\section{Related Work}

The work done in the forensic evaluation of trace evidence, particularly glass has been prolific. Researchers such as Curran, Zadora, Aitken, Lucy and Koons \cite{03Curran, 98Curran, 99Curran, 09Zadora, 09aZadora, 04Aitken, 06Aitken, 02Koons, 99Koons} have led the way with work specifically focused on creating likelihood ratio (LR) models for multivariate data - usually the elemental composition of glass elements. While work by Seheult, Grove, Evett and Walsh \cite{Seheult78, Grove80, Evett84, Evett90, Evett94, Walsh94} were focused on univariate data - Refractive Index (RI) of glass.

Although the input data for the model vary in terms of the number of characteristics or variables used, there is one common thread. Each of them generates a LR measure based on the Bayesian approach described in the following section. A verbal scale for interpretating a LR measure is shown in \fig{table3}. This was created by Evett in 1998 [ref] in an effort to create a uniform framework for reporting opinions in forensic science casework. 

\begin{figure}[h!]
\small
\begin{center}
\begin{tabular}{ | l | p{5cm} |}
\hline
LR & Verbal Statement \\ \hline
0.1-1 & The evidence slightly supports that suggestion that the defendent was not near the car window when it broke. \\ \hline
1 & The evidence is inconclusive. \\ \hline
1-10 & The evidence slightly supports that suggestion that the breaker was within 0.2 meters of the window car  when it broke. \\ \hline
10-100 & ...supports... \\ \hline
100-1,000 & ...strongly supports... \\ \hline
1000- & ...very strongly supports... \\ \hline
    \end{tabular}
\end{center}
\caption{A verbal scale for reporting LRs from Evett[ref]}\label{fig:table3}
\end{figure}


\subsection{Bayesian Approach}

The Bayesian approach refers to Bayes' theorem from probability theory. According to Curran \cite{03Curran}, it's application was advocated as early as 1904 by Henri Poincare and colleagues following the miscarriage of justice in the Dreyfus case \cite{98Taroni} which involved evidence from an examination of handwriting. It was Lindley in 1977 \cite{77Lindley} who first applied the Bayesian approach to glass evidence. In the literature for forensic science Bayes theorem is genearally used as follows:

\begin{itemize}
\item Let \emph{E} denote the evidence in question.
\item Let Hp represent the prosections argument.
\item Let Hd represent the defense argument.
\end{itemize}

Therefore, with the above in mind, Bayes theorem states:

\begin{equation} 
\frac{Pr(H_p / E)}{Pr(H_d / E)} = \frac{Pr(E / H_p)}{Pr(E / H_d)} \times \frac{Pr(H_p)}{Pr( H_d)}
\label{bayes} 
\end{equation}

Which can also be written as the posterior odds = \emph{Liklihood Ratio (LR)} x \emph{prior odds}. The likelihood ratio here is an indication of the strength of the evidence. In other words, instead of deciding whether evidence from a crime scene matches evidence found on a suspect, the LR simply indicates the probability that E comes from a common source vs a different source.

To put Bayesian models into practice, a forensic scientist would in most cases need to make the following assumptions and decisions:
\begin{itemize}
\item is there ready access to a survey database which will provide ``...a means of estimating the \emph{frequency of occurrence} of glass of a given set of properties on the clothing of members of the population'' \cite{Evett90}?
\item did the sample come from one source or multiple sources \cite{98Curran}?
\item should the distribution of the data be treated as a Gaussian?
\item independence of multivariate data \cite{Koons02}?
\end{itemize}

One of the goals of this work is to show that the forensic models created using the Bayesian approach are brittle. We say a model exhibits brittleness when a small change to it's data leads to sudden and massive changes to the interpretation. Although many of the current models try to reduce brittleness by considering all or some of the three(3) major errors (see \fig{errors}) in the creation of LR formulars, inappropriate assumptions still persists thoughout these models, and so brittleness remains. 

\begin{figure}[h!]
\small
\begin{center}
\begin{tabular}{ | l | p{5cm} |}
\hline
Errors & Description \\ \hline
1 & Measurement errors with instruments. \\ \hline
2 & Variation of measurements made on the same object. \\ \hline
3 & Differences between measurements made on different objects of the same evidential type. \\ \hline
    \end{tabular}
\end{center}
\caption{Errors considered in creating LR formulars \cite{06Aitken}}\label{fig:errors}
\end{figure}


\subsection{Examples of Brittleness in Models}

The following forensic models for glass evidence use the Bayesian approach:
\begin{itemize}
\item The 1978 Seheult model (see \fig{tableseh})
\item The 1980 Grove model (see \fig{tablewgro})
\item The 1995 Evett model (see \fig{tableeve})
\item The 1996 Walsh model (see \fig{tablewal})
\end{itemize}

