\documentstyle[epsf,jair,twoside,11pt,theapa]{article}
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\jairheading{3}{1995}{119-145}{3/95}{8/95}
\ShortHeadings{Using Qualitative Hypotheses to Identify Inaccurate Data}
{Zhao \& Nishida}
\firstpageno{119}

\begin{document}

\title{Using Qualitative Hypotheses to Identify Inaccurate Data}

\author{\name Qi Zhao \email qi-zhao@is.aist-nara.ac.jp \\
        \name Toyoaki Nishida \email nishida@is.aist-nara.ac.jp \\
        \addr Graduate School of Information Science\\
              Nara Institute of Science and Technology\\
              8916-5, Takayama-cho, Ikoma-shi, Nara 630-01, Japan}

\maketitle


\begin{abstract}
Identifying inaccurate data has long been regarded as a significant and
difficult problem in AI. In this paper, we present a new method for
identifying inaccurate data on the basis of qualitative correlations among
related data. First, we introduce the definitions of related data and
qualitative correlations among related data. Then we put forward a new
concept called support coefficient function ($SCF$). $SCF$ can be used to
extract, represent, and calculate qualitative correlations among related
data within a dataset. We propose an approach to determining dynamic shift
intervals of inaccurate data, and an approach to calculating possibility of
identifying inaccurate data, respectively. Both of the approaches are based
on $SCF$. Finally we present an algorithm for identifying inaccurate data by
using qualitative correlations among related data as confirmatory or
disconfirmatory evidence. We have developed a practical system for interpreting
infrared spectra by applying the method, and have fully tested the system
against several hundred real spectra. The experimental results show that the
method is significantly better than the conventional methods used in many
similar systems.

\end{abstract}

\section{Introduction}
\label{Introduction}

In many problems of artificial intelligence, inferences are drawn on the
basis of interpretation or analysis of measured data. However, when measured
data are inaccurate, interpreting or analyzing them is very difficult. In
diagnosis or signal analysis, for example, the general reasoning method is
to compare measured data with reference values (Reiter, 1987; Shortliffe \&
Buchanan, 1975). When measured data are not accurate due to noise or other
unforeseen reasons, the comparison between measured data and reference values
can not lead to any useful conclusion. A rule like ``{\em if there is a
strong peak in 3000 $cm^{-1}$- 3100 $cm^{-1}$ on the infrared spectrum of an
unknown compound, then the unknown compound may contain at least one
benzene-ring}'' may work in ideal cases.
However, the rule can not work in general cases.
For example, when the spectral data
are inaccurate, e.g., the measured peak in 3000 $cm^{-1}$- 3100 $cm^{-1}$ is
not a strong peak but a medium one, or a measured strong peak is not exactly
located in 3000 $cm^{-1}$- 3100 $cm^{-1}$ but is slightly shifted, the rule may
not be applied.

In practical problems, especially in data rich problems such as diagnosis
and interpretation, measured data are often inaccurate. One reason is that the
measuring methods are error-prone. For example, a patient's temperature
or blood-pressure may be inaccurately measured or entered, and a witness may
inaccurately describe the features of a criminal. The other reason is that the
real data are not noise-free. For example, among the received signals,
there may be some noise mixed up, and what is worse, infrared spectral
data (peaks) themselves may be noisy, i.e., some peaks may be affected by noise
or other factors. 

Identifying inaccurate data has long been regarded as a significant and
difficult problem in AI. Many methods have been proposed to deal with
the problem. Fuzzy logic provides a mathematical framework for
representation and calculation of inaccurate data (Zadeh, 1978). By fuzzy
logic, reference value $x_{0}$ is associated with a fuzzy interval
$\triangle x$. If a measured data item falls into
[$x_{0}-\triangle x, x_{0}+\triangle x$], then it can be identified as the
reference value with a corresponding membership degree.
Probability theory and possibility theory are also widely used for handling
inaccuracy and uncertainty (Dempster, 1968; Duda, Hart, \& Nilsson, 1976;
Pearl, 1987; Shafer, 1976; Shortliffe \& Buchanan, 1975).
The above methods are commonly used in AI
systems. The way of applying them, however, depends on the nature of domain
problems, and there is not yet a standard and generally
accepted method thus far.

We present a method for identifying inaccurate data on the basis of qualitative
correlations among related data. The method is based on the essential
consideration that some data items within a dataset are qualitatively
dependent: a set of data may describe the same phenomenon, or
refer to the same behavior. For example, a patient's temperature, blood
pressure and other symptomatic data reflect the patient's disease, and a
couple of peaks on an infrared spectrum indicate the presence of a partial
component. We call the dependency among data within a dataset {\em qualitative
correlations among related data}{\footnote{Detailed definitions will be given
later.}}. By considering qualitative correlations among related data, we can
obtain confirmatory or disconfirmatory evidence to identify inaccurate
data. In general, related data should be simultaneously present or absent, so
if most of the related data have been completely identified, these data will
enhance the identification of the rest. For example, a benzene-ring can create
many other peaks besides the strong peak in 3000 $cm^{-1}$- 3100 $cm^{-1}$. All
the peaks created by the benzene-ring are related data which have qualitative
correlations. If all the peaks except that in 3000 $cm^{-1}$- 3100 $cm^{-1}$
have been completely identified, the benzene-ring
is quite likely to be contained by the unknown compound. Therefore, the
inaccurate peak around 3000 $cm^{-1}$- 3100 $cm^{-1}$ may still be
identified. In fact, spectroscopists frequently
use the following knowledge in addition to the rules given at the beginning of
this section:

\begin{quote}
{\em If there is a strong peak around 3000 $cm^{-1}$- 3100 $cm^{-1}$, then the
spectrum may be partially created by benzene-rings ----- check peaks around
1650 $cm^{-1}$, 1550 $cm^{-1}$ and 700 $cm^{-1}$- 900 $cm^{-1}$ to make sure
because a benzene-ring may have other peaks there at the same time.}
\end{quote}

The central idea of our method is to find evidence for identifying
inaccurate data by considering qualitative
correlations among related data. The idea is very common in human thinking.
When all the data except blood pressure of a patient show that the patient has
a certain disease, we would naturally suspect that the blood pressure of the
patient was inaccurately entered. Similarly, when all the peaks except one
indicate that a partial component is present, we would naturally suspect that
the unmatched peak was inaccurately measured or the peak was affected by noise
or something else. If acceptable solutions can be made by assuming an
inaccurate data item to be a reference value based on qualitative correlations
between the data item and its related data, the inaccurate data item may be
compensated and hence identified.

\vspace*{1mm}
Our contributions include: (1) a method which assumes an inaccurate data item
to be a certain reference value based on the qualitative correlations between
the inaccurate data item and all of its related data, (2) an algorithm which
crystallizes the method, and (3) a practical system which uses the
algorithm to interpret infrared spectra.

\vspace*{1mm}
The key point is a new concept called support coefficient function ($SCF$)
for extracting, representing, and calculating qualitative correlations among
related data. When measured data are inaccurate, the qualitative correlations
among related data can provide evidence for confirming or disconfirming the
hypothesis that the measured data are the same as the reference values.
An approach to determining dynamic shift intervals of inaccurate data, an
approach to calculating possibility of identifying inaccurate data,
and an algorithm for identifying inaccurate data are proposed on the basis
of $SCF$, respectively.

\vspace*{1mm}
The method requires few assumptions in advance, so it can avoid inconsistency
in knowledge and data bases. The method identifies inaccurate data by
considering qualitative correlations among related data, so it is quite
effective and efficient, especially in the case of problems where
dependencies among
data apparently exist. In general, qualitative correlations among data can
always, more or less, be extracted. In the worst case where qualitative
correlations are not known a priori, the method degenerates to a conventional
fuzzy method\footnote{We refer to the fuzzy methods which use an empirical
fuzzy interval for each inaccurate data item as conventional fuzzy
methods.}.

\vspace*{1mm}
We have developed a practical system for interpreting infrared spectra by
using the method (Zhao \& Nishida, 1994). The primary task of the system is to
identify unknown compounds by interpreting their infrared spectra. We have
fully tested the system against several hundred real spectra. The experimental
results show that the method is significantly better than the traditional
methods used in many similar systems. The rate of correctness ($RC$) and the
rate of identification ($RI$) which are two important standards for evaluating
the solutions of infrared spectrum interpretation are near 74\% and 90\%
respectively, and the former is the highest among known systems.

\vspace*{1mm}
In the following sections, we first describe the problem of identifying
inaccurate data in Section 2. In Section 3 we give some definitions including
the concept of support coefficient function ($SCF$) and other concepts based on
$SCF$. In Section 4 we introduce our method for identifying inaccurate data by
considering qualitative correlations among related data. Section 5 demonstrates
the application of the method to a knowledge-based system for infrared spectrum
identification, and shows the experimental results of the system. Related work
is discussed in Section 6. Conclusions are addressed in Section 7.

\vspace*{5mm}
\section{Problem Description}

\vspace*{2mm}
In practical problems, measured data can be represented as a finite set:

\newpage

\[
MD=\{d_{1}, d_{2}, ..., d_{n}\},
\]

\noindent and reference values can also be represented as a finite set:

\[
RV=\{r_{1}, r_{2}, ..., r_{N}\}.
\]

Suppose interpreting or analyzing measured data is carried out on the basis
of so-called ``if-then'' rules in which the premises are comparisons between
$MD$ and $RV$ like ``{\em if $d_{i}=r_{j}$ then ...}'', or
``{\em if $(r_{i}\in MD)\wedge (r_{j}\in MD)$ then ...}''.
When $MD$ is accurate, the main operation implied by these premises is usually
to find a corresponding reference value from $RV$ for each data item in $MD$.
However, when $MD$ is inaccurate, the operation becomes complicated. In this
case, it is difficult to determine which reference value an inaccurate data
item corresponds to, e.g., for some measured data no reference value
may be simply identified, while for others more than one may be available.

