The Evaluation of Fibre Transfer Evidence in Forensic Science: A Case Study in Statistical Modelling Author(s): J. C. Wakefield, A. M. Skene, A. F. M. Smith and I. W. Evett Source: Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 40, No. 3 (1991), pp. 461-476 Published by: Wiley for the Royal Statistical Society Stable URL: http://www.jstor.org/stable/2347526 . Accessed: 20/05/2013 12:30

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AppL Statist. (1991) 40, No. 3, pp. 461-476

The Evaluation of FibreTransfer Evidence in ForensicScience: a Case Studyin StatisticalModelling

By J. C. WAKEFIELD and A. M. SKENEt, UK of Nottingham, University A. F. M. SMITH ImperialCollege of Science, Technologyand Medicine, London, UK and 1. W. EVETT Home OfficeForensic Science Service CentralResearch and SupportEstablishment, Aldermaston, UK [Received February1990. Revised November 19901 SUMMARY Frequently, when a crimeis committed, fibres are left at the scene. This paperexamines the modelling aspects of evaluatingthe evidentialcontentof such fibresby using a Bayesian approach. Inferences are made via the likelihood bivariatecolour measurements.Modellingthe distriburatio,derivedfrom a particular tionof colourwithin garmentis discussed indetail. Inaddition,a largedatabase allows an distribution to be incorporated, kerneldensityestimation.Data fromactual empiricalprior utilizing casework are analysed. Keywords: Bayesian inference;Colour measurements; Forensicscience; Kerneldensity estimation; Likelihoodratio; Modellingwithin-garment variability

1. Introduction The potentialfor the use of Bayesian statistical analysisin generalproblemsof the evidentialcontentof trace materialsin criminalcases has been quantifying establishedby Lindley(1977), Evett (1983, 1987) and Evett et al. (1987), the last withthecase of fibre transfer. dealingspecifically A crimehas been committed Considerthe following. during whichseveralfibres have been left at the scene of the crime by the offender.A suspect has been and a garment apprehendedas a resultof a police investigation belongingto the suspect has been taken for examination.Afterthe examinationof the suspect's theavailable forensic evidence of bivariate consists colourmeasurements, garment, y = (Yi . ., yn), fromthe fibresleftat the scene of the crime, togetherwith a set x =

NG7 2RD, UK. Nottingham,

1.1. TheProblem

tAddress for correspondence: Departmentof Mathematics,Universityof Nottingham,UniversityPark,

0035-9254/91/40461 $2.00

? 1991 RoyalStatisticalSociety

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WAKEFIELD, SKENE, SMITH AND EVETT

sample of m fibres froma representative (xl, . . ., Xm)of bivariatemeasurements as a pairof fibre is expressed The colourof a single thesuspect'sgarment. takenfrom are co-ordinates. For detailsofhowthesemeasurements chromaticity complementary made, see Laing et al. (1986). againsteach are to be weighed hypotheses and exhaustive exclusive Two mutually other: fibres came from the suspect'sgarment; (a) C-the recovered came from fibres some othersource. (b) C-the recovered from theactualcolourmeasurements) (distinct IfIdenotes therestoftheinformation to thecrime, use of Bayes's theorem elementary has beenassembledin relation which on I, throughout establishes that,conditioning = p(xyIC,I) p(CII) p(Cx,y, I) p(x,yj C,I) p(CjI) odds on C = likelihoodratioforC x priorodds on C. posterior p(CIx,y,I) (1.1)

of thewayin whichthe on thelikelihoodratioas thesummary Focusingattention byI, we see overthatprovided x, y provideadditionalevidence colourmeasurements that p(x,yIC,I)

_

p(x,yIC,I)

p(xIC,I)p(yIC,I) p(x,ylC,I) of x. if we make the obvious assumptionthat C impliesthat y is independent of measurements import therefore givestheevidential Evaluationof thisexpression I). x, y (giventhe otherinformation 1.2. Database Information general in thelikelihood ratiorequires involved of thedistributions The modelling from (1987): those derived one case. To Evett et al. quote any specific beyond inputs to have the need for data collections background scientists long recognized 'Forensic in UK forensic of evidence'. In 1982, scientists science assist in the interpretation At each on fibres. a collaborative projectto amass information started laboratories fabric a small representative sampleof clothwas takenfromeveryfifth laboratory such butotherarticles forexamination. Manyof theseweregarments itemsubmitted was completed questionnaire and beddingwereincluded.A comprehensive as carpets College of Textiles,Galashiels, foreach sample,whichwas thensentto theScottish wereobtained.In thisway,bymid-1987, data including wherecolourmeasurements had been collectedon nearly8000 samples. Whether such a colour measurements of thegeneral sourcesis can everbe truly populationof fibre collectiQn representative here.Certainly, thedata collection however, a mootpointwhichwe do not consider thatis currently scientist. available to the forensic the best information represents to be madeof howrareor commona particular fibre Thesedata enablean assessment the evaluationof pO(y in C, For each (1.2). equation I) colour is and thisunderlies I fibresamplein the collectionfivepairs of chromaticity were co-ordinates synthetic fibres.For each naturalfibresample, 10 measureobtainedby usingfivedifferent weremade with10different made fibres. Colour measurements from garments ments

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EVALUATION OF FIBRE TRANSFER EVIDENCE

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of the data For a full description fromnatural fibresexhibitgreatervariability. see Laing et al. (1987). methodology, collection of of whichsixtypes makeup 98Wo fibre 13 different types, The database identifies typeis cotton,of whichthereare 2340 samples all samples.The mostcommonfibre are of which there fibre is polyester, thedatabase. The mostcommonsynthetic within 1512 samples. in detailthecase wherea singlefibre was recovered Evettet al. (1987) considered as thatequation (1.2) can be written fromthe sceneof thecrime.It is recognized p(x,yIC,I) p(yIC,x,I)p(xIC,I) p(yIC,x,I)

Statistical Analysis 1.3. Previous

p(X,yIC,I)

p(yIC,x,I)p(xIC,I)

P(yIC,I)

(1.3)

underC and C. It was assumed on I, x is equallylikely that,conditional by assuming of fibrecolour withina particulargarmentwas adequately that the distribution Thus the numerator of the likelihood by a bivariatenormaldistribution. described ratio(1.3) can be written

