Fuzzy Logic and D-S Theory Detect Disease Risk

Fuzzy Logic and Dempster-Shafer Theory
to Detect The Risk of Disease Spreading
Andino Maseleno
6 April 2015
Computer Science Program,
Faculty of Science, Universiti Brunei Darussalam
Jalan Tungku Link, Gadong BE1401, Negara Brunei Darussalam

This research aims to combine the mathematical theory of evidence with the rule based logics to refine
the predictable output. Integrating Fuzzy Logic and Dempster-Shafer theory by calculating the similarity
between Fuzzy membership function in the context to detect the risk of disease spreading and finally to
develop a realistic and useful Web mapping for displaying maps on a screen to locate the risk of disease
spreading. The risk of disease spreading include Highly Pathogenic Avian Influenza H5N1, African
Trypanosomiasis and skin disease. The risk of disease spreading is not classified according to higher
density which is equal to higher risk. This research has considered population changes in an area to detect
the risk of disease spreading. Population density in the area include very low, low, medium, high and very
high. The result reveals that the system has successfully identified the risk of disease spreading, moreover
the maps can be displayed as the visualization.
The novelty aspect of this work is that basic probability assignment is proposed based on the similarity
measure between membership function. The similarity between Fuzzy membership function is calculated
to get a basic probability assignment. This work is recommended to human experts and physicians who
specializes in diagnosing and treatment of diseases. The human experts will find it useful as an aid in the
decision making process and confirmation of suspected cases. The highest percentage of the risk of
disease spreading of Highly Pathogenic Avian Influenza H5N1 is 20 %. The highest percentage of the risk of
disease spreading of African Trypanosomiasis is 17 %. The highest percentage of the risk of disease
spreading of skin disease is 22 %. In this research it is Fuzzy Logic and Dempster-Shafer theory, which
resulted in a 0 % rejection.
Keywords: Fuzzy Logic, Dempster-Shafer theory, the risk of disease spreading, Highly Pathogenic Avian
Influenza H5N1, African Trypanosomiasis, skin disease.
Abstract

1. Introduction
2. Literature Review
3. Methodology
4. Implementation
5. Result and Discussion
6. Conclusions
Contents

Dealing with uncertainty is a fundamental issue for the human-like
intelligence exhibited by machines or software. Evidence theory is an
important tool of uncertainty modeling when both uncertainty origins from
human's lack of knowledge of the physical world and uncertainty derives
from the natural variability of the physical world are present in the problem
under consideration. In decision making processes with human's lack of
knowledge of the physical world and lack of the ability of measuring and
modeling the physical world, the Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence have gained prominence as the methods
of choice over traditional probabilistic methods. The fundamental and
important object of the mathematical theory of evidence is a primitive
function called a basic probability assignment. However, how to obtain basic
probability assignment is still an open issue. The membership function of a
Fuzzy set is a generalization of the indicator function in classical sets. In
Fuzzy Logic, it represents the degree of truth as an extension of valuation.
1.1 Motivation

Dempster-Shafer mathematical theory of evidence is a powerful tool for
combining accumulative evidence and changing prior knowledge in the
presence of new evidence. It also allows for the direct representation of
uncertainty of system responses where an imprecise input can be
characterized by a set or an interval and the resulting output is a set or an
interval. Fuzzy Logic is a logic operations method based on many-valued
logic rather than binary logic or two-valued logic. Two-valued logic often
considers 0 to be false and 1 to be true. Fuzzy Logic deals with truth values
between 0 and 1, and these values are considered as intensity or degrees of
truth. Dempster-Shafer mathematical theory of evidence, a probabilistic
reasoning technique, is designed to deal with uncertainty and
incompleteness of available information. Dempster-Shafer mathematical
theory of evidence allows one to combine evidence from different sources
and arrive at a degree of belief which is represented by a belief function that
takes into account all the available evidence. Degree of belief is the
expected truth value which is the relation between Fuzzy Logic and
Dempster-Shafer mathematical theory of evidence.
1.1 Motivation (cont.)

In the absence of empirical data, experts in related fields provide necessary
information. The fundamental objects of this theory of evidence are called focal
elements, and the primitive function associated with it is called basic probability
assignment. Focal elements are usually crisp subsets of some universal set. However,
in certain situations focal elements may also be represented by Fuzzy numbers.
There are many situations where human often face at the same time Fuzzy and non
Fuzzy uncertainties. This suggests to combine Dempster-Shafer mathematical theory
of evidence and Fuzzy sets frameworks. Thus, the goal of this work is to estimate
basic probability assignments using Fuzzy membership functions which capture
vagueness. The advantage of this method is a new method to obtain basic
probability assignment proposed based on the similarity measure between
membership function. This thesis proposes Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence by integrating Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence by calculating the similarity between Fuzzy
membership function. This thesis implements Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence and Web mapping. Fuzzy Logic and Dempster-
Shafer mathematical theory of evidence implement to detect the risk of disease
spreading and Web mapping to locate the risk of disease spreading.

