\documentclass[12pt]{article}
\usepackage{graphicx}
\title{A Concise Summary of "Further Observations on Glass Evidence Interpretation"}
\author{Matthew Bartenschlag}
\date{October 1, 2008}

\begin{document}
\maketitle
\clearpage

\section{Summary}
Evett et al use forensic surveys to use a Bayesian approach in
determining the statistical significance of finding glass fragments and groups
of glass fragments on individuals associated with a crime.

\section{Analysis}
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 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 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 probabilites
that are of great interest in calculating Evett's proposed likelihood ratio.
These probabilites are S, the probability of finding N fragments per group,
and P, the probability of finding M groups on an individual.

Evett bases his likelihood ratio on two additional parameters: T, the 
probability of finding a given number of glass fragments after an arbitrary
amount of time has passed; and f, the probability that a group of fragments
found on the suspect's clothes would match a control sample. For the purpose of
generalization, Evett uses a value for f that was provided in the Evett-
Buckleton paper. The values for T are based off of a Poisson distribution. The
formula for obtaining the T value is as follows:
\begin{equation} 
T_j = \frac{e^{-\lambda}{\lambda^j}}{j!}
\end{equation}

The formula used to determine the likelihood ratio is given as:

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

where n is the number of fragments recovered from the suspect.

\clearpage

\section{LISP Implementation}
\begin{verbatim}
;;;; Evett's 1994 algorithm for determining LRs


(defun calculate-t (lam j)
  "Calculates the value of Tj given lamda and j"
  (/ (* (exp (- lam)) (expt lam j)) (fact j)))

(defun plot-t ()
  "Plots the T values for Evett94"
  (let* ((path (make-pathname :name "t-values"))
	 (str (open path :direction :output :if-exists :supersede)))
    (format str "#~10A ~10A ~10A~%" "Lambda" "J" "TsubJ")
    (dotimes (lam 8)
      (dotimes (j 8)
	(format str "~10A ~10A ~10A ~%" lam j (calculate-t lam j))))
    (close str)))

  
(defun plot-naive-evett94 ()
  "Plots the naive Evett94 algorithm using different lambas."
  (let* ((path (make-pathname :name "evett-values"))
	 (str (open path :direction :output :if-exists :supersede)))
    (format str "#~10A ~10A ~10A ~%" "LAMBDA" "N" "RESULT")
    (dotimes (lam 7)
      (dotimes (n 5)
	(format str "~10A ~10A ~10A ~%" lam n (naive-evett94 :lam lam :n n))))
    (close str)))

(defun naive-evett94 
    (&optional &key
     (lam 4)
     (n 4)
     (p-values #(.25 .22 .14 .14 .069 .064 .033 .033 .022 .02 .013))
     (s-values #(.728 .166 .065 .018 .018 .006))
     (f-value .03)
     )
  "Performs the naive algorithm for determining the LR using
data supplied by the LHS survey."
  (+ (/ (* (elt p-values 0) (calculate-t lam 1)) (* (elt p-values 1) (elt s-values n) f-value)) (calculate-t lam 0)))

(defun fact (f)
  (if (< f 2) 1 (* f (fact (- f 1)))))

(defun evett94
    (
     &optional
     &key
     (z 1)
     (n 1)
     (p-values #(.41 .28 .13 .1 .05 .02 .01 .005))
     (s-values #(.70 .18 .08 .025 .015))
     (t-values #(.02 .075 .145 .195 .195 .155 .105 
.06 .03 .01 .005))
     (f-value .03)
     )
  "Performs the algorithm for determining the LR using
data supplied by the LHS given a factor of z, and a
number of fragments on clothing n."
  (/ (+ (* (elt t-values 0) (elt p-values 1) (elt s-values n) 
f-value) (* (elt t-values n) (elt p-values 0))) (* (elt p-values 1) (elt 
s-values n) f-value))
  )
\end{verbatim}
\clearpage
\section{Modifying the Data}
Modifying the number of fragments found on a suspect and the lambda factor 
greatly alters the likelihood ratio.

\begin{verbatim}
#LAMBDA     N          Likelihood Ratio     
1          0          19.509127  
1          1          84.31264   
1          2          214.74986  
1          3          774.52496  
1          4          774.52496  
2          0          14.2186775 
2          1          61.898434  
2          2          157.86877  
2          3          569.7283   
2          4          569.7283   
3          0          7.8212447  
3          1          34.131844  
3          2          87.09012   
3          3          314.36206  
3          4          314.36206  
4          0          3.830262   
4          1          16.73577   
4          2          42.712116  
4          3          154.19037  
4          4          154.19037  
5          0          1.7596587  
5          1          7.6942463  
5          2          19.639452  
5          3          70.90264   
5          4          70.90264   
6          0          0.7763151  
6          1          3.3961709  
6          2          8.669446   
6          3          31.299858  
6          4          31.299858  
\end{verbatim}
\clearpage

Similarly, modifying the lambda modifies the T values.
\begin{verbatim}
Lambda	   J		  Probability
0          0          1.0        
0          1          0.0        
0          2          0.0        
0          3          0.0        
0          4          0.0        
0          5          0.0        
0          6          0.0        
0          7          0.0        
1          0          0.36787945 
1          1          0.36787945 
1          2          0.18393973 
1          3          0.06131324 
1          4          0.01532831 
1          5          0.003065662 
1          6          5.109437e-4 
1          7          7.2991956e-5 
2          0          0.13533528 
2          1          0.27067056 
2          2          0.27067056 
2          3          0.18044704 
2          4          0.09022352 
2          5          0.03608941 
2          6          0.012029802 
2          7          0.0034370865 
3          0          0.049787067 
3          1          0.1493612  
3          2          0.2240418  
3          3          0.2240418  
3          4          0.16803135 
3          5          0.100818805 
3          6          0.050409406 
3          7          0.021604031 
4          0          0.01831564 
4          1          0.07326256 
4          2          0.14652511 
4          3          0.19536681 
4          4          0.19536681 
4          5          0.15629345 
4          6          0.10419564 
4          7          0.059540365 
5          0          0.006737947 
5          1          0.033689734 
5          2          0.084224336 
5          3          0.1403739  
5          4          0.17546737 
5          5          0.17546736 
5          6          0.1462228  
5          7          0.10444486 
6          0          0.0024787523 
6          1          0.014872514 
6          2          0.04461754 
6          3          0.08923509 
6          4          0.13385262 
6          5          0.16062315 
6          6          0.16062315 
6          7          0.13767698 
7          0          9.11882e-4 
7          1          0.0063831736 
7          2          0.022341108 
7          3          0.052129254 
7          4          0.09122619 
7          5          0.12771668 
7          6          0.14900279 
7          7          0.14900278
\end{verbatim}

\section{Graphical Models}
\includegraphics[width=6.5in]{filename.png}
\clearpage
\includegraphics[width=6.5in]{filename2.png}
\clearpage
\end{document}
