Consistency of Incomplete Data
Patrick G. Clark
1
and Jerzy Grzymala-Busse
1,2
1
Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045, U.S.A.
2
Institute of Computer Science, Polish Academy of Sciences, 01-237 Warsaw, Poland
Keywords:
Consistency, Incomplete Data, Missing Attribute Values, Rough Set Theory.
Abstract:
In this paper we introduce an idea of consistency for incomplete data sets. Consistency is well-known for
completely specified data sets, where a data set is consistent if for any two cases with equal all attribute
values, both cases belong to the same concept. We generalize the definition of consistency for incomplete
data sets using rough set theory. For incomplete data sets there exist three definitions of consistency. We
discuss two types of missing attribute values: lost values and “do not care” conditions. We illustrate an idea of
consistency for incomplete data sets using experiments on many data sets with missing attribute values derived
from five benchmark data sets. Results of our paper may be applied for increasing the efficiency of mining
incomplete data.
1 INTRODUCTION
A complete data set, i.e., a data set with specified all
attribute values, is consistent if for any two cases with
the same attribute values, both cases belong to the
same concept (class). Yet another definition of con-
sistency is based on rough set theory: a complete data
set is consistent if for any concept its lower and up-
per approximations are equal (Pawlak, 1982; Pawlak,
1991). Consistency for incomplete data sets, i.e., data
sets with some missing attribute values, was not de-
fined in the accessible literature.
The main objective of this paper is to study con-
sistency for incomplete data sets using rough set the-
ory. For incomplete data sets there exist three defini-
tions of approximations, called singleton, subset and
concept. Additionally, for complete data sets an idea
of the approximation was generalized by introducing
probabilistic approximations, with an additional pa-
rameter, interpreted as a probability (Grzymala-Busse
and Ziarko, 2003; Pawlak et al., 1988; Yao, 2007;
Yao and Wong, 1992; Yao et al., 1990; Ziarko, 1993;
Ziarko, 2008). Probabilistic approximations were ex-
tended to incomplete data sets by introducing sin-
gleton, subset and concept probabilistic approxima-
tions in (Grzymala-Busse, 2011). First results on ex-
periments on probabilistic approximations were pub-
lished in (Clark and Grzymala-Busse, 2011).
We discuss singleton, subset and concept consis-
tency for incomplete data sets and study their basic
properties. Additionally, we conducted experiments
on five benchmark data sets converted to many incom-
plete data sets. In incomplete data sets, there are two
kinds of missing attribute values: lost values and “do
not care” conditions (Grzymala-Busse, 2003). Lost
values indicate the original attribute values were not
known and processing this kind of missing attribute
values is conducted by taking into account only exist-
ing, specified attribute values. On the other hand, “do
not care” conditions indicate an actual value was one
of the attribute values and we assumed that the miss-
ing attribute values may be replaced by any attribute
value.
2 COMPLETE DATA SETS
Our basic assumption is that the data sets are pre-
sented in the form of a decision table. An example
of a decision table is shown in Table 1. Rows of the
decision table represent cases, while columns are la-
beled by variables. The set of all cases is denoted by
U. In Table 1, U = {1, 2, 3, 4, 5, 6, 7, 8}. Some vari-
ables are called attributes while one selected variable
is called a decision and is denoted by d. The set of
all attributes will be denoted by A. In Table 1, A =
{Temperature, Headache, Cough} and d = Flu. For
an attribute a and case x, a(x) denotes the value of the
attribute a for case x. For example, Temperature (1) =
high.
80
Clark P. and Grzymala-Busse J..
Consistency of Incomplete Data.
DOI: 10.5220/0004490300800087
In Proceedings of the 2nd International Conference on Data Technologies and Applications (DATA-2013), pages 80-87
ISBN: 978-989-8565-67-9
Copyright
c
2013 SCITEPRESS (Science and Technology Publications, Lda.)
Table 1: A complete decision table.