More than one billion people, one-sixth of the world's population, are affected by
neglected diseases. There is an urgent need to develop better, more affordable
systems for diseases that are devastating developing countries. Based on
Cumulative Number of Confirmed Human Cases of Avian Influenza (H5N1)
Reported to World Health Organization (WHO) in the 2013 from 15 countries,
Indonesia has the largest number of death because of Avian Influenza which are
160 deaths (WHO, 2013). Based on World Health Organization (WHO) report in the
2013 skin diseases still remain common in many rural communities in developing
countries, with serious economic and social consequences as well as health
implications. Directly or indirectly, skin diseases are responsible for much disability
and loss of economic potential, disfigurement, and distress due to symptoms such
as itching or pain. Neglected tropical diseases kill an estimated 534,000 people
world wide every year (WHO, 2009). World Health Organization reports that
African Trypanosomiasis affects mostly poor populations living in remote rural
areas of Africa that can be fatal if not properly treated. Sustainable elimination of
African Trypanosomiasis as a public-health problem is feasible and requires
continuous efforts and innovative approaches.

In this research, a novel combination of Fuzzy Logic and Dempster-Shafer mathematical theory of evidence are applied to
detect the risk of disease spreading. Dempster-Shafer mathematical theory of evidence handles uncertain and
incomplete information through the definition of two dual non additive measures which include plausibility and belief.
These measures are derived from a density function, $m$, called a basic probability assignment or mass function.
Glanville et al (Glanville, 2009) using map layer of Indonesian major cities represents those cities in Indonesia that have a
population greater than 50,000. It is expected that such cities are likely to have at least one poultry market. The density
of cities was calculated over an area of 100km, and the risk of disease spreading classified according to this density
(higher density=higher risk). In this research, the risk of disease spreading is not classified according to higher density
which is equal to higher risk. However, this research has considered population changes in areas to detect the risk of
disease spreading. The risk of disease spreading describes five interpretations which include the risk of disease spreading
should be very low, the risk of disease spreading should be low, the risk of disease spreading should be medium, the risk
of disease spreading should be high, and the risk of disease spreading should be very high. These probabilities assign
evidence to a proposition or hypothesis. The derivation of the basic probability assignment is the most crucial step since
it represents the knowledge about the application as well as the uncertainty incorporates in the selected information
source. Basic probability assignment definition remains a difficult problem to apply Dempster-Shafer theory to practical
applications. Therefore, the experts provide opinion in terms of basic probability assignment for interval or crisp focal
elements. Those basic probability assignments are proposed based on the similarity measure between membership
function. Furthermore, the similarity between Fuzzy membership function is calculated to get basic probability
assignment. Finally, this research shows how they can be effectively used in prediction of the risk of disease spreading.

1. To combine the mathematical theory of evidence
with the rule based to improve the predictable
output. Integrating Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence in the context to
detect the risk of disease spreading.
2. To develop a realistic and useful Web mapping for
displaying maps on a screen to locate the risk of
disease spreading that can be of real usage.
3. To encounter the most important and unexpected
enemies of the human been the epidemic diseases
through the prediction of the risk of disease
spreading.
1.2 Research Objectives

The thesis contributes novel approaches to detect the
risk of disease spreading. The application has also been
developed as research outcome. The contributions of
the thesis are shown as follows:
1. The thesis proposes Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence in the context to
find the highest basic probability assignment of the
risk of disease spreading.
2. The thesis proposes a Web mapping for displaying
maps on a screen to locate the risk of disease
spreading that can be a real usages.
1.3 Research Contributions

Large amount of literature is available on Fuzzy Logic and its
applications. Fuzzy Logic can handle problems with imprecise data
and give more accurate results. Professor L.A. Zadeh introduced the
concept of Fuzzy Logic (Zadeh, 1965); soon after, researchers used
this theory for developing new algorithms and decision analysis.
Fuzzy Logic has been applied successfully in hundreds of application
domains including image processing, mobile robot navigation,
distance relation, high energy physics, natural numbers, medicinal
chemistry, robot manipulators, optimization of machining processes,
power converters, control of permanent-magnet synchronous
motors, and electric power systems.
2.1 Introduction

Several researchers have investigated the relationship between Fuzzy sets and
Dempster-Shafer mathematical theory of evidence and suggested different ways of
integrating them.
Integration within symbolic, rule-based models have been used for control and
classification purposes (Yen, 1990), (Binaghi, 2000).
Yager and Filev attempted to present a Fuzzy inference system based on Fuzzy
Dempster-Shafer mathematical theory of evidence which integrated the
probabilistic information in the output (Yager, 1990). In their works, the consequent
is shaped as a Dempster-Shafer belief structure, where each focal element has the
same membership function.
Binaghi et al. (Binaghi, 2000) proposed a structure for classification tasks similar to
that of Yager (Yager, 1990), where the focal element is a set representing the class
label.
2.1 Introduction (cont.)

Dymova (Dymova, 2012) proposed a critical analysis of conventional
operations on intuitionistic Fuzzy values and their applicability to the
solution of multiple criteria decision making problems in the intuitionistic
Fuzzy setting. Two sets of operations on intuitionistic Fuzzy values based on
the interpretation of intuitionistic Fuzzy sets in the framework of the
Dempster-Shafer theory of evidence are proposed and analysed.
Walijewski (Walijewski, 2002) concentrated on the role of Fuzzy operators,
and on the problem of discretization of continuous attributes.
Dutta et al. (Dutta, 2011) studied Dempster-Shafer theory of evidence by
considering focal elements as triangular Fuzzy number. The authors have
devised a method for obtaining belief and plausibility measure from basic
probability assignments assigned to Fuzzy focal elements.
Boudra et al. (Boudraa, 2004) estimated basic probability assignments using
Fuzzy membership functions.

