FUZZY STATISTICAL DATABASE AND ITS PHYSICAL ORGANIZATION

International Journal of Database Management Systems ( IJDMS ) Vol.5, No.4, August 2013
DOI : 10.5121/ijdms.2013.5403 27
FUZZY STATISTICAL DATABASE AND ITS
PHYSICAL ORGANIZATION
S.Guglani1
, C.P. Katti2
and P.C. Saxena3
School of Computer & System Sciences,
JawaharLal Nehru University, Delhi, India
1
seema_guglani@yahoo.co.in
2
cpkatti@yahoo.com
3
premchand_saxena@yahoo.com
ABSTRACT
Today most of the database systems in used are based on the precise data. However, in the real world
application, it is often partially known or imprecise in nature giving rise to fuzzy relational database which
contains imprecise micro data. Aggregation or statistical operation applied on micro data generates macro data
called fuzzy statistic. In fuzzy statistics apart from fuzzy mean, fuzzy median primitive operation of counting of
tuples in fuzzy relational database is an important aspect. In this context, we investigated the various approaches
to cardinality of fuzzy sets which have been studied in the literature (e.g. sigmacount) and introduce the concept
of fuzzy cardinality in fuzzy relational database. The concept of fuzzy statistical database is developed, which
allows us to deal with the storage of vagueness or impreciseness associated with the statistical data. Fuzzy
statistical database is a collection of fuzzy statistical tables. Depending upon the complexity of domain of fuzzy
attributes in fuzzy statistical table, type-1 fuzzy statistical table and type-2 fuzzy statistical table are defined. We
discuss the physical organization of fuzzy statistical table and algorithm for its implementation is also developed.
KEYWORDS
Fuzzy sets, Fuzzy attribute, Fuzzy relation, Fuzzy statistics, Statistical database, Fuzzy cardinality
1. INTRODUCTION
Most conventional databases in use today are based on the relational model. Each relation in the
database represents a proposition and each record in a relation is a statement such that it evaluates to
'true' for that proposition. It could be argued, however, that this required precision actually gives an
insufficient representation of the world. The model is grounded in binary black-and-white but much of
reality actually exists in shades of grey. As such, the conventional relational database model has
limited usefulness. One area that illustrates this limitation is in the everyday, subjective language
generally used to describe people. For instance, a person might be described as being “tall, with a
wide face and very dark brown eyes”. This description would be difficult to represent under the
conventional relational model because it uses descriptive words that are inherently imprecise, and also
because differing communities (i.e. groups with internal agreement on subjective meanings of these
terms) may describe the same person differently. Many authors have made proposals to model and
handle databases involving uncertain data giving rise to fuzzy relational database. The last two
decades have witnessed a blossoming of research on this topic ([1],[7],[9],[15],[17],[20],[34-39]).Even

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though most of the literature about uncertain databases uses probability theory as the underlying
uncertainty model, some approaches rather rest on possibility theory ([7],[21]). The initial idea
consisting in applying possibility theory to this issue goes back to the early 80‟s[15].Possibility theory
has following advantages over probability theory.
• the qualitative nature of the model makes easier the elicitation of the degrees attached to
candidate values.
• in probability theory , the fact that the sum of the degrees from a distribution must equal 1
makes it difficult to deal with incompletely known distributions.
Our aim is not to demonstrate that the possibility theory based framework is better than the
probabilistic one at modeling uncertain, imprecise databases, but that it constitutes an interesting
alternative in as much as it captures a different kind of uncertainty (of a qualitative nature). For
example, a person who witnesses a car accident and is not sure about the model of the car involved. In
such a case, it seems reasonable to model the vagueness by means of a possibility distribution .e.g.
{1/Santro, 0.8/Verna.0.7/Honda} rather than with a probability distribution. Fuzzy relational database
theory extends the relational model to allow for the representation of imprecise data and thus, it
provides a more accurate representation of the world, that it models. Statistics calculated over such
type of fuzzy relational database leads to imprecise statistic. Such statistical inferences are no longer
confined to the boundaries of classical statistics. Some authors ([30-31]) have extended the concept of
classical statistics in fuzzy environment leading to fuzzy statistics ([26],[31]).Cardinality of a fuzzy set
is an important aspect. For example, a teacher is asked how many tall students he has in one of his
classes. Some possible answers are (a) about 4 (b) 4 (c) 3.52. This is about counting the elements of a
fuzzy set. It can, probably, be argued in different ways, but it is quite clear that (c) is not the best
possible answer. The best possible answer is, perhaps (a) and the next answer is (b). Answers (b) and
(c) are arrived at by using a nonfuzzy concept of cardinality[49] while (a) corresponds to a fuzzy
cardinality concept([40],[42],[44]). Answer (c) is typically obtained by using the concept of sigma
count (which was first defined by [41]). Fuzzy relation is nothing but a fuzzy subset of cartesian
product of domain of its attributes which motivates us to define fuzzy cardinality of a fuzzy relation.
Fuzzy statistics cannot be stored in the existing framework of statistical database
([8],[10],[13],[14],[16],[18],[19],[22], [23],[28]) as these are designed for the efficient storage and
convenient retrieval of precise statistical data. This motivates us to develop fuzzy statistical database.
The fuzzy relation[21] and fuzzy set theory proposed by [1] provide a requisite mathematical
framework for dealing with such extended data values (precise and imprecise data values). Such type
of fuzzy statistical database is quite useful when it is used as a decision, in areas such as management,
decision making, health care, economic planning and census data evaluation. In the quest for
providing proper logical model to fuzzy statistical database, we introduce fuzzy statistical tables
(where partially as well as imprecise (fuzzy) known data values are permitted). Depending upon the
type of fuzzy attributes [33] in fuzzy statistical table, type-1 and type-2 fuzzy statistical tables are
defined. The statisticians may not need to use the entire fuzzy statistical tables during the preliminary
stage of data analysis of certain operations. To enhance responsiveness the statistician may base their
preliminary analysis on a part of fuzzy statistical table which motivates us to define fuzzy primitive
statistical table. The physical organization of data has a major impact on any database system
performance because it is the level at which actual implementation takes place in physical storage.
Fuzzy statistical table is a logical model for fuzzy statistical database, so in view of its storage in the
system we propose its physical organization technique and algorithm for its implementation is
developed.
The paper is organized as follows. In section 2, we introduce the preliminaries which include fuzzy
sets and fuzzy relations. In section 3, fuzzy count is defined. In section 4, fuzzy statistical database is
proposed and its logical model, fuzzy statistical table is introduced. In section 5, fuzzy primitive