\subsubsection{Seheult 1978}
Seheult \cite{Seheult78}, examines and simplifies Lindley's \cite{77Lindley} 6th equation for real-world application of Refractive Index (RI) Analysis. According to Seheult: 
\begin{quote}A measurement $x$, with normal error having known standard deviation $\sigma$, is made on the unknown refractive index $\Theta_1$ of the glass at the scene of the crime. Another measurement $y$, made on the glass found on the suspect, is also assumed to be normal but with mean $\Theta_2$ and the same standard deviation as $x$. The refractive indices $\Theta$ are assumed to be normally distributed with known mean $\mu$ and known standard deviation $\tau$. If $I$ is the event that the two pieces of glass come from the same source($\Theta_1$ = $\Theta_2$) and $\bar{I}$ the contrary event, Lindley suggests that the odds on identity should be multiplied by the factor
\begin{equation}
\frac{p(x,y|I)}{p(x,y|\bar{I})} \label{eq:lin1}
\end{equation}
In this special case, it follows from Lindley's equation (6) that the factor is
\begin{equation}
\frac{1+\lambda^2}{\lambda(2+\lambda^2)^{1/2}}^{-\frac{1}{2(1+\lambda^2)}\cdot(u^2-v^2)} \label{eq:lin2}
\end{equation}
Where
\begin{equation*}
\lambda = \frac{\sigma}{\tau}, \newline
u = \frac{x-y}{\sigma\sqrt{2}} ,  \newline
v = \frac{z-\mu}
{\tau(1+\frac{1}{2}\lambda^2)^{\frac{1}{2}}} , \newline
z = \frac{1}{2}(x+y) 
\end{equation*}
\end{quote}

\subsubsection{Grove 1980}
By adopting a model used by Lindley and Seheult, Grove proposed a non-Bayesian approach based on likelihood ratios to solve the forensic problem. The problem of deciding whether the fragments have come from common source is distinguished from the problem of deciding the guilt or innocence of the suspect. To explain his method, Grove first reviewed Lindley's method. He argued that we should, where possible, avoid parametric assumptions about the underlying distributions. Hence, in discussing the respective roles of $\theta_1$ and $\theta_2$ Grove did not attribute any probability distribution to an unknown parameter without special justification. So when considering ($\theta_1$ != $\theta_2$), $\bar{I}$ can be interpreted as saying that the fragments are present by chance entailing a random choice of value for $\theta_2$. The simplified likelihood ratio obtained from the Grove's derivation is:
\begin{equation}
\frac{\tau}{\sigma}\cdot e^{\big\{\frac{-(X-Y)^2}{4\sigma^{2}} + \frac{(Y-\mu^2)}{2\tau^2}\big\}} \label{eq:gro2} 
\end{equation}

We are of course only concerned with the evidence about \I and $\bar{I}$ so far as it has the bearing on the guilt or innocence of the suspect. Grove also considered the Event of Guilty factor \b{G} in the calculation of likelihood ratio (LR). Therefore the LR now becomes 
\begin{equation} p(X,Y|G)/p(X,Y|\bar{G})\end{equation}
Here in the expansion event \b{T}, that fragments were transferred from the broken window to the suspect and persisted until discovery and event \b{A},that the suspect came into contact with glass from other source. Here p(A/{G})=p(A/$\bar{G}$)=Pa and p(T/G)= Pt. The resulting expression is %\newline

  \begin{equation} 
    \frac{P(X,Y,S|G)}{P(X,Y,S|\bar{G})} = 1+Pt\Big\{(\frac{1}{Pa}-1)\frac {p(X,Y|I)}{p(X,Y|\bar{I})}-1\Big\}\label{eq:gro4}  
  \end{equation} 
  
\subsubsection{Evett 1995}
Evett et al use data from forensic surveys to create a Bayesian approach in determining the statistical significance of finding glass fragments and groups of glass fragments on individuals associated with a crime.\cite{Evett94}

I. W. Evett has been published in numerous journals investigating the usefulness of glass fragments as evidence. In 1990, Evett and Buckleton examined the results of a survey performed by McQuillan and Edgar which analyzed the discovery of glass fragments on persons unrelated to a crime. This paper examines a survey performed by Lambert,  Satterthwaite and Harrison[LSH] titled "A survey of glass fragments recovered from clothing of persons suspected of involvement in crime."

Evett proposes that likelihood ratios are well suited for explaining the existence of glass fragments on a person suspected of a crime. A likelihood ratio is defined in the context of this paper as the ratio of the probability that the suspected person is guilty given the existing evidence to the probability that the suspected person is innocent given the existing evidence. The given evidence, as it applies to Evett's approach, includes the number of different sets of glass and the number of fragments in each unique group of glass.