Ghasemi et al. (Ghasemi, 2013) studied the main characteristic of the proposed
method where is that the rules of Fuzzy inference system are considered as
evidences in which the firing level of each rule and Fuzzy Naive Bayes method are
employed for calculating the basic probability assignment of focal element. In the
Naive Bayes classifier, all variables are assumed to be nominal variables, which
means that each variable has a finite number of values and also assumes
independence of features. However, in large databases, the variables often take
continuous values or have a large number of numerical values.
Binaghi et al. (Binaghi, 2000) presented a supervised classification model integrating
Fuzzy reasoning and Dempster-Shafer propagation of evidence has been built on top
of connectionist techniques to address classification tasks in which vagueness and
ambiguity coexist. The approach is the integration within a Neuro-Fuzzy system of
knowledge structures and inferences for evidential reasoning based on Dempster-
Shafer theory. The common weakness of neural network, however, is a problem of
determination of the optimal size of a network configuration, as this has a significant
impact on the effectiveness of its performance.

The Dempster-Shafer theory originated from the concept of lower and
upper probability induced by a multivalued mapping by Dempster
(Dempster, 1967), (Dempster, 1968). Following this work his student
Glenn Shafer (Shafer, 1976) further extended the theory in his book "A
Mathematical Theory of Evidence", a more thorough explanation of
belief functions. Dempster-Shafer mathematical theory of evidence is
one of the important tool for decision making under uncertainty.
Dempster-Shafer theory makes inferences from incomplete and
uncertain knowledge, provided by different independent knowledge
sources. A first advantage of Dempster-Shafer theory is its ability to
deal with ignorance and missing information. In particular, it provides
explicit estimation of imprecision and conflict between information
from different sources and can deal with any unions of hypotheses.

Dempster-Shafer mathematical theory of evidence is a formal framework
for plausible reasoning which provides techniques for characterizing the
evidences by considering all the available evidences. Dempster-Shafer
theory has been used in decision making.
Lyu et al. (Lyu, 2010) studied Dempster-Shafer theory for the reasoning
with imprecise context.
Yao et al. (Yao, 2011) used Dempster-Shafer theory for the multi-attribute
decision making problems with incomplete information by identify all
possible focal elements from the incomplete decision matrix, and then
calculate the basic probability assignment of each focal element and the
belief function of each decision alternative.

Yu et al. (Yu, 2012) used Dempster-Shafer Theory as an
applied approach to scenario forecasting based on imprecise
probability.
Uphoff et al. (Uphoff, 2013) studied application of Dempster-
Shafer theory to task mapping under epistemic uncertainty.
Dempster-Shafer mathematical theory of evidence implies a
type of uncertainty associated with conditions of ambiguity
through the data by dealing with ignorance and missing
information. This characteristic is due to using a combination
of evidence weight from different sources to obtain a new
evidence weight.

Population density and urbanization are two major factors affecting disease
spreading. People who live in close proximity to one another spread diseases
more quickly and easily (Jones, 2008). Also affecting the spread of
encountering new diseases is migration: this increases as humans move into
previously uninhabited lands because of population growth, or as humans
migrate into areas where they do not have resistance to certain diseases
(Anonymous, 2012). Disease in a population increases with the density of
that population. High densities makes it easier for parasites to find hosts and
spread the disease. Population density is a measurement of the number of
people in an area. It is an average number. Population density is calculated
by dividing the number of people by an area. Population density is usually
shown as the number of people per square kilometer (Geography, 2013). In
this research, population density is a parameter in the algorithm.
Population density of an area may change over time. In recent decades,
some cities have seen their urban centers lose population density, as
residents spread farther out to suburbs and exurbs (Fee, 2011).
2.2 Risk Factor Identification

Population density affect the disease spreading within
a population and other populations because a
population that is very dense will generally see a faster
disease spreading due to the larger amount of contact
between individuals. In a population that is not very
dense, close contact is much less likely to occur, thus
halting the disease spreading. Disease spreading and
population density are highly correlated (Yang, 2012),
(Yoneyama, 2012), (Padua, 2012). Population density is
measured by the average of the number of contacts
with susceptible individuals by each individual in the
population during a fixed length time period (Tarwater,
2001).
2.2 Risk Factor Identification (cont.)

Population density and growth are significant drivers for the emergence
of different categories of infectious diseases (Jones, 2008). Most diseases
require that their host organisms come into close contact with another
compatible organism in order to spread, meaning that denser populations
of suitable hosts promote faster spread of a disease and that less-dense
populations inhibit disease communication. Because disease prevention
relies so heavily on contact between potential carriers, lower population
densities have an increased chance of controlling disease spreading. In this
study, a novel combination of Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence are applied to detect the risk of disease
spreading. The risk of disease spreading is not classified according to higher
density which is equal to higher risk. This research has considered
population changes in an area to detect the risk of disease spreading.
Population density in an area can be very low, low, medium, high and very
high.
2.2 Risk Factor Identification (cont.)

