-Needs an awesome title-

Matthew Bartenschlag, Pavan Reddy, Peter Santiago October 1, 2008

Contents

1 List of Symbols 3

2 Overview 4

2.1 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.1 Lawson . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Seheult . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 1980 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Grove . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 Evett . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 1990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5.1 Evett . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.6 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.6.1 Evett . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.7 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.7.1 Walsh . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.8 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.8.1 Koons . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.9 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.9.1 Koons . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Grove 9

4 Evett 15 4.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.3 Mathematical Formulae . . . . . . . . . . . . . . . . . . . . . 16 4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.4.1 Consequences of Manipulating Inputs λ and n . . . . . 17 4.5 Source Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1 List of Symbols

The following symbols are referenced repeatedly throughout the following papers:

• μ is the known mean of a normal distribution

• σ is the known standard deviation of a normal distribution

• Θ_{n} is the refractive index of a sample n.

• τ is the standard deviation of a normal distribution centered at μ about which Θ, a refractive index, is distributed

• λ_{k} is \frac{σ}{τ}, a contributing factor in Seheult’s equation

• P_{n} is the probability of finding n groups of glass on the surface of a person’s clothes

• 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

• S_{n} is the probability that a group of glass fragments on a person’s clothing consists of n fragments

• f is the probability that a group of fragments on person’s clothing would match the control sample

• λ is the expected number of glass fragments remaining at a time, t

2 Overview

This document traces through the developments of glass research according to the timeline illustrated in Figure 1.

Figure 1: A brief timeline showing major papers written by important contributors in research.

2.1 1973

2.1.1 Lawson

Numerical Correlation and Evaluation in the Comparison of Evidentiary Materials

Lawson mentions the use of emission spectroscopy as an identifying characteristic of glass samples. He makes use of ten automobile head lamp glass fragments as the basis of his theory.

Lawson developed a technique for ”Sorting the results or getting the smallest numerical index so that the closest match of the parameters involved can be easily evaluated” and ”establish[ing] a threshold or level of reproducibility of the particular technique and material involved.”

This paper appeared to be promising, but the decision procedures created in this document were not further considered.

2.2 1978

2.2.1 Seheult

On a Problem in Forensic Science

This paper builds upon Lindley and Evett’s previous papers for using a Bayes approach in determining the likelihood that two glass samples are from the same source. In particular, Seheult defends Lindley’s algorithm using a Neyman-Pearson test.

This paper is not exceptionally valuable to our research because no new decision procedures are provided.

2.3 1980

2.3.1 Grove

Interpretation of Forensic Evidence Using a Likelihood Ratio Waiting for Pavan’s paper.

2.4 1984

2.4.1 Evett

A Quantitative Theory for Interpreting Transfer Evidence in Criminal Cases

A general approach, based on a Bayesian inference, is presented for providing quantitative interpretation of scientific forensic evidence in cases where material is transferred during the commission of crime. An expression is derived which has a potentially wide applicability in forensic casework and the principles of its use are illustrated by examples.

Evett’s paper is outside the scope of our project, but the use of likelihood ratios continues to be a recurring theme in Evett’s work.

2.5 1990

2.5.1 Evett

The Interpretation of Glass Evidence. A Practical Approach

Evett’s 1990 paper attempts to make use of a Bayesian approach even though there are still many considerations are crude.

Evett’s 1990 paper is used as the basis for his 1994 paper. There are graphs that appear in both of these papers, but they are discussed in more detail in Matt Bartenschlag’s research summary of the paper.

2.6 1994

2.6.1 Evett

Further Observations on Glass Evidence Interpretation

Evett uses this paper as a continuation of his 1990 research paper, A Quantitative Theory for Interpreting Transfer Evidence in Criminal Cases. The majority of this paper is focused on previous problems, but he introduces a new factor which strengthens or weakens the conclusions from his previous equations.

This research paper, along with Evett’s previous paper, was analyzed in depth in A Concise Summary of Further Observations on Glass Evidence Interpretation. Significant insight was not gained from this research; it simply confirms that a suspect is more likely to be guilty if a significantly large number of matching glass fragments were recovered from his or her clothing.

2.7 1994

2.7.1 Walsh

A Practical example of the interpretation of glass evidence

The equation presented in this article 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.

2.8 1999

2.8.1 Koons

The Forensic Significance of Glass Composition and Refractive Index Measurements

Koons examines qualities of glass including RI (at D line) and chemical composition to determine how accurately glass samples can be distinguished.

