For data sets with missing attribute values, the corresponding functionρ is incompletely specified (partial). A decision table with incompletely specified function? will be calledincompletely specified, orincomplete.
In the sequel we will assume that all decision values are specified, i.e., they are not missing. Also, we will assume that all missing attribute values are denoted by “?”, by “*” or by “–”, lost values will be denoted by “?”, “do not care” conditions will be denoted by “*”, and attribute-concept values by “–”. Additionally, we will assume that for each case at least one attribute value is specified.
Incomplete decision tables are described by characteristic relations instead of indiscernibility relations. Also, elementary sets are replaced by characteris- tic sets. The characteristic set was called a (binary) neighborhood in [16–18]. An example of an incomplete table is presented in Table 2.
For incomplete decision tables the definition of a block of an attribute- value pair must be modified.
• If an attributeathere exists a casexsuch thatρ(x, a) = ?, i.e., the corre- sponding value is lost, then the casexshould not be included in any block [(a, v)] for all valuesv of attributea.
• If for an attributeathere exists a casexsuch that the corresponding value is a “do not care” condition, i.e.,ρ(x, a) =∗, then the corresponding casex
should be included in blocks [(a, v)] for all specified valuesvof attributea.
Table 2.An incomplete decision table
Attributes Decision Case Temperature Headache Nausea Flu
1 High – No Yes
2 Very high Yes Yes Yes
3 ? No No No
4 High Yes Yes Yes
5 High ? Yes No
6 Normal Yes No No
7 Normal No Yes No
• If for an attributeathere exists a casexsuch that the corresponding value is a attribute-concept value, i.e., ρ(x, a) =−, then the corresponding case
xshould be included in blocks [(a, v)] for all specified valuesv of attribute
athat are members of the setV(x, a), where
V(x,a) ={ρ(y,a)|ρ(y,a)is specified,y ∈U, ρ(y,d) =ρ(x,d)},
anddis the decision.
These modifications of the definition of the block of attribute-value pair are consistent with the interpretation of missing attribute values, lost, “do not care” conditions, and attribute-concept values. Also, note that the attribute- concept value is the most universal, since ifV(x, a) =∅, the definition of the attribute-concept value is reduced to the lost value, and if V(x, a) is the set of all values of an attributea, the attribute-concept value becomes a “do not care” condition.
For Table 2, for case 1,ρ(1, Headache) =−, and V(1, Headache) ={yes}, so we add the case 1 to [(Headache, yes)]. For case 3,ρ(3, T emperature) = ?, hence case 3 is not included in either of the following sets: [(Temperature, high)], [(Temperature, very high)], and [(Temperature, normal)]. Similarly,
ρ(5, Headache) = ?, so the case 5 is not included in [(Headache, yes)] and [(Headache, no)]. Also, ρ(8, T emperature) = −, and V(8, T emperature) =
{high, very high}, so the case 8 is a member of both [(Temperature, high)] and [(Temperature, very high)]. Finally, ρ(8, N ausea) = ∗, so the case 8 is included in both [(Nausea, no)] and [(Nausea, yes)]. Thus,
[(Temperature, high)] ={1, 4, 5, 8}, [(Temperature, very high)] ={2, 8}, [(Temperature, normal)] = {6, 7}, [(Headache, yes)] ={1, 2, 4, 6, 8}, [(Headache, no)] ={3, 7},
[(Nausea, no)] = {1, 3, 6, 8}, [(Nausea, yes)] ={2, 4, 5, 7, 8}.
For a casex∈U, thecharacteristic setKB(x) is defined as the intersection
of the setsK(x, a), for alla∈B.
If ρ(x, a) is specified, then K(x, a) is the block [(a, ρ(x, a)] of attributea
and its value ρ(x, a). If ρ(x, a) =∗ or ρ(x, a) = ? then the setK(x, a) =U. If ρ(x, a) = − and V(x, a) is nonempty, then the corresponding set K(x, a) is equal to the union of all blocks of attribute-value pairs (a, v), where v ∈ V(x, a). IfV(x, a) is empty, thenK(x, a) ={x}.
The way of computing characteristic sets needs a comment. For both “do not care” conditions and lost values the corresponding setK(x, a) is equal to
U because the corresponding attributea does not restrict the set KB(x): if ρ(x, a) =∗, the value of the attributeais irrelevant; ifρ(x, a) = ?, only existing values need to be checked. However, the case when ρ(x, a) = −is different, since the attributearestricts the setKB(x). Furthermore, the description of
KB(x) should be consistent with other (but similar) possible approaches to
missing attribute values, e.g., an approach in which each missing attribute value is replaced by the most common attribute value restricted to a concept. Here the setV(x, a) contains a single element and the characteristic relation is an equivalence relation. Our definition is consistent with this special case in the sense that if we compute a characteristic relation for such a decision table using our definition or if we compute the indiscernibility relation as for complete decision tables using definitions from Sect. 2, the result will be the same. For Table 2 andB=A,
KA(1) ={1,4,5,8} ∩ {1,2,4,6,8} ∩ {1,3,6,8}={1,8}, KA(2) ={2,8} ∩ {1,2,4,6,8} ∩ {2,4,5,7,8}={2,8}, KA(3) =U∩ {3,7} ∩ {1,3,6,8}={3}, KA(4) ={1,4,5,8} ∩ {1,2,4,6,8} ∩ {2,4,5,7,8}={4,8}, KA(5) ={1,4,5,8} ∩U∩ {2,4,5,7,8}={4,5,8}, KA(6) ={6,7} ∩ {1,2,4,6,8} ∩ {1,3,6,8}={6}, KA(7) ={6,7} ∩ {3,7} ∩ {2,4,5,7,8}={7},and KA(8) = ({1,4,5,8} ∪ {2,8})∩ {1,2,4,6,8} ∩U ={1,2,4,8}.
The characteristic set KB(x) may be interpreted as the smallest set of
cases that are indistinguishable fromxusing all attributes fromB, and using given interpretation of missing attribute values. Thus,KA(x) is the set of all
cases that cannot be distinguished fromxusing all attributes. Also, note that the previous definition is an extension of a definition of KB(x) from [7–9]:
for decision tables with only lost values and “do not care” conditions, both definitions are identical.
Thecharacteristic relationR(B) is a relation onU defined forx, y∈U as follows
(x,y)∈R(B)if and only if y∈KB(x).
The characteristic relationR(B) is reflexive but – in general – it does not need to be symmetric or transitive. Also, the characteristic relationR(B) is known if we know characteristic setsKB(x) for all x∈U. In our example,
R(A) ={(1,1),(1,8),(2,2),(2,8),(3,3),(4,4),(4,8),(5,4),
(5,5),(5,8),(6,6),(7,7),(8,1),(8,2),(8,4),(8,8)}
For decision tables, in which all missing attribute values are lost, a special characteristic relationLV(B) was defined by Stefanowski and Tsoukias in [24], see also [23, 25]. Characteristic relationLV(B) is reflexive, but – in general – it does not need to be symmetric or transitive.
For decision tables where all missing attribute values are “do not care” con- ditions a special characteristic relationDCC(B) was defined by Kryszkiewicz in [14], see also, e.g., [15]. RelationDCC(B) is reflexive and symmetric but – in general – is not transitive.
Obviously, characteristic relationsLV(B) and DCC(B) are special cases of the characteristic relationR(B). For a completely specified decision table, the characteristic relationR(B) is reduced to IND(B).