The LSH survey used empirical evidence to supply probabilities relevant to Evett's proposal. The LSH survey inspected individuals and collected glass fragments from each of them. These fragments were placed into groups based on their refractive index (RI) and other distinguishing physical properties. The number of fragments and the number of sets of fragments were recorded, and the discrete probabilities were published.  In particular, there are two unique probabilities that are of great interest in calculating Evett's proposed likelihood ratio.
\begin{itemize}
\item S, the probability of finding N glass \emph{fragments} per group
\item P, the probability of finding M \emph{groups} on an individual.
\end{itemize}

%\clearpage
%\subsection{Mathematical Formulae}
The following symbols are used by Evett to express his equations:
\begin{itemize}
\item $P_n$ is the probability of finding $n$ groups of glass on the surface of a person's
clothes
\item $T_n$ is the probability that $n$ fragments of glass would be transferred, retained
and found on the suspect's clothing if he had smashed the scene window
\item $S_n$ is the probability that a group of glass fragments on a person's clothing
consists of $n$ fragments
\item $f$ is the probability that a group of fragments on person's
clothing would match the control sample
\item $\lambda$ is the expected number of glass fragments remaining at a time, $t$
\end{itemize}
Evett utilizes the following equations to determine the likelihood ratio for the first case described in his 1994 paper. In this case, a single window is broken, and a single group of glass fragments is expected to be recovered.

\begin{equation}
LR = \frac{{P_0}{T_n}}{{P_1}{S_n}{f}}+{T_0}
\end{equation}

\begin{equation} 
T_n = \frac{e^{-\lambda}{\lambda^n}}{n!}
\end{equation}

\subsubsection{Walsh 1996}
The equation presented by Walsh \cite{Walsh94} is similar to one of Evett's. The difference is that Walsh argues that instead of incorporating grouping	and matching, only grouping should be included. Walsh says this is because match/non-match is really just an arbitrary line. He examines the use of a technique in interpreting glass evidence of a specific case. This technique is as follows:
\begin{equation}
\frac{T_L P_0 p(\bar{X},\bar{Y}|S_y,S_x)}{P_1S_Lf_1}
\end{equation}
Where 

\begin{itemize}
\item
$T_L$ = the probability of 3 or more glass fragments being transferred from the crime scene to the person.
\item
$P_0$ = the probability of a person having no glass on their clothing
\item
$P_1$ = the probability of a person having one group of glass on their clothing
\item
$S_L$ = the probability that a group of glass on clothing is 3 or more fragments
\item
$\bar{X}$ and $\bar{Y}$ are the mean of the control and recovered groups respectively
\item
$S_x$ and $S_y$ are the sample standard deviations of the control and recovered groups respectively
\item
$f_1$ is the value of the probability density for glass at the mean of the recovered sample
\item
$p(\bar{X},\bar{Y}|S_y,S_x)$ is the value of the probability density for the difference between the sample means
\end{itemize} 
%%%%

\subsubsection{The Brittleness in the Seheult Model}
It is easy enough to demonstrate ``brittle'' interpretation models for glass. Consider the 1978 Seheult \cite{Seheult78} glass interpretation model. This model returns the likelihood ratio that the crime scene sample is the same as the same from the suspect. \fig{sehfig} shows how the likelihood ratio changes (on the vertical axis) as we try different values from the refractive index of from glass from two sources (x and y). This model could lead to incorrect interpretations if minor errors are made when measuring the refractive index of glass samples taken from a suspect's clothes. Note how, near the region where x=y, how tiny changes in the x or y refractive indexes can lead to dramatic changes in the likelihood ratio (from zero to one). %need to fix

\begin{figure}[h!]
\begin{center}
\includegraphics{seheult}
\end{center}
\caption{Brittleness in the 1978 Seheult interpretation models. Very small changes to RI near critical value of x=y can result in dramatic increases in the likelihood that the samples are reported to be similar}\label{fig:sehfig}
\end{figure}

%%%%  

\subsubsection{Illustrating Brittleness with Contrast Set Learning}
Brittleness in models can be further illustrated with Contrast Set Learning (CSL) (see Figures \ref{fig:tableseh} - \ref{fig:tablewal}). Generally CSL refers to a set of learning algorithms which compare sets of data. These algorithms will then determine which attributes and which ranges of attributes of a set are important for differentiating it from the other set(s). To figure out the importance level of each attribute value range for predicting the LR let us look at an example. 