By comparing all of the above attributes, it is extraordinarily unlikely to find two unrelated pieces of glass that match between 10^{−5} and 10^{−13} (this would be the denominator in the likelihood ratio.

2.9 2002

2.9.1 Koons

Interpretation of Glass Composition Measurements: The Effects of Match Criteria on Discrimination Capability

3 Grove

In this article,Grove adopted a model used by Lindley and Seheult and proposed a nonBayesian solution based on likelihood ratio.According to Grove:

The measurements are normally distributed with known standard deviation σ about a mean θ. A measurement x, with normal error having known standard deviation σ, is made on the glass of refractive index Θ_{1} 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 θ_{2} and the same standard deviation as x. The refractive indices θ are assumed to be normally distributed with known mean μ and known standard deviation τ. If I is the event that the two pieces of glass come from the same source(θ_{1} = θ_{2}) and Ī the contrary event, Lindley suggests that the odds on identity should be mutliplied by the factor

\frac{p(x, y|I)}{p(x, y|Ī)}  (1)

Grove followed non Bayesian approach, so in discussing the respective roles of θ_{1} and θ_{2} he did not attribute any probability distribution to an unknown parameter without special justification.So when considering (θ_{1} != θ_{2}),Ī can be interpreted as saying that the fragments are present by chance entailing a random choice of value for θ_{2}.The simplified likelihood ratio obtained from the Grove’s derivation is:

\frac{τ}{σ} · e^{\{\frac{−(X−Y )^{2}}{4σ^{2}} + \frac{(Y −μ^{2})}{2τ^{2}}\}}  (2)

The Figure2 demonstrates the implementation ot the grove equation for likelihood calculation in lisp.The function takes the refractive indices of the samples from two places and compares them only when their standard deviation is same.Here in this fuction the mu and tau are the given parameters.The Figure3 demonstrates algorithhm to comapre two set of fragments here the likelihood ration is specified as a limit for the no of fragments to be considered as the proof in the investigation

(defun  variance (lst)
              (let ((mean1 (mean  lst))
                    (n (length  lst)))
  (/ (reduce #'+  (map ' list #'( lambda (x) (square (- mean1 x)))  lst))
                 n)))
defmacro  square (x)
   �(*  ,x ,x))
(defun  mean (lst)
  (/ (reduce #'+  lst) (length  lst)))
;funtion  to  calculate  groveidentity(when G,T,A,S are  considered)
(defun  groveidentity(pa pt x y sd mu tau)
                   (let*((l1 (- (/ 1 pa) 1))
                                (l2 (grovelr x y sd mu tau))
                                     (l3 (* l1 l2))
                                          (factor  (+ 1 (* pt
(- l3  1)))))
                                           factor))
;function  to  calculate  grove  LR
(defun  grovelr(x y sd mu tau)
            (let* ((u (/ tau sd))
           (v (/ (* (- x y)(- x y) )(* 4 sd sd) ))
(w (/ (* (- y mu)(- y mu) )(* 2 tau  tau)))
             (lr (* u (+ (exp v) (exp w)))))
                                    lr))
(defun  grove(&optional (x-values  #(1.518255  1.518199  1.518273  1.518292))
                            (y-values  #(1.518273   1.518347
1.518119   1.518255))
                            (mu  1.5182599)
                            (tau  0.00367)
                            (pa  0.01)
                            (pb  0.9))
        (let((X (mean x-values))
        (Y (mean y-values))
        (sd (sqrt (variance x-values))))
       (format t "grove  lr is ~A    " (grovelr X Y sd mu tau))
      (format t "grove  identity  is ~A" (groveidentity  pa pb X Y sd mu tau))))

                Figure 2: CODE: groveidentities

;aglorithm  written  by  peter  for  comparing  the  two  data  arrays
(defun  applygrovelr (datarray  filename)
(let ((lm (make-array (list (array-dimension  datarray  0)
(array-dimension  datarray  0))
:initial-element  nil))
(tempval  0)
(totalpairs  0)
(totalmatches  0))
(with-open-file (out  filename
:direction :output
:if-exists :supersede)
(dotimes (i (array-dimension  lm 0))
;(format t "Now on: ~a~%" i)
(dotimes (j (array-dimension  lm 0))
(if (equalp (aref lm i j) nil)
(cond  ((= (aref  datarray i 2) (aref  datarray j 2))
(handler-case
(setf  tempval
(grovelr1(aref  datarray i 1)
  (aref  datarray j 1)
  (* (aref  datarray i 2)
      (expt 10  -5))))
(floating-point-overflow
(setf  tempval ' inf)))
(setf (aref lm i j) tempval)
(setf (aref lm j i) tempval)
(if (not (= i j))
(setf  totalpairs  (+  totalpairs  1)))
(cond ((and (or (> tempval  500)
(equalp  tempval ' inf))
(not (= i j)))
(setf  totalmatches  (+  totalmatches  1))
(with-standard-io-syntax
(format  out
"~a   ~a    ~a    ~a   ~a
~a~%"
(aref  datarray i 2)
(aref  datarray i 0)
(aref  datarray j 0)
(aref  datarray i 1)
(aref  datarray j 1)
 tempval)))