A map is a visual representation of an area, a symbolic
depiction highlighting relationships between elements of that
space such as objects, regions, and themes (Njue, 2010).
People have used maps for centuries to represent their
environment. Maps are used to show locations, distances,
directions and the size of areas. Maps also display geographic
relationships, differences, clusters and patterns. Maps are
used for navigation, exploration, illustration and
communication in the public and private sectors (Nations,
2000). Nearly every area of scientific enquiry uses maps in
some form or another. Maps, in short, are an indispensable
tool for many aspects of professional and academic work.
2.3 Web Mapping

Cartography is the art and science of making maps. To create modern, high-quality
cartography requires the use of appropriate technology. This provides efficient
processes that amplify human creative and expressive skills in order to
communicate the essential spatial message (ESRI, 2004). Cartography has been
affected by the information revolution somewhat later than other fields. Early
computers were good at storing numbers and text. Maps, in contrast, are complex,
and digital mapping requires large data storage capacity and fast computing
resources. Furthermore, mapping is fundamentally a graphical application, and
early computers had limited graphical output capabilities. The earliest mapping
applications implemented on computers in the 1960s did not therefore find wide
application beyond a few government and academic projects. It took until the
1980s for commercial geographic information systems to reach a level of capability
that would lead to their rapid adoption, for example, in local and regional
government, urban planning, environmental agencies, mineral exploration, the
utility sectors and commercial marketing and real estate firms.
2.3 Web Mapping (cartography...)

Web mapping is the process of designing, implementing, generating and delivering
maps on the World Wide Web (Kresse, 2012). While Web mapping primarily deals with
technological issues, Web cartography additionally studies theoretic aspects: the use of
Web maps, the evaluation and optimization of techniques and workflows, the usability of
Web maps, social aspects, and more. The Web mapping applications are based on the
common client-server concept, which describes a relationship between two programs: a
client program, Web browser, that sends a request to a server program that then
executes the request and returns the requested information (Havlickova, 2009). Clients
and Servers often operate on separate hardware over computer network. A client is
typically a Web browser. Common Web browsers on the market include Internet Explorer,
Mozilla Firefox, and Google Chrome. Server, on the other hand, is a high-performance
host that provides services and sharing resources with multiple clients. A map rendering
engine is usually built on top of a Web server and linked with a back end GIS database
ready to retrieve and render data. The client communicates with server by sending HTTP
request, the server in turn process the request, respond and send back HTML documents,
maps and images for client display. The client communicates with server by sending HTTP
request, the server in turn process the request, respond and send back HTML documents,
maps and images for client display (Jiang, 2010). In the case of Web mapping applications
2.3 Web Mapping (web mapping is..)

The Web mapping server is the engine behind the maps. The mapping
server or Web mapping program needs to be configured to communicate
between the Web server and assemble data layers into an appropriate
image. A map is not possible without some sort of mapping information for
display. Mapping data is often referred to as spatial or geospatial data and
can be used in an array of desktop mapping programs or Web mapping
servers. Mapping data to outbreak of the disease uses spatial and non-spatial
data in ArcGIS format. There are variety of health surveillance systems that
help supplement existing public health system by focusing on Web mapping
(Turton, 2008), (Freifeld, 2014). Several researchers have investigated the
maps to contribute to meet international targets for reduced malaria illness
and death (Hay, 2009), (Gething, 2011). Those methods are not to locate the
risk of disease spreading. This research is aimed to develop a realistic and
useful Web mapping for displaying maps on a screen to locate the risk of
disease spreading so that action may be taken.
2.3 Web Mapping (The Web Mapping server is..)

Fuzzy Logic and Dempster-Shafer mathematical theory of evidence
implement to detect the risk of disease spreading and Web mapping to
locate the risk of disease spreading. The set of linguistic variables and their
meanings is compatible and consistent with the set of conditional rules
used, the overall outcome of the qualitative process is translated into
objective and quantifiable results. Fuzzy mathematical tools and the
calculus of Fuzzy IF-THEN rules provide a most useful paradigm for the
automation and implementation of an extensive body of human knowledge
heretofore not embodied in the quantitative modeling process. These
mathematical tools provide a means of sharing, communicating, and
transferring this human subjective knowledge of systems and processes.
The next chapter, Chapter 3, discusses methodology which is used in the
thesis. As part of this, Chapter 3 explains uncertainty reasoning, Dempster-
Shafer mathematical theory of evidence, Fuzzy Logic theory, Fuzzy Logic and
Dempster-Shafer mathematical theory of evidence, system processes and
database.
2.4 Summary

3.1 Introduction
This chapter explains uncertainty reasoning, Dempster-Shafer
mathematical theory of evidence, Fuzzy Logic, Fuzzy Logic and
Dempster-Shafer mathematical theory of evidence, system processes,
and database. Precise and certain rules usually can not model all
concepts with generality and conciseness at the same time on the
contrary, some degree of imprecision allows to formalize real-world
criteria with a limited number of rules. Imperfection is a general term
used to describe rules softened in different ways: from gradualness,
where a known property may be present at different levels, to
uncertainty, where some information is missing either because of a lack
of awareness and understanding of a set of information or due to an
intrinsic aleatory in the context being analyzed (Sottara, 2010).
Probability theory provides a consistent framework for dealing with
uncertain knowledge for a robust and reliable recognition of complex
event (Romdhane, 2010).