p(yIC,x,I) =p(yI,u,

dE, I) p(y, Ix, I) d,u

of a bivariate withmean i and normaldistribution where pO(y 'I i, E, I) is thedensity for theseparameters, density covariancematrixE, and p(it, Ix, I) is a posterior ofvaguepriorknowledge regarding representations distribution. Usingconventional is shownby Evettet al. (1987) to be a bivariate ,uand E, the formof thenumerator mathematical form. Studentt-type havingan explicit density, in theknowledge of thetypeof fibre For thedenominator, givenI (whichincludes thedensity thegeneral distribution describing p(y IC, I) is simply question),theform fibre thatwe wouldsee in whatever byI, populationis defined of colourco-ordinates estimator. kernel density whichEvettet al. evaluatedby usinga bivariate in thispaper results the froma collaborative projectbetween The workdescribed Home OfficeForensicScienceServiceCentralResearchand SupportEstablishment of Nottingham Statistics Group. and theUniversity and modelling numerical issuesassociatedwith theextension The projectaddressed fibres. The approach described by Evett of the above problemto severalrecovered are et al. (1987) is not tractableformorethan a singlefibre.Numericalprocedures formsare assumed and thisprovidedan opporwhatever mathematical necessary of the distributional assumptions to examinemore closelythe conventional tunity original work. In addition, in the contextof the possible modellingstrategies the requiredlikelihoodratio pose available, the numericaldemandsof calculating theBayesianinference framework. challenging problemswithin via a case study in an important paperis to exhibit, The mainpurposeofthepresent and novel modelling and newlyemerging applicationsfield,a class of interesting and some possible problems,involvingboth likelihood and prior specification, willbe on thestatistical aspectsof theproblem. solutions.Our emphasisthroughout to 'single-colour' attention and 'single-source' problems.It we restrict In particular,

assuming x = (xl, . . ., xm) to have been a sample from the same bivariate normal

andStructure Paper of this 1.4. Motivation

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WAKEFIELD, SKENE, SMITH AND EVETT

has to considermulticoloured scientist garments and that is clear thatthe forensic severalsources.The models fibres attached to anyscenemayhavecomefrom foreign to encompassthesesituations but no new statistical herecan be extended described of theHome OfficeForensicScienceService,there issuesarise. In thewidercontext the statisticalmethodologyinto intelligent are importantissues of integrating for laboratoryuse. We shall not discuss such problems knowledge-based systems here. thevariability coIn Section2, we deal withmodelling of observedchromaticity In Section3, we deal withmodelling from thesame garment. ordinates amongfibres of the parameters involvedin modelling withinthe prior(population)distribution garmentvariability.Computational problems are discussed in Section 4 and of the methodology and refinements are examinedin Section 5. Two extensions in Section6. illustrative examples,takenfromactual case work,are presented Variability 2. ModellingWithin-garment 2.1. GeneralApproach Fig. 1 displays the 'horseshoe'-shaped region within which complementary to lie. In thisfigure, N denotesthe co-ordinates (u, v) are constrained chromaticity neutral,or achromatic, point. The positionsof typicalhues, i.e. colours, are also hue and saturation but does not take indicated.In essence,thecolour space reflects Thus all whites, and blackmap to theneutral intoaccountlightness. point.If a greys line weredrawnfromthe neutralpointto theboundaryof the colour space, movetheboundary an increase in saturation ment (colours represents alongthelinetowards lie on the boundary)-the hue is constant.Most describedby a singlewavelength is closeto theneutral clothhas a pale yellowcolourwhich point.In all dyeing undyed in uptakeof thedye.Whena singledyeis used the is somevariability processesthere boththeundyed colourand thedyecolourwith saturation reflects colourofthefibres more Measurements rather thusappear close to a linejoining variability. exhibiting of theundyed pointto thedyecolour.Manycoloursare achievedvia thecombination In this case colour measurements for individual or more fibres are scattered two dyes. about a linejoiningthetwo predominant dyesis dyecolours.The need formultiple forcertain colours.For example, blues can be achieved also morecommonplace many witha singledyewhereasgreensoftenrequiremultiple dyes. of fibres froma particular A typicalscatter sample diagramforthe co-ordinates of manysuchscatterplots within thedatabase is also shownin Fig. 1. The inspection withthelong axis frequently orientation indicateda preferred towardsthe pointing with the above of the consistent discussion. The direct modelling yellow area, distribution is thuscomplicated whichvariesstochastically bivariate bya correlation foreach sampleof a particular fibre withlocation.As an initialsteptherefore, type, sample means were calculated and principal component axes were identified. valueson to theseaxes werethenstudied of standardized to try to identify Projections formsin the long and shortdirections.Fig. 2 shows the possible distributional standardized plots forwool fibres. This in turnleads to the following strategy forwithin-garment generalmodelling variability. (a) two orthogonal axes u' and v'. Identify

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EVALUATION OF FIBRE TRANSFER EVIDENCE

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0.8

0.6 -

0.4 -

0.2 N-

00 0.2 0.4 0.6 0.8

in colour space (approximatepositionsof the main Fig. 1. Scatterof fivebivariatemeasurements coloursare indicated:1, yellow;2, red; 3, blue; 4, green)

I-

illI

-2

-1

0 (a)

1

2

-2

0 (b)

2

Fig. 2.

in (a) thelong direction and (b) theshortdirection Standardizedvalues forwool fibres

theorientation of the u '-axis relative to theoriginalu-axis. (b) Identify locationsand spreads,thatthe distributions of (c) Assume, givenorientations, of thechromaticity on to theseaxes are statistically projections co-ordinates independent. thebivariate Giventhisstrategy, describing fibre colourwithin a garment is a density mathematical two for five-parameter familywith one parameterfor orientation, location and two for spread. The orientation is definedas the angle of the u '-axis relativeto the u-axis. Additionallythe spreads are constrainedso that the long of scatter is identified as havingthelargerspreadparameter. direction made in Evettet at. (1987) was thatthescatter was wellmodelled The assumption theprojections However,considering separately by a bivariatenormaldistribution. axes as inFig. 2, thenormalhypothesis on to thelongand short is clearly rejected bya fornearly all fibre conventional Thisfinding prompts a searchforother X2-test, types. distributional fitto the observedunivariate shapes providinga more satisfactory distributions.

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WAKEFIELD, SKENE, SMITH AND EVETT

2.2. ExponentialPower Family The so-called exponential power family is a general univariate family of by withlocation r, spreada and shape parameter3. It is defined distributions I p(zIor,u,3ocuoexp( exp(- - 2Z z-r a = P(zlT a|

2/(1+3))

to the normal alternatives and leptokurtic This familyincludes both platykurtic Box and Tiao for (1973). example, See, distribution. was made to findvalues of a an attempt fibre typeand direction For a particular to thosein Fig. 2. It was not via similar obtained the plots mimicked profiles which 3 value of which adequatelymodelledthe a to identify particular however, possible, in thelong the observed distribution For direction. example, in either behaviour tail tailed. and light is heavyshouldered direction 2.3. 'Roof' Distribution via A second approach considered was to describe the joint distributions which were simple combinationsof rectangularand conditional distributions See Fig. 3. It was found that a good fitto the data was distributions. triangular to wereconsidered of fibre measurements thata smallproportion obtained,provided in a skirt aroundthebase of theroof. be lying In addition to the usual five parameters-(It1,112), mean locations; (a,, a2), constantswere required to standard deviation spreads; X, orientation-further of theroof,e.g. thedistancein thelong of thedifferent thelengths portions describe theroofridgebeginsto slope downand from themean,to thepointwhere direction, are easilyestimated fora givenfibre type constants These extra oftheskirt. theextent in each of thelong and shortdirections, measurements standardized by considering forthetotalprobability to equal 1. Although withtheconstraints necessary together fits to muchof thefibre data, itproved,subsethismodelprovided verysatisfactory

(a)

