Open Computer Science 2022; 12: 123–133
Research Article
Tamilarasu Sangeetha and Amalanathan Geetha Mary*
Rough set-based entropy measure with weighted
density outlier detection method
https://doi.org/10.1515/comp-2020-0228
received June 23, 2020; accepted May 10, 2021
1 Introduction
Abstract: The rough set theory is a powerful numerical
model used to handle the impreciseness and ambiguity of
data. Many existing multigranulation rough set models
were derived from the multigranulation decision-theoretic rough set framework. The multigranulation rough
set theory is very desirable in many practical applications
such as high-dimensional knowledge discovery, distributional information systems, and multisource data processing. So far research works were carried out only for
multigranulation rough sets in extraction, selection of
features, reduction of data, decision rules, and pattern
extraction. The proposed approach mainly focuses on
anomaly detection in qualitative data with multiple granules. The approximations of the dataset will be derived
through multiequivalence relation, and then, the rough
set-based entropy measure with weighted density method
is applied on every object and attribute. For detecting outliers, threshold value fixation is performed based on the
estimated weight. The performance of the algorithm is
evaluated and compared with existing outlier detection
algorithms. Datasets such as breast cancer, chess, and
car evaluation have been taken from the UCI repository
to prove its efficiency and performance.
Data may be in the form of text, numbers, or mixed types
where a system can be easily identified and processed.
Data have different structures and dimensions. Data mining
is a technology that is used to obtain information from
larger databases. Objects that deviate from others based
on characteristics or behavior are anomalies [1]. When a
machine fails to work properly, a system that does not
respond properly to given inputs, manmade faults, a simple
diversion in population, and fraudulent activities are the
causes for arising outliers. Outliers are identified through
a different kind of pattern generation in application areas
like medical databases, cyber security systems, drastic
effects of climatic variations, and also in military systems.
Some of the different types of outliers are point outliers, contextual outliers, and collective outliers. The
point anomaly is also known as a global outlier where
an object deviates from the rest of the objects [2]. For
example, the broadcasting of packets in a very short period
by a computer is identified as the victim of hacking. Contextual outliers are objects that deviate only from a particular
situation [3]. For example, 28°C in Chennai is considered
normal during summer, but the same will be taken as an
outlier during winter. A group of objects, in particular, which
deviates from the whole dataset, are called a collective outlier. A group of students who are irregular in their studies
among the class is termed as outliers.
The proven theories, methods, and techniques for granular computing use granules in the form of classes, groups,
or clusters in the universe. Some of the application domains
such as analysis of intervals, clusters, retrieval of information from the databases, machine learning algorithms,
Dempster–Shafer theories, and divide–conquer methods
use granular computing technique. Sometimes, the available
data are incomplete and unclear [4]. So granular computing
is needed in this context to make the problem simple.
Acquiring precise information costs very high, and granulation of data reduces the cost. The granularity of data can be
achieved through proximity relation, the similarity between
data. The granularity of data can be achieved through the
Keywords: approximations, entropy, granules, outliers,
rough sets
* Corresponding author: Amalanathan Geetha Mary, School of
Computer Science and Engineering, Vellore Institute of Technology,
Vellore 632 001, Tamil Nadu, India,
e-mail: geethamary.a@gmail.com
Tamilarasu Sangeetha: School of Computer Science and
Engineering, Vellore Institute of Technology, Vellore 632 001,
Tamil Nadu, India, e-mail: sangee_arasu05@yahoo.co.in
Open Access. © 2022 Tamilarasu Sangeetha and Amalanathan Geetha Mary, published by De Gruyter.
Creative Commons Attribution 4.0 International License.
This work is licensed under the
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Tamilarasu Sangeetha and Amalanathan Geetha Mary
approximation concept. There exists a connection between
granular computing and rough sets. Partitions can be made
from the attributes, and approximations are defined.
This article provides a proposed method for detecting
outliers in multigranular rough sets with a multiequivalence
relation. The definitions for the positive and negative
regions for multigranular rough sets are discussed in
Section 2. A detailed literature review about the multigranular rough sets is discussed in Section 3. The proposed
model and proposed algorithm are provided in Section 4.
The empirical study and experimental analysis of a proposed system are presented in Sections 5 and 6.