\fig{tableseh1} shows a snippet of the result of applying CSL to a data set generated from the Seheult model. The attribute values of $x$ have been binned using equal frequency binning, and so the $Range$ row represents the minimum and maximum values of each bin. The $Rank$ values represents the level of importance for each range of $x$ values. Here a rank of 1.0 is optimum indicating that any $x$ value in the range of 0.95 to 0.98 is more likely to have a higher LR than $x$ values from 0.94 to 0.95 and 0.98 to 1.0. %need to change to actual values

For the purpose of this work, \fig{tableseh1} and Figures \ref{fig:tableseh} - \ref{fig:tablewal}, offer a perfect visual of brittleness in these forensic models. Recall our definition for brittleness as a tiny change in data can lead to a change in interpretation i.e. the rank value can change from 0.23 to 1.0 if x=0.945 changes to x=0.955. Additionally, without even looking at the numbers, a casual glance at the CSL charts in the paper shows the clear contrasts of neighboring ranges where a dark block (low LR) sits next to a light block (high LR).

%The Seheult data set contains 1000 instances and six(6) attributes which report on the LR class (see \fig{tableseh}). Paying particlualr attention to attributes $x$ and $y$ (the refractive indices from a crime scene and a suspect respectively), whose values are randomly selected from a range of 1.4 to 1.6.   In this case brittleness is indicated when these ranges of attributes lie next to each other. The representation for this can be see clearly in \fig{tableseh1} when a dark block (low probability of match) sit next to a light block (high probability of match).

\begin{figure}[h!]
\begin{center}
\resultbox{
\begin{tabular}{cc|c|c|c|c|l}
\hline
\multicolumn{1}{|c|}{\multirow{2}{*}{x}} &
\rg & \br{0.23} \cl{0.94 - 0.95} & \br{1.0} \cb{0.95 - 0.96} & \br{0.64} \cl{0.96 - 0.97} & \br{0.64} \cl{0.97 - 0.98} & \br{0.36} \cl{0.98 - 1.0}  \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.23} & \br{1.0} \cb{1.0} & \bb{0.64} & \bb{0.64} & \bb{0.36}  \\ 
\cline{1-7}
\end{tabular}
}
\caption{Brittleness in Seheult for x}
\label{fig:tableseh1}
\end{center}
\end{figure}

For Seheult \fig{tableseh} any change in the x attribute from the ranges 0.94 - 0.95 to 0.95 - 0.96 or the mu attribute from 0.10 - 0.18 to 0.18 - 0.26 can lead to a significant difference to the forensic interpretation. Other variables in this and other models also show the same brittleness. For instance, the brittleness of the Grove model \fig{tablewgro} can be seen clearly with the y attribute with the rank values jumping up and down with successive ranks of 0.25, 1.0, 0.33, 0.86 and 0.18. While with the Walsh model \fig{tablewal}, although there are regions of brittleness they are not as abundant as the three other statistical models.

Our solution to solving the problem of brittleness in these models is to first move away from the Bayesian approach and apply simple chemometric analysis to evidence. Furthermore, reduce brittleness with Prototype Learning (PL). The following section explores these points further.



