The UIU problem in the knowledge provided by 
human experts refers to the reliability or accuracy of 
human expertise, which is mainly affected by the 
soundness of experts' knowledge. 
Theories for Dealing with Uncertainty 
Numerous theories have been developed to 
accommodate uncertainty problems in knowledge 
based systems. The commonly used methods are 
probability theory and uncertainty theory. In addition, 
a number of other theories, such as the 
Dempster/Schafer theory of evidence, possibility 
theory, and plausibility theory, have also been 
proposed, in order to solve some of the problems 
unable to be solved by probability and uncertainty 
theories. However, it can be found by looking into 
these theories that none of them takes into account the 
reliability of knowledge sources used for deriving 
probabilities or alike certainty measures. An exception 
is found in Neapolitan (1990) where the uncertainty in 
probabilities provided by human experts is mentioned, 
and a method for dealing with this uncertainty is 
proposed. However, as discussed previously, the UIU 
problem not only exists in the knowledge provided by 
human experts, but also in all other knowledge sources 
such as time-serial or non-time-serial databases. 
Therefore, existing theories for dealing with uncertain 
problems in knowledge based reasoning are 
inadequate, and how to solve this adequacy should 
become an issue in the research on uncertainty theory. 
MODELING OF THE UNCERTAINTY IN 
UNCERTAINTY 
Three issues need to be addressed in order to build a 
model to take into account the UIU problem in 
reasoning with inexact knowledge. Firstly, a formal 
definition needs to be given to the UIU concept, so as 
to formulate the scope of the problem. Based on this 
definition, the second step is to formulate a model to 
represent the defined concept. Methods for estimating 
variable values involved in the model then need to be 
developed. Further, the method for integrating the 
UIU measure into the reasoning of inexact knowledge 
should be formulated 
Definition of The UIU Concept 
Although different theories for dealing with 
uncertainty represent the uncertainty concept in 
different ways, they can all be transformed into such a 
syntax that, given certainty evidence, a certainty value 
refers to a measure, such as a likelihood, a probability, 
or a certainty factor, which indicates the certainty of an 
event occurrence. Thus, we can define the UIU 
concept as follows: 
Let CV be a certainty value indicating the certainty of 
an event occurrence, given certain evidence. Then, 
the reliability of the certainty value CV or the 
quantitative measure of the UIU problem is termed as 
Certainty In Uncertainty, and denoted by CIU. If CV is 
provided by experts, CIU is a measure of the reliability 
940 
of the expertise; if CV is extracted from an existing 
non-time series database, CIU is a function of database 
accuracy and sample size; if CV is elicited from a time 
series database, CIU is a function of database accuracy, 
sample size available in the database, the number of 
time periods included in the database, and the standard 
deviation of an event's occurrence over time periods 
in the database. The range of CIU is [0, 1], where 0 
means that a certainty value is completely uncertain, 1 
means that a certainty value is completely certainty, 
while values between 0 and 1 represent the varied 
degrees of certainty of a certainty value. 
Mathematical Modeling of the UIU Problem 
Based on the definition of the uncertainty in 
uncertainty values CIU, we can construct a function 
between CIU and the factors related to CIU as follows: 
CIU zx W(T, S, Sd, A) (1) 
where: 
CIU - the uncertainty in uncertainty values to be 
evaluated; 
T - the number of time periods (year, month, day, etc.) 
involved in the database; 
S - the size of a sample available in the database for 
eliciting the certainty value of an evidence; 
Sd - the standard deviation of occurrence of an event 
over time periods involved in the database; 
A - the accuracy of data in a database or the reliability of 
an expert's statement. 
In order to define the functional relationship "P in 
equation (1), we start with an analysis of the 
differential relationships between (CIU, T), (CIU, S), 
(CIU, Sd), and (CIU, A). Based on the characteristics of 
the variables involved, we can find that a positive AA 
would produce less increase of CIU with the increase of 
A; the same would be true for AS and AT., while 
contrary to these variables, a positive ASd would cause 
larger decrease of CIU with the increase of Sd. In 
addition, the function should have such a 
characteristic that, as CIU is getting closer to its upper 
or lower limits, it becomes very difficult to produce 
any more change in CIU. Thus, we can establish the 
following partial differential equation: 
ACIU = CIU(1-CIU) [ (1/T)AT + (1/S)AS + (1/A)AA -SdASd] (2) 
Applying calculus to the equation, we thus obtain a 
mathematical model for the uncertainty in uncertainty 
values CIU: 
CIU= S*T*A* exp(- Sd2/2 + C)/(1 + S*T*A* expl-Sd2/2+C)) (3) 
where C is a constant. Other variables are as defined in 
equation (1). 
Equation (3) can be applied to the three different 
knowledge elicitation cases (as discussed before) in the 
following ways: 
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