Thuyết trình cơ sở dữ liệu nâng cao fuzzy orderings in flexible query answering systems

Introduction VQS Conditions based on fuzzy orderings The aggregation issue Summary 2... 1/ The vague query system VQSThe syntax of VQL The operator ‘‘IS’’ should be understood in

Advanced Database Systems

Fuzzy Orderings in Flexible Query

Answering Systems

Lê Hồng Dũng – 7140819

Âu Mậu Dương – 7140820

Lê Nguyên Dũng – 7140224 Ngô Đình Dũng – 1570203

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Introduction

VQS

Conditions based on fuzzy orderings

The aggregation issue

Summary

2

1/ The vague query system (VQS)

VQS is an add-on to conventional relational databases which acts as a proxy between the user and the database

Since VQS communicates with the underlying database only on the basis of standard SQL which allows easy integration into existing

1/ The vague query system (VQS)

The syntax of VQL

The operator ‘‘IS’’ should be understood in the sense of ‘‘is similar to’’

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1/ The vague query system (VQS)

There is one single ‘‘IS’’ condition in the query, VQS retrieves all records

from the data source and ranks them according to the distance from the query value

In case that the column contains numeric values, the distance between two values x;

y can easily be computed as the absolute value of the difference |x - y| (Euclidean norm for the one-dimensional real space R).

If the column under consideration is non-numeric, the distance is computed as the

distance of the associated values in the corresponding NCR table

VQS works with normalized distances Every condition, therefore, is

assigned a distance value normalized to the unit interal [0;1].

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1/ The vague query system (VQS)

In case that two or more ‘‘IS’’ conditions are combined with ‘‘AND’’,

a weighted average of the distances in the different columns is used

to rank the results (equal weights are used by default, which can be overridden using the optional ‘‘WEIGHTED BY’’ expression)

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VÍ DỤ 1

Xét bảng dữ liệu

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VÍ DỤ 1

Bảng 2 Ma trận khoảng cách của Location trong bảng 1 (bảng NCR)

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VÍ DỤ 1

Consider the following query:

SELECT FROM HotelTable WHERE Location IS ‘Salzburg Center’

AND Price IS 70 AND Category IS 4 INTO ResultSet

Assume that the distances between locations are given as in Table 2 (as the result of computing a distance measure for corresponding values in an NCR table).

To compute the result set for this query which means that the distance of any two locations is divided by 147.8, each discrepancy in the price is divided by 60, and each discrepancy in the category is divided by 2

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VÍ DỤ 1

Using equal weights, we obtain the result set shown in Table 3 sorted

by the closeness to the query

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NHƯỢC ĐIỂM CỦA VQS

Disadvantages: Example 1 clearly demonstrates two severe shortcomings of

VQS:

1 VQS is restricted to ‘‘IS’’ queries that are interpreted with a certain tolerance for imprecision For the price column, however, this is a painful limitation, as the user is not necessarily interested in a price that is as close to 70 as possible, but more likely in a

price that exceeds 70 as little as possible.

2 The normalization of distances is done for all columns independently solely on the basis of the largest distance between two values in the column The result is that two distance values for different columns may be difficultly comparable.

This paper is to tackle the first problem.

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Ngôn ngữ oVQL (Ordering-enriched vague query language)

Ngôn ngữ oVQL (Ordering-enriched vague query language)

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Ngôn ngữ oVQL (Ordering-enriched vague query language)

It is obvious that oVQL differs from VQL in the respect that there is

an explicit distinction between numeric and non-numeric attributes

For non-numeric ones, only the ‘‘IS’’ condition is defined like in VQL

For numeric ones, three new types of conditions ‘‘IS AT LEAST’’, ‘‘IS AT MOST’’, and ‘‘IS WITHIN’’ are added.

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Introduction

VQS

The aggregation issue

Summary

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Conditions based on fuzzy orderings

The single conditions ‘‘IS’’, ‘‘IS AT LEAST’’, ‘‘IS AT MOST’’, and ‘‘IS WITHIN’’ are modeled

Example: Price IS AT MOST 70

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Conditions based on fuzzy orderings

 Definition 1: Fuzzy equivalence relation

A binary fuzzy relation E: X2 → [0,1] is called fuzzy equivalence relation with respect to a t-norm T, for

brevity T-equivalence, if and only if the following three axioms are fulfilled for all x, y, z X:

1 Reflexivity: E(x,x) = 1.

2 Symmetry: E(x,y) = E (y,x).

3 T- transitivity: T(E(x,y), E(y,z)) E(x,z)



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Conditions based on fuzzy orderings

 Definition 2: fuzzy ordering

 A fuzzy relation L: X2 → [0,1] is called fuzzy ordering with respect to a t-norm T and a T-equivalence E, for brevity T-E-ordering, if and only if it is T-transitive and fulfills the following two axioms for all x, y

1 E-Reflexivity: E(x,y) L(x,y).

2 T-E-antisymmetry: T(L(x,y),L(y,x))



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Conditions based on fuzzy orderings

Definition 3:

A crisp ordering on a domain X and a T-equivalence E: X2 → [0,1] are called compatible if and only if the following holds for all x, y, z X:

x y z => E(x,z)



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Conditions based on fuzzy orderings

Theorem 1: [1, 2] Consider a fuzzy relation L on a domain X and a T-equivalence E Then the following two statements are equivalent:

L is a strongly complete T-E-ordering.

There exists a linear ordering the relation E is compatible with such that L can be represented as follows:

• L(x,y) =

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Conditions based on fuzzy orderings

Theorem 2: Consider a continuous Archimedean t-norm T with additive generator f.

For any pseudo-metric d: X2 → [0,] the mapping Ed: X2 → [0,] defined as Ed(x,y) = f-1 ( min (d(x,y), f(0) ) (1)

Provided that E: X2 → [0,] is a T-equivalence, we can define a pseudo-metric dE: X2 → [0,] as dE (x,y) = f (E(x,y)) (2)



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Conditions based on fuzzy orderings

Proposition 1: Let T be a continuous Archimedean t-norm with an additive generator f and let be an ordering of the domain X.

If a pseudo-metric d: X2 → [0,], fulfills

x y z => d(x,z) (3)

If a fuzzy equivalence relation E: X2 → [0,1] is compatible with , , its induced pseudo-metric

dE defined as in (2), fulfills property (3).



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Conditions based on fuzzy orderings

 Some important formula be deduced from theorem 1&2:

EC(x,y) = max(1 ) (TL-equivalence)

E’C(x,y) = exp() (TP-equivalence) The value C is obviously the maximal distance of two objects x and y.



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Conditions based on fuzzy orderings

Some important formula be deduced from theorem 1&2:

LC(x,y) = (TL-Ed,C-ordering)

L’C(x,y) = (TP-Ed,C-ordering)



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Conditions based on fuzzy orderings

Formula is applied following as:

Introduction

VQS

Conditions based on fuzzy orderings

Summary

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The aggregation issue

- oVQL with two and more conditions ?

- How to set weighted for a condition?

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Problem (1)● VQS query with n condition:

– C = {t1, , tn}

● Mapping: A: [0, 1]n -> [0, 1]

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Problem (2)

Require the following properties:

–If all conditions are perfect fulfilled, the overall degree be 1

–If all degree are 0, the overall degree be 0

–If one degree ti increased, while others are const, the overall degree not

much decrease

–Allow the user to assign relative importance to each condition as these

weights

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Definition 4● Definition 4: A domination T if

Then:

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n n

n n

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x

x

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, ,(

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Example TL (1)● Lukasiewicz t-norm TL :

f

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Example● Lukasiewicz t-norm TP :

33

n n

i

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● If not specify any weight, all weights are equal:

● If specify weight for all conditions:

● If specify weight some conditions, filled up with 1’s

w

w w

1

~

~

Table 4 QueryConsider the following query:

SELECT FROM HotelTable

WHERE Location IS ‘Salzburg Center’

Introduction

VQS

Conditions based on fuzzy orderings

The aggregation issue

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Does not specify any weight at all

If the weight was defined

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W(Location) = 4

W(Price) = 1

W(Star) = 1

W1 = 4/6 = 2/3W2 = 1/6

W3 = 1/6

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The choice of the underlying t-norm

With equal weights

With Weights

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The choice of the underlying t-norm

We have two choices

Leads to the same results, there is no point in choosing

Condition, radius C has a clear and unambiguous interpretation

However, T(L) is a pragmatic and justifiable choice

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Operation of Fuzzy System

Create a new type condition, such as “AT LEAST MEDIUM” or “AT MOST AROUND”

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CÂU HỎI THẢO LUẬN

1/ Trong hệ thống VQS, các thuộc tính phi số được tính như thế nào?

2/ Cách tính khoảng cách lớn nhất giữa các phần tử trong hệ thống VQS?

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