For example, if received signals are known to be accurate, and an
expected signal (reference value) can not be found from the signal
series (measured data), then we can conclude that the expected signal does not
appear. However, if received signals are inaccurate, and an expected signal
can not be identified from the signal series, it is hard to decide whether the
expected signal does not appear or appears but looks
different due to the inaccuracy.

Most currently known approaches for dealing with inaccurate data such as fuzzy
logic and probabilistic reasoning are mainly based on quantitative
similarity or closeness between measured data and reference values. In some
cases, however,
the identity of qualitative features is more effective and reliable
than quantitative similarity or closeness.

Consider signal analysis again. If an inaccurate signal has the same
qualitative features as the expected one such as the interval of frequency, the
signal may still be identified even though its quantitative features are
slightly different from those of the expected one such as strength etc.;
conversely, an inaccurate signal may not be identified if it is quantitatively
similar to an expected signal but does not have the same qualitative features
as the expected one.

We discussed the following points in Section 1, (1) some data items
within a dataset are qualitatively dependent (i.e., they are related data),
(2) there are qualitative correlations among related data, and (3) qualitative
correlations among related data enable us to confirm or disconfirm the identity
of qualitative features.

Therefore, $RV$ and $MD$ can be, explicitly or implicitly, divided into finite
groups on the basis of qualitative dependencies among data, and the data
in each group are related to each other. For example, $RV$ can be divided into
$R_{1}$, $R_{2}$, ... and $R_{k}$:

\[
RV=R_{1}\cup R_{2}\cup ...\cup R_{k},
\]

\noindent where

\[
R_{j}=\{r_{j_{l}} \mid r_{j_{l}}\in RV, 1\leq l \leq m\}.
\]

The qualitative correlations among related data in $R_{j}$ include:
(1) data in $R_{j}$ should be simultaneously present or absent which means
that all reference values in $R_{j}$ should have corresponding data in $MD$,
(2) the presence of $r_{j_{p}}$ may enhance the presence of $r_{j_{q}}$, and
the absence of $r_{j_{p}}$ may depress the presence of $r_{j_{q}}$.
Considering the qualitative correlations among related data will lead to
evidence for the identification of inaccurate data.

The problem of interpreting/analyzing inaccurate data is to make
qualitative hypotheses for $MD$, or in other words, to find a subset
of $RV$ for $MD$, which is corresponding
to $MD$:

\[
IN(MD),\hspace*{5mm} (IN(MD) \subset RV).
\]

The problem can be briefly represented as the following
predicate calculus:

\[
\forall d_{i}\forall R_{j}((d_{i}@R_{j})\wedge(R_{j}@MD)\rightarrow
R_{j}\subset IN(MD))\footnote{Conflicts (overlaps) in $IN(MD)$ should be
eliminated. We will not discuss conflict-resolving in this paper, but will
concentrate on the method for identifying inaccurate data, i.e.,
$?$ $d_{i}@R_{j}$ and $?$ $R_{j}@MD$. Interested readers may refer to
the paper by Zhao (1994) for specific discussion concerning the problem of conflict
resolution.},
\]

\noindent where ``$d_{i}@R_{j}$'' and ``$R_{j}@MD$'' are two essential
qualitative predicates in our method which represent that $d_{i}$ possibly
(qualitatively) belongs to $R_{j}$ (i.e., $?$ $d_{i}\in R_{j}$), and $R_{j}$
possibly (qualitatively) belongs to $MD$ (i.e., $?$ $R_{j}\subset MD$),
respectively. Determining ``$A@B$'' is based on qualitative correlations among
related data. The work presented in this paper is mainly concentrated on
determining ``$d_{i}@R_{j}$'' and ``$R_{j}@MD$'', and realizing the above
predicate calculus.

\section{Preliminaries}
\label{Preliminaries}

Before introducing our method, we first put forward and explain several
new concepts in this section.

\subsection{Qualitative Correlations among Related Data}

\vspace*{2mm}
\noindent{\bf Definition 3.1} {\em Related data: If data $d_{1}$, $d_{2}$,
..., and $d_{m}$ describe a common phenomenon,
or they refer to the same behavior simultaneously, then they
can be treated as related data.}

\vspace*{2mm}
For example, a patient's temperature, blood pressure and other symptomatic
data are related data, and all the features for describing a criminal are also
related data. The phenomenon that some data within a dataset are related data
is more apparent in engineering. For instance, there are two types of related
data in infrared spectrum interpretation as shown in Figure 1.
First, as far as a single peak is concerned, the frequency (position) $f_{i}$,
strength (height) $s_{i}$, and width (shape) $w_{i}$ of the peak
are related data. Second, a partial component may create numerous peaks at the
same time. If we consider all the peaks that a partial component may create,
all of these peaks are related data.

\vspace*{2mm}
\noindent{\bf Definition 3.2} {\em Qualitative correlations among related
data: If $d_{i}$ and $d_{j}$ are two related data items, then the presence
of $d_{i}$ enhances the presence of $d_{j}$,
and the absence of $d_{i}$ depresses the presence
of $d_{j}$. This kind of effect is called qualitative correlations among
related data.}

\centerline{\epsfile{file=fq1.ps}}

\vspace*{2mm}
\centerline{Figure 1: Example of related data in spectrum interpretation}

\vspace*{3mm}

Consider the above example of spectrum interpretation again. If spectral data
are inaccurate (i.e., some measured peaks look like but are not exactly
the same as reference peaks), considering qualitative correlations among
related data may lead to qualitative evidence for the identification of
inaccurate data.
For example, suppose the frequency of a peak is slightly different from the
reference value, and both the strength and width of the peak are the same
as the reference values. Then the frequency of the peak may still be identified
since both of its related data support it. Similarly, if peaks at low
frequency sections are inaccurate, considering related peaks at high frequency
sections may help identify these peaks, and vice versa.

\subsection{Support Coefficient Function}

\vspace*{2mm}
\noindent{\bf Definition 3.3} {\em Support coefficient function (SCF): If
there are $m-1$ data related to $d_{i}$, then the support coefficient function
of $d_{i}$ calculates the total effects from the related data by considering
the qualitative correlations between $d_{i}$ and each of its related data.}

\vspace*{2mm}
Suppose $\pi(d_{i},$ $d_{j})$ represents the qualitative correlation between
$d_{i}$ and $d_{j}$, then the support coefficient function of $d_{i}$ can be
defined as:

\[
SCF_{i} = \beta(\sum_{j=1, j\neq i}^{m}\pi(d_{i}, d_{j}), m).
\]

$SCF_{i}$ should directly depend on how many and how much
related data support $d_{i}$. When $SCF_{i}$ is greater than a certain value
given by domain experts, the related data tend to support $d_{i}$; otherwise,
the related data tend to depress $d_{i}$.

\subsection{Evidence Based on SCF}

In Section 2, we used ``$d_{i}@R_{j}$'' to express that $d_{i}$ can be
qualitatively identified from $R_{j}$. Realizing ``$d_{i}@R_{j}$''
requires a definition of a shift interval $\triangle$ for $R_{j}$ such as:

\[
R_{j}\pm\triangle=\{(r_{j_{l}}\pm\triangle) \mid l=1,2,...,m\},
\]

\noindent and a definition of the possibility of
``$d_{i} \in R_{j}\pm\triangle$''.

The above formula is similar to that in fuzzy logic, but contains completely
different meanings. The primary difference is that the shift
intervals are dynamically determined by $SCF_{i}$, while in fuzzy logic, the
fuzzy intervals are usually provided by domain experts in advance or calculated
with quantitative criteria.

\vspace*{2mm}
\noindent{\bf Definition 3.4} {\em Shift interval: Shift interval is a dynamic
region for inaccurate data. Given a standard fuzzy interval for inaccurate
data, the shift interval of $d_{i}$ varies around the standard fuzzy
interval on the basis of $SCF_{i}$. When $SCF_{i}$ shows that the related data
support $d_{i}$, the shift interval of $d_{i}$ becomes wider than the
standard fuzzy interval. On the other hand, when $SCF_{i}$ shows that the
related data do not support $d_{i}$, the shift interval of $d_{i}$ becomes
narrower than the standard fuzzy interval.}

\vspace*{2mm}
\noindent{\bf Definition 3.5} {\em Evidence based on $SCF_{i}$: $SCF_{i}$
determines the shift interval of $d_{i}$, that is, $SCF_{i}$ determines how
widely $d_{i}$ is allowed to shift. The wider the shift interval, the more
easily $d_{i}$ is identified. Therefore, $SCF_{i}$ provides confirmatory or
disconfirmatory evidence for identifying $d_{i}$.}

\section{Making Qualitative Hypotheses for Inaccurate Data}
\label{Qualitative Hypotheses}

In this section, we introduce and analyze our method for identifying inaccurate
data. We first discuss the processes of realizing two essential predicates in
our method, ``$d_{i}@R_{j}$'' and ``$R_{j}@MD$'' respectively. Then, we
present an algorithm for making qualitative hypotheses for inaccurate data
(i.e., for realizing the predicate calculus described in Section 2).