Attributes Decision
Case Temperature Headache Cough Flu
1 high yes yes yes
2 normal yes no yes
3 high no no yes
4 high no no yes
5 high no no no
6 normal yes no no
7 normal yes no no
8 normal no yes no
A significant idea used for scrutiny of data sets is
a block of an attribute-value pair. Let (a, v) be an
attribute-value pair. For complete data sets, i.e., data
sets in which every attribute value is specified, a block
of (a, v), denoted by [(a, v)], is the following set
{x | a(x) = v}. (1)
For Table 1, blocks of all attribute-value pairs are
[(Temperature, normal)] = {2, 6, 7, 8},
[(Temperature, high)] = {1, 3, 4, 5},
[(Headache, yes)] = {1, 2, 6, 7},
[(Headache, no)] = {3, 4, 5, 8},
[(Cough, yes)] = {1, 8},
[(Cough, no)] = {2, 3, 4, 5, 6, 7}.
A special block of a decision-value pair is called
a concept. In Table 1, the concepts are [(Flu, yes)] =
{1, 2, 3, 4} and [(Flu, no)] = {5, 6, 7, 8}.
Let B be a subset of the set A of all attributes.
Complete data sets are characterized by the indis-
cernibility relation IND(B) defined as follows: for
any x, y ∈ U ,
(x, y) ∈ IND(B) if and only if a(x) = a(y)
for any a ∈ B
(2)
Obviously, IND(B) is an equivalence relation.
The equivalence class of IND(B) containing x ∈ U
will be denoted by [x]
B
and called B-elementary set.
A-elementary sets will be called elementary. We have
[x]
B
= ∩{[(a, a(x))] | a ∈ B}. (3)
The set of all equivalence classes [x]
B
, where x ∈
U, is a partition on U denoted by B
∗
. For Table 1, A
∗
= {{1}, {2, 6, 7}, {3, 4, 5}, {8}}. All members of A
∗
are elementary sets.
The data set, presented in Table 1, contains con-
flicting cases, for example cases 2 and 6: for any at-
tribute a ∈ A, a(2) = a(6), yet cases 2 and 6 belong to
two different concepts. A data set containing conflict-
ing cases will be called inconsistent. We may recog-
nize that a data set is inconsistent comparing the par-
tition A
∗
with a partition of all concepts: there exist an
elementary set that is not a subset of any concept. For
Table 1, the elementary set {2, 6, 7} is not a subset
of any of the two concepts {1, 2, 3, 4} and {5, 6, 7,
8}. There exists yet another way to recognize incon-
sistency of data sets, based on ideas of B-lower and
B-upper approximations. Let X be a subset of U. The
B-lower approximation of X, denoted by appr
B
(X),
is defined as follows
{x | x ∈ U, [x]
B
⊆ X }. (4)
The B-upper approximation of X, denoted by
appr
B
(X), is defined as follows
{x | x ∈ U, [x]
B
∩ X 6=
/
0}. (5)
For the data set from Table 1 and the concept
[(Flu, yes)] = {1, 2, 3, 4} = X,
appr
A
(X) = {1}
and
appr
A
(X) = {1, 2, 3, 4, 5, 6, 7}.
A data set is inconsistent if and only if there exists
a concept X for which appr
A
(X) 6= appr
A
(X).
3 INCOMPLETE DATA SETS
An example of the incomplete data set is presented in
Table 2. Some attribute values are missing. Such val-
ues are denoted either by ?, denoting a lost value (the
original attribute value was not known, we will try to
use only existing, specified attribute values) or by ∗,
denoting a “do not care” condition (we are assuming
that the missing attribute value may be replaced by
any attribute value).
For incomplete decision tables the definition
of a block of an attribute-value pair is modified
(Grzymala-Busse, 2003; Grzymala-Busse, 2004a;
Grzymala-Busse, 2004b).
