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sions, just one row is kept with the same values of
conditional attributes and a single decision attached
with this row that have the value of the most common
decision for rows from the group. The third approach
is well known in the rough set theory (Pawlak, 1991;
Skowron and Rauszer, 1992) and is called generalized
decision – GD approach. In this case, an inconsistent
decision table is transformed into the table with many-
valued decisions and after that each set of decisions is
encoded by a number (decision) such that equal sets
are encoded by equal numbers and different sets are
encoded by different numbers.
In literature, often, problems that are connected
with multi-label data are considered for classifica-
tion (multi-label classification problem) (Clare and
King, 2001; Comit´e et al., 2003; Loza Menc´ıa and
F¨urnkranz, 2008; Tsoumakas and Katakis, 2007;
Tsoumakas et al., 2010; Zhou et al., 2012). How-
ever, in this paper the aim is to study decision trees
for multi-label decision tables for knowledge repre-
sentation. Furthermore, decision tree has been used as
the model to represent such knowledge and the goal
is to compare the decision tree structure for different
heuristics and different approaches of representation
of inconsistent decision table.
In this paper, a greedy heuristic ‘misclassifica-
tion error’ has been introduced for decision table with
many-valued decisions. Its performance has been
compared with the heuristic ‘number of boundary
subtables’ from the paper (Azad et al., 2013). Data
sets have been used from UCI ML repository as well
as KEEL repository (Alcal-Fdez et al., 2009). The
advantages of the KEEL data sets are twofolds: (1)
they have big amount of data, and (2) they are real
life examples of decision tables with many-valued de-
cisions. The heuristic ‘numberof boundarysubtables’
is complex in terms of time, and memory require-
ment. Hence, it is essential to design new heuristic
which gives equal or better results but with less time
complexity, and memory requirement. At the end, re-
sults have been presented which show that the use of
MCD and, especially, MVD approaches can reduce
the complexity of trees in comparison with GD ap-
proach. The goal is to find all decisions for the case
GD whereas a fixed decision for the case of MCD and
arbitrary decision for the case of MVD for a particu-
lar row. That means we are moving from highly re-
stricted decision constraint to less restricted decision
constraint. Hence we usually get less complex tree
for the last case than others. This comparison is cru-
cial for knowledge representation since we can get the
useful knowledge in the form of less complicated de-
cision trees.
This paper consists of five sections. Section 2 con-
tains main definitions. In Sect. 3, the greedy algo-
rithm for construction of decision trees is presented.
Section 4 contains results of experiments and Sect. 5
concludes the paper.
2 MAIN DEFINITIONS
A decision table is a rectangular table T filled by
non-negative integers. Columns of this table are la-
beled with conditional attributes f
1
, . . . , f
n
. If we have
strings as values of attributes, we have to encode the
values as nonnegativeintegers. In addition, if we have
real valued data, we have to discretize the value to use
in this format. Rows of the table are pairwise differ-
ent, and each row is labeled with a natural number
(decision) which is interpreted as a value of the de-
cision attribute. To differentiate with decision table
with many-valued decisions, we sometimes call it de-
cision table with one-valued decisions.
It is possible that T is inconsistent, i.e., contains
equal rows with different decisions. The table T can
contain also equal rows with equal decisions. The
most frequent decision attached to rows from a group
of rows in a decision table T with one-valued decision
is called the most common decision for this group of
rows. For approach called most common decision –
MCD, we transform inconsistent decision table T into
consistent decision table T
MCD
with one-valued deci-
sion. Instead of a group of equal rows with different
decisions, we consider one row from the group and
we attach to this row the most common decision for
the considered group of rows.
For approach called generalized decision – GD,
we transform inconsistent decision table T into con-
sistent decision table T
GD
with one-valued decisions.
Instead of a group of equal rows with different de-
cisions, we consider one row from the group and we
attach to this row the set of all decisions for rows from
the group. Then instead of a set of decisions we attach
to each row a code of this set – a natural number such
that the codes of equal sets are equal and the codes of
different sets are different.
For approach called many-valued decisions –
MVD, we transform an inconsistent decision table T
into a decision table T
MVD
with many-valued deci-
sions. Instead of a group of equal rows with different
decisions, we consider one row from the group and
we attach to this row the set of all decisions for rows
from the group (Moshkov and Zielosko, 2011).
Note that each decision table with one-valued de-
cisions can be interpreted also as a decision table with
many-valued decisions. In such table, each row is la-
beled with a set of decisions which has one element.
'MisclassificationError'GreedyHeuristictoConstructDecisionTreesforInconsistentDecisionTables
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