Dempster-Shafer theory is a mathematical theory of
evidence that assigns probabilities to sets.
The Dempster-Shafer mathematical theory of evidence is
based on probability theory. The Dempster-Shafer evidential
theory is a method about uncertainty reasoning, and this
theory reduces the requirements of the knowledge of prior
probability and conditional probability. In the process, it has
able to synthesize the evidence from different sources and
dealing with uncertainty. One of the most important
features of Dempster-Shafer mathematical theory of
evidence is that the model is designed to deal with varying
levels of precision regarding the information and no further
assumptions are needed to represent the information.

Fuzzy Logic is based on the theory of Fuzzy sets, where an object's
membership of a set is more gradual rather than just member or not a
member. Fuzzy Logic uses the whole interval of real numbers between
zero or False and one or True to develop a logic as a basis for rules of
inference. In this research, a novel combination of Fuzzy Logic and
Dempster-Shafer mathematical theory of evidence are applied to the
system. This chapter presents a model integrating Tsukamoto Fuzzy
reasoning and Dempster-Shafer mathematical theory of evidence. The
salient aspect of the approach is the integration within Tsukamoto Fuzzy
reasoning and inferences for evidential reasoning based on Dempster-
Shafer mathematical theory of evidence. The system is constructed
using open source components that allow for easy customization, and
by utilizing open standards the server can be accessed by other clients
with very little effort on the part of expert users who require the ability
to carry out more advanced analysis than is possible using the Web
based client.

Uncertainty is a general concept that reflects human lack of sureness about something or someone,
ranging from just short of complete sureness with an almost complete lack of conviction about an
outcome. The fundamental structure of uncertainty model contains following three components which
include the description of information uncertainty or rules; the description of evidence uncertainty or
facts; and the spread of uncertainty (Li, 2010).
Reasoning theories are divided into certainty reasoning theories and uncertainty reasoning theories.
Certainty thinking was once and will be still prevailing in different disciplines. In the Cartesian
philosophy, mathematics was the only accurate knowledge learning to provide. With the combination of
mathematics and physics, all sorts of natural and social phenomena could be explained in science.
Leibniz, philosopher and mathematician, was convinced that symbolic language of science could
construct the universal logic and logical calculus, and all phenomena could be clearer. Newton's
absolute time-space sure that all the observable physical quantity in principle could be in infinite
accurate measurement and its foundation was the uncertainty of physical laws. An unknown world was
deterministic for perfectly rational policy makers in the traditional decision science view. A man could
get the reflect of maximization as long as according to the principle which marginal benefit equals
marginal cost decision. However, the world is uncertain (Yanfei, 2013).
3.2 Uncertainty reasoning

Decisions are often taken on the basis of imperfect information and knowledge
(imprecise, uncertain, incomplete) provided by several more or less reliable
sources and depending on the states of the world: decisions can be taken in
certain, risky or uncertain environment (Tacnet, 2011). The lack of certainty is
ubiquitous and happens in every single event people encounter in the real
world. Whether it rains or not tomorrow is uncertain; whether there is a flight
delay is uncertain. Just as Socrates, was a classical Greek (Athenian) philosopher
credited as one of the founders of Western philosophy, in ancient Greek said,
"as for me, all I know is I know nothing." Uncertainty distinguishes from a
certainty in the degree of belief or confidence. If certainty is referred to as a
perception or belief that a certain system or phenomenon can experience or
not, uncertainty indicates a lack of confidence or trust in an article of
knowledge or decision. Uncertainty is a term used in subtly different ways in a
number of fields, including philosophy, physics, statistics, economics, finance,
insurance, psychology, sociology, engineering, and information science. It
applies to predictions of future events, to physical measurements that are
already made, or to the unknown. Uncertainty arises in partially observable
and/or stochastic environments, as well as due to ignorance and/or indolence
(Norvig, 2013). According to the Cambridge Dictionary, "uncertainty is a
situation in which something is not known, or something that is not known or
certain (Cambridge, 2013).
3.2 Uncertainty reasoning (cont.)