2 Background
2.1 Rough set theory
Rough set concepts were developed by Pawlak [5] in
1980s. It is a powerful mechanism that used to handle
the uncertainty and vagueness of data. It can be applied
in all domains, particularly in artificial intelligence. The
primary benefit of rough sets is that all the variables
needed for computation are retrieved from the available
dataset. The imprecise information can be obtained by
the concept of approximations because there is no need
of knowing preliminary information about the data. A
dataset has been taken to represent knowledge. Columns
of the dataset are labeled as an attribute, and rows of
a dataset are labeled as objects. The subset formed
from attributes associates a single or multiequivalence
relation, whereas a group of objects forms a set. The
knowledge of data can be derived from attributes, and
the concepts can be derived from objects. The basic ideas
of rough sets are as follows: a relational database has to
be classified to generate rules and to make ideas, and
knowledge can be obtained through equivalence classes
and approximation concept.
The rough set theory provides a solution to all the
major domains in artificial intelligence. Rough set concepts are used to determine the model and activities of
the drug; attitude control of satellites, iron, and steel
blast furnace; diagnosis of machines, in the area of
neural network and also in decision support systems
[6]. With the help of algorithms, the rough set theory
detects unseen data, recognizes the link between data
that cannot be analyzed by statistical methods, allows
both measurable and quantifiable data, can obtain decision rules, leads to reduction of data, analyses data very
significantly, can be learned easily, and the output can be
interpreted directly without any prior knowledge.
In the classical rough set model, a lower approximation can be determined by equivalence classes that are a
subset of the objective set, and in upper approximation,
equivalence classes should be nonempty and overlapped
with the objective set [7]. There is no error tolerance in
this classical approach. In probabilistic rough sets, the
approximation concepts are based on rough membership
function and inclusion method. The decision-theoretic
rough set model uses α and β for acceptance and rejection, and the values are defined between 0 and 1 [8].
In the Pawlak rough set model, the equivalence
classes should be reflexive, symmetric, and transitive. If
the transitive relation does not obtain equivalence relation, it has to be replaced with tolerance relation. Mostly,
rough set concepts are used in the classification of data
[5]. Pawlak’s rough set model is very sensitive to noisy
data. It can be identified by fixing a probabilistic threshold
value β within the range of 0–0.5 based on the level of
noisy data. This can be achieved by a variable precision
rough set model. In the multigranulation rough set model,
the multiequivalence classes are derived to achieve the
goal [9]. To make intelligent decisions in critical situations,
the game-theoretic rough set model has been used. Each
player should possess the value of α or β based on the
parameter region. Their probabilistic values should be
either increased or decreased slightly. Decreased value of
α indicates a positive region, and an increased value of β
indicates a negative region. The granular structure has
been introduced through equivalence relation, not by
rough set data analysis. This article proposes outlier detection of categorical data in multigranular rough sets using
the rough entropy-based weighted density method.
2.2 Multigranulation rough set
Any information system will have n number of attributes
and objects. It can hold sometimes missing or null values,
which are termed to be irregular. If a universe U contains
regular objects and attributes, they are termed to be complete information system otherwise if they have irregular
objects and attributes are known as incomplete information system [10]. Pawlak’s rough set lower approximation
was fully dependent on single binary relation, whereas,
in multigranulation rough set, the lower approximation
will be derived by using multiequivalence relation. In
both cases, the upper approximation will be derived
based on the complementary set of lower approximation.
Rough set-based entropy measure
Let us consider T to be the universe and A ⊆ Ŷ , Ŷ be the
partition of T. The approximation concept of SGRS is
characterized as follows:
A̲ =
∪ {B ∈ Yˆ
: B ⊆ A} ,
: B ∩ A ≠ ∅} .
A¯ = ∪{B ∈ Y
(1)
(2)
Also, optimistic multigranularity of the rough set model
provides many individual granular structures that need
minimum one granular structure to satisfy the inclusion
condition between equivalence class and objective set,
while in pessimistic multigranulation rough set model, the
granular structure of minimum one should be a nonempty
intersection of the objective. Let us consider a complete
information system T = (U, Atr, fn) and M̂ , N̂ be two segments over the universe U, A ⊆ U . The lower and upper
approximation of MGRS is defined by the following
formulae:
A̲ = {a : Mˆ (a) ⊆ A or Nˆ (a) ⊆ A} ,
(3)
A¯ = ∼(A) Mˆ + Nˆ .