• If for an attribute a there exists a case x such that
a(x) = ?, i.e., the corresponding value is lost, then
the case x should not be included in any blocks
[(a, v)] for all values v of attribute a,
• If for an attribute a there exists a case x such that
the corresponding value is a “do not care” condi-
tion, i.e., a(x) = ∗, then the case x should be in-
ConsistencyofIncompleteData
81
Table 2: An incomplete decision table.
Attributes Decision
Case Temperature Headache Cough Flu
1 ∗ yes yes yes
2 normal ? no yes
3 ? no ∗ yes
4 high no no yes
5 high ∗ ? no
6 normal yes ∗ no
7 normal yes no no
8 normal ? yes no
cluded in blocks [(a, v)] for all specified values v
of attribute a.
Thus, for Table 2, the blocks of all attribute-value
pairs are
[(Temperature, normal)] = {1, 2, 6, 7, 8},
[(Temperature, high)] = {1, 4, 5},
[(Headache, yes)] = {1, 5, 6, 7},
[(Headache, no)] = {3, 4, 5},
[(Cough, yes)] = {1, 3, 6, 8},
[(Cough, no)] = {2, 3, 4, 6, 7}.
Let B be a subset of the set A of all attributes. For
a case x ∈ U the characteristic set K
B
(x) is defined as
the intersection of the sets K(x,a), for all a ∈ B, where
the set K(x, a) is defined in the following way:
• If a(x) is specified, then K(x, a) is the block
[(a, a(x))] of attribute a and its value a(x),
• If a(x) =? or a(x) = ∗ then the set K(x, a) = U.
Characteristic set K
B
(x) may be interpreted as the
set of cases that are indistinguishable from x using
all attributes from B, with a given interpretation of
missing attribute values. Thus, K
A
(x) is the set of all
cases that cannot be distinguished from x using all at-
tributes.
For Table 2 and B = A,
K
A
(1) = U ∩ {1, 5, 6, 7} ∩ {1, 3, 6, 8} = {1, 6},
K
A
(2) = {1, 2, 6, 7, 8} ∩U ∩ {2, 3, 4, 6, 7} = {2, 6, 7},
K
A
(3) = U ∩ {3, 4, 5} ∩U = {3, 4, 5},
K
A
(4) = {1, 4, 5} ∩ {3, 4, 5} ∩{2, 3, 4, 6, 7} = {4},
K
A
(5) = {1, 4, 5} ∩U ∩U = {1, 4, 5},
K
A
(6) = {1, 2, 6, 7, 8} ∩ {1,5, 6, 7} ∩U = {1, 6, 7},
K
A
(7) = {1, 2, 6, 7, 8} ∩ {1, 5, 6, 7} ∩ {2, 3, 4, 6, 7} =
{6, 7}, and
K
A
(8) = {1, 2, 6, 7, 8} ∩U ∩ {1, 3, 6, 8} = {1, 6, 8}.
For incomplete data sets there exist three distinct
definitions of approximations. Let X be a subset of U.
The B-singleton lower approximation of X, denoted
by appr
singleton
B
(X), is defined as follows
{x | x ∈ U, K
B
(x) ⊆ X }. (6)
The singleton lower approximations were stud-
ied in many papers, see, e.g., (Grzymala-Busse,
2003; Grzymala-Busse, 2004b; Kryszkiewicz, 1995;
Kryszkiewicz, 1999; Lin, 1989; Lin, 1992; Slowinski
and Vanderpooten, 2000; Stefanowski and Tsoukias,
1999; Stefanowski and Tsoukias, 2001; Yao, 1998).
The B-singleton upper approximation of X, de-
noted by appr
singleton
B
(X), is defined as follows
{x | x ∈ U, K
B
(x) ∩ X 6=
/
0}. (7)
The singleton upper approximations, like single-
ton lower approximations, were also studied in many
papers, e.g., (Grzymala-Busse, 2003; Grzymala-
Busse, 2004b; Kryszkiewicz, 1995; Kryszkiewicz,
1999; Slowinski and Vanderpooten, 2000; Ste-
fanowski and Tsoukias, 1999; Stefanowski and
Tsoukias, 2001; Yao, 1998).
