shortcomings. First, the hurdle values for attribute
segmentation are crisp, which is inconsistent with human information processing.
Using our previous example of credit analysis, an applicant making US$30 000
annually is considered good, but another person making US$29 999 will be
considered bad by our rule. It is obvious that the difference is not so sharp in
the real world. Second, the crisp nature of the hurdle values also affects the
robustness of the induced multi-dimensional decision trees. Because attribute
segmentation is determined by the training cases, the resulting knowledge model
based on crisp rules is more sensitive to the noises in the training data.
In this paper, we propose an approach called the multi-dimensional fuzzy
inductive learning method (MFILM) that integrates the fuzzy set theory into the
tree induction process to overcome these limitations. A major advantage of the fuzzy
approach is that it allows the classification process to be more flexible and the
resulting tree to be more accurate due to the reduced sensitivity to slight changes
of hurdle points. Our empirical studies confirm this hypothesis by showing that
MFILM can outperform than existing methods.
The remainder of the paper is organized as follows. Basic concept of the
inductive learning process is introduced in section 2. Section 3 will introduce
MDTI techniques. This is followed by section 4, which is an introduction to the
268 Y.-T. Chen and B. Jeng
fuzzy set concepts. Then, the MFILM is presented in section 5. Finally, empirical
results and conclusions are presented in sections 6 and 7 respectively.
2. SDTI learning
Induction is a process by which a knowledge structure can be created from a set of
data to explain or predict the behaviour of the data set. Early work of inductive
learning can be traced back to 1966 when Hunt et al. (1996) developed a method for
induction. This method was later implemented and expanded by Paterson and
Niblett (1982) to create ACLS (A Concept Learning System) and by Quinlan
(Quinlan 1979, 1986) to develop