Uncertainty arises from different sources in various forms, and is classified in different ways by
different communities. According to the origin of uncertainty, it is categorized into aleatory
uncertainty or epistemic uncertainty. Aleatory uncertainty derives from the natural variability of the
physical world. It reflects the inherent randomness in nature. It exists naturally regardless of human
knowledge. For example, in an event of flipping a coin, the coin comes up heads or tails with some
randomness. Even if researchers do many experiments and know the probability of coming up heads,
researchers still cannot predict the exact result in the next turn. Aleatory uncertainty cannot be
eliminated or reduced by collecting more knowledge or information. No matter whether people know
it or not, this uncertainty stays there all the time. Aleatory uncertainty is sometimes also referred to as
natural variability, objective uncertainty, external uncertainty, random uncertainty, stochastic
uncertainty, inherent uncertainty, irreducible uncertainty, fundamental uncertainty, real world
uncertainty, or primary uncertainty. Epistemic uncertainty origins from human's lack of knowledge of
the physical world and lack of the ability of measuring and modelling the physical world. Unlike
aleatory uncertainty, given more knowledge of the problem and proper methods, epistemic
uncertainty can be reduced and sometimes can even be eliminated. For example, the estimation of
the distance between Bandar Seri Begawan and Kuala Belait can be more precise if people have
known the distance from Bandar Seri Begawan to Tutong. Epistemic uncertainty is sometimes also
called knowledge uncertainty, subjective uncertainty, internal uncertainty, incompleteness, functional
uncertainty, informative uncertainty, or secondary uncertainty. Dempster-Shafer mathematical theory
of evidence can deal with both aleatory and epistemic uncertainty.
3.2 Uncertaintiy reasoning (cont.)

It is difficult to avoid uncertainty when attempting to make models of the real world.
Uncertainty is inherent to natural phenomena, and it is impossible to create a perfect
representation of reality. Classic mathematics deals with ideal worlds where perfect
geometric figures exist and can verify extraordinary conditions. The formalisation of Fuzzy
sets started in the 1960s with the works of Zadeh (Zadeh, 1965) in Fuzzy sets and Dempster
(Dempster, 1968) in belief functions. Belief functions offer a non Bayesian method for
quantifying subjective evaluations by using probability. In the 1970s, it was further developed
by Shafer, whose book Mathematical Theory of Evidence (Shafer, 1976) remains a classic in
belief functions, or the so-called Theory of Evidence. This theory has been also called the
Dempster-Shafer Mathematical Theory of Evidence. In the 1980s, the scientific community
working with Artificial Intelligence got involved in using the theory of evidence in
applications. The Dempster-Shafer theory or the theory of belief functions is a mathematical
theory of evidence which can be interpreted as a generalization of probability theory in
which the elements of the sample space to which nonzero probability mass is attributed are
not single points but sets. The sets that get nonzero mass are called focal elements. The sum
of these probability masses is one, however, the basic difference between Dempster-Shafer
mathematical theory of evidence and traditional probability theory is that the focal elements
of a Dempster-Shafer structure may overlap one another. The Dempster-Shafer mathematical
theory of evidence also provides methods to represent and combine weights of evidence.
3.3 Dempster Shafer mathematical theory of evidence

3.3.1 Representation of evidence

In decision making processes with human's lack of knowledge of the physical world and lack of the ability
of measuring and modelling the physical world, the Fuzzy Logic and Dempster-Shafer mathematical
theory of evidence have gained prominence as the methods of choice over traditional probabilistic
methods. The fundamental and important object of the mathematical theory of evidence is the primitive
function called a basic probability assignment.
In the absence of empirical data, experts in related field provide necessary information. However how to
obtain basic probability assignment is still an open issue. The membership function of a Fuzzy set is a
generalization of the indicator function in classical sets. In Fuzzy Logic, it represents the degree of truth as
an extension of valuation. Fuzzy Logic is a logic operation method based on many-valued logic rather than
binary logic or two-valued logic.
Dempster-Shafer mathematical theory of evidence, a probabilistic reasoning technique, is designed to
deal with uncertainty and incompleteness of available information. Dempster-Shafer mathematical theory
of evidence allows one to combine evidence from different sources and arrive at a degree of belief which
is represented by a belief function that takes into account all the available evidence. The degree of belief
is expecting a truth value which is the relation between Fuzzy Logic and Dempster-Shafer mathematical
theory of evidence.
3.3.1 Representation of evidence

3.3.2 Evidence combination (!)

3.5 Fuzzy Logic and Dempster-Shafer mathematical theory of
evidence

The membership function of a Fuzzy set is a generalization of the indicator function in
classical sets. In Fuzzy Logic, it represents the degree of truth as an extension of valuation.
Properties of membership function are:
1. The membership function should be strictly monotonically increasing, or strictly
monotonically decreasing, or strictly monotonically increasing then strictly
monotonically decreasing with the increasing value of elements in the universe of
discourse X. This term is given by the equation
2. The membership function should be continuous or piecewise continuous.
3. The membership function should be differentiable to provide smooth results.
4. The membership function should be of simple straight segments to make the process of
fuzzy models easy and to high accuracy.
3.5 Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence (cont.)

A new method to obtain basic probability
assignment is proposed based on the similarity
measure between membership function. This
thesis proposes Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence by calculating the
similarity measure between Fuzzy membership
function. Method to integrate Fuzzy Logic and
Dempster-Shafer mathematical theory of evidence
as follows:
3.5 Fuzzy Logic and Dempster-Shafer mathematical
theory of evidence

3.5 Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence

Flowchart of Fuzzy Logic and Dempster-Shafer

People often have to make decisions based on uncertain
awareness and understanding of a set of information, not
only in their private lives, but also in professional activity.
Therefore, any reasoning method that tries to replicate
human reasoning must be able to draw conclusions from
uncertain models and uncertain data. Models may be
uncertain because of indeterminism in the real world or
because of human lack of knowledge. Furthermore, data
may be incomplete because of pieces of information may be
not available in a diagnostic case, ambiguous because of a
pronoun in a sentence may refer to different subjects,
erroneous because of patients may lie to their doctors, or
imprecise because of the limited precision of measuring
devices, subjective estimations, or natural language.
Summary of Chapter 3