\subsection{Predicate ``$d_{i}@R_{j}$''}

\vspace*{1mm}
When $d_{i}$ is accurate, ``$d_{i}@R_{j}$'' is equal to ``$d_{i}\in R_{j}$''.
If there is a reference value in $R_{j}$ which corresponds to $d_{i}$
(i.e., $r_{j_{p}}\in R_{j}$ and $r_{j_{p}}=d_{i}$), then $d_{i}@R_{j}=T$. If
there is no reference value corresponding to $d_{i}$, then $d_{i}@R_{j}=F$.
When $d_{i}$ is inaccurate, however, it is not sure whether $r_{j_{p}}$
corresponds to $d_{i}$. In this case, ``$d_{i}@R_{j}$'' means that $d_{i}$
possibly (qualitatively) belongs to $R_{j}$, or in other words, $r_{j_{p}}$
possibly (qualitatively) corresponds to $d_{i}$. The value of
``$d_{i}@R_{j}$'' is not $T$ or $F$, but the possibility of
``$r_{j_{p}}=d_{i}$'' or ``$d_{i}\in R_{j}$''.

\vspace*{1mm}
We discussed in Section 2 that in some cases
the identity of qualitative features is
more robust and reliable than quantitative similarity or closeness.
We have also discussed that qualitative correlations among related
data can lead to evidence for the identity of qualitative
features in diagnosis or interpretation. So if $r_{j_{p}}$
($r_{j_{p}}\in R_{j}$) is assumed to correspond to $d_{i}$, and there
are $m-1$ reference values ($r_{j_{1}}$, $r_{j_{2}}$, ..., $r_{j_{p-1}}$,
$r_{j_{p+1}}$, ..., $r_{j_{m}}$) related to $r_{j_{p}}$, then each of the
$m-1$ reference values should correspond to a certain data item in $MD$,
and the $m-1$ data items in $MD$ are also related to each other. Therefore,
qualitative correlations between $d_{i}$ and its $m-1$ related data items in
$MD$ should be considered.

Our method first determines the possibility of ``$r_{j_{p}}=d_{i}$'' by
calculating the similarity or closeness between $r_{j_{p}}$ and $d_{i}$ like
conventional fuzzy methods, then considers qualitative correlations among
related data to obtain evidence for updating the possibility. When the
qualitative correlations show that the related data support
``$r_{j_{p}}=d_{i}$'', the possibility of ``$r_{j_{p}}=d_{i}$'' will increase.
When the qualitative correlations show that the related data do not support
``$r_{j_{p}}=d_{i}$'', the possibility will decrease.

\subsubsection{Defining Support Coefficient Function}

Suppose $r_{j_{q}}$ ($r_{j_{q}}\in R_{j}$) corresponds to
$d_{t}$. Because $r_{j_{q}}$ is related to $r_{j_{p}}$, $d_{t}$ is related
to $d_{i}$. As we have discussed, the qualitative correlation between
$d_{i}$ and $d_{t}$ means that
if $d_{t}$ exists, then $d_{i}$ is enhanced; otherwise, $d_{i}$ is depressed.

We first define the qualitative correlation between two related data items,
$d_{i}$ and $d_{t}$, as:

\[
c_{i}(d_{t}) = \left\{\begin{array}{cl}
       1 & {\mbox{if $d_{t}$ can be found from $MD$ which satisfies:}}
\hspace*{1mm}r_{j_{q}}-d_{o}\leq d_{t}\leq  r_{j_{q}}+d_{o}\\
       0 & {\mbox{if $d_{t}$ can not be found from $MD$ which satisfies:}}
\hspace*{1mm}r_{j_{q}}-d_{o}\leq d_{t}\leq r_{j_{q}}+d_{o}
                      \end{array}
               \right.
\]

\noindent where $d_{o}$ is a standard fuzzy interval of inaccurate data, and
$c_{i}(d_{t})$ expresses the qualitative correlation between $d_{i}$ and
$d_{t}$. $c_{i}(d_{t})$=1 means that $d_{i}$ is enhanced since its related
data item $d_{t}$ can be found from the measured dataset, and $c_{i}(d_{t})$=0
means that $d_{i}$ is depressed since its related data item $d_{t}$ can
not be found from the measured dataset. The definition of $c_{i}(d_{t})$ is
simply based on the consideration that if a data item is identified, then the
data item will support its related data items (i.e., the coexisting data
items).

As there are $m$ reference values in $R_{j}$, we can define the support
coefficient function $SCF_{i}$ for $d_{i}$ based on $c_{i}(d_{t})$
$(t=1, 2, ..., m, \hspace*{1mm}t\neq i)$:

\[
SCF_{i} = \frac{1 + \sum_{t=1, t\neq i}^{m}c_{i}(d_{t})}{m}
\]

\noindent where $0<SCF_{i}\leq 1$, and $SCF_{i}$ expresses the total
qualitative correlations between $d_{i}$ and all of its related data. In other
words, $SCF_{i}$ reflects the support coefficient of $r_{j_{p}}$
corresponding to $d_{i}$.

If $m=1$, then $SCF_{i}=1$. When $m>1$, $SCF_{i}$ is in the direct ratio to
the number of the related data which may be identified from $MD$.

\subsubsection{Determining Dynamic Shift Interval}

Suppose $d_{o}$ is a standard fuzzy interval of inaccurate data, we define
the dynamic shift interval of $d_{i}$ based on $SCF_{i}$ as:

\[
\triangle d_{i} = \frac{(2m-1)d_{o}}{m}\times SCF_{i}
\]

%\vspace*{1mm}
\noindent where $0<\triangle d_{i}<2d_{o}$, and $\triangle$$d_{i}$ is
in the direct ratio to $SCF_{i}$.

If $m=1$, then $SCF_{i}=1$, and $\triangle$$d_{i}=d_{o}$. In other words, when
qualitative
correlations among data are not known a priori, $SCF_{i}=1$ and
$\triangle$$d_{i}=d_{o}$. In this case, the method degenerates to a
conventional fuzzy method.

When $m$ is fixed, the more the related data are identified, the greater
$SCF_{i}$ is, therefore the greater $\triangle$$d_{i}$ is. When $SCF_{i}$ is
fixed, $\triangle$$d_{i}$ depends on the number of related data.

\vspace*{2mm}
Table 1 shows the relation among $\triangle$$d_{i}$, $m$ and $SCF_{i}$.

\vspace*{6mm}

\centerline{\begin{tabular}{|c|c|c|c|c|c|c|c|}     \hline
\multicolumn{2}{|c|}{$\triangle$$d_{i}$} & \multicolumn{6}{c|}{m} \\ \cline{3-8}
\multicolumn{2}{|c|}{} & 1 & 10 & 50 & 100 & 500 & 1000 \\ \hline
 & 1 & $\hspace*{5mm} d_{o}\hspace*{5mm} $ & 1.9000$d_{o}$ & 1.9800$d_{o}$ &1.9900$d_{o}$ & 1.9980$d_{o}$ & 1.9990$d_{o}$ \\ \cline{2-8}
 & 0.8 & / & 1.5200$d_{o}$ & 1.5840$d_{o}$ & 1.5920$d_{o}$ & 1.5984$d_{o}$ & 1.5992$d_{o}$ \\ \cline{2-8}
$SCF_{i}$ & 0.5 & / & 0.9500$d_{o}$ & 0.9900$d_{o}$ & 0.9950$d_{o}$ & 0.9990$d_{o}$ & 0.9995$d_{o}$ \\ \cline{2-8}
 & 0.3 & / & 0.5700$d_{o}$ & 0.5940$d_{o}$ & 0.5970$d_{o}$ & 0.5994$d_{o}$ & 0.5997$d_{o}$ \\ \cline{2-8}
 & 0.1 & / & 0.1900$d_{o}$ & 0.1980$d_{o}$ & 0.1990$d_{o}$ & 0.1998$d_{o}$ & 0.1999$d_{o}$ \\ \hline
\end{tabular}
}

\vspace*{4mm}

\centerline{Table 1: Relation among $\triangle$$d_{i}$, $m$ and $SCF_{i}$}

\vspace*{5mm}

We can draw the following properties from the above formulas.

\vspace*{1mm}
\noindent{\bf Property 1}: {\em With the same $m$, the more the related data
are
identified, the greater $SCF_{i}$ is; otherwise, the smaller $SCF_{i}$ is.}

\vspace*{1mm}
\noindent{\bf Property 2}: {\em With the same m, the greater the $SCF_{i}$,
the greater is $\triangle$$d_{i}$. In other words, the more the related data
support $d_{i}$, the more widely $d_{i}$ is allowed to shift.}

\vspace*{1mm}
\noindent{\bf Property 3}: {\em With the same $SCF_{i}$, the greater the m,
the less $\triangle$$d_{i}$ varies along with m. In other words, the greater
the number of related data, the less a single related
data item can affect $d_{i}$.}

\vspace*{2mm}
Property 2 and Property 3 are illustrated in Figure 2.

\vspace*{5mm}

\centerline{\epsfile{file=fq2.ps}}

\vspace*{2mm}
\centerline{Figure 2: $\triangle$$d_{i}$ versus $m$ with different
$SCF_{i}$}

\vspace*{5mm}

\noindent{\bf Property 4}: {\em $\triangle$$d_{i}$ is in linear relation
to $SCF_{i}$. The slope is equal to, or greater than 1.5, which means
that $\triangle$$d_{i}$ heavily depends on $SCF_{i}$.}

%\newpage
\noindent{\bf Property 5}: {\em Along with the increase of $m$, the slope
increases very slightly. In other words, $\triangle$$d_{i}$ depends on the
number of the related data which support $d_{i}$, rather than the total number
of related data.}

\vspace*{1mm}
Property 4 and Property 5 are illustrated in Figure 3.