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statistical tables are defined. In section 6, physical organization of fuzzy statistical table is discussed
and finally in section 7, algorithm for implementation of fuzzy statistical table is developed.
2. PRELIMINARIS
2.1. Fuzzy Set
Fuzzy sets have been introduced by [1] as an extension of the classical notion of set. In classical set
theory, the membership of elements in a set is assessed in binary terms according to a bivalent
condition –an element either belongs or does not belong to the set. By contrast, fuzzy set theory ([26]
,[27],[29])permits the gradual assessment of the membership of elements in a set, which is described
with the aid of a membership function valued in the real unit interval [0,1]. Fuzzy sets generalize
classical sets, since the indicator functions of classical sets are special cases of membership functions
of fuzzy sets if the latter only take values 0 or 1.Let X be a classical set of objects called universe of
discourse whose elements are denoted by x. A fuzzy set A in X is characterized by membership
function ߤ‫ܣ‬:ܺ→[0,1] where ߤ‫ܣ‬(‫)ݔ‬ for each ‫ܺ∈ݔ‬ denotes the grade of membership of x in the fuzzy set
A. Such type of fuzzy sets are known as type-1. Following the notations used in fuzzy set theory, we
write
A= {ߤ‫ܣ‬(‫)1ݔ‬ / ‫1ݔ‬ ,……,ߤ‫ܣ‬ (‫ݔ‬݊ ) / ‫ݔ‬݊ }
where ‫ݔ‬݅∈ܺ,1≤݅≤݊. A membership function to the value 0 describes that the member is not included in
the fuzzy set, 1 describes a fully included member of the fuzzy set. Value strictly between 0 and 1
characterize the fuzzy members, which belong to the fuzzy set partially. Depending upon the
complexity of underlying domain, fuzzy sets may be of type-2.Type-2 fuzzy sets are defined to be
fuzzy sets whose grades of membership are themselves fuzzy.
2.2 Fuzzy Relations
Fuzzy relations were developed to capture various types of imprecise information occuring in the real
world. The imprecise values in a database system can be broadly classified into two types-
• Attribute Uncertainity : The situation where some attribute values are uncertain.
• Existential Uncertainity : The situation where the existence of some tuples is uncertain.
Fuzzy relation can very well incorporate both of these impreciseness. Out of the various approaches
proposed in the fuzzy database literature to represent impreciseness in attribute values, we now
discuss some of the significant ones. Broadly two approaches have been most popular. First is the
similarity based approach, which characterizes the impreciseness by using linguistic terms e.g. poor,
fair, good etc. and the degree of similarity between a pair of linguistic terms is characterized by a
similarity matrix.The second approach is based on Zadeh‟s possibility theory[2]which uses possibility
distribution as a value of an attribute to capture the impreciseness of the first type. The second type of
impreciseness i.e. the partial membership of a tuple in a relation allows a tuple to be a partial member
of a relation, for e.g. the animals which are considered “somewhat endangered” are partial members of
the Endangered-Species relation. A tuple with a partial membership in a relation is referred as
weighted tuple ([20],[21]). The possibility based approach [9] described above is believed to be more
general and popular; the important reason being that it handles all types of imprecise information.
We now provide the basic notations and concepts in two possibility based fuzzy relational data models
which we treat in this paper. The models are known as type-1 & type-2 fuzzy relational data models

30
[20].While introducing them, we will show how the models capture all types of impreciseness in a
better way. Adhering to the notations of classical relational database theory, a relation schema R in
fuzzy relational data model is defined as a finite set of attributes {‫ܣ‬1,‫ܣ‬2,….‫ܣ‬݊} and is denoted as ܴ
(‫ܣ‬1,‫ܣ‬2,….‫ܣ‬݊) . Corresponding to each attribute ‫ܣ‬݅,݅=1…݊ , is a set ݀‫ܣ(݉݋‬݅) called the domain of
‫ܣ‬݅.Along with each attribute ‫ܣ‬݅, a set ܷ݅, called universe of discourse ܷ݅ is associated for the domain
values of ‫ܣ‬݅. A fuzzy relation r on a relation scheme ܴ (‫ܣ‬1,‫ܣ‬2,….‫ܣ‬݊) is a fuzzy subset of
݀‫ܣ(݉݋‬1)×݀‫ܣ(݉݋‬2)×……×݀‫.)݊ܣ(݉݋‬ Mathematically, fuzzy relation r is defined as a fuzzy subset of
ܷ1×ܷ2×…..ܷ݊ and is characterized by the n-variate membership function ߤ‫ݎ‬; ܷ1×ܷ2×…..ܷ݊→[0,1].
Like classical relation, fuzzy relation r will be represented as a table with an additional column for
ߤ‫ݎ‬(‫)ݐ‬ denoting the membership value of the tuple t in r. Depending upon the complexity of domain of
attributes in fuzzy relation it is classified into two types : type-1 fuzzy relation and type-2 fuzzy
relation.
In type-1 fuzzy relations, dom(‫ܣ‬݅) is a fuzzy set(or a classical set). A type-1 fuzzy relation may be
considered as a first-level extension of classical relations,where impreciseness in the association
among entities can be captured.The type-2 fuzzy relations provide further generlization by allowing
dom(‫ܣ‬݅) to be a set of fuzzy sets.Table 1 shows an instance of type-2 fuzzy relation
EMPLOYEE(Name,State,Sex,Experience,Salary,Incometax) in an organization which contains the
information about name, state where the employee lives, sex(male or female), his expérience, his
status of salary and the incometax paid by the employee. In the EMPLOYEE relation dom(Name),
dom(State) and dom(Sex) are assumed to be crisp set while dom(Experience), dom(Salary) and
dom(Incometax) are sets of fuzzy sets in their universes ܷ‫݁ܿ݊݁݅ݎ݁݌ݔܧ‬, ܷ݈ܵܽܽ‫ݕݎ‬ and ܷ‫ݔܽݐ݁݉݋ܿ݊ܫ‬ respectively.
ܷ‫݁ܿ݊݁݅ݎ݁݌ݔܧ‬, ܷ݈ܵܽܽ‫ݕݎ‬ and ܷ‫ݔܽݐ݁݉݋ܿ݊ܫ‬ are assumed to be sets of positive integers in the range 0-30,10,000-
100,000 and 0-10,000 respectively. The fuzzy set descriptors High, Low, Little and Moderate have
been used to represent fuzzy data values over respective domains. The membership functions of the
fuzzy set descriptors High, Low, Little and Moderate are domain dependent and are as given below :
For ‫∈ݔ‬ ܷ‫,݁ܿ݊݁݅ݎ݁݌ݔܧ‬
ߤ‫݁ݐܽݎ݁݀݋ܯ‬(‫)ݔ‬ = (1+ |‫)|8−ݔ‬ −1
for ‫ݔ‬ > 1
= 0 otherwise
ߤ‫݈݁ݐݐ݅ܮ‬(‫)ݔ‬ = (1+12‫)ݔ‬ −1
for ‫ݔ‬ > 0
= 0 otherwise
Similarly,
ߤHigh(y) = (1 + a|y-c|)-1 for y ≤ c
= 1 for y > c
where ܽ = 1/20,000 , ܿ = 60,000 for ‫ݕ‬ ∈ ܷ݈ܵܽܽ‫ݕݎ‬ and ܽ = 1/1,000 , ܿ = 5,000 for ‫ݕ‬ ∈ܷ‫ݔܽݐ݁݉݋ܿ݊ܫ‬
Also ߤ‫ݓ݋ܮ‬ ( ‫ݕ‬ ) = 1−ߤHigh(‫.)ݕ‬
3. FUZZY COUNT
As is often the case when extending a classical set theoretical notion to fuzzy set theory, various
alternative definitions for the notion of the cardinality of a fuzzy set (on a finite universe) have been
proposed. These proposals can be roughly subdivided in two categories:

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i. scalar cardinalities, where the cardinality of a fuzzy set is a positive real number.
([34],[41],[45-47]).
ii. fuzzy cardinalities, where the cardinality of a fuzzy set is defined as a fuzzy quantity (not
necessarily convex) ([1],[40],[47-49]).
Scalar cardinality of a fuzzy set is the sum of the membership values of all elements of the fuzzy set.
In particular, scalar cardinalities of a fuzzy set associate to each fuzzy set a positive real number. In
the similar way, fuzzy cardinalities of a fuzzy set associate to any fuzzy set a convex fuzzy natural
number. The fuzzy cardinality of fuzzy sets is itself also a fuzzy set on the universe of natural
numbers. The first definition of fuzzy cardinality of fuzzy sets by means of mapping from the set of
natural numbers to the interval [0,1], was proposed by Zadeh ([3-4].As fuzzy relation r on a relation
scheme ܴ (‫ܣ‬1,‫ܣ‬2,….‫ܣ‬݊) is a fuzzy subset of ݀‫݉݋‬ (‫ܣ‬1) × ݀‫݉݋‬ (‫ܣ‬2) ×……× ݀‫݉݋‬ (‫ܣ‬݊) which motivates
us to define the fuzzy cardinality of a fuzzy relation. Let ‫ݐ‬݅ be the ݅‫ݐ‬h
tuple of fuzzy relation r and ߤ‫ݎ‬(‫ݐ‬݅)
denote the membership degree of ݅‫ݐ‬h
tuple in r, then fuzzy cardinality of fuzzy relation r , called as
fuzzy count (denoted by FC) is defined by
with membership degree denoted by ߤ‫ݐ݊ݑ݋ܥ‬ and defined as
ߤ‫ݐ݊ݑ݋ܥ‬=ߤ‫ݎ‬ (‫ݐ‬‫ܥܨ‬) ∧ (1−ߤ‫ݎ‬ (‫ݐ‬‫ܥܨ‬+1) )
using the representation of r as
where ߤ‫ݎ‬ (‫ݐ‬1) ,…..ߤ‫ݎ‬ (‫ݐ‬݊) arranged in decreasing order of magnitude and ߤ‫ݎ‬( ‫ݐ‬0) =1, ߤ‫ݎ‬( ‫ݐ‬݊+1) =0.
The following simple procedure can be used to calculate fuzzy count of fuzzy relation r.
Algorithm
i. Input : a fuzzy relation r.
ii. Take fuzzy projection([3],[4]) and fuzzy selection ([3],[4]) on r over the specified predicates
yielding fuzzy relation sr.
iii. If sr=∅ , set FC = 0 with ߤ‫ݐ݊ݑ݋ܥ‬= ߤ‫ݎ‬(‫ݐ‬0) ∧(1−ߤ‫ݎ‬(‫ݐ‬1) ) . Stop.
iv. Arrange the tuples of sr such that ߤ‫ݎ‬(‫ݐ‬1) ,…..ߤ‫ݎ‬(‫ݐ‬݊) are arranged in decreasing order of magnitude.
v. Set pointer p to the first tuple of sr.
vi. If p ≠ ݈݊݅
initialize A and B array
i←1
while ( p ≠ ݈݊݅ )
A[i]←ߤ‫ݎݏ‬(‫ݐ‬݅−1)+ߤ‫ݎݏ‬(‫ݐ‬݅)
i++
p++

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k←1
largest←0
for j←1 ‫݋ݐ‬ (݅−1)
if (A[j]>1) then
B[k]←j
k++
Store the largest element of B in largest
vii. If ߤ‫ݎݏ‬(‫ݐ‬݈ܽ‫ݐݏ݁݃ݎ‬) ≥ 0.5 then ‫ܥܨ‬ = largest.
viii. If ߤ‫ݎݏ‬(‫ݐ‬݈ܽ‫ݐݏ݁݃ݎ‬) < 0.5 then ‫ܥܨ‬ = largest − 1.
ix. ߤc‫ݐ݊ݑ݋‬=ߤ‫ݎݏ‬ (‫ݐ‬‫ܥܨ‬) ∧ (1−ߤ‫ݎݏ‬ (‫ݐ‬‫ܥܨ‬+1) ).
x. Stop.
Example. Consider a fuzzy relation EMPLOYEE shown in table 1 as discussed in section 2.2. If the
query is: find the number of male people in Delhi who are Little experienced and paying low
incometax then the answer to it is obtained by using the algorithm of fuzzy count in EMPLOYEE
fuzzy relation as follows:
1. Take fuzzy projection on EMPLOYEE over the attributes State, Sex, Exp and Incometax.
2. Select the tuples(fuzzy selection) in the fuzzy relation obtained in step 1 where State=Delhi Sex=M,
Exp=little and Incometax=low.
3. Represent sr as
4. Using step (v)-(viii) of above algorithm ,we get FC=3
5. Using step (ix), we get ߤ‫ݐ݊ݑ݋ܥ‬=0.05
The answer to the query is 3 with membership degree 0.05. We can also say that the number of male
people in Delhi who are little experienced and paying low incometax is 3 with truth value 0.05.
Our définition of fuzzy count is in accordance with that of classical cardinality
Preposition 1. If r is a classical relation with k tuples then fuzzy count of r is k.
Proof. If r is a classical relation with k tuples then r can be represented as
where ߤ‫ݎ‬(‫ݐ‬1) ,…..ߤ‫ݎ‬ (‫ݐ‬݇) arranged in decreasing order of magnitude and ߤ‫ݎ‬(‫ݐ‬0) =1,ߤ‫ݎ‬ (‫ݐ‬݇+1) =0. Using
the above algorithm,
FC=k
ߤ‫ݐ݊ݑ݋ܥ‬ = ߤ‫ݎݏ‬ (‫ݐ‬݇) ∧ (1−ߤ‫ݎݏ‬ (‫ݐ‬݇+1)).
=1 ∧ (1−0) = 1.
Hence fuzzy count of r is 1 with truth value 1.
Proposition 2. Fuzzy cardinality of a fuzzy relation is a fuzzy convex set.
Proof. Let r be a fuzzy relation.Then,