The Dempster-Shafer mathematical theory of evidence has
attracted considerable attention as a promising method of
dealing with some of the basic problem arising in combination
of evidence and data fusion. Tsukamoto Fuzzy reasoning does
the mapping from given input to an output using Fuzzy Logic.
Tsukamoto Fuzzy reasoning models have a number of rules
based on if-then conditions. In fact, these rules are easy to
learn and use and can be modified according to the situation.
It helps to make decisions and can be used in decision
analysis. The next chapter discusses the implementation of
Fuzzy Logic and Dempster-Shafer mathematical theory of
evidence to the risk of disease spreading and Web mapping to
locate the risk of disease spreading.
Summary of Chapter 3 (cont.)

Fuzzy logic and Dempster-Shafer mathematical theory of evidence contribute new
ideas to detect the risk of disease spreading. The risk of disease spreading is not
classified according to higher density which is equal to higher risk. This thesis
considers population changes in an area to detect the risk of disease spreading.
Population density in areas which include very low, low, medium, high and very
high. The result reveals that in areas which are in close proximity to Kendal and
Temanggung, the highest basic probability assignment value of the risk of disease
spreading of Highly Pathogenic Avian Influenza H5N1 is very low which is equal to
0.20. In areas which are in close proximity to Angola and Zambia, the highest
basic probability assignment value of the risk of disease spreading of African
Trypanosomiasis is very low which is equal to 0.173. In areas which are in close
proximity to Bandung and Purwakarta, the highest basic probability assignment
value of the risk of disease spreading of skin disease is low which is equal to 0.22.
In chapter 5, results and discussions are presented, which include the risk of
disease spreading of Highly Pathogenic Avian Influenza H5N1, the risk of disease
spreading of African Trypanosomiasis, and the risk of disease spreading of skin
disease.

Chapter 5
Result and Discussion

In this chapter, the result reveals that in areas which are in close proximity to Kendal
and Temanggung that the risk of disease spreading of Highly Pathogenic Avian
Influenza H5N1 is very low, which means the risk of disease spreading of Highly
Pathogenic Avian Influenza H5N1 is very rare but cannot be excluded. In areas
which are in close proximity to Angola and Zambia that the risk of disease spreading
of African Trypanosomiasis is very low, which means the risk of disease spreading of
African Trypanosomiasis is very rare but cannot be excluded. In areas which are in
close proximity to Bandung and Purwakarta that the risk of disease spreading of
skin disease is low, which means the risk of disease spreading of skin disease is rare
but does occur. An implementation of applying Fuzzy Logic and Dempster-Shafer
theory in solving a decision problem in the risk of disease spreading shows that it
does improve the decision results.

This research visualizes the risk of disease spreading
considering the connections between regions for the
global spread of infection and population density.
Furthermore, the vagueness present in the definition of
terms is consistent with the information contained in
the conditional rules when observing some complex
process. While this work has been done in applying
Fuzzy Logic and Dempster-Shafer theory in solving real
world disease problems, the preceding discussion of the
Fuzzy Logic and Dempster-Shafer theory together with
the implementation given, indicates a very promising
new starting point in the application of this theory.

Introduction
This chapter summarises the conclusions of the research, research
contributions and directions for future work suggested. Early warning
system of the risk of disease spreading is important in interrupting the
transmission cycle of the parasite and progress of the disease to the
late stage. Early warning is the provision of timely and effective
information, through identifying institutions, that allows individuals
exposed to a hazard to take action to avoid or reduce their risk and
prepare for effective response. Therefore, cost effective, simple, rapid,
robust and reliable methods, are urgently needed. There is also an
urgent need for accurate tools for the diagnosis of the disease
spreading, a new initiative for the development of new diagnostic tests
to support the control of disease.

Population density is a major factor affecting disease spreading within a
population and other populations because a population that is very dense will
generally see a faster disease spreading due to the larger amount of contact
between individuals. In a population that is not very dense, close contact is much
less likely to occur, thus halting the disease spreading. Disease in a population
increases with the density of that population. High population density makes it
easier for parasites to find hosts and the disease spreading. Population density is
a measurement of the number of people in an area. It is an average number.
Population density is calculated by dividing the number of people by an area.
Population density is usually shown as the number of people per square
kilometer. This research has considered Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence to detect the risk of disease spreading. The risk
of disease spreading is not classified according to higher density which is equal to
higher risk. This research has considered population changes in an area to detect
the risk of disease spreading. Population density in areas which include very low,
low, medium, high and very high. The result reveals that the system has
successfully identified the existence of disease and the risk of disease spreading,
moreover the maps can be displayed as the visualization. Web mapping is also
used for displaying maps on a screen to visualize the result of the identification
process.
Introduction (cont.)