(b)

tX '-~~~~~~~~~~~(b)

(c)

(a)

(c)

threesectionsand a plan view Fig. 3. Roof distribution:

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EVALUATION OF FIBRE TRANSFER EVIDENCE

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alternative and so yetanother to operatewith numerically to be very difficult quently, was sought. 2.4. Beta Distribution which wereoutlying discarded fibres procedure thatthemeasurement It was known of a particular themaincluster thiswasthought sample.Initially whencomparedwith to have little influence on tail shape but subsequentlythis fact motivated Bivariate densities defined of of finite byproducts consideration rangedistributions. scaled beta distributions providedan adequate fitto thedata and proved symmetric moretractable computationally. considerably on (i - at, beta distribution A univariaterandom variable Z has a symmetric of W = (z - A)/2at + 2 is givenby it+ at) ifthedensity

p(w)

o

Wa-l(1-W)a-l

0<

W <

1.

forthetwoorthogonal foreach fibre type. directions Thus a and t mustbe estimated with and a minimum a searchprocedure chi-squared criterion, Thiswas accomplished studiessuggested thatprecisevalues werenot critical. but subsequentsensitivity 2.5. Comment Althoughthe large amount of data available makes it possible to distinguish modelsforthewithin-garment alternative by usinggoodness-ofvariability between thattheoverallobjectiveis thecomputation itmustbe remembered of the fit criteria, of thisratioto modelling which likelihood ratio(1.2). It is thesensitivity assumptions and thustheeventual choiceof modelis mediated concern bysensitivity is of primary

issues.

3.

ModellingthePriorDistribution

variation involvingfive parameters,the Given a model for within-garment of a priordistribution evaluationof the likelihoodratio requiresthe specification it is size of the database Given the technically possibleto obtain al, a2, 4). 2, P(pA, fromeach of the samplesin the database and thento of the parameters estimates kernel estimate. is computaa five-dimensional construct However,suchan estimate in Section4 to be described and thenumerical intensive techniques integration tionally of such evaluations. requirea largenumber Anotherapproach would be to evaluatethe priorin advance at a regulargridof thefive-dimensional However,the space and thento use interpolation. pointswithin to characterize sucha surface is largeand five-dimensional of pointsrequired number is difficult. interpolation The discussionin Section 2.1 combinedwithan empiricalexaminationof the thatthepriorcould be wellapproximated database suggested by a form

(3.1) p(q I X), P(A,l 2, a1,a2, 0 = P(A,l 2) p(al I r) P(a21 r) the wherer is the radial distancefromthe neutralpointto (A,, 2) and X describes relative to theneutral point(Fig. 4), i.e. thespreadson the radialangleof thecluster given position, and depend only on principalcomponentaxes are independent, thedistancefromthe neutral point.

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468

v

WAKEFIELD, SKENE, SMITH AND EVETT

2

X~~~~x x

XX (1 .U2)

/ It

N

u

data cloud Fig. 4. r and X fora particular

of thelong axis is independent of saturation but dependson hue, The orientation For example, ofdyesinvolved. intheblueregion ofthecolour i.e. on thecombination themajority ofgarments to havea density the alignedtowards space,we wouldexpect undyedpoint whereaswithgreens,whichcan arise frompure greendyes or from combinationsof blues and yellows,we would expectthe prior for 0 to be more of possibilities. thismixture dispersereflecting Notingthat

p (ori Ir) = p (ai. r)/p(r),i

and

P(o IX) = P(og X)/P(X)g

= 19

29

can be evaluated with four bivariatedensities, we see that a prior distribution and r), r) X), each of whichcan be constructed p(a2, p(0, by using P(19 A2). P(arl, methods. kernel timethata Ratherthanhavingto evaluatethefourtwo-dimensional kernels every was required,it was decided to set up four reference tables in prior probability the advance. If the database wereto be updatedit would thenbe simpleto recreate requiredgridsof points. When the priorfora particular pointwas required,cubic to the desiredpoint. The reference tables were splines were used to interpolate achieved as follows: for each of the seven quantities of interest(five model theradial distanceand theradial angle) sampleestimates wereobtained parameters, flbre foreach samplein thedatabase of a particular type.Fromthese,kernel density foreach of thebivariate densities wereevaluatedat each pointon a 20 x 20 estimates and r, thekerneldensities werecreatedin terms of log ai and log r. The grid.For (ri to obtain the requiredformin termsof the original densitieswere then inverted so thatdensity This operationwas undertaken estimates close to 0 were parameters. see Silverman notunderestimated; were (I1986).For theangles0 and X. theestimates to the kernel 'wrapped around' so that, for example,X close to 360? contributed foranglesclose to 0? as wellas to 360?. estimate Bivariate normal kernelsused afterinitial experimentation indicatedthat the

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EVALUATION OF FIBRE TRANSFER EVIDENCE

Y2 r

469

..40. 0

0.4]

3

0.

3 0.3

0 LI20.

81

.GP~ ~~~~~~~~~02

1

0.2

0.3

0.4

0.5

0.0 0.00

T 0.02

0.04

(a)

r

(b)

0.2-

-o005ooo-0 0.3

0.605 0.010

200

0.1

100

o

-50 0

5

5

0.0 0.000

0-

(c)

(d)

forwool fibres estimates for(a)p (Al1, (b) p (a,, r), (c) p(a2, r) and (d) p(4, Fig. 5. Kerneldensity ,tt2), 0): 1, yellow;2, red; 3, blue; 4, green

to the choice of kernelshape. Initially, window were insensitive estimates density to n - "I byusingguidelines suggested by Silverman widths werechosenproportional where theamountof data was largethisresulted, for types (1986). However,forfibre the finaldensity estimate. in a veryspikydensity Consequently P(I.', /.2) especially, estimatesused slightlylarger window widths giving a smoothnesswhich was withthe considerable to us by the Home communicated consistent priorknowledge Contourplotsofthefourbivariate estimates forwool colourchemists. Office density are at iWo, are shownin Fig. 5. Displayedcontours 5%, 10%, 25%o,500o, 75/o and Fig. 5(a) showsthata largeamountof thedata liesin theblue and theredregions. and is similar forotherfibre types. plotis typical shape of thecontour The triangular modesrepresents modeoccurs popularcolours.The largecentral Each ofthemultiple thatthestandarddeviations increase at theneutral point.Figs 5(b) and 5(c) confirm show morevariability as we move away fromthe neutral withradius-the clusters thisand displaysthe priorforp(G, Ir) fortwo particular point. Fig. 6(a) confirms

95%/o of the mode.