\begin{figure*}[h!]
\begin{center}
\resultbox{
\begin{tabular}{cc|c|c|c|c|c|c|c|c|c|c|l}
\cline{3-12}
& & \model{Seheult} \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{x}} &
\rg & \br{0.45} \cl{0.88 - 0.89} & \br{0.28} \cl{0.89 - 0.90} & \br{0.22} \cl{0.90 - 0.91} & \br{0.36} \cl{0.91 - 0.93} & \br{0.54}\cl{0.93 - 0.94} & \br{0.23} \cl{0.94 - 0.95} & \br{1.0} \cb{0.95 - 0.96} & \br{0.64} \cl{0.96 - 0.97} & \br{0.64} \cl{0.97 - 0.98} & \br{0.36} \cl{0.98 - 1.0}  \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.45} & \bb{0.28} & \bb{0.22} & \bb{0.36} & \bb{0.54} & \bb{0.23} & \br{1.0} \cb{1.0} & \bb{0.64} & \bb{0.64} & \bb{0.36}  \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{y}} &
\rg & \br{0.33} \cl{0.88 - 0.89} & \br{0.33} \cl{0.89 - 0.90} & \br{0.62} \cl{0.90 - 0.91} & \br{0.41} \cl{0.91 - 0.93} & \br{0.41} \cl{0.93 - 0.94} & \br{0.33} \cl{0.94 - 0.95} & \br{0.51} \cl{0.95 - 0.96} & \br{1.0} \cb{0.96 - 0.98} & \br{0.62} \cl{0.98 - 0.99} & \br{0.73} \cl{0.99 - 1.0}  \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.33} & \bb{0.33} & \bb{0.62} & \bb{0.41} &  \bb{0.41} & \bb{0.33} & \bb{0.51} & \br{1.0} \cb{1.0} & \bb{0.62} & \bb{0.73}  \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{mu}} &
\rg & \br{0.72} \cl{0.10 - 0.18} & \br{0.29} \cl{0.18 - 0.26} & \br{0.72} \cl{0.26 - 0.36 }& \br{0.48} \cl{0.36 - 0.45} & \br{0.59} \cl{0.45 - 0.55} & \br{0.48} \cl{0.55 - 0.65} & \br{0.59} \cl{0.65 - 0.73} & \br{1.0} \cb{0.73 - 0.83} & \br{0.59} \cl{0.83 - 0.91} & \br{0.59} \cl{0.91 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.72} & \bb{0.29} & \bb{0.72} & \bb{0.48} & \bb{0.59} & \bb{0.48} & \bb{0.59} & \br{1.0} \cb{1.0} & \bb{0.59} & \bb{0.59}      \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{tau}} &
\rg & \br{0.48} \cl{0.10 - 0.18} & \br{0.77} \cl{0.18 - 0.27} & \br{0.19} \cl{0.27 - 0.37} & \br{0.19} \cl{0.37 - 0.47} & \br{0.39} \cl{0.47 - 0.55} & \br{0.47} \cl{0.55 - 0.64} & \br{0.19} \cl{0.64 - 0.72} & \br{0.39} \cl{0.72 - 0.82} & \br{0.19} \cl{0.82 - 0.91} & \br{1.0} \cb{0.91 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.48} & \bb{0.77} & \bb{0.19} & \bb{0.19} &  \bb{0.39} & \bb{0.47} & \bb{0.19} & \bb{0.39} & \bb{0.19} & \br{1.0} \cb{1.0}    \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{lr}} &
\rg & \br{1.0} \cb{-1 - 0} & \br{0.1} \cl{0 - 1} & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \br{1.0} \cb{1.0} & \bb{0.1} & N/A & N/A & N/A  & N/A & N/A & N/A & N/A & N/A      \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{time}} &
\rg & \br{1.0} \cb{0.0 - 0.11} & \br{0.85} \cb{0.11 - 0.20} & \br{0.47} \cl{0.20 - 0.31} & \br{0.05} \cl{0.31 - 0.40} & \br{0.0} \cl{0.40 - 0.49} & \br{0.0} \cl{0.49 - 0.61} & \br{0.0} \cl{0.61 - 0.72} & \br{0.0} \cl{0.72 - 0.83} & \br{0.0} \cl{0.83 - 0.92} & \br{0.0} \cl{0.92 - 1.0}     \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \br{1.0} \cb{1.0} & \br{0.85} \cb{0.85} & \bb{0.47} & \bb{0.05} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0}      \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{cost}} &
\rg & \br{1.0} \cb{0.0 - 0.10} & \br{0.86} \cb{0.10 - 0.17} & \br{0.23} \cl{0.17 - 0.30} & \br{0.15} \cl{0.30 - 0.39} & \br{0.0} \cl{0.39 - 0.50} & \br{0.0} \cl{0.50 - 0.60} & \br{0.0} \cl{0.60 - 0.67} & \br{0.0} \cl{0.67 - 0.80} & \br{0.0} \cl{0.80 - 0.90} & \br{0.0} \cl{0.90 - 1.0}     \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \br{1.0} \cb{1.0} & \br{0.86} \cb{0.86} & \bb{0.23} & \bb{0.15} &  \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0}     \\ 
\cline{1-12}

\end{tabular}
}
\caption{CSL on Seheult Showing Ranks of Value Ranges}
\label{fig:tableseh}
\end{center}
\end{figure*}