Conclusions of the research
In Chapter 3, this research has described Fuzzy Logic and Dempster-Shafer mathematical
theory of evidence. Dempster-Shafer mathematical theory of evidence is a theory of
uncertainty which was first proposed by Arthur Dempster and extended by Glenn Shafer.
The Dempster-Shafer theory is a mathematical theory of evidence based on belief
functions and plausible reasoning, which is used to combine separate pieces of
information, evidence, to calculate the probability of an event. Dempster-Shafer
mathematical theory of evidence allows us to combine evidence from different sources
and arrive at a degree of belief which is represented by a belief function that takes into
account all the available evidence. In Dempster-Shafer mathematical theory of evidence,
evidence can be associated with multiple possible events. There are two basic
components in Dempster-Shafer mathematical theory of evidence which include the
basic probability assignment and the rule of combination. Dempster-Shafer
mathematical theory of evidence can make inferences from the incomplete and
uncertain knowledge, provided by different independent knowledge sources. This
research presented a novel combination of Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence which are applied to the system. A new method to
obtain basic probability assignment is proposed based on the similarity measure
between membership function. The integration within Tsukamoto Fuzzy reasoning and
inferences for evidential reasoning based on Dempster-Shafer mathematical theory of
evidence by calculating the similarity between Fuzzy membership function.

Next, in chapter 4, this research focused on implementation using Fuzzy Logic and Dempster-Shafer
mathematical theory of evidence to the risk of disease spreading and Web mapping to locate the risk of
disease spreading.
In chapter 5, results and discussions of the research are presented. In Kendal and Temanggung, the
highest basic probability assignment value of the risk of disease spreading of Highly Pathogenic Avian
Influenza H5N1 is very low which is equal to 0.20. It means the risk of disease spreading of Highly
Pathogenic Avian Influenza H5N1 is very rare but cannot be excluded. The risk of disease spreading of
Highly Pathogenic Avian Influenza H5N1 in areas which include Batang, Kendal, Kota Magelang, Kota
Salatiga, Kota Semarang, Magelang, Semarang, Temanggung, and Wonosobo. In Angola and Zambia,
the highest basic probability assignment value of the risk of disease spreading of African
Trypanosomiasis is very low which is equal to 0.173. It means the risk of disease spreading of African
Trypanosomiasis is very rare but cannot be excluded. The risk of disease spreading of African
Trypanosomiasis in areas which include Angola, Botswana, Congo, Congo DRC, Malawi, Mozambique,
Namibia, Tanzania, Zambia and Zimbabwe. In Bandung and Purwakarta, the highest basic probability
assignment value of the risk of disease spreading of skin disease is low which is equal to 0.22. It means
the risk of disease spreading of skin disease is rare but does occur. The risk of disease spreading of skin
disease in areas which include Bandung, Cianjur, Garut, Karawang, Kota Bandung, Kota Cimahi,
Purwakarta, Subang and Sumedang. Figure~ref{fig:bpaconclusions} shows the basic probability
assignment of the disease.
Conclusions of the research

Research Contributions
This section states the contributions of the thesis. This research proposed Fuzzy
Logic and Dempster-Shafer mathematical theory of evidence to detect the risk of
disease spreading and Web mapping to locate the risk of disease spreading. A
novel combination of Fuzzy logic and Dempster-Shafer mathematical theory of
evidence are Integrating Fuzzy Logic and Dempster-Shafer mathematical theory of
evidence by calculating the similarity between Fuzzy membership function in the
context to detect the risk of disease spreading. This detection system of disease
spreading is applied to detect the risk of disease spreading include Highly
Pathogenic Avian Influenza H5N1, African Trypanosomiasis and skin disease. A
computer system that can make decision-making. The computer can make
inferences and arrive at a specific conclusion. The system provides powerful and
flexible means for obtaining solutions to a disease spreading detection problem
that often cannot be dealt with by other, more traditional and orthodox methods.
Thus, their use is proliferating to many sectors of human social and technological
life, where their applications are proving to be critical in the process of decision
support and problem solving.

The simplest possible method for using probabilities to
quantify the uncertainty in a database is that of attaching a
probability to every member of a relation, and to use these
values to provide the probability that a particular value is the
correct answer to a particular query. An expert in providing
knowledge is uncertain in the form of rules with the
possibility, the rules are probability value. The knowledge is
uncertain in the collection of basic events can be directly used
to draw conclusions in simple cases, however, in many cases
the various events associated with each other. Reasoning
under uncertainty that used some of mathematical
expressions, gave them a different interpretation which is
each piece of evidence may support a subset containing
several hypotheses.

This is a generalization of the pure probabilistic framework in
which every finding corresponds to a value of a variable. This
research has presented integrating Fuzzy Logic and Dempster-
Shafer mathematical theory of evidence in the context to detect
the risk of disease spreading. The highest percentage of the risk of
disease spreading of Highly Pathogenic Avian Influenza H5N1 is 20
%. The highest percentage of the risk of disease spreading of
African Trypanosomiasis is 17 %. The highest percentage of the risk
of disease spreading of skin disease is 22 %. In this research it is
Fuzzy Logic and Dempster-Shafer theory, which resulted in a 0 %
rejection. And also this research has developed Web mapping for
displaying maps on a screen to locate the risk of disease spreading.
Finally, Fuzzy Logic and Dempster-Shafer mathematical theory of
evidence have shown good results.

Fuzzy Logic and D-S Theory Detect Disease Risk

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