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470 p(ral

=

WAKEFIELD, SKENE, SMITH AND EVETT .02)

(a,rI r.10)

0.00 Fig. 6.

O.01 (a)

0.02

0.03

1T

-50

0 (b)

50

0

distributionp (a) p (u,Ir) and (b) marginal (0)

nowtoFig. 5(d),themodeoccurring andonelarge. Referring ofr,onesmall choices towards alltend topoint which X= 3000 corresponds totheyellows, atapproximately ofthecolour to corresponds space,thisorientation theneutral region point.In this the blues, which again are prek= 500. The other mode at X= 1300 represents sides ofthe the andyellows neutral blues lieonopposite they point andso even though at X= 2400, havea flatter reflectof k.Greens, distribution, values lying havesimilar of two different often ariseas mixtures colours;see the ingthefactthatgreens tended to point towards theneutral we in Section 2.1. If thegreens point discussion distribution themarginal of bluesand yellows. Fig. 6(b) shows combinations p(O). to the was obtainedfrom approximations p(k, X) via discrete This distribution 500 reflects Themodeat approximately theprevious discussion of integrals. required andthegreens. theyellows theblues, 4. Implementation

4.1. Evaluation of LikelihoodRatio would expect the mode of p(k IX= 2400) to lie at k= - 600. The mode of this liesat X = 500 indicating are often thatthegreens distribution conditional actually towardsthe neutral between- 900 and 900 point. kis defined oriented dominantly

in equation thejointdistribution of a setofbivariate Each term (1.2) represents be from a single source. Reference to C and C can therefore colourmeasurements measurements, a collection and,ifz denotes dropped (zl,..*, Zk) ofk bivariate p(zI)=

k _

p(ziI) p(O I) dO,

(4.1)

discussed in Section 2 andassumed to be with byoneoftheforms p (ziI0) modelled of the form in 3. and a discussed Section of prior p (O II) specifying I, independent stilldepends on I, theinformation additional to theactual Notethattheintegral fibre type. colour measurements, including

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EVALUATION OF FIBRE TRANSFER EVIDENCE

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4.2. Computation

the For anychoiceofp (zi IO) and p (O I), theevaluationof equation(4.1) requires numerical integral. evaluationof a five-dimensional severalnovel et al. (1985, 1987)describe Naylorand Smith(1982, 1983)and Smith Among of Bayesian methods. for the implementation strategies integration numerical very we outline which methods Gauss-Hermite Cartesian product theseare adaptive here. briefly if that,in one dimension, noting byfirst is motivated strategy The Gauss-Hermite the then x normal density a by polynomial be well approximated can an integrand an rule(withan exactanswerfrom willbe wellestimated bya Gauss-Hermite integral is actuallyof polynomial x normal form,withthe n-pointrule if the integrand polynomialcomponentof degreeup to 2n - 1). We thennote that,possiblyafter many of the marginalposteriordensitiesarisingin suitable reparameterization, can plausiblybe regardedas of polynomial x normalform.A Bayesianinference however,is likelyto exhibitseveral joint posteriordensityfor many parameters, whichwould make the applicationof Gauss-Hermiteproductrules dependencies, rulein each of theGauss-Hermite inefficient. Moreover,theimplementation highly for that and scale of location a the direction specification requires parameter values fortheseare not known. direction, and correct the gridat which a product forfinding iterative strategy thefollowing This suggests providethe basis for the accurate evaluationsof the likelihood x priorfunction of thejoint and henceany aspectsof interest constant evaluationof thenormalizing density. posterior Reparameterize individual parameters so that the resulting working by a polynomialx normalform.Often can be well represented parameters to take anyvalue on willallow theworking parameters thisparameterization thereal line. (b) Using initialestimatesof the joint posteriormean vectorand covariance scaled, transform further to a centred, parameters, matrixforthe working set of parameters. moreorthogonal forthese orthogonal (c) Using the derivedinitiallocation and scale estimates of interest of functions Cartesianproductintegration perform parameters, dimensioned grids. usingsuitably untilstable themeanand covarianceestimates, successively updating (d) Iterate, dimension. and between gridsof specified results are obtainedboth within (a) here, adopted fortheproblemdescribed The parameterization

V1 = 1si 2 = A2,

13

=

log

/

U2

U2 /

2

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472

WAKEFIELD, SKENE, SMITH AND EVETT

However,further powerand roofdensities. provedadquate fornormal,exponential byzl, . . ., Zk whenthebivariate intherangeofIt, A2,i and u2areimplied constraints is considered.This pointis discussedin more detailin Wakefield beta distribution The mean parameters pointis worthmentioning. et al. (1989). A further i, and A2 the horseshoe-shaped regionof Fig. 1 and so could be constrained mustlie within In practice,however,theyneverfall close to the boundary.Fig. 5(a) accordingly. butagain,inpractice, constrained bearsthisout. The variancesU2 and q2 are similarly thisis not a problem.

Studies 4.3. Sensitivity

of combinaElementsof the likelihoodratio (1.2) werecalculatedwitha variety roofand model(normal,double exponential, variability tionsof thewithin-garment in our uncertainty the choice of window (reflecting beta) and priorspecifications of the sensitivity are not givenhere. The resultsshowed, width).Detailed findings the values of individual wereclearly influenced that components although however, theresulting likelihood ratioswerewithin of a factor bythechoiceofmodeland prior, 2 in mostcases. feasible,the computational are computationally Althoughall such combinations with being bivariate normality costis influenced bythechoiceofmodelto someextent theuse oftheiterative Gauss-Hermite strategy. Furthermore, themostefficient given a factor of 10are currently to viewedas sufficient likelihood ratiosaccurateto within content.Thus themodelling and sensitivity exerciseshowsthat evidential represent The as thedefaultmodelis adequate at present. theadoptionof bivariate normality Impleexampleswhichfollowin Section6 havebeenanalysedunderthisassumption. is possibleon thecurrent rangeof personal of theintegration procedures mentation computer systemsalthough more powerful single-userworkstationsare more appropriate. Furtherdevelopmentsin computertechnologywill minimizethe overa morerealistic form. forchoosingbivariate distributional normality argument choice. are our preferred Bivariatebeta distributions and Refinements 5. Extensions eventI appearingin We now considerin a littlemore detail the conditioning all additional information availableaboutthefibres apart equation(1. 1). Irepresents co-ordinate measurements. theactual complementary from chromaticity of a typicaleventI, considerthe situation whereforeign As a simpleillustration of thefibres revealsthat fibres have been foundon a victim.An initialexamination they type(e.g. wool or nylon), (a) are of a particular such as delustrant etc. and (b) mayhave characteristics can be colour(e.g. redorblue) and possibly (c) to thenakedeyeare ofa particular as lightor dark etc. described fibres containing Then I is the eventthata suspecthas been foundwitha garment whichmatchthisdescription. of thecolourof bothrecovered fibres is then and control The precisemeasurement undertaken and y and x are obtained.