\vspace*{3mm}
\centerline{\epsfile{file=fq3.ps}}

\vspace*{0mm}
\centerline{Figure 3: $\triangle$$d_{i}$ versus $SCF_{i}$ with different $m$}

\vspace*{2mm}

\subsubsection{Calculating Value of Predicate ``$d_{i}@R_{j}$''}

The value of ``$d_{i}@R_{j}$'' is equal to the possibility of
``$r_{j_{p}}=d_{i}$'' which can be calculated by using the following formula:

\[
\mu_{i} = 1 - \frac{\mid d_{i} - r_{j_{p}}\mid}{\triangle d_{i}}
\]

\noindent where $\mu_{i}\leq 1$.

At a glance, the representation of $\mu_{i}$ looks like the membership degree
of ``$r_{j_{p}}-\triangle d_{i}\leq d_{i}\leq r_{j_{p}}+\triangle d_{i}$'' in
fuzzy logic. However, the meaning is completely different, for
$\triangle d_{i}$ is neither provided by domain experts nor determined by
quantitative similarity or closeness. Here $\triangle d_{i}$ is determined
on the basis of
qualitative correlations among related data. When qualitative correlations
among related data are not considered, $\triangle d_{i}$ is $d_{o}$, and the
possibility is $1 - \frac{\mid d_{i} - r_{j_{p}}\mid}{d_{o}}$. With the
consideration of qualitative correlations, the possibility is updated.

Two new properties can be drawn from the above formula for calculating
$\mu_{i}$.

\vspace*{1mm}
\noindent{\bf Property 6}: {\em With the same $d_{i}$, the greater the
$\triangle$$d_{i}$, the greater is $\mu_{i}$. In other words, the wider the
dynamic shift interval, the greater is the value of ``$d_{i}@R_{j}$''.
Formally, if $\triangle$$d_{i}''$$\geq$
$\triangle$$d_{i}'$$\geq$$\triangle$$d_{i}$, then
$\mu_{i}''$$\geq$$\mu_{i}'$$\geq$$\mu_{i}$.}

\vspace*{1mm}
\noindent{\bf Property 7}: {\em $SCF_{i}$ provides qualitative evidence for
accepting or rejecting $d_{i}$ as $r_{j_{p}}$ since $\mu_{i}$ is in the
direct ratio to $\triangle$$d_{i}$, and $\triangle$$d_{i}$ is in the direct
ratio to $SCF_{i}$.}

Property 6 and Property 7 are illustrated in Figure 4.

\vspace*{3mm}

\centerline{\epsfile{file=fq4.ps}}

\vspace*{0mm}
\centerline{Figure 4: Value of ``$d_{i}@R_{j}$'' versus various
$\triangle$$d_{i}$}

\vspace*{3mm}
The above process of realizing ``$d_{i}@R_{j}$'' and calculating the value of
``$d_{i}@R_{j}$'' can be expressed by the following procedure.

\vspace*{1.5mm}
\hspace*{10mm}
     {\bf Procedure} $d_{i}@R_{j}$

\vspace*{1mm}

\hspace*{20mm}
         $select$ $r_{j_{p}}$ $from$ $R_{j};$

\hspace*{20mm}
         $SCF_{i} = 0;$

\hspace*{20mm}
         $if$ $d_{i} = r_{j_{p}}\{$

\hspace*{30mm}
             $SCF_{i} = 1;$

\hspace*{30mm}
             $\mu_{i} = 1;$

\hspace*{20mm}
         $\}$

\hspace*{20mm}
         $else\{$

\hspace*{30mm}$for$ $each$ $r_{j_{l}}\in R_{j}$ $(l=1, ..., m,$ $l\neq p)\{$

\hspace*{41mm}
                   $calculate$ $c_{i}(d_{t})${\footnote{$d_{t}$ stands for the
data item in $MD$ which corresponds to $r_{j_{l}}$.}};

\hspace*{41mm}
                   $SCF_{i} =  SCF_{i} + c_{i}(d_{t});$

\hspace*{30mm}
              $\}$

\hspace*{30mm}
              $SCF_{i} = (1 + SCF_{i})/m;$

\hspace*{30mm}
              $\triangle d_{i} = d_{o}\times SCF_{i}\times(2m-1)/m;$

\hspace*{30mm}
              $\mu_{i} = 1 - \mid d_{i} - r_{j_{p}}\mid /\triangle d_{i};$

\hspace*{20mm}
         $\}$

\hspace*{20mm}
         $if$ $\mu_{i}>0$

\hspace*{30mm}
             $return$ $\mu_{i};$

\hspace*{20mm}
         $else$

\hspace*{30mm}
             $return$ $NIL$

\hspace*{10mm}
         {\bf end procedure}

\vspace*{1.5mm}

When $d_{i}$ can be identified with a certain possibility (i.e., $\mu_{i}>0$),
the procedure returns $T$ (i.e., the value of $\mu_{i}$); otherwise, the
procedure returns $F$.

\subsection{Predicate ``$R_{j}@MD$''}

When $MD$ is accurate, ``$R_{j}@MD$'' is equal to ``$R_{j}\subset MD$''. If all
the $m$ reference values in $R_{j}$ can be identified from $MD$, then
$R_{j}@MD=T$; otherwise $R_{j}@MD=F$. When $MD$ is inaccurate, however,
``$R_{j}@MD$'' means that $R_{j}$ is possibly (qualitatively) a subset
of $MD$. The value of ``$R_{j}@MD$'' is not $T$ or $F$, but the
possibility that all the reference values in $R_{j}$ can be identified from
$MD$.

If $\mu_{l}>0$ $(l=1, 2, ..., m)$, then $R_{j}$ can be regarded as a subset of
$MD$ with a certain possibility. Let $s_{1}$, $s_{2}$, ..., and $s_{m}$
be the priorities of the reference values in $R_{j}$, then the value of
``$R_{j}@MD$'' can be calculated based on $\mu_{1}$, $\mu_{2}$, ..., and
$\mu_{m}$ by using the following formula:

\vspace*{2mm}
\[
R_{j}@MD = \frac{\sum_{l=1}^{m}s_{l}\times \mu_{l}}{\sum_{l=1}^{m}s_{l}}, \hspace*{10mm}s_{l}>0, \hspace*{5mm}\mu_{l}>0.
\]

\vspace*{2mm}
Suppose $\mu_{i}$ has been calculated by using {\bf procedure $d_{i}@R_{j}$},
then the process of realizing ``$R_{j}@MD$'' and calculating the value of
``$R_{j}@MD$'' can be expressed by a simple procedure.

\vspace*{1.5mm}
\hspace*{10mm}
     {\bf Procedure} $R_{j}@MD$

\vspace*{1mm}
\hspace*{20mm}
         $P = s_{i}\times\mu_{i};$

\hspace*{20mm}
         $S = s_{i};$

\hspace*{20mm}
         $for$ $l = 1$ $to$ $m$ $(l\neq p)$$\{$

\hspace*{30mm}
             $\mu_{l}=d_{t}@R_{j};$

\hspace*{30mm}
             $if$ $\mu_{l}>0$$\{$

\hspace*{40mm}
             $P = P + s_{l}\times\mu_{l};$

\hspace*{40mm}
             $S = S + s_{l};$

\hspace*{30mm}
             $\}$

\hspace*{30mm}
             $else\{$

\hspace*{40mm}
             $P =0;$

\hspace*{40mm}
             $exit;$

\hspace*{30mm}
             $\}$

\hspace*{20mm}
         $\}$

\hspace*{20mm}
         $if$ $P > 0$

\hspace*{30mm}
             $return$ $P/S;$

\hspace*{20mm}
         $else$

\hspace*{30mm}
             $return$ $NIL$

\hspace*{10mm}
       {\bf end procedure}

\newpage
When $R_{j}$ can be identified as a subset of $MD$ with a certain
possibility (i.e., $P/S$), the procedure returns $T$ (i.e., the value of
$P/S$); otherwise, the procedure returns $F$.

\subsection{Algorithm for Making Qualitative Hypotheses for Inaccurate Data}

We give the following algorithm for interpreting/analyzing measured data based
on {\bf procedure $d_{i}@R_{j}$} and {\bf procedure $R_{j}@MD$}.
When measured data are not accurate, the algorithm can identify inaccurate
data items by considering qualitative correlations among related data.