%%%%%%%%

\begin{figure*}[h!]
\begin{center}
\resultbox{
\begin{tabular}{cc|c|c|c|c|c|c|c|c|c|c|l}
\cline{3-12}
& & \model{Grove} \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{x}} &
\rg & \br{0.70} \cl{0.88 - 0.89} & \br{0.56} \cl{0.89 - 0.90} & \br{0.84} \cl{0.90 - 0.91} & \br{0.56} \cl{0.91 - 0.93} & \br{0.69}\cl{0.93 - 0.94} & \br{1.0} \cb{0.94 - 0.95} & \br{0.84} \cl{0.95 - 0.96} & \br{0.44} \cl{0.96 - 0.97} & \br{1.0} \cb{0.97 - 0.99} & \br{0.44} \cl{0.99 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.70} & \bb{0.56} & \bb{0.84} & \bb{0.56} & \bb{0.69} & \br{1.0} \cb{1.0} & \bb{0.84} & \bb{0.44} & \br{1.0} \cb{1.0} & \bb{0.44}   \\ 
\hr
\cline{1-12}
\multicolumn{1}{|c|}{\multirow{2}{*}{y}} &
\rg & \br{0.62} \cl{0.88 - 0.89} & \br{0.51} \cl{0.89 - 0.90} & \br{0.25} \cl{0.90 - 0.91} & \br{1.0} \cb{0.91 - 0.93} & \br{0.33} \cl{0.93 - 0.94} & \br{0.86} \cl{0.94 - 0.95} & \br{0.18} \cl{0.95 - 0.96} & \br{0.41} \cl{0.96 - 0.97} & \br{0.73} \cl{0.97 - 0.99} & \br{0.51} \cl{0.99 - 1.0}    \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.62} & \bb{0.51} & \bb{0.25} & \br{1.0} \cb{1.0} & \bb{0.33} & \bb{0.86} & \bb{0.18} & \bb{0.41} & \bb{0.73} & \bb{0.51}      \\ \cline{1-12}
\cline{1-12}
\hr 
\multicolumn{1}{|c|}{\multirow{2}{*}{mu}} &
\rg & \br{0.34} \cl{0.10 - 0.20} & \br{0.51} \cl{0.20 - 0.30} & \br{0.86} \cl{0.30 - 0.38 }& \br{0.18} \cl{0.38 - 0.47} & \br{1.0} \cb{0.47 - 0.56} & \br{0.41} \cl{0.56 - 0.66} & \br{0.41} \cl{0.66 - 0.74} & \br{1.0} \cb{0.74 - 0.83} & \br{0.51} \cl{0.83 - 0.91} & \br{0.25} \cl{0.91 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.34} & \bb{0.51} & \bb{0.86} & \bb{0.18} & \br{1.0} \cb{1.0} & \bb{0.41} & \bb{0.41} & \br{1.0} \cb{1.0} & \bb{0.51} & \bb{0.25}     \\ \cline{1-12}
\cline{1-12}
\hr 
\multicolumn{1}{|c|}{\multirow{2}{*}{tau}} &
\rg & \br{0.54} \cl{0.10 - 0.20} & \br{0.22} \cl{0.20 - 0.30} & \br{0.44} \cl{0.30 - 0.39} & \br{0.36} \cl{0.39 - 0.48} & \br{1.0} \cb{0.48 - 0.57} & \br{0.11} \cl{0.57 - 0.66} & \br{0.44} \cl{0.66 - 0.75} & \br{0.44} \cl{0.75 - 0.83} & \br{1.0} \cb{0.83 - 0.93} & \br{0.28} \cl{0.93 - 1.0}  \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.54} & \bb{0.22} & \bb{0.44} & \bb{0.36} & \br{1.0} \cb{1.0} & \bb{0.11} & \bb{0.44} & \bb{0.44} & \br{1.0} \cb{1.0} & \bb{0.28}  \\ \cline{1-12}
\cline{1-12}
\hr 
\multicolumn{1}{|c|}{\multirow{2}{*}{lr}} &
\rg & \br{1.0} \cb{-1 - 0} & \br{0.1} \cl{0 - 1} & N/A & N/A & N/A & N/A & N/A & N/A & N/A & N/A      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \br{1.0} \cb{1.0} & \bb{0.1} & N/A & N/A & N/A  & N/A & N/A & N/A & N/A & N/A      \\ \cline{1-12}
\cline{1-12}
\hr 
\multicolumn{1}{|c|}{\multirow{2}{*}{time}} &
\rg & \br{0.69} \cl{0.0 - 0.11} & \br{1.0} \cb{0.11 - 0.22} & \br{0.22} \cl{0.22 - 0.32} & \br{0.04} \cl{0.32 - 0.42} & \br{0.0} \cl{0.42 - 0.52} & \br{0.0} \cl{0.52 - 0.62} & \br{0.0} \cl{0.62 - 0.73} & \br{0.0} \cl{0.73 - 0.83} & \br{0.0} \cl{0.83 - 0.92} & \br{0.0} \cl{0.92 - 1.0}     \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.69} & \br{1.0} \cb{1.0} & \bb{0.22} & \bb{0.04} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0}     \\ \cline{1-12}
\cline{1-12}
\hr %
\multicolumn{1}{|c|}{\multirow{2}{*}{cost}} &
\rg & \br{0.62} \cl{0.0 - 0.10} & \br{0.85} \cl{0.10 - 0.21} & \br{1.0} \cb{0.21 - 0.30} & \br{0.17} \cl{0.30 - 0.41} & \br{0.0} \cl{0.41 - 0.52} & \br{0.0} \cl{0.52 - 0.61} & \br{0.0} \cl{0.61 - 0.71} & \br{0.0} \cl{0.71 - 0.81} & \br{0.0} \cl{0.81 - 0.91} & \br{0.0} \cl{0.91 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.62} & \bb{0.85} & \br{1.0} \cb{1.0}  & \bb{0.17} &  \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0}     \\ \cline{1-12}

\end{tabular}
}
\caption{CSL on Grove Showing Ranks of Value Ranges}
\label{fig:tablewgro}
\end{center}
\end{figure*}