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EVALUATION OF FIBRE TRANSFER EVIDENCE

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oddsonthe totheposterior contribution (1.1),theoverall equation As weseefrom of the garment is a combination thesuspect's comefrom fibres having recovered onI. just conditioned (1.2),basedony andx andI, andan oddsratio ratio likelihood on the depends term, p(CI I)/p(CI I), clearly of thelatter Detailedquantification For ofa suspect anda garment. theidentification surrounding circumstances specific or was the becauseof thegarment, identified primarily was thesuspect example, We on other grounds? ofthe suspect tothe identification found subsequently garment the the effect of concentrate on but instead shall not pursuesuch issueshere, in (a)-(c) onp (OII). information though intheory, (b) isstraightforward ofanIbased on (a) and/or account Taking inpractice. ofthe Suchinformation thesubset simply identifies someissues itraises of 3 which the kernel Section is to be applied. on methodology database informano additional is usually there suchas woolandcotton, fibres Fornatural good to obtain and thus the database is sufficient available present tionoftype (b) of rare fibre or where a combination For estimates. situations types density kernel leadsto a cross-sectional and delustrant shapeand radiusof fibres type, synthetic is notyet toyield database precise thepresent sufficiently large initial match, precise ratiocan stillbe In suchcases thelikelihood of colourdistribution. descriptions obtained datapoints available. thenear uniform from thefew priors using calculated willthusbe smallbuttheinitial valueof thecolourmeasurements The evidential willusually havehigh evidential content. ofthekind Information matching process colour that an apparent match has beenmadeis not described by(c) andbythefact which In effect, it imposes a further to incorporate. conditioning, straightforward for theappropriate ofthedatabase thekernel, by(a) and(b), implied that part implies entire horseshoeofthe colour rather than over the a subset space, isdefined over only is precisely to quantify earlier described (seeFig. 1). Whatis difficult region shaped in (c) andthefact ofthecolourspaceis defined bythedescription which subregion has beenobtained. a prima that facievisualmatch istodefine theappropriate ofthecolour subset space tothis Oursolution problem Thesizeofthis circle location ofthe recovered fibres. centred atthe mean tobea circle science caseofficers. Theregion ofa match forensic does, after waschosen consulting butouranalyses tobereasonably onthemean location proved tosome depend extent, modified The implied form ofp(OII) is then to theexactvaluechosen. insensitive in Section3 by replacing estimate of the the kernel density from thatdiscussed circle use of a The to the circular one constrained by region. p(Al ', A2) component

has negligible impacton thevalues of thansome othershapedneighbourhood rather case the likelihoodratioobtainedand thisapproach has been used in the following studies. 6. Illustrative Analyses

I 6.1. Example Thevictim intoa garden andwasrobbed ofherhandbag. wasdragged Thevictim

fibre jumper. The assailantwas wearinga was wearinga greenwool and synthetic Threegreen werefoundon thesuspect's wool fibres checkedcottonshirt. grey-green werefoundon thevictim's cottonfibres jumper. 10 control shirt.Seven grey-green and thevictim's weretakenfromboththe suspect'sshirt jumper. fibres