(4)
In Figure 1, the small circles with shaded region [a]x
and [a]y are lower approximation under MGRS and the
big circles with shaded region [b]x and [b]y are lower
approximation under SGRS.
2.3 Related work
Outlier objects are defined as a single object's anomalous
behaviour or small groups of objects that are more inconsistent than the rest. There is a chance of abnormal occurrences in spatial or temporal locality that forms a cluster
Figure 1: Difference between SGRS and MGRS [17].
125
known as anomalies or outliers. They used the LDBSCAN
algorithm for clustering and LOF to detect the inconsistency of a single object [11]. Detecting outliers is the primary step in the applications of data mining. They have
proposed many outlier detection algorithms for parametric
and nonparametric, univariate, and multivariate. Outlier
detection techniques are also based on spatial, distancebased, and density-based clustering methods. If outliers
exist in the dataset, individual observation should be taken
to maintain robustness by providing suitable estimators.
An object that is dissimilar from the rest of the objects
is an anomaly. First, frequent patterns of a dataset are
generated. Items that are having lower frequent patterns
are outliers. They have designed the frequent pattern outlier factor (FPOF) to detect transactions that are outliers,
and to identify outliers alone, the find FPOF method
needs to be used. Researchers show their interest when
trying to find rare events than frequent patterns. Existing
works show that being an outlier object is a binary property. Each object is assigned with a degree of score to be
an outlier [12]. By using the local outlier factor (LOF), the
neighborhood of an object with its surroundings and how
much it is isolated from others are calculated. It is crucial
to detect outliers in many application areas. The topic
of determining outlier scores was an extension of objects
in terms of clusters. The individual cluster has its outlier
factor, which is the clustering-based outlier method [13].
It has two stages: the first stage forms clusters based on
the clustering algorithm, and the second stage detects
outliers based on the outlier factor.
A new definition was given to identify outliers based
on the local outlier factor, which shows the importance of
the behavior of data, which is local. Cluster-based local
outlier factor (CBLOF) was defined to measure and represent the natural quality of outliers. Outliers were presented based on k nearest neighbor graph with outlier
indegree factor [14]. Also, they have extended the work
of k nearest neighbor clustering work. Existing outlier
detection methods are not suitably fit for scattered realworld data due to parameter issues and data patterns,
which are implicit. So, they had proposed the local density outlier factor (LDOF) to measure the distance around
its neighbor [15]. If the distance is farther, then the isolated objects or small clusters are known as outliers. k
means is the most popular clustering algorithm to form
clusters on a dataset. However, it works only for a fixed
data stream, which fails when data streams are dynamic
[16]. The mean of previous clusters is compared with the
current cluster to detect candidate outliers effectively.
Neural network-based learning technique uses SOM
and ART. The SOM algorithm builds to map a high-
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Tamilarasu Sangeetha and Amalanathan Geetha Mary
dimensional input space to low-dimension output space
by assuming that the topological structure exists in the
input space.
Classification based on an optimistic multigranulation rough set model was proposed for the medical diagnosis system [18]. From the generated patterns, the initial
cause for the disease and its symptoms can be diagnosed.
They proved that a single granular method provides
effective results when compared with the multigranular
method [19]. The approximation concept of the classical
approach was based on a single binary relation on the
universe. But multiple granularities of approximations
were based on multiequivalence relation. The decision
rule of SGRS follows the “AND” rule, and MGRS follows
the “OR” rule [20]. The tolerance of incomplete rough sets
could not be determined with the help of single granulation, but it can be achieved with the help of multiple
granularities. Plenty of algorithms are available for attribute reduction and for making decision rules. In both
data mining and machine learning, the test cost is not
taken into consideration, which can be solved by the
multigranulation rough set [21]. This test cost method is
a generalization of three methods such as optimistic, pessimistic, and β MGRS [22]. The approximation concept of
optimistic MGRS forms a lattice. The formed lattice
should not be distributive and complemented, and it is
equivalent to a single granulation model. But pessimistic
multigranulation forms a clopen topology on the available dataset, which forms a normal Boolean algebra. The
MGRS model has been generalized into fuzzy sets. The
single relationship of the fuzzy rough set and optimistic
and pessimistic MGRS model was discussed. By combining
the idea of SGRS and MGRS, a Bayesian decision-theoretic
rough set [23] was developed using a probabilistic theory
that converts the parameters into rough sets.