))))))

' done)))

Figure 3: CODE: grovecomparision

10

We are of course only concerned with the evidence about I and Ī so far as it has the bearing on the guilt or innocence of the suspect. Grove also considered the Event of Guilty factor Ḡ in the calculation of likelihood ratio.Therefore the LR now becomes

p(X, Y |G)/p(X, Y |Ḡ)  (3)

Here in the expansion event T̄, that fragments were transferred from the broken window to the suspect and persisted until discovery and event Ā,that the suspect came into contact with glass from other source.Here p(A/G)=p(A/Ḡ)=Pa and p(T/G)= Pt.The resulting expression is

\frac{P (X, Y, S|G)}{P (X, Y, S|Ḡ)} = 1 + P t\{(\frac{1}{P a} − 1)\frac{p(X, Y |I)}{p(X, Y |Ī)} − 1\} (4)

The Figure 2 demonstrates the calculation of grove LR when the event Ḡ is considered.Here in the equation Pa and Pt values are fixed an passed to the function directly.

Figure 4 shows the results of running Grove on crime scene data.Note that the graph is concentrated only at the higer ends because the likelihood ration criteria is high resulting in the matching of fragments which have closer refractive indices.Figure 5 shows runnig Grove on survey data.

Figure 4: crimescene Data 12 Figure 5: Survey Data

13

4 Evett

4.1 Summary

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.

4.2 Introduction

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 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 probabilites that are of great interest in calculating Evett’s proposed likelihood ratio. These probabilites are S, the probability of finding N glass fragments per group, and P, the probability of finding M groups on an individual.

4.3 Mathematical Formulae

Let these symbols be defined for the following equations:

• P_{n} is the probability of finding n groups of glass on the surface of a person’s clothes

• 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

• S_{n} is the probability that a group of glass fragments on a person’s clothing consists of n fragments

• f is the probability that a group of fragments on person’s clothing would match the control sample

• λ is the expected number of glass fragments remaining at a time, t Evett utilizes the following equations to determine the likelihood ratio.

LR = \frac{P_{0}T_{n}}{P_{1}S_{n}f} + T_{0}  (5)
T_{n} = \frac{e^{−λ}λ^{n}}{n!}  (6)

4.4 Analysis

From the previous equations, there are two decidingly influencial factors which may alter the resulting likelihood ratios. The first component, T_{n} is dependent upon the factor λ. Evett uses a seemingly arbitrary λ for illustrative purposes, so it is necessary to investigate how choosing a different λ modifies the resulting likelihood ratio. Another parameter which greatly affects the likelihood ratio is n, the number of fragments recovered on a suspect. The following subsection illustrates the consequences of modifying n and λ.

4.4.1 Consequences of Manipulating Inputs λ and n

Evett claims in his paper that an expert said that an expected value of 4 glass fragments would be found on a suspect’s clothes if they were at the crime scene when the glass was broken at a given time. This value is used as λ in a Poisson distribution to determine T_{n}, so it is reasonable to assume that λ could be a number near 4. Altering λ yields a plane without peaks or valleys (the only exception being when λ = 0 and n = 0). The plot of n and λ is given in Figure 6.

Figure 6: Plot showing different values of T_{n} based off of modifications to λ, n

Because T_{n} appears in Evett’s formula, λ and n and plotted against the resulting likelihood ratio in Figure 7. This figure shows very interesting results when λ = 2. When λ = 2 and n is 4, the likelihood ratio reaches a maximum; intuitively, this makes sense. If only 2 fragments were expected to be found on a suspect, but 4 fragments were recovered, there’s a much greater likelihood that the suspect was present when the glass was shattered.

Figure 7: Plot showing the likelihood of guilt to innocence with inputs to λ, n

4.5 Source Code

Figure 8 shows a lisp implementation of Evett’s algorithm. The figure also shows the procedures used to calculate the source data for Figures 7 and 6.

;;;;  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" "N" "TsubN")
  (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)))))

Figure 8: CODE: evett94.lisp