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474

V

WAKEFIELD, SKENE, SMITH AND EVETT

V

0.32 -

x

0.335 0.3350.332** x

x

0.31-~~~~

0.31-

x

X

0.3-

~~~x

0.3250.32 -

x I 0.36 I 0.38 I 0.4 (a) l 0.42 0.44

u

0.315 I

x 0.315 l 0.32 (b) I 0.325

u

Fig. 7. Control (x) cotton grey-green

and recovered (X) fibre measurements for (a) the green wool and (b) the

thegreenwool. Fig. 7(a) givesa scatterplot measurements showing Considerfirst and controlfibres.The value of the likelihoodratiobased on fromboth recovered thesedata alone is 700. ratiois 30. The thevalue of thelikelihood cotton, Considering onlythegrey-green fibre are shownin Fig. 7(b). measurements controland recovered ratiofortwoon I, thelikelihood In mostcases, it can be arguedthat,conditional thelikelihood is the productof the separateratios.Overall,therefore, way transfer ratioin thiscase is 21000. 6.2. Example 2 theforensic was asked to scientist Followinga complexseriesof armedrobberies, ofgetaway linksbetween three and a number it cars. In particular, suspects investigate a knife to investigate thelinkbetween foundin thepossessionof one of was required cashbagswhich and several had beensplitopen. We shall thesuspects purplenylon-66 consider from thecash bags and the knife and (a) purplefibres foundinone ofthecarsand matched with a suitownedbyone (b) bluewool fibres of the suspects. 6.2.1. Purple cash bags had werecontacted and onlytwo batchesof thistypeof fibres The manufacturers werefoundon theknife and chromaticity were co-ordinates everbeenmade. 73 fibres all thebags werevisually a randomly selected15ofthese.Fibresfrom calculatedfrom butthey could be splitintotwosetsbyvisiblespectroscopy and thin indistinguishable 13 fibres matched the first batch and theremaining twofibres layerchromatography. for each of the matchedthe second batch. There were fivecontrolmeasurements of thecontroland recovered fibresforthe second batch is batches.The scatterplot there are onlytworecovered ratio thelikelihood fibres, shownin Fig. 8(a). Although moments of modelparameters indicated that,in thisexample,both is 217. Posterior

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EVALUATION OF FIBRE TRANSFER EVIDENCE

V V

475

0.41 -

x x 0.3750.376035 0.36 x x . X~~~~~~~ x X x

0.4 -

x

x

x

0.38 I 0.35 I 0.355 (a) I 0.36 0.365

u

0.355 I

0.34

I

0.345

I

l 0.35 (b)

I 0.355

I 0.36

u

for(a) thepurplecash bags and (b) theblue measurements Fig. 8. Control( x ) and recovered (-) fibre wool suit

and thedispersion theorientation oftherecovered and control data werevery similar. The highvalue is not a consequenceof therarity of thepurplecolour as theratiois on I. The factthata knife conditional was recovered with fibres thatmatched therare colourof thecash bags would normally increasethestrength of evidence.However, thisaspectof theevidencemustbe modelledviap (CI I)/p(Cl I). 6.2.2. Blue wool suit 15 blue wool fibreswerefoundand thesematchedfibres takenfroma blue suit In thiscase 20 control ownedbyone ofthesuspects. measurements weretakenand the subsequentmeasurements, along withthe recovered co-ordinates, are displayedin Fig. 8(b). In this case a likelihoodratio of 0.8 was obtained,the data being sufficiently in dispersionbetweenthe recoveredand control numerousto suggestdifferences samples. Acknowledgements herewas madepossiblebya Home Office The workreported grant supporting J.C. Wakefield. We also acknowledgeinvaluable assistance fromP. E. Cage, A. W. Hartshorneand D. K. Laing of the Home OfficeCentral Research and Support and R. Cook oftheMetropolitan Establishment Police ForensicScienceLaboratory. References

Box, G. E. P. and Tiao, G. C. (1973) Bayesian Inferencein Statistical Analysis,p. 157. New York: Addison-Wesley. Evett, I. W. (1984) A quantitative theoryfor interpreting transfer evidencein criminalcases. Appl. Statist.,33, 25-32. and forensicscience: problemsand perspectives. (1987) Bayesian inference Statistician, 36, 99-106. C. G. G. (1987) Evaluationofthelikelihood Evett,I. W., Cage, P. E. and Aitken, ratioforfibre transfer cases. Appl. Statist.,36, 174-180. evidencein criminal

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WAKEFIELD, SKENE, SMITH AND EVETT

A. W., Cook, R. and Robinson,G. (1987)A fibre Laing,D. K., Hartshorne, data collection forforensic and examination methods.J. Forens. Sci., 32, 364-369. scientists-collection Laing, D. K., Hartshorne, A. W. and Harwood, R. J. (1986) Colour measurements on singletextile fibres. Forens. Sci. Int., 30, 65-77. Lindley,D. V. (1977) A problemin forensic science.Biometrika, 64, 207-213. Naylor,J. C. and Smith,A. F. M. (1982) Applicationsof a methodforthe efficient computation of distributions. posterior Appl. Statist.,31, 214-225. modelin clinicalchemistry: (1983) A contamination an illustration of a methodfortheefficient of posterior distributions. computation Statistician, 32, 82-87. B. W. (1986) DensityEstimation and Data Analysis.London: Chapman and Silverman, for Statistics Hall. and Smith,A. F. M., Skene,A. M., Shaw, J. E. H. and Naylor,J. C. (1987) Progresswithnumerical graphicalmethodsforpracticalBayesianstatistics. Statistician, 36, 75-82. Smith, A. F. M., Skene, A. M., Shaw, J. E. H., Naylor, J. C. and Dransfield, M. (1985) The of theBayesianparadigm.CommunsStatist.Theory implementation Meth., 14, 1079-1102. A. F. M. and Evett,I. W. (1989)A case study in forensic J. C., Skene,A. M., Smith, science: Wakefield, transfer evidence.ResearchReport.Nottingham theevaluationof fibre Statistics Group,University of Nottingham.

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AppL Statist. (1991) 40, No. 3, pp. 461-476

The Evaluation of FibreTransfer Evidence in ForensicScience: a Case Studyin StatisticalModelling

By J. C. WAKEFIELD and A. M. SKENEt, UK of Nottingham, University A. F. M. SMITH ImperialCollege of Science, Technologyand Medicine, London, UK and 1. W. EVETT Home OfficeForensic Science Service CentralResearch and SupportEstablishment, Aldermaston, UK [Received February1990. Revised November 19901 SUMMARY Frequently, when a crimeis committed, fibres are left at the scene. This paperexamines the modelling aspects of evaluatingthe evidentialcontentof such fibresby using a Bayesian approach. Inferences are made via the likelihood bivariatecolour measurements.Modellingthe distriburatio,derivedfrom a particular tionof colourwithin garmentis discussed indetail. Inaddition,a largedatabase allows an distribution to be incorporated, kerneldensityestimation.Data fromactual empiricalprior utilizing casework are analysed. Keywords: Bayesian inference;Colour measurements; Forensicscience; Kerneldensity estimation; Likelihoodratio; Modellingwithin-garment variability

1. Introduction The potentialfor the use of Bayesian statistical analysisin generalproblemsof the evidentialcontentof trace materialsin criminalcases has been quantifying establishedby Lindley(1977), Evett (1983, 1987) and Evett et al. (1987), the last withthecase of fibre transfer. dealingspecifically A crimehas been committed Considerthe following. during whichseveralfibres have been left at the scene of the crime by the offender.A suspect has been and a garment apprehendedas a resultof a police investigation belongingto the suspect has been taken for examination.Afterthe examinationof the suspect's theavailable forensic evidence of bivariate consists colourmeasurements, garment, y = (Yi . ., yn), fromthe fibresleftat the scene of the crime, togetherwith a set x =

NG7 2RD, UK. Nottingham,

1.1. TheProblem

tAddress for correspondence: Departmentof Mathematics,Universityof Nottingham,UniversityPark,

0035-9254/91/40461 $2.00

? 1991 RoyalStatisticalSociety

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462

WAKEFIELD, SKENE, SMITH AND EVETT

sample of m fibres froma representative (xl, . . ., Xm)of bivariatemeasurements as a pairof fibre is expressed The colourof a single thesuspect'sgarment. takenfrom are co-ordinates. For detailsofhowthesemeasurements chromaticity complementary made, see Laing et al. (1986). againsteach are to be weighed hypotheses and exhaustive exclusive Two mutually other: fibres came from the suspect'sgarment; (a) C-the recovered came from fibres some othersource. (b) C-the recovered from theactualcolourmeasurements) (distinct IfIdenotes therestoftheinformation to thecrime, use of Bayes's theorem elementary has beenassembledin relation which on I, throughout establishes that,conditioning = p(xyIC,I) p(CII) p(Cx,y, I) p(x,yj C,I) p(CjI) odds on C = likelihoodratioforC x priorodds on C. posterior p(CIx,y,I) (1.1)

of thewayin whichthe on thelikelihoodratioas thesummary Focusingattention byI, we see overthatprovided x, y provideadditionalevidence colourmeasurements that p(x,yIC,I)

_

p(x,yIC,I)