Outlier detection, as well as background knowledge
about the domain, was obtained by applying the scheme
of multilevel approximation. With the help of data, table
outliers were detected by using the granularity method.
Outliers are detected by assigning scores to local outlier
factors and class outlier factors. Also, the rough membership function is used to determine outliers [24]. A multigranulation rough set model was developed based on
SCED (seeking common ground while eliminating differences), which was also termed as pessimistic multigranulation rough set model. From this, attribute reduction,
approximations, and decision rule were induced.
On multiple granulations, a characteristic function
and parameter, which is called the level of information,
are added to determine an object, to support the precise
information. When the size of the neighborhood is zero,
a new rough set model has degenerated to normal multiple granularities [25]. The neighborhood-based multiple
granular approaches extend the application in different
domains. The approximation was built on combined relation, and the groups of equivalence relations is brought
into an equivalence relation through the union and intersection sets.
The intuitionistic fuzzy multiple granularities were
developed to generalize the existing three intuitionistic
fuzzy rough set model and their extensions to remove
redundancy in multigranular structures [26]. The differences and relationships between multiple granularities
and multiple granular covering rough set models are
determined [13]. The constraints of two MGCRS form a
lattice. In real life, there may exist two different universes
map under a multigranulation rough set, and a decision
has been made with optimistic and pessimistic multiplegranulation method [27]. The standard rough set model's
idea of approximation is based on a single equivalence
relation, but the multigranulation rough set model uses
several equivalence relations, and multi granulation
fuzzy approximation spaces (MGFAS) identifies six types
of rough approximations. Hence, they proved that fuzzy
binary relations are the nearest pair to the undefined set,
and pessimistic-based multigranulation upper and lower
approximations are the farthest pair to the undefined
set [28].
A new approach called rough topology has been
developed to analyze many medical problems. The rough
lower and upper approximation, boundary region, and
core reduct are made to find out the key element, which
is the cause of disease occurrence [29]. By using the concept of lower approximation reduct, a matrix with discernibility and theorem for judgment has been developed to
make a fuzzy decision. For a fuzzy-based incomplete information system, the multigranulation rough set has been
applied based on dominance relation. Hence, the dominance multigraulation rough set [17] has been established,
and the three kinds of “OR” decision rules are obtained.
3 Attribute reduction
The attribute reduct concepts have been given more
importance in rough sets. To maintain a good accuracy
level, the original dataset is reduced into different subfragments. The attribute selection process uses a reduct
method to remove attributes that are considered to be
weak (less strength) or unnecessary [30]. The indispensable attribute of the dataset defined in Table 1 is determined based on the association rule strength or
Rough set-based entropy measure
Table 1: Hiring dataset
Objects
Degree
Experience
Reference
Decision
E1
E2
E3
E4
E5
E6
E7
E8
MTech
MSc
MSc
MSc
MSc
MSc
MTech
ME
High
High
Medium
Medium
High
Medium
Low
Low
Big
Big
Small
Small
Small
Medium
Medium
Medium
Yes
Yes
No
No
No
Yes
No
Yes
confidence [13]. The rule is defined as the ratio of several
samples Ei, which contains Ei U decision to the number of
samples that contain Ei. Table 1 presents the hiring
dataset [31], with conditional attributes such as {degree,
experience, reference} and one {decision} attribute.
The strength of rules for attribute degree is as follows:
● (Degree = MTech) → (Decision = Yes), rule strength
→ 50%.
● (Degree = MSc) → (Decision = Yes), rule strength → 40%.
● (Degree = MSc) → (Decision = No), rule strength → 60%.
● (Degree = MTech) → (Decision = No), rule strength
→ 50%.
● (Degree = ME) → (Decision = Yes), rule strength →
100%.