The B-subset lower approximation of X , denoted
by appr
subset
B
(X), is defined as follows
∪ {K
B
(x) | x ∈ U, K
B
(x) ⊆ X }. (8)
The subset lower approximations were introduced
in (Grzymala-Busse, 2003; Grzymala-Busse, 2004b).
The B-subset upper approximation of X, denoted
by appr
subset
B
(X), is defined as follows
∪ {K
B
(x) | x ∈ U, K
B
(x) ∩ X 6=
/
0}. (9)
The subset upper approximations were introduced
in (Grzymala-Busse, 2003; Grzymala-Busse, 2004b).
The B-concept lower approximation of X , denoted
by appr
concept
B
(X), is defined as follows
∪ {K
B
(x) | x ∈ X, K
B
(x) ⊆ X }. (10)
The concept lower approximations were intro-
duced in (Grzymala-Busse, 2003; Grzymala-Busse,
2004b).
The B-concept upper approximation of X , de-
noted by appr
concept
B
(X), is defined as follows
∪ {K
B
(x) | x ∈ X, K
B
(x) ∩ X 6=
/
0}
= ∪ {K
B
(x) | x ∈ X}.
(11)
The concept upper approximations were studied
in (Grzymala-Busse, 2003; Grzymala-Busse, 2004b;
Lin, 1992).
For Table 2 and X = {1, 2, 3, 4}, all A-singleton,
A-subset and A-concept approximations are:
appr
singleton
A
(X) = {4},
DATA2013-2ndInternationalConferenceonDataManagementTechnologiesandApplications
82
appr
singleton
A
(X) = {1, 2, 3, 4, 5, 6, 8},
appr
subset
A
(X) = {4},
appr
subset
A
(X) = U,
appr
concept
A
(X) = {4},
appr
concept
A
(X) = {1, 2, 3, 4, 5, 6, 7}.
0
0.2
0.4
0.6
0.8
1
1.2
0 5 10 15 20 25 30 35
Number of distinct approximations
Percentage of missing attribute values
1, subset
2, subset
1, singleton
2, singleton
1, concept
2, concept
Figure 1: Bankruptcy data set with lost values.
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30 35
Number of distinct approximations
Percentage of missing attribute values
1, subset
2, subset
1, singleton
2, singleton
1, concept
2, concept
Figure 2: Bankruptcy data set with “do not care” conditions.
4 PROBABILISTIC
APPROXIMATIONS
By analogy with lower and upper approximations de-
fined using characteristic sets, we will introduce three
kinds of probabilistic approximations: singleton, sub-
set and concept. Again, let B be a subset of the at-
tribute set A and X be a subset of U.
A B-singleton probabilistic approximation of X
with the threshold α, 0 < α ≤ 1, denoted by
appr
singleton
α,B
(X), is defined as follows
{x | x ∈ U, Pr(X | K
B
(x)) ≥ α}, (12)
0
5
10
15
20
25
0 10 20 30 40
Number of distinct approximations
Percentage of missing attribute values
recurrence-events, subset
no-recurrence-events, subset
recurrence-events, singleton
no-recurrence-events, singleton
recurrence-events, concept
no-recurrence-events, concept
Figure 3: Breast cancer data set with lost values.
0
20
40
60
80
100
120
140
160
180
200
0 10 20 30 40
Number of distinct approximations
Percentage of missing attribute values
recurrence-events, subset
no-recurrence-events, subset
recurrence-events, singleton
no-recurrence-events, singleton
recurrence-events, concept
no-recurrence-events, concept
Figure 4: Breast cancer data set with “do not care” condi-
tions.
0
1
2
3
0 5 10 15 20 25 30 35 40
Number of distinct approximations
Percentage of missing attribute values
1, subset
0, subset
1, singleton
0, singleton
1, concept
0, concept
Figure 5: Echocardiogram data set with lost values.
where Pr(X | K
B
(x)) =
|X ∩ K
B
(x)|
|K
B
(x)|
is the conditional
probability of X given K
B
(x) and |Y | denotes the car-
dinality of set Y .