%%%%%%%%%%%%%%

\begin{figure*}[h!]
\begin{center}
\resultbox{
\begin{tabular}{cc|c|c|c|c|c|c|c|c|c|c|l}
\cline{3-12}
& & \model{Evett} \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{lamb}} &
\rg & \br{0.02} \cl{0.09 - 0.20} & \br{0.58} \cl{0.20 - 0.29} & \br{0.66} \cl{0.29 - 0.36} & \br{1.0} \cb{0.36 - 0.46} & \br{0.38} \cl{0.46 - 0.55} & \br{0.08} \cl{0.55 - 0.64} & \br{0.18} \cl{0.64 - 0.72} & \br{0.15} \cl{0.72 - 0.80} & \br{0.02} \cl{0.80 - 0.90} & \br{0.04} \cl{0.90 - 1.0}     \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.02} & \bb{0.58} & \bb{0.66} & \br{1.0} \cb{1.0} & \bb{0.38} & \bb{0.08}  & \bb{0.18} & \bb{0.15} & \bb{0.02} & \bb{0.04}  \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{n}} &
\rg & \br{0.0} \cl{0.0 - 0.20} & \br{0.02} \cl{0.20 - 0.40} & \br{0.35} \cl{0.40 - 0.60} & \br{0.61} \cl{0.60 - 0.80} & \br{1.0} \cb{0.80 - 1.0} & N/A & N/A & N/A & N/A & N/A    \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.0} & \bb{0.02} & \bb{0.35} &  \bb{0.61} & \br{1.0} \cb{1.0} & N/A & N/A & N/A & N/A & N/A      \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{f-values}} &
\rg & \br{1.0} \cb{0.04 - 0.14} & \br{0.99} \cb{0.14 - 0.24} & \br{0.44} \cl{0.24 - 0.36} & \br{0.20} \cl{0.36 - 0.45} & \br{0.37} \cl{0.45 - 0.55} & \br{0.20} \cl{0.55 - 0.64} & \br{0.15} \cl{0.64 - 0.73} & \br{0.25} \cl{0.73 - 0.81} & \br{0.11} \cl{0.81 - 0.91} & \br{0.03} \cl{0.91 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \br{1.0} \cb{1.0} & \br{0.99} \cb{0.99} & \bb{0.44} & \bb{0.20} & \bb{0.37} & \bb{0.20} & \bb{0.15} & \bb{0.25} & \bb{0.11} & \bb{0.03}      \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{lr}} &
\rg & \br{0.0} \cl{N/A} & \br{0.0} \cl{N/A} & \br{0.0} \cl{N/A} & \br{0.0} \cl{0.01 - 0.04} & \br{0.0} \cl{0.04 - 0.09} & \br{0.0} \cl{0.09 - 0.16} & \br{0.0} \cl{0.16 - 0.28} & \br{0.02} \cl{0.28 - 0.47} & \br{0.15} \cl{0.47 - 0.71} & \br{1.0} \cb{0.71 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.0}  & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.02} & \bb{0.15} & \bb{1.0} \cb{1.0}     \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{time}} &
\rg & \br{0.91} \cl{0.0 - 0.10} & \br{1.0} \cb{0.10 - 0.20} & \br{0.57} \cl{0.20 - 0.30} & \br{0.18} \cl{0.30 - 0.39} & \br{0.77} \cl{0.39 - 0.50} & \br{0.11} \cl{0.50 - 0.58} & \br{0.10} \cl{0.58 - 0.71} & \br{0.0} \cl{0.71 - 0.82} & \br{0.0} \cl{0.82 - 0.91} & \br{0.0} \cl{0.91 - 1.0}     \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.91}  & \br{1.0} \cb{1.0} & \bb{0.57} & \bb{0.18} & \bb{0.77} & \bb{0.11} & \bb{0.10} & \bb{0.0} & \bb{0.0} & \bb{0.0}      \\ 
\hr%
\multicolumn{1}{|c|}{\multirow{2}{*}{cost}} &
\rg & \br{0.92} \cl{0.0 - 0.10} & \br{0.88} \cl{0.10 - 0.20} & \br{0.85} \cl{0.20 - 0.31} & \br{1.0} \cb{0.31 - 0.40} & \br{0.15} \cl{0.40 - 0.52} & \br{0.20} \cl{0.52 - 0.62} & \br{0.04} \cl{0.62 - 0.73} & \br{0.0} \cl{0.73 - 0.82} & \br{0.0} \cl{0.82 - 0.92} & \br{0.0} \cl{0.92 - 1.0}     \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.92} & \bb{0.88} & \bb{0.85} & \br{1.0} \cb{1.0} &  \bb{0.15} & \bb{0.20} & \bb{0.04} & \bb{0.0} & \bb{0.0} & \bb{0.0}      \\ 
\cline{1-12}

\end{tabular}
}
\caption{CSL on Evett Showing Ranks of Value Ranges}
\label{fig:tableeve}
\end{center}
\end{figure*}