\vspace*{2mm}
\hspace*{10mm}
     {\bf Algorithm} $Making$-$Qualitative$-$Hypotheses$

\vspace*{1.5mm}
\hspace*{20mm}
         $IN(MD)$ = $\emptyset;$

\vspace*{1mm}
\hspace*{20mm}
         $for$ $i$ = $1$ to $n$ $\{$

\hspace*{30mm}
         $for$ $j$ = $1$ to $k$ $\{$

\hspace*{40mm}
         $P(R_{j})$ = $0;$

\hspace*{40mm}
         $if$ $d_{i}@R_{j}$ $(i.e.,$ {\bf Procedure} $d_{i}@R_{j})$

\hspace*{50mm}
         $if$ $R_{j}@MD$ $(i.e.,$ {\bf Procedure} $R_{j}@MD)$ $\{$

\hspace*{60mm}
         $R_{j}\rightarrow IN(MD);$

\hspace*{60mm}
         $P(R_{j})$ = $R_{j}@MD;$

\hspace*{50mm}
         $\}$

\hspace*{50mm}
         $end$ $if$

\hspace*{40mm}
         $end$ $if$

\hspace*{30mm}
        \}

\hspace*{30mm}
        $end$ $for$

\hspace*{20mm}
        \}

\hspace*{20mm}
        $end$ $for$

\hspace*{10mm}
       {\bf end algorithm}

\vspace*{1.5mm}

In the algorithm, $P(R_{j})$ represents the value of ``$R_{j}@MD$''.
The algorithm is actually the realization of the predicate calculus:
$\forall d_{i}\forall R_{j}((d_{i}@R_{j})\wedge(R_{j}@MD)\rightarrow
R_{j}\subset IN(MD))$.

For each measured data item in \{$d_{1}$, $d_{2}$, ..., $d_{n}$\}, the
algorithm searches \{$R_{1}$, $R_{2}$, ..., $R_{k}$\} once. For each
$R_{j}$ ($R_{j}=\{r_{j_{1}}, r_{j_{2}}, ..., r_{j_{m}}\}$), the algorithm
checks other $n-1$ measured data items for $m$ times, and other $m-1$
reference values for $n$ times. Therefore, with blind search, the number of
operations is about (at worst):
$n\times k\times[m\times(n-1)+n\times(m-1)]
=2\times k\times m\times n^{2}-k\times n^{2}-k\times m\times n$.
Since $k$ and $m$ are two constants, the complexity of the algorithm is
$O(n^{2})$.

\section{Application to Infrared Spectrum Interpretation}
\label{Application}

We have developed a knowledge-based system for interpreting infrared spectra
by applying the proposed method, and have fully tested the system against
several hundred real spectra. The experimental results show that the proposed
method is significantly better than the conventional methods used in many
similar systems.

\subsection{Infrared Spectrum Interpretation}

The primary task of infrared spectrum interpretation is to identify unknown
objects by interpreting their infrared spectra. In this paper, we will limit
the problem to interpretation of infrared spectra of compounds to determine
composition of unknown compounds without loss of generality.

Selecting infrared spectrum interpretation as the domain of application is 
out of the following reasons:

\begin{enumerate}

\item Interpreting infrared spectra is a very significant problem in
both academic research and industrial application. For example, in
chemical science and engineering, interpreting infrared spectra of
compounds is the most effective way to identify unknown compounds, and
to analyze the composition and purity of compounds (Colthup, Daly, \&
Wiberley, 1990).

\item Interpreting infrared spectra is a very difficult problem.
First, spectral data are huge in quantity, and complex in representation.
Second, both symbolic reasoning and numerical analysis are needed to interpret
infrared spectral data (Puskar, Levine, \& Lowry, 1986; Sadtler, 1988).

\item Interpreting infrared spectra is a typical problem dealing with
inaccurate data since spectral data are often inaccurate. They
often shift from their theoretical values due to various reasons. For example,
the following is an assertion for spectrum interpretation:

\hspace*{5mm}{\em The high frequency peak of partial component $PC_{\alpha}$
is located at $F_{i}$.}

In practice, however, the peak of $PC_{\alpha}$ may irregularly shift around
$F_{i}$ due to noise or other unforeseen reasons. When the above assertion is
used to identify real spectra, uncertainty arises.

\end{enumerate}

\subsection{Applying the Proposed Method to Infrared Spectrum Interpretation}

Interpreting infrared spectra is a special problem of diagnosis. Suppose
the infrared spectrum of an unknown compound can be thresholded
and represented as a finite set of peaks (i.e., the measured dataset $MD$):

\[
Sp = \{p_{1}, p_{2}, ..., p_{n}\},
\]

\noindent where every peak consists of the frequency (position) $f$,
strength (height) $s$, and width (shape) $w$, respectively:

\[
p_{i} = (f_{i}, s_{i}, w_{i}) \hspace*{10mm}i = 1, 2, ..., n.
\]

Because $f_{i}$, $s_{i}$ and $w_{i}$ refer to the same peak $p_{i}$, they
are related data. This is the first kind of related data
in infrared spectrum interpretation.

Suppose there are finite partial components (i.e., reference values $RV$):

\vspace*{2mm}
\hspace*{30mm}$PC =\{PC_{1}, PC_{2}, ..., PC_{k}\}$

\vspace*{1.5mm}
\hspace*{37mm}$=\{\{p_{j_{1}}, p_{j_{2}}, ..., p_{j_{m}}\}\mid j=1,2,...,k\}$

\vspace*{1.5mm}
\hspace*{37mm}$=\{\{(f_{j_{p}}, s_{j_{p}}, w_{j_{p}})\mid p=1,2,...,m\}\mid j=1,2,...,k\}$.

%\newpage
\vspace*{4mm}

Because $f_{j_{p}}$, $s_{j_{p}}$ and $w_{j_{p}}$ also refer to the same
reference peak $p_{j_{p}}$, they are the
first kind of related data as well.

The spectroscopic knowledge for interpreting infrared spectra is usually
expressed as ``{\em if $p_{i}$ is equal to $p_{j_{p}}$, then $p_{i}$ may
be created by partial component $PC_{j}$}''. Here ``$p_{i}$ is equal to
$p_{j_{p}}$'' represents that $f_{i}$, $s_{i}$, and $w_{i}$ are equal to
$f_{j_{p}}$, $s_{j_{p}}$, and $w_{j_{p}}$ respectively.

The first kind of related data has the following qualitative correlations:

\begin{enumerate}

\item $f_{i}$, $s_{i}$ and $w_{i}$ should be identified simultaneously, that
is,

\hspace*{10mm}$\bullet$ if $f_{i}$ is $f_{j_{p}}$, then $s_{i}$ is $s_{j_{p}}$
and $w_{i}$ is $w_{j_{p}}$, and

\hspace*{10mm}$\bullet$ if $s_{i}$ is $s_{j_{p}}$, then $f_{i}$ is $f_{j_{p}}$
and $w_{i}$ is $w_{j_{p}}$, and

\hspace*{10mm}$\bullet$ if $w_{i}$ is $w_{j_{p}}$, then $f_{i}$ is $f_{j_{p}}$
and $s_{i}$ is $s_{j_{p}}$.

\item related data support each other. For example, if both $f_{i}$ and
$s_{i}$ have been identified, then they will enhance the
identification of $w_{i}$. Conversely, if $f_{i}$ and $s_{i}$ have not been
identified, then they will weaken the identification
of $w_{i}$.

\end{enumerate}

Our method for identifying $f_{i}$, $s_{i}$ and $w_{i}$ based on the
qualitative correlations among them can be formalized as the following
predicate calculi, respectively:

\vspace*{4mm}
\hspace*{20mm}$\forall f_{i}\forall p_{j_{p}}((f_{i}@p_{j_{p}})\wedge(p_{j_{p}}@p_{i})\rightarrow p_{i}\hspace*{1mm} is\hspace*{1mm} created\hspace*{1mm} by\hspace*{1mm} PC_{j})$, and

\vspace*{1mm}
\hspace*{20mm}$\forall s_{i}\forall p_{j_{p}}((s_{i}@p_{j_{p}})\wedge(p_{j_{p}}@p_{i})\rightarrow p_{i}\hspace*{1mm} is\hspace*{1mm} created\hspace*{1mm} by\hspace*{1mm} PC_{j})$, and

\vspace*{1mm}
\hspace*{20mm}$\forall w_{i}\forall p_{j_{p}}((w_{i}@p_{j_{p}})\wedge(p_{j_{p}}@p_{i})\rightarrow p_{i}\hspace*{1mm} is\hspace*{1mm} created\hspace*{1mm} by\hspace*{1mm} PC_{j})$,

\vspace*{4mm}

\noindent where ``$p_{i}$ $is$ $created$ $by$ $PC_{j}$'' means that
$f_{i}$, $s_{i}$ and $w_{i}$ can be qualitatively identified to be
$f_{j_{p}}$, $s_{j_{p}}$ and $w_{j_{p}}$.

In general, each partial component may create finite peaks at the same time.
So if $p_{i}$ is created by $PC_{j}$, then $Sp$ is partially created by
$PC_{j}$; if $Sp$ is partially created by $PC_{j}$, then all the peaks that
$PC_{j}$ may create should be contained by $Sp$ simultaneously. Therefore, all
the peaks created by a partial component are also related data. This is the
second kind of related data in infrared spectrum interpretation.

The second kind of related data has the following qualitative correlations:

\begin{enumerate}

\item all the peaks of a partial component should be identified simultaneously,
that is,

\vspace*{1mm}
\hspace*{10mm}if $p_{i}$ is $p_{j_{p}}$, then $p_{j_{l}}\in Sp \hspace*{2mm}(l=1,
2, ..., m, l\neq p)$.

\vspace*{1mm}

\item the peaks created by the same partial component support each other. For
example, if most of the peaks of a partial component have been identified,
these peaks will enhance the identification of the rest peaks. Conversely, if
most of the peaks of a partial component can not be identified, then the
identification of the rest peaks will be depressed.

\end{enumerate}

Our method for identifying related peaks based on the qualitative correlations
can be formalized as the following predicate calculus:

\vspace*{4mm}
\hspace*{20mm}$\forall p_{i}\forall PC_{j}((p_{i}@PC_{j})\wedge(PC_{j}@Sp)\rightarrow PC_{j}\subset IN(Sp))$.