The strength of rules for attribute experience is as
follows:
● (Experience = High) → (Decision = Yes), rule strength
→ 66%.
● (Experience = Medium) → (Decision = No), rule
strength → 66%.
● (Experience = High) → (Decision = No), rule strength
→ 33%.
● (Experience = Medium) → (Decision = Yes), rule
strength → 33%.
● (Experience = Low) → (Decision = No), rule strength
→ 50%.
● (Experience = Low) → (Decision = Yes), rule strength
→ 100%.
The strength of rules for attribute reference is as
follows:
● (Reference = Big) → (Decision = Yes), rule strength
→ 100%.
● (Reference = Small) → (Decision = No), rule strength
→ 100%.
● (Reference = Medium) → (Decision = Yes), rule
strength → 66%.
● (Reference = Medium) → (Decision = No), rule strength
→ 33%.
127
By rule generation, it can be easily identified that
attribute Degree and reference have maximum strength
when compared with attribute experience. Table 2 shows
the indispensable attributes of the hiring dataset.
4 Proposed model
Outlier detection plays a key role in all application domains.
The missing values and incomplete data presented in the
data table provide ambiguousness, while compiling the
data results in the erroneous output [32]. To avoid such
kind of scenario, outlier detection is needed. Several
methods are employed for outlier detection in the qualitative, quantitative, and mixed types of data. The proposed model detects outlier in the multigranulation
rough set with lower and upper approximated values.
Approximations are derived through multiequivalence
relations with segments of attributes. The given input
should be categorical.
In the preprocessing stage, through multiequivalence
relation, the lower and upper approximation of the dataset
is derived. Then, at the postprocessing stage, the rough
set-based entropy measure outlier detection method is
applied to the approximation sets. By fixing the appropriate value for the threshold, outliers are identified. The
steps are clearly shown in Figure 2.
4.1 Rough set-based entropy measure with
weighted density outlier detection
method
A dataset may incorporate missing information, some
negative and invalid data. So the dataset is characterized
to be unclear and deficient. To handle this context, a rough
Table 2: Indispensable attributes
Objects
Degree
Reference
E1
E2
E3
E4
E5
E6
E7
E8
MTech
MSc
MSc
MSc
MSc
MSc
MTech
ME
Big
Big
Small
Small
Small
Medium
Medium
Medium
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Tamilarasu Sangeetha and Amalanathan Geetha Mary
Figure 2: The proposed model for outlier detection in MGRS.
set-based entropy measure with a weighted density outlier
detection method is proposed for multigranulation rough
sets [33]. Based on the multiequivalence relation, the
upper and lower approximation will be derived. In the
postprocessing stage, the indiscernibility relation concerning objects is determined, the objects that are having
uncertain values are calculated by complement entropy
measure, and then the weighted density values will be calculated for every attribute and object [34]. The threshold
value will be fixed from obtained values. Values lower
than the threshold value are denoted as outliers. A higher
threshold value is fixed for stable data, and the lower
threshold value for unstable data is required to detect outliers. Sometimes we need prior knowledge from experts to
fix proper values for the parameter [35]. It is not easy to
maintain uniformity in fixing threshold values that can be
applied to all data sets. The following definitions will be
used to detect outliers, which are discussed below:
q
|Q| |Q| j
,
|T| |P|
j=1
n
CPME(RY) =
∑
where Qjq denotes complement set of Q j, which is
Qjq = P − Q .
Definition 4. Let DST = (T , P , Q), the weight of every
attribute for Q is defined as follows:
Weight of attribute(Q) =
1 − CPME(RY)
.
∑nj = 1(Qj )
Definition 5. The average density of each attribute will be
determined as follows:
∣[Pj]Q ∣
Average density of each attribute(Pj ) =
.
∣T ∣
From that, the weighted density of each object will be
determined as follows:
Weighted density of object (P ) =
∑ (Avg density(Pj )⋅ T (Q)).
pi ∈ P
Definition 1. A dataset DST is defined by the triplet DST =
(T, P, Q), where T represents the universe, P represents
the objects, and Q represents the attributes in a dataset.
Definition 2. Let DST = (T , P, Q) and RY⊆Q . The indiscernible relation of RY for pi in P or qi in Q is represented
as follows:
{T|ind(RY)} = {[ pi ]RY |pi ∈ T } .