A B-subset probabilistic approximation of the set
X with the threshold α, 0 < α ≤ 1, denoted by
appr
subset
α,B
(X), is defined as follows
∪{K
B
(x) | x ∈ U, Pr(X | K
B
(x)) ≥ α}. (13)
ConsistencyofIncompleteData
83
0
5
10
15
20
25
0 5 10 15 20 25 30 35 40
Number of distinct approximations
Percentage of missing attribute values
1, subset
0, subset
1, singleton
0, singleton
1, concept
0, concept
Figure 6: Echocardiogram data set with “do not care” con-
ditions.
0
1
2
3
4
5
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
yes, subset
no, subset
yes, singleton
no, singleton
yes, concept
no, concept
Figure 7: Hepatitis data set with lost values.
0
20
40
60
80
100
120
140
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
yes, subset
no, subset
yes, singleton
no, singleton
yes, concept
no, concept
Figure 8: Hepatitis data set with “do not care” conditions.
A B-concept probabilistic approximation of the
set X with the threshold α, 0 < α ≤ 1, denoted by
appr
concept
α,B
(X), is defined as follows
∪{K
B
(x) | x ∈ X, Pr(X | K
B
(x)) ≥ α}. (14)
Let type ∈ {singleton, subset, concept}. Note that
appr
type
1,B
(X) = appr
type
B
(X) (15)
and for the smallest possible positive α (in our exper-
iments such α = 0.001)
appr
type
α,B
(X) = appr
type
B
(X). (16)
For Table 2, all distinct A-singleton, A-subset and
A-concept approximations of the set X = {1, 2, 3, 4}
are
appr
singleton
0.333,A
(X) = {1, 2, 3, 4, 5, 6, 8}
appr
singleton
0.5,A
(X) = {1, 3, 4, 5},
appr
singleton
0.667,A
(X) = {3, 4, 5},
appr
singleton
1,A
(X) = {4},
appr
subset
0.333,A
(X) = U,
appr
subset
0.5,A
(X) = {1, 3, 4, 5, 6},
appr
subset
0.667,A
(X) = {1, 3, 4, 5},
appr
subset
1,A
(X) = {4},
appr
concept
0.333,A
(X) = {1, 2, 3, 4, 5, 6, 7},
appr
concept
0.5,A
(X) = {1, 3, 4, 5, 6},
appr
concept
0.667,A
(X) = {3, 4, 5},
appr
concept
1,A
(X) = {4}.
5 CONSISTENCY
Let X be a concept of the incomplete data set, i.e., a
block of a decision-value pair. Let B be a subset of the
set A of all attributes. Let type ∈ {singleton, subset,
concept}. The concept X is B-type consistent in the
data set if and only if for any α > 0
appr
type
α,B
(X) = X . (17)
If the concept X is A-type consistent in the data set
then it will be called type consistent.
For completely specified data sets, if B(X) = X
or B(X) = X then B(X) = X = B(X). An analogous
result is not true for incomplete data sets.
A data set will be called B-type consistent when
for any concept X the number of all distinct B-type
probabilistic approximations of X is equal to one. If
the data set is A-type consistent then it will be called
type consistent.
DATA2013-2ndInternationalConferenceonDataManagementTechnologiesandApplications
84
6 EXPERIMENTS
We conducted experiments on eight benchmark data
sets, taken from the University of California at Irvine
Machine learning Repository, see Table 3. For any
data set, a family of new data sets was derived, with
randomly placed lost values, starting from 0%, with
the number of lost values gradually increasing, with
the increment of 5%, until in the process of adding
lost values, for some case, all attribute values were
missing. If so, we conducted an additional two ran-
dom attempts and, if there was still a case with miss-
ing all attribute values, the process of deriving new
data sets with lost values was terminated. For any
derived data set with lost values we created a corre-
sponding data set with “do not care” conditions by
replacing all lost values by “do not care” conditions.