p(xIC,I)p(yIC,I) p(x,ylC,I) of x. if we make the obvious assumptionthat C impliesthat y is independent of measurements import therefore givestheevidential Evaluationof thisexpression I). x, y (giventhe otherinformation 1.2. Database Information general in thelikelihood ratiorequires involved of thedistributions The modelling from (1987): those derived one case. To Evett et al. quote any specific beyond inputs to have the need for data collections background scientists long recognized 'Forensic in UK forensic of evidence'. In 1982, scientists science assist in the interpretation At each on fibres. a collaborative projectto amass information started laboratories fabric a small representative sampleof clothwas takenfromeveryfifth laboratory such butotherarticles forexamination. Manyof theseweregarments itemsubmitted was completed questionnaire and beddingwereincluded.A comprehensive as carpets College of Textiles,Galashiels, foreach sample,whichwas thensentto theScottish wereobtained.In thisway,bymid-1987, data including wherecolourmeasurements had been collectedon nearly8000 samples. Whether such a colour measurements of thegeneral sourcesis can everbe truly populationof fibre collectiQn representative here.Certainly, thedata collection however, a mootpointwhichwe do not consider thatis currently scientist. available to the forensic the best information represents to be madeof howrareor commona particular fibre Thesedata enablean assessment the evaluationof pO(y in C, For each (1.2). equation I) colour is and thisunderlies I fibresamplein the collectionfivepairs of chromaticity were co-ordinates synthetic fibres.For each naturalfibresample, 10 measureobtainedby usingfivedifferent weremade with10different made fibres. Colour measurements from garments ments

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EVALUATION OF FIBRE TRANSFER EVIDENCE

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of the data For a full description fromnatural fibresexhibitgreatervariability. see Laing et al. (1987). methodology, collection of of whichsixtypes makeup 98Wo fibre 13 different types, The database identifies typeis cotton,of whichthereare 2340 samples all samples.The mostcommonfibre are of which there fibre is polyester, thedatabase. The mostcommonsynthetic within 1512 samples. in detailthecase wherea singlefibre was recovered Evettet al. (1987) considered as thatequation (1.2) can be written fromthe sceneof thecrime.It is recognized p(x,yIC,I) p(yIC,x,I)p(xIC,I) p(yIC,x,I)

Statistical Analysis 1.3. Previous

p(X,yIC,I)

p(yIC,x,I)p(xIC,I)

P(yIC,I)

(1.3)

underC and C. It was assumed on I, x is equallylikely that,conditional by assuming of fibrecolour withina particulargarmentwas adequately that the distribution Thus the numerator of the likelihood by a bivariatenormaldistribution. described ratio(1.3) can be written

p(yIC,x,I) =p(yI,u,

dE, I) p(y, Ix, I) d,u

of a bivariate withmean i and normaldistribution where pO(y 'I i, E, I) is thedensity for theseparameters, density covariancematrixE, and p(it, Ix, I) is a posterior ofvaguepriorknowledge regarding representations distribution. Usingconventional is shownby Evettet al. (1987) to be a bivariate ,uand E, the formof thenumerator mathematical form. Studentt-type havingan explicit density, in theknowledge of thetypeof fibre For thedenominator, givenI (whichincludes thedensity thegeneral distribution describing p(y IC, I) is simply question),theform fibre thatwe wouldsee in whatever byI, populationis defined of colourco-ordinates estimator. kernel density whichEvettet al. evaluatedby usinga bivariate in thispaper results the froma collaborative projectbetween The workdescribed Home OfficeForensicScienceServiceCentralResearchand SupportEstablishment of Nottingham Statistics Group. and theUniversity and modelling numerical issuesassociatedwith theextension The projectaddressed fibres. The approach described by Evett of the above problemto severalrecovered are et al. (1987) is not tractableformorethan a singlefibre.Numericalprocedures formsare assumed and thisprovidedan opporwhatever mathematical necessary of the distributional assumptions to examinemore closelythe conventional tunity original work. In addition, in the contextof the possible modellingstrategies the requiredlikelihoodratio pose available, the numericaldemandsof calculating theBayesianinference framework. challenging problemswithin via a case study in an important paperis to exhibit, The mainpurposeofthepresent and novel modelling and newlyemerging applicationsfield,a class of interesting and some possible problems,involvingboth likelihood and prior specification, willbe on thestatistical aspectsof theproblem. solutions.Our emphasisthroughout to 'single-colour' attention and 'single-source' problems.It we restrict In particular,

assuming x = (xl, . . ., xm) to have been a sample from the same bivariate normal

andStructure Paper of this 1.4. Motivation

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WAKEFIELD, SKENE, SMITH AND EVETT

has to considermulticoloured scientist garments and that is clear thatthe forensic severalsources.The models fibres attached to anyscenemayhavecomefrom foreign to encompassthesesituations but no new statistical herecan be extended described of theHome OfficeForensicScienceService,there issuesarise. In thewidercontext the statisticalmethodologyinto intelligent are importantissues of integrating for laboratoryuse. We shall not discuss such problems knowledge-based systems here. thevariability coIn Section2, we deal withmodelling of observedchromaticity In Section3, we deal withmodelling from thesame garment. ordinates amongfibres of the parameters involvedin modelling withinthe prior(population)distribution garmentvariability.Computational problems are discussed in Section 4 and of the methodology and refinements are examinedin Section 5. Two extensions in Section6. illustrative examples,takenfromactual case work,are presented Variability 2. ModellingWithin-garment 2.1. GeneralApproach Fig. 1 displays the 'horseshoe'-shaped region within which complementary to lie. In thisfigure, N denotesthe co-ordinates (u, v) are constrained chromaticity neutral,or achromatic, point. The positionsof typicalhues, i.e. colours, are also hue and saturation but does not take indicated.In essence,thecolour space reflects Thus all whites, and blackmap to theneutral intoaccountlightness. point.If a greys line weredrawnfromthe neutralpointto theboundaryof the colour space, movetheboundary an increase in saturation ment (colours represents alongthelinetowards lie on the boundary)-the hue is constant.Most describedby a singlewavelength is closeto theneutral clothhas a pale yellowcolourwhich point.In all dyeing undyed in uptakeof thedye.Whena singledyeis used the is somevariability processesthere boththeundyed colourand thedyecolourwith saturation reflects colourofthefibres more Measurements rather thusappear close to a linejoining variability. exhibiting of theundyed pointto thedyecolour.Manycoloursare achievedvia thecombination In this case colour measurements for individual or more fibres are scattered two dyes. about a linejoiningthetwo predominant dyesis dyecolours.The need formultiple forcertain colours.For example, blues can be achieved also morecommonplace many witha singledyewhereasgreensoftenrequiremultiple dyes. of fibres froma particular A typicalscatter sample diagramforthe co-ordinates of manysuchscatterplots within thedatabase is also shownin Fig. 1. The inspection withthelong axis frequently orientation indicateda preferred towardsthe pointing with the above of the consistent discussion. The direct modelling yellow area, distribution is thuscomplicated whichvariesstochastically bivariate bya correlation foreach sampleof a particular fibre withlocation.As an initialsteptherefore, type, sample means were calculated and principal component axes were identified. valueson to theseaxes werethenstudied of standardized to try to identify Projections formsin the long and shortdirections.Fig. 2 shows the possible distributional standardized plots forwool fibres. This in turnleads to the following strategy forwithin-garment generalmodelling variability. (a) two orthogonal axes u' and v'. Identify

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0.8

0.6 -

0.4 -

0.2 N-

00 0.2 0.4 0.6 0.8

in colour space (approximatepositionsof the main Fig. 1. Scatterof fivebivariatemeasurements coloursare indicated:1, yellow;2, red; 3, blue; 4, green)

I-

illI

-2

-1

0 (a)

1

2

-2

0 (b)

2

Fig. 2.

in (a) thelong direction and (b) theshortdirection Standardizedvalues forwool fibres