Definition 3. Let DST = (T , P , Q), and RY⊆Q and
T
{ Q1 ,
ind(RY)
Q2 … Qm} . The complement entropy (CPME)
with respect to RY is defined as follows:
Definition 6. Let us consider the dataset DST = ( T , P , Q),
and ∅ is a fixed threshold value from the weighted density
objects. If the value of Weighted density (P ) < ∅, then p
is termed to be an outlier.
The algorithm for the proposed model has been
shown below:
Input: Dataset DST (T, P, Q) and ∅ be threshold
value.
Output: Set Y has outlier data.
Step 1: Start.
Step 2: Input the dataset of categorical type.
Step 3: Apply multiequivalence relation over the
dataset to determine upper and lower approximation.
Rough set-based entropy measure
Step 4: Let Y = ∅.
Step 5: For every attribute qi ∈ Q.
Step 6: The indiscernibility relation U/IND (Pi) according to definition 2 will be calculated.
Step 7: The complement entropy function according
to definition 3 will be calculated.
Step 8: For every attribute qi ∈ Q, the weighted density method will be applied by Definition 4.
Step 9: For each object pi ∈ T, the weighted density
method will be applied by Definition 5.
Step 10: If (weighted density (pi) < ∅).
Step 11: Y = Y ∪ {pii}.
Step 12: Return Y.
Step 13: Stop.
129
M ∪ N ̂ = {E 1} , {E 2} , {E 3, E 4, E 5} , {E 6} , {E 7} , {E 8} .Then, by applying equation (3), the lower approximation
of the dataset with the multiequivalence relation is
derived.
Mˆ = {E 1, E 7}{E 2, E 3, E 4, E 5, E 6}{E 8} ,
A = {E 1, E 2, E 6, E 8} = {E 8} ,
Nˆ = {E 1, E 2}{E 3, E 4, E 5}{E 6, E 7, E 8} ,
A = {E 1, E 2, E 6, E 8} = {E 1, E 2} ,
A̲ Mˆ + Nˆ = {E 8} ∪ {E 1, E 2} = {E 1, E 2, E 8} .The upper approximation with multiequivalence relation based on equation (4)
is as follows:
Mˆ = {E 1, E 7}{E 2, E 3, E 4, E 5, E 6}{E 8} ,
A = {E 1, E 2, E 6, E 8} ,
5 An empirical study on hiring
dataset
Let us consider, the hiring data set taken by Komorowski,
which has been used for classification purposes. The proposed algorithm is explained briefly by taking eight samples from the dataset, which is presented in Table 1.
The multigranulation rough set method uses the
multiequivalence relation [36] to derive a lower and
upper approximation for the attribute degree and reference, which are denoted as M and N, respectively.
From Table 1, consider A = {E 1, E 2, E 6, E 8} , and
M ∪ N ̂ = {E 1} , {E 2} , {E 3, E 4, E 5} , {E 6} , {E 7} , {E 8} .
The
lower approximation is A̲ Mˆ + Nˆ = {E 1, E 2, E 8} , and upper
ˆ
ˆ
approximation is A¯ M + N = {E 1, E 2, E 6, E 7, E 8} . While
applying the proposed method, object E8 is detected as
an outlier, and when we extend our approach to the
upper approximation level, object E7 is detected as an
outlier. The obtained values are clearly explained in Sections 6.1 and 6.2.
5.1 Concept of approximation under
multigranulation rough set
Based on decision = “yes,” let A = {E1, E2, E6, E8} and
consider the attributes Degree and Reference, which are
represented as M̂ and N̂ , respectively. Then, the three
segments are procured from Table 2 as follows:
Mˆ = {E 1, E 7}{E 2, E 3, E 4, E 5, E 6}{E 8} ,
Nˆ = {E 1, E 2}{E 3, E 4, E 5}{E 6, E 7, E 8} .
= {E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8} ,
Nˆ = {E 1, E 2}{E 3, E 4, E 5}{E 6, E 7, E 8} ,
A = {E 1, E 2, E 6, E 8} ,
= {E 1, E 2, E 6, E 7, E 8} ,
ˆ ˆ
A¯ M + N = {E 1, E 2, E 3, E 4, E 5, E 6, E 7, E 8} ∩
= {E 1, E 2, E 6, E 7, E 8} ,
= {E 1, E 2, E 6, E 7, E 8} .