\subsection{System for Interpreting Infrared Spectra}

Our system is implemented with C and MS-WINDOWS. Figure 5 shows the data
flow diagram of the system.

\vspace*{1mm}

\centerline{\epsfile{file=fq5.ps}}

\vspace*{1mm}

\centerline{Figure 5: Data flow diagram of the system}

\vspace*{3mm}

The input data of the system are infrared spectra of unknown compounds, and
the solutions are partial components that the input spectra may contain.
Because inferences are based on qualitative features of spectral data and
qualitative correlations among related data, the system can gain
high correct interpretation performance with noisy spectral data.

As we mentioned before, there are two types of related data in infrared
spectrum interpretation: all the features of a single peak (i.e., $f_{i}$,
$s_{i}$ and $w_{i}$ of $p_{i}$), and all the peaks of a single partial
component (i.e., $p_{1}$, $p_{2}$, ... and $p_{m}$). The inference engine of
the system employs the proposed method to both types of the related data when
inaccuracy arises.

\subsection{An Example}

We discuss the performance of the system through the following
example. Figure 6 shows an infrared spectrum of an unknown compound. The
spectrum is very hard to interpret since the peak with an arrow (named $p_{1}$)
shifts substantially. Our system correctly identifies that $p_{1}$ is
created by partial component benzene-ring.

In contrast, many similar systems can not correctly identify the peak (Clerc,
Pretsch, \& Zurcher, 1986; Hasenoehrl, Perkins, \& Griffiths, 1992; Wythoff,
Buck, \& Tomellini, 1989) since the peak of a  benzene-ring at this frequency
position (named $p_{b_{1}}$) should be a strong peak (i.e., $s_{b{1}}>1.000$)
according to spectroscopic knowledge, not a medium one ($s_{1}=0.510$) as the
case in this example. Systems based on conventional
fuzzy methods usually assume a fuzzy interval for each
inaccurate peak, then determine the membership degree that the inaccurate
peak is in the fuzzy interval. Suppose the reference value for a strong peak
is $1.000$, and the fuzzy interval for a strong peak is $0.300$ (Colthup, Daly,
\& Wiberley, 1990), then only peaks with strength of $1\pm 0.300$ can be
regarded as strong peaks. Obviously, by conventional fuzzy methods, the
possibility of $p_{1}$ being a strong peak is zero, i.e.,
$\mu_{benzene-ring}(s_{1})=0$.

Inferring on the basis of qualitative correlations among related data, our
system makes
a correct interpretation of the spectrum. Through the following two cases, we
introduce the inference process of the system, and at the same time
demonstrate the use of our method for identifying inaccurate data.

\vspace*{3mm}
\centerline{\epsfile{file=fq6.ps}}

\vspace*{2mm}

\centerline{Figure 6: An example of infrared spectrum}

\vspace*{1mm}
\subsubsection{Case I: Considering the First Kind of Related Data}

Because the frequency (position) and width (shape) of $p_{1}$ are both the
same as those of benzene-ring, the possibility of $f_{1}$ being identified
as $f_{b{1}}$ is 100\% (i.e., $\mu_{benzene-ring}(f_{1})=1$), and the
possibility of $w_{1}$ being identified as $w_{b{1}}$ is also 100\% (i.e.,
$\mu_{benzene-ring}(w_{1})=1${\footnote{$\mu_{*}(d)$ means the possibility
of $d$ being identified by conventional fuzzy methods, i.e., $SCF$ is not
considered.}}.

As we have discussed before, $f_{1}$, $s_{1}$ and $w_{1}$ are related data,
so we can obtain confirm evidence for identifying $s_{1}$ by considering
qualitative correlations among $s_{1}$, $f_{1}$ and $w_{1}$:

\vspace*{2mm}

\hspace*{5mm}$\mu_{benzene-ring}(f_{1})=1$,

\vspace*{2mm}
\hspace*{11mm}so, $c_{s_{1}}(f_{1})=1$ ($c_{s_{1}}(f_{1})$ represents the
qualitative correlation between $s_{1}$ and $f_{1}$),

\vspace*{2mm}
\hspace*{5mm}$\mu_{benzene-ring}(w_{1})=1$,

\vspace*{2mm}
\hspace*{11mm}so, $c_{s_{1}}(w_{1})=1$ ($c_{s_{1}}(w_{1})$ represents the
qualitative correlation between $s_{1}$ and $w_{1}$)

\vspace*{3mm}
\hspace*{5mm}so, $SCF_{s_{1}}=\frac{1+2}{3}=1$, and

\vspace*{3mm}
\hspace*{11mm}$\triangle s_{1}=\frac{(6-1)\times 0.300}{3}\times 1=0.500$, and

\vspace*{3mm}
\hspace*{11mm}$s_{1}@p_{b_{1}}=1-\frac{1-0.510}{0.500}=0.02$.

\newpage
%\vspace*{2.5mm}
By considering $SCF_{s_{1}}$, the possibility of $p_{1}$ being regarded as
a strong peak of benzene-ring increases from $0$ to $0.02$. As possibility,
$0.02$ may not be different from $0.04$ or $0.06$, but $0.02$ is significantly
different from $0$. Many near-misses may be handled by the negligible
possibility. For example, in most systems based on fuzzy and other
methods (Clerc, Pretsch, \& Zurcher, 1986), it is impossible to identify
$p_{1}$ to be ``strong'' (i.e., $\mu_{benzene-ring}(s_{1})=0$), but
considering qualitative correlations among related data makes it possible
although the possibility is only $0.02$.

As mentioned before, $f_{1}$ and $w_{1}$ are both the same as the reference
values, so $f_{1}@p_{b_{1}}=1$, and $w_{1}@p_{b_{1}}=1$.

Suppose the priorities of $f_{1}$, $s_{1}$ and $w_{1}$ are $2$, $1$ and $1$
respectively, then the possibility of $p_{1}$ being identified as
$p_{b_{1}}$ is:

\vspace*{1mm}
\[
\mu_{1} = p_{b_{1}}@p_{1} = \frac {2 \times 1 + 0.02 + 1} {4} = 0.755.
\]

\subsubsection{Case II: Considering the Second Kind of Related Data}

The process of considering the second kind of related data is quite similar.

We have got that the possibility of $p_{1}$ being created by a benzene-ring
is $\mu_{1}$ ($\mu_{1}=0.755$). Suppose the benzene-ring can create $m$ peaks:
\{$p_{b_{1}}$, $p_{b_{2}}$, ..., $p_{b_{m}}$\}, then the $m$ peaks are related
to each other. If $p_{1}$ is created by the benzene-ring, then $Sp$ is
partially created by the benzene-ring, i.e., the benzene-ring is contained by
the unknown spectrum; if $Sp$ is partially created by the benzene-ring, then
the other $m-1$ peaks of the benzene-ring should also be identified.

By using the same procedure as obtaining $\mu_{1}$, we can get $\mu_{2}$,
$\mu_{3}$, ... and $\mu_{m}$ as well. According to our method, the qualitative
correlation between two related peaks, $p_{i}$ and $p_{j}$, is defined as:

\vspace*{-1mm}
\[
c_{i}(p_{j}) = \left\{\begin{array}{cl}
       1 & \hspace*{10mm}if\hspace*{3mm} \mu_{j}\geq 0.5\\
       0 & \hspace*{10mm}if\hspace*{3mm} \mu_{j}<0.5.
                      \end{array}
               \right.
\]

\vspace*{-1mm}
So

\[
SCF_{i} = \frac{1 + \sum_{j=1, j\neq i}^{m}c_{i}(p_{j})}{m},\hspace*{5mm}0<SCF_{i}\leq 1.
\]

\vspace*{-1mm}
Let $d_{o}=1$, then

\[
\triangle d_{i} = \frac{2m-1}{m}\times SCF_{i},\hspace*{5mm}0<\triangle d_{i}<2,
\]

and

\vspace*{-1mm}
\[
p_{i}@benzene-ring=1-\frac{1-\mu_{i}}{\triangle d_{i}}, \hspace*{5mm}p_{i}@benzene-ring\leq 1.
\]

Roughly, when $SCF_{i}>0.5$, related peaks tend to support $p_{i}$.
When related peaks support $p_{i}$, $\triangle d_{i}>1$.
When $\triangle d_{i}>1$, $p_{i}@benzene-ring>\mu_{i}$.

\vspace*{4mm}
Table 2 shows the relation among $p_{i}@benzene-ring$, $\mu_{i}$ and
$\triangle d_{i}$.