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Then there is a unique value j such that following inequalities hold :
Let FC = k, then
The pattern of membership degrees of being increasing up to a certain value i.e. 1−ߤ‫ݎ‬(‫ݐ‬݆+1) or ߤ‫ݎ‬(‫ݐ‬݆)
and then decreasing signifies that fuzzy cardinality of a fuzzy relation is a fuzzy convex set.
4. FUZZY STATISTICAL DATABASE
A statistical database is a database used for statistical analysis purposes. Over the last several years
there has been growing interest in statistical database. This interest is due to the inadequacy of
commercial database management systems to support statistical applications. A similar situation exists
in other application areas such as CAD/CAM, VLSI design, and knowledge-based systems. The main
reason for this situation is that most of the commercial systems available today were designed
primarily to support transactions for business applications (such as marketing and banking), while
other applications has different data characteristics and processing requirements. Another reason for
the interest is the large amount of data that exist in statistical database applications. Many practical
databases collected for statistical purposes, such as trade data between countries, or the various census
data, are too large to be handled with conventional data management techniques efficiently. But, in
addition, there is a large amount of data that was not collected originally for experimental or statistical
purposes, that has tremendous potential when used for statistical purposes. For example, routine
patient records in hospitals can be used for statistical "cause and effect" studies. Business transactions
can be statistically analyzed for policy setting and econometric models. For the most part, such
sources of routine collections of data are left unused because adequate data management and analysis
facilities do not exist. Statistical database applications differ from commercial applications both in the
properties of the data and in the operations over the data. For example, statistical database often
contain sparse data (usually in a multidimensional matrix form) that need to be efficiently compressed.
They often contain special' data types such as vectors or time series. Similarly, statistical analysis
requires different types of operations over the data. Typically, only a few variables are examined over
a large number of the cases (as is the case in determining cross-correlation). In addition to statistical
operators, such as sampling and aggregation, the access of the data is of a different nature. For
example, it is quite common to access a region in multidimensional space, such as finding materials
with certain approximate properties, or cases that fit a statistical pattern. For such cases
multidimensional data structures and search methods are desirable. A statistical database can be
thought of as a collection of data sets which gives summary information on a certain population of
individuals or objects. It contains macro data, a collection of micro data, obtained by statistical
inferences over relational databases ([12],[19],[32]). Many researchers have explored the
fundamentals of statistical database([6],[8],[10-14],[16],[18],[22-24],[28]) could not be stored in the

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existing framework of statistical database as these are designed for storing the precise statistical data.
It is therefore imperative to provide a data model[5] for the storage of fuzzy statistics which motivates
us to develop the concept of fuzzy statistical database. A fuzzy statistical database gives the user the
ability to obtain fuzzy statistics information directly from the fuzzy statistical database which relieves
the user from the task of calculating fuzzy statistics from the fuzzy relation. It is important that the
fuzzy statistical database which incorporates imprecision be able to appropriately propagate the level
of uncertainty associated with the data to the level of uncertainty associated with answers or
conclusions based on data.
A fuzzy statistical scheme in fuzzy statistical data model is defined as ‫)ܥ,ܿܨ,ݎܨ(ܵܨ‬ where
• ‫ݎܨ‬ denote the fuzzy row attribute forest consisting of finite set of fuzzy attributes[33].
• ‫ܿܨ‬ denote the fuzzy column attribute forest consisting of finite set of fuzzy attributes and
• C is the fuzzy statistics.
Fuzzy statistical table on fuzzy statistical scheme ‫)ܥ,ܿܨ,ݎܨ(ܵܨ‬ is a tabular representation of fuzzy
statistics. It is represented as a table with additional two-dimensional array of cells for representing the
membership degree of fuzzy statistics, denoted by ߤ .The row header and column header of a fuzzy
statistical table are labeled by fuzzy attributes. Fuzzy attributes are structured in the form of an
ordered set of trees i.e. ‫ݎܨ‬ and ‫ܿܨ‬ .Each cell in a fuzzy statistical table has an associated set of fuzzy
row and fuzzy column attributes. The set of fuzzy row and fuzzy column attributes of a cell forms a
path from the root to a leaf in a fuzzy row and fuzzy column attribute tree. Each cell in a fuzzy
statistical table is labeled by an attribute called cell attribute. We now turn to a more formal
explanation of our definition.
Fuzzy statistical database is a collection of fuzzy statistical tables. A parenthesized expression to
specify an attribute tree which is a preorder enumeration of the tree (i.e. first the root then the sub trees
from left to right) is used. Suppose for accessing the fuzzy statistic C in fuzzy statistical table,
ܴ‫ܣ‬1,….,ܴ‫ܣ‬݊ be fuzzy row attributes with their appropriate universes ܴܷ1,….,ܴܷ݊ respectively which
forms a path from root to a leaf in fuzzy row attribute tree of ‫ܨ‬‫ݎ‬ and ‫ܤܥ‬1,….‫ܤܥ‬݉ be fuzzy column
attributes with their appropriate universes ‫ܷܥ‬1,…..,‫ܷܥ‬݉ which forms a path from root to a leaf in a
fuzzy column attribute tree of ‫ܿܨ‬ . Then fuzzy statistical table on fuzzy statistical scheme
FS(ܴ‫ܣ‬1(ܴ‫ܣ‬2(ܴ‫ܣ‬3…… ܴ‫ܣ‬݊)),(‫ܤܥ‬1(‫ܤܥ‬2 ‫ܤܥ‬3…..‫ܤܥ‬݉ )),(C)) is a fuzzy subset of
ܴܷ1×ܴܷ2×……×ܴܷ݊×‫ܷܥ‬1×‫ܷܥ‬2×…..×‫ܷܥ‬݉.
A fuzzy statistical table instance is a collection of cell instances structured as specified by the fuzzy
statistical scheme. A cell instance consists of value of its fuzzy row and fuzzy column attributes and a
value for its fuzzy statistic along with its membership degree. Depending upon the complexity of
domain of fuzzy row and fuzzy column attributes of fuzzy statistical table, it can be classified into two
categories-
(a) Type-1 Fuzzy Statistical Table
(b) Type-2 Fuzzy Statistical Table
Type -1 Fuzzy Statistical Table: A type-1 fuzzy statistical table enables us to capture impreciseness
of the statistics associated with the fuzzy row and fuzzy column attributes. In type-1 fuzzy statistical
table the domain of fuzzy row and fuzzy column attributes may be a classical subset or a fuzzy subset
of their appropriate universes. In other words, here fuzzy attributes [33] are of type-1.Let c be an
instance of fuzzy statistics in two dimensional array along with its membership degree ߤ(ܿ) and
ܴ‫ܣ‬1,….,ܴ‫݊ܣ‬ be fuzzy row attributes with their appropriate universes ܴܷ1,….,ܴܷ݊ respectively which