5.2 Outlier detection in multigranulation
rough set
Through multiequivalence relations, lower and upper
approximations are derived. Then, the rough set-based
entropy measure with weighted density outlier detection
method has been applied on the lower approximation set
values to detect outliers that are presented in Table 3.
The indiscernibility relation for each attribute is calculated. Objects with similar values based on attributes
are defined as follows:
U
= {E 1, E 2}{E 8} .
Degree
U
= {E 1, E 2}{E 8} .
Experience
U
= {E 1, E 2}{E 8} .
Reference
The complement entropy function is calculated for
each attribute with the obtained indiscernible relation.
CE (Degree) =
1
4
2
1
2⎛
1 − ⎞ + ⎛1 − ⎞ = .
3⎠ 9
3⎠ 3⎝
3⎝
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Tamilarasu Sangeetha and Amalanathan Geetha Mary
Table 3: Lower approximation
Table 4: Upper approximation
Objects
Degree
Experience
Reference
Objects
Degree
Experience
Reference
E1
E2
E8
MTech
MTech
ME
High
High
High
Big
Big
Medium
E1
E2
E6
E7
E8
MTech
MTech
MSc
MTech
M.E
High
High
High
High
High
Big
Big
Medium
Medium
Medium
CE (Experience) =
4
3
, CE (Experience) = .
9
9
The weight of each attribute should be calculated by
adding the total number of attributes with the complement entropy function.
Weight of attribute (Degree) =
5
.
12
6
Weight of attribute (Experience) =
.
12
5
Weight of attribute (Reference) =
.
12
The weight of each object should be calculated by the
summation of the product of the weight of attributes with
indiscernible objects.
W (E1) =
6
2
5
2
5
×
+1×
+ ×
= 1.05,
12
3
12
3
12
W (E 2) = 1.05, W (E 8) = 0.91,
If θ < 1.05, then object E 8 is an outlier.
The same method has to be followed upon the upper
approximation set to detect outliers, which are shown
in Table 4. The rough set-based entropy measure with
weighted density value will be calculated for each object
and attribute to detect outlier. Then, object E7 is detected
as an outlier.
6 Experimental analysis
The benchmark datasets, from the UCI repository, have
been taken to illustrate the working procedure of the
proposed method. The breast cancer dataset has 286
objects and 9 attributes with one class attribute. Among
the 286 objects, 9 missing values exist, and so to make the
dataset balanced, some of the majority classes are removed
randomly and made equal to the minority classes by the
undersampling method. Also, data sets such as chess and
car have been taken for the analysis. The chess dataset
has 3,196 objects and 36 attributes with no missing values.
The car dataset has 1,728 objects and 6 attributes with
no missing values and is compared with other machine
learning outlier detection algorithms. The implementation
was carried out with Intel Pentium Processor, 1GigaByte
RAM, and Windows10 operating system. The rough set-based
entropy measure with weighted density outlier detection
Table 5: Comparison between proposed and existing method
Dataset
Feature type No. of features
considered
Feature extraction and Classification
selection method
algorithm
Accuracy
(RSBEMOD) (%)
Accuracy
(LOF) (%)
Breast
cancer
Categorical
Reduct (rough sets)
Support vector
machine (SVM)
95.71
93.72
Chess
Categorical
Reduct (rough sets)
SVM
93.24
91.23
Car
Categorical
Reduct (rough sets)
SVM
93.55
92.67
1. Nodecap
2. Breast
3. Quad
4. Irradiat
1. bkspr
2. Bkxbq
3. Bkxcr
4. bxqsq
1. Buying
2. Maint
3. lug_boot
4. Safety
Rough set-based entropy measure
131
Figure 3: Comparison chart for proposed and existing outlier detection methods.
method is applied on the dataset and compared with existing
outlier detection methods to prove its efficiency.