\vspace*{9mm}
\centerline{\begin{tabular}{|c|c|c|c|c|c|c|}     \hline
\multicolumn{2}{|c|}{$p_{i}@benzene-ring$} & \multicolumn{5}{c|}{$\mu_{i}$} \\ \cline{3-7}
\multicolumn{2}{|c|}{} & 1 & 0.8 & 0.5 & 0.3 & 0 \\ \hline
 & 1.3 &\hspace*{5mm}1\hspace*{5mm}&\hspace*{2mm}0.846\hspace*{2mm}&\hspace*{2mm}0.615\hspace*{2mm}&\hspace*{2mm}0.462\hspace*{2mm}&\hspace*{2mm}0.231\hspace*{2mm}\\ \cline{2-7}
 & 1.1 &1& 0.818 & 0.545 & 0.364 & 0.091 \\ \cline{2-7}
\hspace*{3mm}$\triangle d_{i}$\hspace*{3mm}& 1 & 1             & 0.8   & 0.5   & 0.3   & 0     \\ \cline{2-7}
 & 0.9 &1& 0.778 & 0.444 & 0.222 & -0.111 \\ \cline{2-7}
 & 0.7 &1& 0.714 & 0.286 & 0     & -0.429 \\ \hline
\end{tabular}
}

\vspace*{8mm}

\centerline{Table 2: Relation among $p_{i}@benzene-ring$, $\mu_{i}$ and
$\triangle d_{i}$}

%\newpage
\vspace*{11mm}

In the above example, $SCF_{1}=0.850$, and $\triangle d_{1}=1.658$, so

\vspace*{7mm}

\[
p_{1}@benzene-ring=1-\frac{1-0.755}{1.658}=0.852.
\]

\vspace*{6mm}
Therefore, the possibility of $p_{1}$ being identified as
$p_{b_{1}}$ increases from $0.755$ to $0.852$ due to qualitative
correlations among related peaks. The process is similar to the probability
propagation in probabilistic reasoning. Here identifying $p_{1}$ is a
hypothesis, and qualitative correlations among related data of $p_{1}$
are pieces of evidence.

\vspace*{2mm}
After all the peaks of the benzene-ring are identified, the possibility that
the benzene-ring is contained by $Sp$ can be finally calculated by employing
the same method as described in Section 5.4.1.

\vspace*{8mm}
\subsection{Analysis of Experimental Results}

\vspace*{7mm}
We compare two methods in the experiments. The first method (called
``{\em\bf $AF$}'') is a conventional fuzzy method which is used by most
similar systems (Clerc, Pretsch, \& Zurcher, 1986; Wythoff, Buck, \& Tomellini,
1989). To use {\em\bf $AF$}, each reference value must be associated with a
fuzzy interval for dealing with inaccuracy. Both reference values and fuzzy
intervals are empirically determined (Colthup, Daly, \& Wiberley, 1990).

\vspace*{3mm}
Table 3 lists some reference values and their fuzzy intervals used by
{\em\bf $AF$}.

\newpage
\vspace*{1mm}
\centerline{\begin{tabular}{llll}     \hline \hline \\
$CH_{3}$ & $2960\pm 15cm^{-1}$ & $strong\pm 0.3$ & $sharp\pm 1$ \\
         & $2870\pm 15cm^{-1}$ & $strong\pm 0.3$ & $sharp\pm 1$ \\
         & $1450\pm 10cm^{-1}$ & $medium\pm 0.3$ & $sharp\pm 0.5$ \\
         & ...                 &                 &                \\
$benzene-ring$ & $3055\pm 25cm^{-1}$ & $strong\pm 0.3$ & $sharp\pm 1.5$ \\
         & $1645\pm 10cm^{-1}$ & $medium\pm 0.3$ & $sharp\pm 0.5$ \\
         & $1550\pm 30cm^{-1}$ & $medium\pm 0.3$ & $sharp\pm 1$ \\
         & $1450\pm 3cm^{-1}$ & $medium\pm 0.3$ & $sharp\pm 0$ \\
         & ...                 &                 &                \\
$-CH_{2}-OH$ & $3635\pm 5cm^{-1}$ & $strong\pm 0.3$ & $broad\pm 1$ \\
         & $3550\pm 25cm^{-1}$ & $strong\pm 0.3$ & $sharp\pm 1$ \\
         & ...                 &                 &       \\         \\ \hline \hline
\end{tabular}
}

\vspace*{6mm}

\centerline{Table 3: Some reference values and their fuzzy intervals}

\vspace*{5mm}

The membership function of {\em\bf $AF$} is:

\[
\mu_{r}(d) =max\{0, 1-\frac{\mid d-r\mid}{\triangle d}\},
\]

\noindent where $d$ is a measured data item, $r$ is a reference value,
$\triangle d$ is the fuzzy interval of $r$, and $0\leq\mu_{r}(d)\leq 1$.

The second method (called ``{\em\bf $AF^{*}$}'') is the proposed method.
{\em\bf $AF^{*}$} uses the same reference values and fuzzy intervals as
{\em\bf $AF$}, but the fuzzy intervals in {\em\bf $AF^{*}$} are only used as
standard fuzzy
intervals based on which dynamic shift intervals are determined by considering
qualitative correlations among related data.

\vspace*{1mm}
{\em\bf $AF$} and {\em\bf $AF^{*}$} use the same reference values and empirical
fuzzy intervals. The formula for calculating membership degrees in
{\em\bf $AF$}
(i.e., $\mu_{r}(d) =max\{0, 1-\frac{\mid d-r\mid}{\triangle d}\}$) is also
similar to the formula for calculating possibility in {\em\bf $AF^{*}$} (i.e.,
$\mu_{i}=1-\frac{\mid d_{i}-r_{j_{p}}\mid}{\triangle d_{i}}$). However,
in {\em\bf $AF$}, $\triangle d$ is simply an empirical fuzzy interval, while
in {\em\bf $AF^{*}$}, $\triangle d_{i}$ is a dynamic shift interval based on
qualitative correlations among related data.

We have tested the system against several hundred real infrared spectra of
organic compounds. The experimental results show that {\em\bf $AF^{*}$} is
significantly better than {\em\bf $AF$}.

Table 4 lists part of the experimental results in which the first column
indicates the solutions obtained by {\em\bf $AF$}; the second column indicates
the solutions obtained by {\em\bf $AF^{*}$}; and the third column shows the
correct solutions.

\centerline{\epsfile{file=tq4.ps}}

\vspace*{5mm}

\centerline{Table 4: Experimental results with {\em\bf $AF$} and
{\em\bf $AF^{*}$}}

\newpage
%\vspace*{5mm}

There are two important standard metrics for evaluating solutions of infrared
spectrum interpretation:

\vspace*{1mm}
\noindent{\bf Definition 5.1} {\em Rate of correctness (RC): the rate that
the identified partial component set is exactly the same as the 
partial component set in the correct solutions.}

\vspace*{0.5mm}
\noindent{\bf Definition 5.2} {\em Rate of identification (RI): the rate that
the partial components in the correct solutions are identified.}

\vspace*{0.5mm}
Table 5 shows the comparison between {\em\bf $AF$} and {\em\bf $AF^{*}$} with
the two standard metrics.

\vspace*{4mm}

\centerline{\begin{tabular}{|c||cc|cc|} \hline
 & RC &(error-rate)& RI &(error-rate)\\ \hline
{\em\bf $AF$} & 0.455 &(0.545)& 0.812 &(0.188)\\ \hline
\hspace*{3mm}{\em\bf $AF^{*}$}\hspace*{3mm}& 0.736 &(0.264)& 0.894 &(0.106)\\ \hline
\end{tabular}
}

\vspace*{3mm}

\centerline{Table 5: Evaluation of {\em\bf $AF$} \& {\em\bf $AF^{*}$} with
$RC$ and $RI$}

\vspace*{4mm}

Table 5 demonstrates that both the $RC$ and $RI$ increase by integrating $SCF$,
but the $RC$ increases more significantly. The reason is that although
{\em\bf $AF$} can identify most partial components of unknown compounds, the
rate that it can identify all partial components of unknown compounds is low
because there are always some partial components whose measured peaks seriously
shift from the reference values.

\subsection{Comparison with Related Systems}

Related systems mainly fall into the following four categories:
(1) Systems based on Y/N classification,
(2) Systems based on fuzzy logic,
(3) Systems based on pattern recognition, and
(4) Systems based on neural networks.

\subsubsection{Systems Based on Yes/No Classification}

The method commonly used by spectroscopists in practice is numerical
analysis (Colthup, Daly, \& Wiberley, 1990). Numerical analysis is primarily
based on comparison between spectral data and reference values. Reference
values are usually some regions like $frequency:3615\pm 5cm^{-1}$ or
$strength:1.000\pm 0.300$. If spectral data are in certain regions, the answer
of classification is yes; otherwise, the answer is no.

Most systems for interpreting infrared spectra use this method (Hasenoehrl,
Perkins, \& Griffiths, 1992; Puskar, Levine, \& Lowry, 1986; Wythoff, Buck, \&
Tomellini, 1989). For example, in Wythoff's system, rules for comparing
spectral data are in the following forms.

\vspace*{4mm}
{\small\em
\centerline{\begin{tabular}{lll} \hline
ANY PEAK(S)&FREQUENCY:1700-1707&STRENGTH:0.7-1.0\\
           &WIDTH:SHARP TO BROAD\\
\hspace*{10mm}ANSWER -YES-& &\\
ACTION - ***& &\\ \hline
\end{tabular}
}
}

%\newpage
\vspace*{4mm}
The advantage of these systems is that they are very easy to develop
because they can directly use spectroscopic knowledge, and do not need further
computation. However, the problem is that each of these systems is only
applicable to a class of compounds, or pure compounds because
in the case of seriously inaccurate spectral data,
the reference values (regions) can not reflect
the inaccuracy. For example, Hasenoehrl's system is only for distinguishing
compounds containing at least one carbonyl functionality from other
compounds, although the $RI$ of the system is about 98\% (naturally,
the $RC$ is
not available), and Puskar's system is only for identifying hazardous
substances.