%%%%%%%%%%%%%%%%

\begin{figure*}[h!]
\begin{center}
\resultbox{
\begin{tabular}{cc|c|c|c|c|c|c|c|c|c|c|l}
\cline{3-12}
& & \model{Walsh} \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{fl}} &
\rg & \br{1.0} \cb{0.0 - 0.10} & \br{0.49} \cl{0.10 - 0.21} & \br{0.10} \cl{0.21 - 0.32} & \br{0.02} \cl{0.32 - 0.42} & \br{0.01} \cl{0.42 - 0.52} & \br{0.0} \cl{0.52 - 0.59} & \br{0.0} \cl{0.59 - 0.69} & \br{0.0} \cl{0.69 - 0.78} & \br{0.0} \cl{0.78 - 0.90} & \br{0.0} \cl{0.90 - 1.0}      \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \br{1.0} \cb{1.0} & \bb{0.49} & \bb{0.10} & \bb{0.02} & \bb{0.01} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0}    \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{p}} &
\rg & \br{0.03} \cl{0.20 - 0.27} & \br{0.04} \cl{0.27 - 0.36} & \br{0.14} \cl{0.36 - 0.43} & \br{0.18} \cl{0.43 - 0.51} & \br{0.47} \cl{0.51 - 0.60} & \br{0.28} \cl{0.60 - 0.68} & \br{0.80} \cl{0.68 - 0.76} & \br{0.10} \cl{0.76 - 0.84} & \br{0.47} \cl{0.84 - 0.93} & \br{1.0} \cb{0.93 - 1.0}   \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.03} & \bb{0.04} & \bb{0.14} & \bb{0.18} &  \bb{0.47} & \bb{0.28} & \bb{0.80} & \bb{0.10} & \bb{0.47} & \br{1.0} \cb{1.0}  \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{lr}} &
\rg & \br{0.0} \cl{0.04 - 0.08} & \br{0.0} \cl{0.08 - 0.11} & \br{0.0} \cl{0.11 - 0.14} & \br{0.0} \cl{0.14 - 0.17} & \br{0.0} \cl{0.17 - 0.20} & \br{0.0} \cl{0.20 - 0.25} & \br{0.002} \cl{0.25 - 0.33} & \br{0.15} \cl{0.33 - 0.49} & \br{0.37} \cl{0.49 - 0.92} & \br{1.0} \cb{0.92 - 1.0}   \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.0} & \bb{0.0} & \bb{0.0} & \bb{0.0} &  \bb{0.0} & \bb{0.0} & \bb{0.002} & \bb{0.15} & \bb{0.37} & \br{1.0} \cb{1.0}       \\ 
\hr
\multicolumn{1}{|c|}{\multirow{2}{*}{time}} &
\rg & \br{0.51} \cl{0.0 - 0.11} & \br{0.59} \cl{0.11 - 0.23} & \br{0.57} \cl{0.23 - 0.33} & \br{1.0} \cb{0.33 - 0.44} & \br{0.18} \cl{0.44 - 0.55} & \br{0.06} \cl{0.55 - 0.63} & \br{0.05} \cl{0.63 - 0.72} & \br{0.0} \cl{0.72 - 0.81} & \br{0.0} \cl{0.81 - 0.90} & \br{0.0} \cl{0.90 - 1.0}  \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \bb{0.51} & \bb{0.59} & \bb{0.57} & \br{1.0} \cb{1.0} & \bb{0.18} & \bb{0.06} & \bb{0.05} & \bb{0.0} & \bb{0.0} & \bb{0.0}   \\ 
\hr%
\multicolumn{1}{|c|}{\multirow{2}{*}{cost}} &
\rg & \br{1.0} \cb{0.0 - 0.10} & \br{0.64} \cl{0.10 - 0.20} & \br{0.38} \cl{0.20 - 0.31} & \br{0.27} \cl{0.31 - 0.41} & \br{0.31} \cl{0.41 - 0.52} & \br{0.08} \cl{0.52 - 0.61} & \br{0.06} \cl{0.61 - 0.73} & \br{0.0} \cl{0.73 - 0.82} & \br{0.0} \cl{0.82 - 0.92} & \br{0.0} \cl{0.92 - 1.0}   \\ 
\multicolumn{1}{|c|}{}                        &
\rk & \br{1.0} \cb{1.0} & \bb{0.64} & \bb{0.38} & \bb{0.27} &  \bb{0.31} & \bb{0.08} & \bb{0.06} & \bb{0.0} & \bb{0.0} & \bb{0.0}   \\ 
\cline{1-12}

\end{tabular}
}
\caption{CSL on Walsh Showing Ranks of Value Ranges}
\label{fig:tablewal}
\end{center}
\end{figure*}