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forms a path from root to a leaf in fuzzy row attribute tree for accessing c alongwith their membership
function denoted by ߤܴ‫݅ܣ‬,
ߤܴ‫݅ܣ‬:ܴܷ݅→ [0,1] ,݅ = 1….݊.
Let ‫ܤܥ‬1,….‫ܤܥ‬݉ be fuzzy column attributes with their appropriate universes ‫ܷܥ‬1,…..,‫ܷܥ‬݉ which
forms a path from root to a leaf in a fuzzy column attribute tree for accessing c along with their
membership function denoted by ߤ‫݆ܤܥ‬ for j=1….m,
ߤ‫݆ܤܥ‬ :‫ܷܥ‬݆ → [0,1]
Consider the Cartesian product, ݀‫݉݋‬ (ܴ‫ܣ‬1) ×݀‫݉݋‬ (ܴ‫ܣ‬2) ×….×݀‫)݊ܣܴ(݉݋‬ ×݀‫ܤܥ(݉݋‬1) ×݀‫݉݋‬
(‫ܤܥ‬2)×……….×݀‫.)݉ܤܥ(݉݋‬
It is a fuzzy subset of ܴܷ1×ܴܷ2×……×ܴܷ݊×‫ܷܥ‬1×‫ܷܥ‬2×…..×‫ܷ݉ܥ‬ . In type-1 fuzzy statistical table
the membership function of fuzzy statistic satisfies the following inequality:
ߤ ܿ ≤ min( ߤܴ‫ܣ‬1 (‫)1ݑݎ‬ ,….ߤܴ‫݊ܣ‬ (‫ݑݎ‬݊) ,ߤ‫ܤܥ‬1 (ܿ‫ݑ‬1) ,..ߤ‫݉ܤܥ‬ (ܿ‫ݑ‬݉))
where ‫ݑݎ‬݅ ∈ ܴܷ݅, ܿ‫ݑ‬݆ ∈ ‫,݆ܷܥ‬ ݅=1…݊, ݆ =1….݉. We can also treat ߤ(ܿ) as a fuzzy truth value
belonging to [0,1] i.e. ߤ(ܿ) is a truth value of a fuzzy predicate associated with c when the fuzzy
attributes which are occurring in path from root to leaf to access c are replaced by their values. We
illustrate type-1 fuzzy statistical table by an example.
Example. Consider a fuzzy statistical scheme
2000COUNT(State(Sex(Exp,Sal)),(Incometax),(Count))
of highly salaried, highly paying incometax and highly experienced people in a sample of a
population. The fuzzy statistic being measured is the fuzzy count represented by cell attribute count
where ‫ܨ‬‫ݎ‬ is fuzzy row attribute forest consisting of single tree with fuzzy attributes State, Sex,
Experience and Salary. Experience and Salary are denoted by Exp and Sal respectively, ‫ܨ‬ܿ is fuzzy
column attribute forest consisting of single tree with fuzzy attribute Incometax. Count is the fuzzy
count of male and female people in a state who are highly experienced and are paying high incometax
or having high salary in a sample of a population. Table 2 shows an instance of fuzzy statistics table
2000COUNT. In 2000COUNT there are 128 instances for the cell attribute count with corresponding
128 instances characterizing their fuzziness. Suppose the Universe of discourse for the Exp, ܷ‫݌ݔܧ‬ is
the set of positive integers in the range 0-30, Universe of discourse for Sal ,ܷ݈ܵܽ is the set of integers in
the range 10,000-100,000, Universe of discourse for Incometax, ܷ‫ݔܽ݁݉݋ܿ݊ܫ‬ is the set of integers in the
range 0-10,000, Universe of discourse for State is {Delhi, Bombay} , Universe of discourse for Sex is
{M,F}.Here domain of State and Sex are crisp sets whereas the domain of Experience, Incometax and
Salary are fuzzy sets High-Exp, High-Sal and High-Incometax in their appropriate universes. i.e.
݀‫)݁ݐܽݐܵ(݉݋‬ = {‫݈݁ܦ‬h݅,‫}ݕܾܽ݉݋ܤ‬
݀‫݉݋‬ (ܵ݁‫)ݔ‬ = {‫,ܯ‬ ‫}ܨ‬
݀‫݉݋‬ (‫)݌ݔܧ‬ = High-Exp
݀‫݉݋‬ (݈ܵܽ) = High-Sal
݀‫݉݋‬ (‫ݐ݁݉݋ܿ݊ܫ‬a‫)ݔ‬ =High-Incometax
The membership function ߤ‫ܺܪ‬,ߤ‫ܵܪ‬ and ߤ‫ܫܪ‬ of the fuzzy sets High-Exp, High-Sal and High-Incometax,
are as given below:

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Type-2 Fuzzy Statistical Table: Although type-1 fuzzy statistical table enable us to represent
impreciseness of the statistics associated with fuzzy attributes, its role in capturing uncertainty in data
values is rather limited. For example, in a type-1 fuzzy statistical table 2000COUNT, one is not
permitted to specify the salary of male people in Delhi to be in the range Rs. 40,000-Rs.50,000 or low
because of type -1 fuzzy attributes. So in order to accommodate a wider class of data ambiguities, a
further generalization of the type-1 fuzzy statistical table is considered in the form of type-2 fuzzy
statistical table. In type-2 fuzzy statistical table domain of fuzzy row and fuzzy column attributes are
sets of fuzzy sets in their appropriate universes. In other words it consists of type-2 fuzzy attribute.
Suppose c be an instance of fuzzy statistics in type-2 fuzzy statistical table with membership degree ߤ
ܿ and ܴ‫ܣ‬1,…. ܴ‫ܣ‬݊ be fuzzy row attributes with their appropriate universes ܴܷ1,….,ܴܷ݊ which forms a
path from root to a leaf in a fuzzy row attribute tree for accessing c along with their membership
function denoted by ߤܴ‫݅ܣ‬ for ݅=1…݊, ߤܴ‫݅ܣ‬:ܴܷ݅→ 0,1 and ‫݉ܤܥ.,1ܤܥ‬ be fuzzy column attributes with
their appropriate universes ‫ܷ݉ܥ,..…,1ܷܥ‬ respectively which forms a path from root to a leaf in a
fuzzy column attribute tree for accessing c along with their membership function denoted by ߤ‫݆ܤܥ‬ for
j=1…m , ߤ‫→݆ܷܥ:݆ܤܥ‬ 0,1 . Here, dom(R‫)݅ܣ‬ is the set of fuzzy sets in Rܷ݅, i=1….n and dom(‫)݆ܤܥ‬ is
the set of fuzzy sets in ‫,݆ܷܥ‬ j=1….m. Consider a tuple ‫ܽݎ(=ݐ‬1,‫ܽݎ‬2,….,‫ܽݎ‬݊,ܾܿ1,ܾܿ2….,ܾܿ݉) in cartesian
product ݀‫݉݋‬ ܴ‫ܣ‬1 ×݀‫݉݋‬ ܴ‫ܣ‬2 ×..×݀‫݉݋‬ ܴ‫݊ܣ‬ ×݀‫݉݋‬ ‫ܤܥ‬1 ×݀‫݉݋‬ ‫ܤܥ‬2 ×……݀‫.)݉ܤܥ(݉݋‬ It is a fuzzy
subset of ܴܷ1×ܴܷ2×…ܴܷ݊×‫ܷܥ‬1×‫ܷܥ‬2×….‫.ܷ݉ܥ‬ In type-2 fuzzy statistical table, the membership
function of fuzzy statistic satisfies the following inequality:
ߤ(ܿ) may be regarded as a fuzzy truth value belonging to [0,1] i.e. ߤ (ܿ) is a truth value of a fuzzy
predicate associated with c when the fuzzy attributes which are occurring in path from root to leaf to
access c are replaced by their values. We illustrate type-2 fuzzy statistical table by an example.
Example . Consider a type-2 fuzzy statistical scheme
2001COUNT(State(Sex(Exp,Sal)),(Incometax),(Count))
in a sample of a population shown in table 3. As in above example , the Universe of discourse for the
Exp ܷ‫݌ݔܧ‬ is the set of positive integers in the range 0-30, Universe of discourse for Sal ܷ݈ܵܽ is the set
of integers in the range 10,000-100,000, Universe of discourse for Incometax ܷ‫ݔܽ݁݉݋ܿ݊ܫ‬ is the set of
integers in the range 0-10,000, Universe of discourse for State is {Delhi, Bombay} , Universe of
discourse for Sex is {M, F}. Domain of state and sex are crisp sets whereas the domain of Experience,
Incometax and Salary are set of fuzzy sets in their respective universes .i.e.
݀‫݉݋‬ (‫)݌ݔܧ‬ = set of fuzzy sets in ܷ‫݌ݔܧ‬ ={Little,Mod,10,15-20}
݀‫݉݋‬ (݈ܵܽ) =set of fuzzy sets in ܷ݈ܵܽ ={30,000,High,Low,40,000−60,000}
݀‫݉݋‬ (‫)ݔܽݐ݁݉݋ܿ݊ܫ‬ =set of fuzzy sets in ܷ‫݁݉݋ܿ݊ܫ‬tܽ‫ݔ‬ ={High,Low,3,000,4,000−7,000}