The performance of rough set-based entropy measure
with weighted density values has been compared with
several existing outlier detection algorithms like k nearest
neighbor (KNN), average KNN (Avg KNN), histogrambased outlier sequence (HBOS), feature bagging (FB), isolation forest (IF), and local outlier factor (LOF). The local
outlier factor algorithm detects outliers by calculating the
distances of the neighbors with their density. It forms a
group based on proximal values, and deviated values are
considered as outliers. In feature bagging, the base estimator is fixed and divides the dataset into subsamplings.
The accurate prediction can be calculated by taking an
average of all base estimators. In most cases, the local
outlier factor is used as the base estimator. In an isolation
forest, the dataset is divided into multiple subtrees. The
isolated objects from others are considered outliers. The
algorithm particularly suits well for multidimensional
data. By constructing histograms, outliers are easily identified by applying an unsupervised histogram-based outlier sequence algorithm. The regression and classification
problems are handled by the k nearest neighbor algorithm. Based on the distance measure, calculate the
vote of each neighbor. Average knn creates a super
sample for all classes and a particular class average is
calculated by its training samples. Rough set-based
entropy measure outlier detection algorithm (RSBEMOD)
determines all objects and attributes weighted density
value by considering its indiscernible relation, complement entropy, and an average weight of attributes and
objects. Table 5 shows the performance comparison
between the proposed method and the local outlier
factor (existing method).
Also, Figure 3 shows the comparison chart for a
rough set-based entropy measure weighted density over
existing methods.
Table 6: Presence of outliers
Table 7: Removal of outliers
6.1 Metrics used to evaluate the
performance
To measure the performance of the algorithm, precision
(P), recall (R), accuracy (A), and F1 measure are
Measures
Precision
Recall
Accuracy (%)
F1 measure
Measures
Precision
Recall
Accuracy (%)
F1 measure
Breast cancer
Chess
Car
1.0
1.0
1.0
0.9167
0.9155
0.9294
91.67
91.55
92.94
0.9565
0.9559
0.9634
Breast cancer
Chess
Car
1.0
1.0
1.0
0.9571
0.9324
0.9355
95.71
93.24
93.55
0.9781
0.9650
0.9667
132
Tamilarasu Sangeetha and Amalanathan Geetha Mary
calculated. The formula used to calculate these measures
are as follows:
P=
True positive
,
True positive + False positive
detection for a mixed dataset using the multigranulation
rough set and also for dynamic inputs can be developed.
Funding information: The authors state no funding involved.
True positive
,
True positive + False negative
Author contributions: Tamilarasu Sangeetha: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration,
True positive + True negative
A=
. resources, software, supervision, validation, visualization,
Frue positive + False positive + False negative + True negative
roles/writing – original draft, and writing. Amalanathan
F1 measure clearly labels valid objects without any false Geetha Mary: writing – review and editing.
alarms. It lowers the threat caused by false positive and
false negative values.
Conflict of interest: The authors state no conflict of
R=
F 1 measure =
2×P×R
.
P+R
Precision or positive predictive value represents the percentage of relevant objects from the total objects, whereas
recall does the same function of sensitivity. F1 score clearly
labels valid objects without any false alarms. It lowers the
threat caused by false positive and false negative values.
Table 6 shows the performance of the algorithm over the
datasets in the presence of outliers, and Table 7 shows the
performance after the removal of outliers.
7 Conclusion
In this article, outlier detection for categorical data using
multiple granules has been developed. The classical rough
set concept uses single binary relation, and the multigranulation rough set uses multiequivalence relation to derive
approximations over the universe. Then, rough set-based
entropy measure with a weighted density outlier detection
method has been applied to detect outliers. So far, the
single granular method uses the “AND” rule, whereas
the multiple granulations use the “OR” rule. The proposed method applies multiequivalence relation to derive
approximations, and then, a rough set-based entropy measure with weighted density value for objects and attributes
is calculated. From that, a threshold value will be fixed.
The values that are smaller than the threshold value are
identified as anomalies. So, a proper object will not be
detected as an outlier anymore. Datasets, which are taken
from UCI repositories such as breast cancer, chess, and car
evaluation datasets, are compared with rough set-based
entropy measure weighted density outlier detection method
and the existing outlier detection algorithms. The proposed
method is very accurate in detecting outliers when compared with other existing methods. In the future, outlier
interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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