Our objective was to test, for any data set from such
set family, how many distinct singleton, subset and
concept probabilistic approximations may be created
when the α parameter changes from 0.001 to 1.
For any data set, the number of distinct single-
ton, subset and concept probabilistic approximations
for all concepts were recorded, see Figures 1–14. On
these figures the numbers of distinct singleton, subset
and concept approximations are shown for all con-
cepts. For example, on Figure 1, “1, subset” means
the concept labeled in the data set by “1” combined
with the subset probabilistic approximation. It is clear
that for data sets with “do not care” conditions all
three numbers were larger than for corresponding data
sets with lost values.
All eight data sets created from the bankruptcy
data, with percentage of lost values starting at 0%
and ending with 35%, are singleton consistent (and
hence subset and concept consistent). Additionally,
the bankruptcy data set with 0% and 5% of “do not
care” conditions are also singleton consistent. On the
other hand, all 19 data sets created from the breast
cancer data set, i.e., the original data set, 9 data sets
with lost values and 9 data sets with “do not care”
conditions are singleton inconsistent.
All data sets with up to 35% of lost values, derived
from the echocardiogram data set, are singleton con-
sistent. The data set derived from the echocardiogram
data set, with 40% of lost values, is an example of the
data set with one concept (labeled by “1”) that is con-
cept consistent but not singleton consistent. Four data
sets derived from the echocardiogram data set, with
0, 5, 10 and 15% of “do not care” conditions, are sin-
gleton consistent.
For the data sets derived from hepatitis data set,
both data sets with lost values and “do not care” con-
ditions with 5% of missing attribute values are sin-
Table 3: Data sets used for experiments.
Data set Number of
cases attributes concepts
Bankruptcy 66 5 2
Breast cancer 277 9 2
Echocardiogram 74 7 2
Hepatitis 155 19 2
Wine recognition 178 13 3
gleton consistent. Moreover, for the concept “yes”,
data sets with lost values derived from the hepatitis
data set, up to 60%, are concept consistent. For data
sets derived from the wine recognition data set, for
the concept “3” with up to 5% of lost values and with
up to 5% of “do not care” conditions, are singleton
consistent.
0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
1, singleton
3, singleton
2, singleton
Figure 9: Singleton probabilistic approximations for wine
recognition data set with lost values.
0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
1, subset
3, subset
2, subset
Figure 10: Subset probabilistic approximations for wine
recognition data set with lost values.
ConsistencyofIncompleteData
85
0
2
4
6
8
10
12
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
1, concept
3, concept
2, concept
Figure 11: Concept probabilistic approximations for wine
recognition data set with lost values.
0
20
40
60
80
100
120
140
160
180
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
1, singleton
3, singleton
2, singleton
Figure 12: Singleton probabilistic approximations for wine
recognition data set with “do not care” conditions.
0
10
20
30
40
50
60
70
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
1, subset
3, subset
2, subset
Figure 13: Subset probabilistic approximations for wine
recognition data set with “do not care” conditions.
7 CONCLUSIONS
In this paper we introduced the idea of consistency
for incomplete data sets. Consistency is defined for
a single concept. For a given type of approximation,
0
5
10
15
20
25
30
35
40
45
50
0 10 20 30 40 50 60
Number of distinct approximations
Percentage of missing attribute values
1, concept
3, concept
2, concept
Figure 14: Concept probabilistic approximations for wine
recognition data set with “do not care” conditions.
it is possible that some concepts are consistent while
other concepts are not consistent. Consistency of the
concept depends on type of approximation. A con-
cept may be subset and concept consistent while it is
singleton inconsistent.
Results of our experiments show that for a given
data set (or a concept) there exist more consistent data
sets, of all types, for data sets affected by lost values
than for data sets affected by “do not care” conditions.
For some data sets all derived incomplete data sets,
for all concepts, are singleton consistent, for some
data sets only some concepts are singleton consistent,
while for some data sets all concepts are not singleton
consistent.
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