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The membership functions of the fuzzy set descriptors High, Low, Little and Mod are domain
dependent and are as given below.
The fuzzy statistics count is the fuzzy count of male and female people in a state who are experienced
or salaried and are paying incometax in a sample of a population.
5. FUZZY PRIMITIVE STATISTICAL TABLE
Fuzzy statistical table provides an efficient logical modelling tool for statisticians to store fuzzy
statistics in a statistical environment. Sometimes statisticians are not interested in the entire fuzzy
statistical table. Instead of taking full tables into account, their analysis may based on a part of fuzzy
statistical table. For example, for making decisions on how many people to be retained in an
organization the statistics of experienced people is to be taken care of. Then there is no need to go
through the entire 2001COUNT fuzzy statistical table, a part of it serves the desired purpose. This
motivates us to propose fuzzy primitive statistical table.
Definition. A fuzzy statistical table ‫ܨ(ܵܨ‬‫ݎ‬,‫ܨ‬ܿ,‫)ܥ‬ is a fuzzy primitive statistics table if |‫ܨ‬‫ݎ‬| =1, |‫ܨ‬ܿ| = 1
and each tree in ‫ܨ‬‫ݎ‬ and ‫ܨ‬ܿ has exactly one leaf .The fuzzy statistical table shown in table III consists of
two fuzzy primitive statistics table as |‫ܨ‬‫ݎ‬| =1, |‫ܨ‬ܿ| =1 and the tree in ‫ܨ‬‫ݎ‬ has two leaves. The instance of
two fuzzy primitive statistics table of above example are shown in table 4 and table 5 respectively.
6. PHYSICAL ORGANIZATION OF FUZZY STATISTICAL TABLE
If for any sub tree in fuzzy row attribute forest ‫ܨ‬‫ݎ‬ or in fuzzy column attribute forest ‫ܨ‬ܿ , there is
always a unique instance in a fuzzy statistical table, we say that the fuzzy statistical table has a full
cross product otherwise it is incomplete. For example, consider the fuzzy statistical table FSINC
instance in table 6. It does not have a full cross product as the instance of Exp as a sub tree of Delhi is
different from the instance of Exp as a sub tree of Bombay. Assuming that the fuzzy statistical table
have a full cross product, we separately store the fuzzy row attribute forest ‫ܨ‬‫ݎ‬, fuzzy column attribute
forest ‫ܨ‬ܿ ,fuzzy statistics and their membership degree. The cell value arrays ‫ܥ‬‫ݏ‬ and ‫ܥ‬ߤ store the fuzzy
statistics and their membership degrees respectively. The cell value array ‫ܥ‬ߤ is in one to one
correspondence with ‫ܥ‬‫ݏ‬.

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6.1. Computing Fuzzy Attributes
For simplicity, we put fuzzy row forest ‫ܨ‬‫ݎ‬ into ordered tree TR by making the root nodes in ‫ܨ‬‫ݎ‬ as
immediate descendents of a dummy attribute ߠ‫ݎ‬ and fuzzy column attribute forest ‫ܨ‬ܿ into ordered tree
TC by making the root nodes in ‫ܨ‬ܿ as an immediate descendents of dummy attribute ߠܿ.Consider an
ordered sub tree of tree TR or TC with root R. The total number of leaves in the instance of the sub
tree with root R denoted by NL(R) is given by
where |ܴ| denotes the number of instance of attributes R and ܵ1,ܵ2,……ܵ݊ are immediate successors of
R
Let attribute S be an immediate successors of attribute R in an ordered tree. The total number of leaves
in the instance of a sub tree with root R and to the left of S, denoted by ܰ‫)ܵ,ܴ(ܮ‬ is given by
where ܵ1,ܵ2,……ܵ݊ are immediate successors of R and they are to the left of S in an ordered tree. Let
ܶܰ‫)ܴ(ܮ‬ denote the total number of leaves in an instance corresponding to one sub tree of R, then
ܶܰ‫)ܴ(ܮ‬ =ܰ‫)ܴ(ܮ‬ / |ܴ|
For efficient representation of fuzzy attributes, we use the total ordering to the values of a fuzzy
attribute and use the relative position of an attribute value as its encoded value. For example, consider
Sex attribute and it’s ordered instances M and F, we would use integers 0 and 1 to encode M and F
respectively. Attributes with their instance and coded value are given in table 7.
Let (ܲ,ܸ) denote the root to leaf path instance such that
1. ܲ = (ܺ1,ܺ2,…..ܺ݊) is a root to leaf path of TR or TC where the root is omitted. ܺ1 is an
immediate successor of the root and ܺ݅+1 is an immediate successor of ܺ݅ for 1 ≤ ݅ <݊
2. ܸ= (‫)݊ݔ……,2ݔ,1ݔ‬ is an ordered set of values where ‫ݔ‬݅ ≥ 0 is an encoded value for the
attribute ܺ݅ , 1 ≤ ݅ ≤݊.
For TR, when root to leaf path instance is given (ܲ,ܸ) the corresponding row leaf number is denoted
by ܴ‫((ܰܮ‬ ܺ1,‫ݔ‬1) , (ܺ2,‫ݔ‬2) ,….. (ܺ݊,‫ݔ‬݊) or simply ܴ‫.ܰܮ‬ It can be computed as
where ܺ0 is the dummy root attribute of the ordered tree. Leaves are numbered beginning at 0.
Similarly, for TC the column leaf number denoted by CLN corresponding to root to leaf instance
(ܲ,ܸ) can be computed from above formula.
Given N (row leaf number or column leaf number), the corresponding root to leaf path instance can be
obtained as follows:
Let {ܵ1,ܵ2,…….ܵ݊} be the ordered set of immediate successors of ܺ0 (the dummy root attribute). Then