In fact, spectroscopists also use qualitative analysis in some specific cases
in addition to the formal spectroscopic knowledge, such as ``{\em if the peaks
in 600 $cm^{-1}$- 900 $cm^{-1}$ look like the peaks of benzene-rings, then the
peaks in 3000 $cm^{-1}$- 3100 $cm^{-1}$ are quite likely to be created by
a benzene-ring}''. Unfortunately, the qualitative analysis was hardly applied
to these systems since it can not be used in usual ways. In contrast, our
system can successfully use the qualitative analysis like spectroscopists.
The way of using it is the method proposed in this paper. As a result, our
system is applicable to all compounds
which exhibit high performance with respect to correctness.

\subsubsection{Systems Based on Fuzzy Logic}

Since spectral data are always inaccurate, and the representation of
spectroscopic knowledge is quite like that in fuzzy logic, some systems
naturally use fuzzy logic or some techniques similar to fuzzy logic
(Clerc, Pretsch, \& Zurcher, 1986). In these systems, fuzzy intervals which
are similar to the regions described in Section 5.6.1 are given for reference
values, and memberships of inaccurate data are calculated on the basis of the
degrees that the inaccurate data are in the fuzzy intervals. These systems are
better than those described in Section 5.6.1 in some cases,
but the degrees that inaccurate data are in fuzzy intervals
do not necessarily reflect the possibility of the inaccurate data being the
reference values. For example, in Figure 7, it is difficult to
determine which peak is closer to the reference value only by considering
the degrees that $peak$ $a$ and $peak$ $b$ are in the fuzzy interval.

\vspace*{3mm}
\centerline{\epsfile{file=fq7.ps}}

\vspace*{3mm}

\centerline{Figure 7: Two peaks in a fuzzy interval}

\vspace*{3mm}

However, by applying the method proposed in this paper, the above problem
can be easily solved. As we discussed in Section 5.6.1, in practice
spectroscopists also frequently use knowledge about correlations among peaks
in addition to the formalizable spectroscopic knowledge. This kind
of knowledge is essential to our method which enables us to use qualitative
correlations among related data as evidence for the identification of
inaccurate data.

We have compared the fuzzy method used by these systems with our method in
Section 5.5. So far as we know, the $RC$ of our system is the highest among
the similar systems, and the $RI$ of our system is higher than that of
most of the systems.

\subsubsection{Systems Based on Pattern Recognition}

Some systems use pattern recognition techniques to interpret infrared
spectra (Jalsovszky \& Holly, 1988; Sadtler, 1988), of which {\sc Sadtler}
is the
most popular commercial system. The system compares known patterns with
unknown ones, and determines the possibility of an unknown pattern being a
known one by calculating the quantitative similarity or closeness between the
two patterns.

Unlike fuzzy techniques, pattern recognition considers a group of
data (i.e., a pattern) at the same time. However, pattern recognition is
primarily based on quantitative analysis. We have discussed that in many cases
especially when the inaccuracy of spectral data is not slight, qualitative
features of spectral data are much more important than quantitative ones. For
example, Figure 8 shows two simple cases. The difference between the two
patterns in (a) is smaller than that in (b). From the viewpoint of
{\sc Sadtler},
the two patterns in (a) are closer than those in (b). However, the two patterns
in (b) may be the same in some cases, while the two patterns in (a) may not
be the same in any case. The reason is that the qualitative features
(frequency positions of peaks) of the two patterns in (a) are different.

\vspace*{3mm}

\centerline{\epsfile{file=fq8.ps}}

\vspace*{3mm}

\centerline{Figure 8: Quantitative differences between patterns}

\vspace*{3mm}
Because quantitative similarity and closeness are not always sound, most
systems based on pattern recognition including {\sc Sadtler}
can not give concrete
solutions. In general, the solutions of these systems are only a series of
candidates from which users have to finally decide the possible one by
themselves. It is difficult to compare these systems with ours because the
solutions of these systems are quite loose, and neither the RC nor the RI
is available. {\sc Sadtler}, for example, usually gives the list of all known
patterns associated with the values of quantitative differences between the
unknown patterns and these known ones.

\subsubsection{Systems Based on Neural Networks}

Recently, neural networks have been applied to infrared spectrum interpreting
systems (Anand, Mehrotra, Mohan, \& Ranka, 1991; Robb \& Munk, 1990).
In Anand's system, a neural network approach is used to analyze the presence
of amino acids in protein molecules. To this specific classification, the $RI$
of Anand's system is about 87\%, and the $RC$ is not available. In Robb's
system, a linear neural network model is developed for interpreting
infrared spectra. The system is for general purpose like our system. Without
prior input of spectrum-structure correlations, the $RC$ of Robb's system is
equal to 53.3\%.

Although the $RC$ and $RI$ of our system are both higher than those of the two
systems, we still think that using neural networks is very promising,
especially when model training or system learning is a must. The research
concerning applying neural networks to our system is left for the future.

\section{Related Work and Discussion}
\label{Related Work}

Identifying inaccurate data has long been regarded as a significant and
difficult problem in AI. Many methods and techniques have been proposed.

Fuzzy logic provides the mathematical fundamentals of representation and
calculation of inaccurate data (Bowen, Lai, \& Bahler, 1992; Negoita \&
Ralescu, 1987; Zadeh, 1978). Our method is primarily based on fuzzy theory. But
compared with conventional fuzzy techniques, the advantages of our method
include: (1) fuzzy intervals of inaccurate data are dynamically determined
so that dynamic information can be used; (2) fuzzy intervals are based on
qualitative features of data and qualitative correlations among related data
so that the solutions are more robust. The limitation of our method is that
when qualitative correlations among related data are not known in advance, the
method degenerates to a conventional fuzzy method. For instance, if $SCF$ is
unavailable, the two methods described in Section 5.5 become the same.

Pattern recognition provides the techniques for interpreting measured data in
group (Jalsovszky \& Holly, 1988). By pattern recognition methods, related
data and connections among data can be considered.
However, there are two preconditions which must be satisfied for complex data
analysis by pattern recognition to be successful. The first precondition is
that we have to obtain adequate data bases from which we can derive the
patterns we need to recognize, and the second precondition is that we have to
demonstrate that there are suitable metrics of similarity between patterns.
When patterns
explicitly exist, and measured patterns are not seriously noisy (e.g.,
fingerprint recognition), pattern recognition methods are effective. However,
if patterns are not explicit, or patterns change irregularly which implies
that there is not a stable metrics for determining the similarity between
patterns (e.g., spectrum interpretation), our method is more practical and
robust.

In identifying inaccurate data, the roles of ``$d_{i}@R_{j}$'' and
``$R_{j}@MD$'' are quite similar to the role of subjective statements
or prior probabilities in other systems (Duda, Hart, \& Nilsson, 1976;
Shortliffe \& Buchanan, 1975).
However, the essential difference is that our method
dynamically calculates the values of ``$d_{i}@R_{j}$'' and ``$R_{j}@MD$''
>from qualitative correlations among related data so that it does not need many
assumptions beforehand, and can avoid inconsistency in knowledge and data
bases. Our method can also handle possibility propagation among
inference networks. Readers may have noticed it from the process of considering
the second kind of related data in spectrum interpretation (see Section 5.4.2).

When statistical samples are sufficient, or subjective statements can be
consistently obtained, probabilistic reasoning
methods can be applied to inaccurate data identification. When statistical
samples of
inaccurate data are not enough and consistent subjective statements are
not available, our method is very effective.

Our ongoing research related to probabilistic reasoning is to consider the
interaction among identified partial components. As we discussed before,
spectroscopists frequently use the knowledge such as {\em ``if $C_{6}H_{6}$
coexists with $CH_{3}$, then the peaks of $CH_{3}$ around 2900 $cm^{-1}$ may
shift''}, or {\em ``if -C-O-C- has been identified, then the strength of the
peaks of $CH_{3}$ may change''}. Therefore, it is possible to update the
possibilities of identified partial components by considering the interaction
among them. Using probabilistic reasoning to analyze the effects among
identified partial components would not only help us identify inaccurate
data, but also provide us with the reason why the data are inaccurate. The
research and experiments will be the subject of our sequel paper.

\section{Conclusions}
\label{Conclusions}

In this paper, we have presented a new method for identifying inaccurate data
on the basis of qualitative correlations among related data. We first
introduced a new concept called support coefficient function ($SCF$). Then, we
proposed an approach to determining dynamic shift intervals of inaccurate
data based on $SCF$, and an approach to calculating possibility of
identifying inaccurate data, respectively. We also presented an algorithm for
using qualitative correlations among related data as confirmatory or
disconfirmatory evidence for the identification of inaccurate data. We have
developed a
practical system for interpreting infrared spectra by applying the proposed
method, and have fully tested the system against several hundred real spectra.
The experimental results show that the proposed method is significantly better
than the conventional methods used in many similar systems. In this paper we
have also described the system and the experimental results.

\vspace*{1mm}
Briefly, our novel work includes:

\begin{enumerate}

\item A method which assumes an inaccurate data item to be a certain reference
value on the basis of qualitative correlations between the inaccurate data item
and all of its related data.

\item An algorithm which crystallizes the method.

\item A practical system which uses the algorithm to interpret
infrared spectra.

\end{enumerate}

\section*{Acknowledgments}

Thanks to the editors and anonymous reviewers of JAIR for their helpful
comments and suggestions, and to Chunling Sui and Mitchell Bradt for
proofreading the manuscript. This research was partially supported by
Horiba Ltd., Kyoto, Japan,  and the first author wishes to thank
ASTEM Research Institute, Kyoto, Japan, where he worked as a researcher
in 1991 - 1994.

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