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6.2. Storage of Cell Attribute Values
The values of cell attributes are stored in two cell value array one is cell value array1 ‫ܥ‬ܵ and the other
is cell value array2 ‫ܥ‬ߤ . The rows of the ‫ܥ‬‫ݏ‬ correspond to the root to leaf path instances of trees in ‫ܨ‬‫ݎ‬
and the column of ‫ݏܥ‬ correspond to the root to leaf path instances of trees in ‫ܨ‬ܿ.They are stored in two
cell value file, cell value file1 and cell value file2 respectively in row order with one value per logical
record. Let < (ܴܲ,ܸܴ), (ܲ‫)ܥܸ,ܥ‬ > be an ordered set of root to leaf path instances where
• ܴܲ=(ܺ1,ܺ2,….ܺ݉) is a root to leaf path of TR and ߠ‫ݎ‬ is omitted from the tree.
• ܸܴ= (‫ݔ‬1,‫ݔ‬2,……‫ݔ‬݉) is an ordered set of values for the attributes in the path ܴܲ, where ‫ݔ‬݅ is an
encoded value for ܺ݅, 1 ≤ ݅ ≤ ݉ .
• ܲ‫=ܥ‬ (ܻ1,ܻ2,…..ܻ݊) is a root to leaf path of TC and ߠܿ is omitted from the path.
• ܸ‫=ܥ‬ (‫ݕ‬1,‫ݕ‬2,….‫ݕ‬݊) is an ordered set of values for the attributes in the path ܲ‫,ܥ‬where ‫ݕ‬݅ is an
encoded value for ܻ݅,1 ≤ ݅ ≤݊.
The relative number, r1 of the record in the cell value file1 containing the cell value associated with <
ܴܲ,ܸܴ ,(ܲ‫)ܥܸ,ܥ‬ > is obtained as
‫ܰܮܴ=1ݎ‬ ((ܺ1,‫ݔ‬1) ,…… (ܺ݉,‫ݔ‬݉)) ܰ‫ߠ(ܮ‬ܿ1) +‫ܻ((ܰܮܥ‬1,‫ݕ‬1) ,….. (ܻ݉,‫ݕ‬݉)) +1
The corresponding record number r2 in cell value file2 characterizing the fuzziness of record in cell
value file1 is given by ‫2ݎ‬ = ‫.1ݎ‬
Given ‫1ݎ‬ the relative number of a record in the cell value file1, the ordered set of root to leaf path
instances < ܴܲ,ܸܴ ,(ܲ‫>)ܥܸ,ܥ‬ corresponding to ‫1ݎ‬ can be obtained as
(1) First, we obtain the row number for (ܴܲ, ܸܴ) and then column number for (ܲ‫)ܥܸ,ܥ‬
ܴܰ= [ (‫)1ܿߠ(ܮܰ)1−1ݎ‬ ]
NC = r1 – 1 – NR (NL (ߠܿ1))
(2) Then, we obtain the root to leaf path instances corresponding to ܴܰ and ܰ‫ܥ‬ as discussed in
section 6.1.
7. ALGORITHM
In this section, we develop the algorithm for creating fuzzy statistical table. The following variables
and data types have been used in explaining the algorithmic details.

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UD Universe of Discourse, two dimensional string array for domain of attributes of row header and column
header of Fuzzy Statistical Table
I Two dimensional string array for Instances of attributes of row header and column header of Fuzzy
Statistical Table
MM Two dimensional integer array(column dimension 4) for storing the variables in the mathematical model
of Instances of I
TR Binary tree created by making the root nodes in ‫ݎܨ‬ as immediate descendent of a dummy attribute rowroot
TC Binary tree created by making the root nodes in ‫ܿܨ‬ as immediate descendent of dummy attributes
columnroot
PR Array to store the nodes of tree TR in row header of Fuzzy Statistical Table
PC Array to store the nodes of tree TC in column header of Fuzzy Statistical Table
VR ,VR1,VR2,V integer array to store the encoded value of Instances of nodes of tree,TR in row header
of Fuzzy Statistical Table
VC ,VC1,VC2,V2 integer array to store the encoded value of Instances of nodes of tree TC in column
header of Fuzzy Statistical Table
‫ݏܥ‬ Two dimensional array to store fuzzy statistics of Fuzzy Statistical Table
‫ߤܥ‬ Two dimensional array to store membership degrees of fuzzy statistics of Fuzzy Statistical Table
x,y Integer arrays
RLN ,RLN1,RLN2 Row leaf number
CLN,CLN1,CLN2 Column leaf number
M Boolean variable
Develop the following structures
structure treenode
{ string : Name_of_Attribute
UD : Universe_of_Discourse
number : Type /* number is either 1 or 2 for Type-1 and Type-2 resp. */
I : Instance
MM : Mathematicalmodel
treenode *Lchild
treenode *Rchild
}
structure Fuzzytable
{ string : Name_of_table
list : ‫ݏܥ‬
list : ‫ߤܥ‬
treenode * TR
treenode * TC
}
structure node
{ number : info
node * next
}
Algorithm for Creating Fuzzy Statistical Table
Create_Fuzzy_Statistical_Table( )
1. integer n
2. Fuzzytable table
3. initialize table ← null
4. print “ Do you want to create a fuzzy statistical table “
5. print “ enter 1 for yes and 0 for no “
6. read n
7. if (n=1) then
8. Input_Fuzzy_Statistical_Table(&table)
9. else

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10. print “ Good Bye”
11. return
Input_Fuzzy_Statistical_Table(table)
1. integer b,n,VR[ ]
2. treenode *PR[ ]
3. print “Enter name of Fuzzy Statistical Table”
4. read table.Name_of_table
5. print “ enter b=1 for creating row tree and b=2 for creating column tree “
6. read b /* variable to store code for the construction of tree TR or TC */
7. If (b=1) then
8. table.TR.Name_of_Attribute⟵”Rowroot” /* initialization of root
attribute of row header */
9. table.TR.Universe_of_Discourse⟵null
10. table.TR.Type⟵1
11. table.TR.Instance⟵null
12. table.TR.Mathematicalmodel⟵null
13. print “ Enter number of children of dummy root attribute in row header “
14. read n /* n is an integer which can take value either 1 or 2 */
15. print “ Enter data for row tree “
16. table.TR.Lchild⟵ Create_tree( )
17. If (n=2) then
18. table.TR.Rchild⟵ Create_tree( )
19. If (b=2) then
20. table.TC.Name_of_Attribute⟵”Columnroot” /* initialization of root
attribute of Column header */
21. table.TC.Universe_of_Discourse⟵null
22. table.TC.Type⟵1
23. table.TC.Instance⟵null
24. table. TC.Mathematicalmodel⟵null
25. print “ Enter number of children of dummy root attribute in column header”
26. read n /* n is an integer which can take value either 1 or 2 */
27. table.TC.Lchild⟵ Create_tree( )
28. If (n=2) then
29. table.TC.Rchild⟵ Create_tree( )
30. initialize PR←null
31. initialize VR← -1
32. If (table.TR.Name_of_Attribute=”rowroot”) then
33. PR[0]⟵table.TR
34. else
35. search table.TR for table.TR.Name_of_Attribute=”rowroot”
36. PR[0]⟵table.TR
37. initialize table. ‫←ݏܥ‬table.‫←ߤܥ‬ null
38. CalculatePRVR(&table,1,PR[0].Lchild,&PR,&VR)
39. return
Create_tree( )
1. treenode *q
2. integer n,len,choice
3. print “Input data for the attributes of Fuzzy Statistical Table “
4. print “Enter the name of attribute--->”
5. read q.Name_of_Attribute
6. print “Enter the Universe of discourse as two dimensional string array--->”
7. read q.Universe_of_Discourse

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8. print “Enter the type of fuzzy data--->”
9. read q.Type
10. print “ Enter the Instances as two dimensional string array--->”
11. read q.Instance
12. len←length(q.Instance)
13. if (q.type=1) then
14. print “membership degrees are to be entered by user if yes enter 1 else 0”
15. read choice
16. if (choice=1) then
17. for i ← 0 to (len-1)
18. print “Enter membership degree of q.Instance[i]--->”
19. read q.Mathematicalmodel[i]
20. else
21. print “Enter data for the mathematical model of q.Instance[i]--->”
22. print “1− 1 sign x−breakvalue /denominator −1"
23. print “If compliment of some mathematical function enter 1 else 0--->”
24. read q.Mathematicalmodel[1][0]
25. print “Enter sign used in mathematical model--->”
27. print “ Enter the break value used in mathematical model--->”
29. print “ Enter the denominator value used in mathematical model --->“
31. if (q.type=2) then
32. for i←0 to (len-1)
33. if (check_linguistic_term(q.Instance[i])=true) then
34. print “Enter data for the mathematical model of q.Instance[i]--->”
35. print “1− 1 sign x−breakvalue /denominator −1"
36. print “If compliment of some mathematical model enter 1 else 0--->”
37. read q.Mathematicalmodel[i][0]
38. print “Enter sign used in mathematical model--->”
40. print “ Enter the break value used in mathematical model--->”
42. print “ Enter the denominator value used in mathematical model --->“
44. else
45. print “Enter membership degree of q.Instance[i]--->”
46. read q.Mathematicalmodel[i]
47. print “Enter number of children”
48. read n
49. If (n=0) then
50. q.Lchild⟵null
51. q.Rchild⟵null
52. return q
53. If (n=1) then
54. q.Lchild ⟵ Create_tree( )
55. q.Rchild⟵null
56. return q
57. If (n=2) then
58. q.Lchild⟵ Create_tree( )

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59. q.Rchild⟵ Create_tree( )
60. return q
CalculatePRVR (table,index1,q,PR,VR) /* index1 is variable to indexed through PR and VR */
1. treenode *P[ ]
2. integer j,m,VC[ ],RLN,index2
3. If (q≠null) then
4. PR[index1]⟵q
5. for i ⟵ 0 to [length(q.Instance)-1]
6. VR[index1]⟵i
7. print “ Enter data for”,q.Name_of_Attribute,”Instance encoded value”,i
8. If (q.Lchild=null) and (q.Rchild=null) then
9. RLN←0
10. for j ←1 to index1 /*calculates row leaf number*/
11. m ← ܰ‫ܮ‬ ܴܲ ݆−1 ,ܴܲ ݆ +ܸܴ ݆ ∗ܶܰ‫ܮ‬ ܴܲ ݆
12. RLN←RLN + m
13. initialize PC← null
14. initialize VC← -1
15. If (table.TC.Name_of_Attribute=”columnroot”) then
16. PC[0]⟵table.TC
17. else
18. search tree table.TC for table.TC.Name_of_Attribute=”columnroot”
19. PC[0]⟵table.TC
20. index2⟵1 /* variable to indexed through PC and VC */
21. CalculatePCVC(&table,index1,&PR,&VR ,index2,PC[0].Lchild,&PC,&VC,RLN)
22. index1⟵index1+1
23. CalculatePRVR(&table,index1,q.Lchild,&PR,&VR)
24. CalculatePRVR(&table,index1,q.Rchild,&PR,&VR)
25. index1⟵index1-1
26. VR[index1]← -1
27. return
CalculatePCVC(table,index1,PR,VR,index2,r,PC,VC,RLN)
1. integer i,CLN,j,m
2. If (r≠null) then
3. PC[index2]⟵r
4. for i ⟵ 0 to [length(r.Instance)-1]
5. VC[index2]⟵i
6. print “ Enter data for”,r.Name_of_Attribute,”Instance encoded value”,i
7. If (r.Lchild=null) and (r.Rchild=null) then
8. CLN←0
9. for j ←1 to index2 /*calculates column leaf number */
10. m ← ܰ‫ܮ‬ ܲ‫ܥ‬ ݆−1 ,ܲ‫ܥ‬ ݆ +ܸ‫ܥ‬ ݆ ∗ܶܰ‫ܮ‬ ܲ‫ܥ‬ ݆
11. CLN←CLN + m
12. print “Enter statistical data--->”
13. read table.‫[ݏܥ‬RLN,CLN]
14. print “Enter corresponding membership degree--->”
15. read table.‫[ߤܥ‬RLN,CLN]
16. index2⟵index2+1
17. CalculatePCVC(&table,index1,&PR,&VR, index2,r.Lchild,&PC,&VC,RLN)
18. CalculatePCVC(&table,index1,&PR,&VR, index2,r.Rchild,&PC,&VC,RLN)
19. index2⟵index2-1
20. VR[index2]← -1
21. Return

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8. CONCLUSION
Although there has been a flurry of research activity in database during recent years, it does not cover
the issue of storage of linguistic terms or ambiguities in statistical data values. In this paper, we have
defined the concept of fuzzy count and addressed the issue of representing fuzziness in statistical
environment giving rise to fuzzy statistical database. It alleviates the user from tedious task of
computing fuzzy statistics again and again from fuzzy relation. Fuzzy statistical database would be
quite useful when used as a decision in areas such as management, decision making and census data
evaluation. A logical view for representation of fuzzy statistical database, fuzzy statistical table is
discussed. Depending upon the type of category attributes in fuzzy statistical table type-1 and type-2
fuzzy statistical tables are explored. The physical organization technique of fuzzy statistical table is
discussed and algorithm for its implementation is also developed. For retrieving the information about
a particular tuple, we are developing a query system where DDL and DML are taken care of so that
our desired information can be displayed.
ACKNOWLEDGEMENTS
We thank the School of Computer & Systems Sciences, Jawaharlal Nehru University for providing us the
resources to conduct this research. We also thank all the researchers whose contribution to the field of
fuzzy databases has helped us to give this paper the present shape.
Table 1. An instance of type-2 fuzzy relation Employee in a sample of a population
Table 2. An instance of type-1 fuzzy statistical table 2000COUNT of highly salaried, highly paying
incometax and highly experienced employees in a sample of a population

45
Table 3. An instance of type-2 fuzzy statistical table 2001COUNT in a sample of a population
Table 4. An instance of fuzzy primitive table FS1 in a sample of a population
Table 5 An instance of fuzzy primitive table FS2 in a sample of a population
Table 6 An instance of fuzzy statistical table with incomplete cross product, FSINC

46
Table 7. Encoded value of attributes
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