theorientation of the u '-axis relative to theoriginalu-axis. (b) Identify locationsand spreads,thatthe distributions of (c) Assume, givenorientations, of thechromaticity on to theseaxes are statistically projections co-ordinates independent. thebivariate Giventhisstrategy, describing fibre colourwithin a garment is a density mathematical two for five-parameter familywith one parameterfor orientation, location and two for spread. The orientation is definedas the angle of the u '-axis relativeto the u-axis. Additionallythe spreads are constrainedso that the long of scatter is identified as havingthelargerspreadparameter. direction made in Evettet at. (1987) was thatthescatter was wellmodelled The assumption theprojections However,considering separately by a bivariatenormaldistribution. axes as inFig. 2, thenormalhypothesis on to thelongand short is clearly rejected bya fornearly all fibre conventional Thisfinding prompts a searchforother X2-test, types. distributional fitto the observedunivariate shapes providinga more satisfactory distributions.

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WAKEFIELD, SKENE, SMITH AND EVETT

2.2. ExponentialPower Family The so-called exponential power family is a general univariate family of by withlocation r, spreada and shape parameter3. It is defined distributions I p(zIor,u,3ocuoexp( exp(- - 2Z z-r a = P(zlT a|

2/(1+3))

to the normal alternatives and leptokurtic This familyincludes both platykurtic Box and Tiao for (1973). example, See, distribution. was made to findvalues of a an attempt fibre typeand direction For a particular to thosein Fig. 2. It was not via similar obtained the plots mimicked profiles which 3 value of which adequatelymodelledthe a to identify particular however, possible, in thelong the observed distribution For direction. example, in either behaviour tail tailed. and light is heavyshouldered direction 2.3. 'Roof' Distribution via A second approach considered was to describe the joint distributions which were simple combinationsof rectangularand conditional distributions See Fig. 3. It was found that a good fitto the data was distributions. triangular to wereconsidered of fibre measurements thata smallproportion obtained,provided in a skirt aroundthebase of theroof. be lying In addition to the usual five parameters-(It1,112), mean locations; (a,, a2), constantswere required to standard deviation spreads; X, orientation-further of theroof,e.g. thedistancein thelong of thedifferent thelengths portions describe theroofridgebeginsto slope downand from themean,to thepointwhere direction, are easilyestimated fora givenfibre type constants These extra oftheskirt. theextent in each of thelong and shortdirections, measurements standardized by considering forthetotalprobability to equal 1. Although withtheconstraints necessary together fits to muchof thefibre data, itproved,subsethismodelprovided verysatisfactory

(a)

(b)

tX '-~~~~~~~~~~~(b)

(c)

(a)

(c)

threesectionsand a plan view Fig. 3. Roof distribution:

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EVALUATION OF FIBRE TRANSFER EVIDENCE

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alternative and so yetanother to operatewith numerically to be very difficult quently, was sought. 2.4. Beta Distribution which wereoutlying discarded fibres procedure thatthemeasurement It was known of a particular themaincluster thiswasthought sample.Initially whencomparedwith to have little influence on tail shape but subsequentlythis fact motivated Bivariate densities defined of of finite byproducts consideration rangedistributions. scaled beta distributions providedan adequate fitto thedata and proved symmetric moretractable computationally. considerably on (i - at, beta distribution A univariaterandom variable Z has a symmetric of W = (z - A)/2at + 2 is givenby it+ at) ifthedensity

p(w)

o

Wa-l(1-W)a-l

0<

W <

1.

forthetwoorthogonal foreach fibre type. directions Thus a and t mustbe estimated with and a minimum a searchprocedure chi-squared criterion, Thiswas accomplished studiessuggested thatprecisevalues werenot critical. but subsequentsensitivity 2.5. Comment Althoughthe large amount of data available makes it possible to distinguish modelsforthewithin-garment alternative by usinggoodness-ofvariability between thattheoverallobjectiveis thecomputation itmustbe remembered of the fit criteria, of thisratioto modelling which likelihood ratio(1.2). It is thesensitivity assumptions and thustheeventual choiceof modelis mediated concern bysensitivity is of primary

issues.

3.

ModellingthePriorDistribution

variation involvingfive parameters,the Given a model for within-garment of a priordistribution evaluationof the likelihoodratio requiresthe specification it is size of the database Given the technically possibleto obtain al, a2, 4). 2, P(pA, fromeach of the samplesin the database and thento of the parameters estimates kernel estimate. is computaa five-dimensional construct However,suchan estimate in Section4 to be described and thenumerical intensive techniques integration tionally of such evaluations. requirea largenumber Anotherapproach would be to evaluatethe priorin advance at a regulargridof thefive-dimensional However,the space and thento use interpolation. pointswithin to characterize sucha surface is largeand five-dimensional of pointsrequired number is difficult. interpolation The discussionin Section 2.1 combinedwithan empiricalexaminationof the thatthepriorcould be wellapproximated database suggested by a form

(3.1) p(q I X), P(A,l 2, a1,a2, 0 = P(A,l 2) p(al I r) P(a21 r) the wherer is the radial distancefromthe neutralpointto (A,, 2) and X describes relative to theneutral point(Fig. 4), i.e. thespreadson the radialangleof thecluster given position, and depend only on principalcomponentaxes are independent, thedistancefromthe neutral point.

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468

v

WAKEFIELD, SKENE, SMITH AND EVETT

2

X~~~~x x

XX (1 .U2)

/ It

N

u

data cloud Fig. 4. r and X fora particular

of thelong axis is independent of saturation but dependson hue, The orientation For example, ofdyesinvolved. intheblueregion ofthecolour i.e. on thecombination themajority ofgarments to havea density the alignedtowards space,we wouldexpect undyedpoint whereaswithgreens,whichcan arise frompure greendyes or from combinationsof blues and yellows,we would expectthe prior for 0 to be more of possibilities. thismixture dispersereflecting Notingthat

p (ori Ir) = p (ai. r)/p(r),i

and

P(o IX) = P(og X)/P(X)g

= 19

29

can be evaluated with four bivariatedensities, we see that a prior distribution and r), r) X), each of whichcan be constructed p(a2, p(0, by using P(19 A2). P(arl, methods. kernel timethata Ratherthanhavingto evaluatethefourtwo-dimensional kernels every was required,it was decided to set up four reference tables in prior probability the advance. If the database wereto be updatedit would thenbe simpleto recreate requiredgridsof points. When the priorfora particular pointwas required,cubic to the desiredpoint. The reference tables were splines were used to interpolate achieved as follows: for each of the seven quantities of interest(five model theradial distanceand theradial angle) sampleestimates wereobtained parameters, flbre foreach samplein thedatabase of a particular type.Fromthese,kernel density foreach of thebivariate densities wereevaluatedat each pointon a 20 x 20 estimates and r, thekerneldensities werecreatedin terms of log ai and log r. The grid.For (ri to obtain the requiredformin termsof the original densitieswere then inverted so thatdensity This operationwas undertaken estimates close to 0 were parameters. see Silverman notunderestimated; were (I1986).For theangles0 and X. theestimates to the kernel 'wrapped around' so that, for example,X close to 360? contributed foranglesclose to 0? as wellas to 360?. estimate Bivariate normal kernelsused afterinitial experimentation indicatedthat the

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EVALUATION OF FIBRE TRANSFER EVIDENCE

Y2 r

469

..40. 0

0.4]

3

0.

3 0.3

0 LI20.

81

.GP~ ~~~~~~~~~02

1

0.2

0.3

0.4

0.5

0.0 0.00

T 0.02

0.04

(a)

r

(b)

0.2-

-o005ooo-0 0.3

0.605 0.010

200

0.1

100

o

-50 0

5

5

0.0 0.000

0-

(c)

(d)

forwool fibres estimates for(a)p (Al1, (b) p (a,, r), (c) p(a2, r) and (d) p(4, Fig. 5. Kerneldensity ,tt2), 0): 1, yellow;2, red; 3, blue; 4, green

to the choice of kernelshape. Initially, window were insensitive estimates density to n - "I byusingguidelines suggested by Silverman widths werechosenproportional where theamountof data was largethisresulted, for types (1986). However,forfibre the finaldensity estimate. in a veryspikydensity Consequently P(I.', /.2) especially, estimatesused slightlylarger window widths giving a smoothnesswhich was withthe considerable to us by the Home communicated consistent priorknowledge Contourplotsofthefourbivariate estimates forwool colourchemists. Office density are at iWo, are shownin Fig. 5. Displayedcontours 5%, 10%, 25%o,500o, 75/o and Fig. 5(a) showsthata largeamountof thedata liesin theblue and theredregions. and is similar forotherfibre types. plotis typical shape of thecontour The triangular modesrepresents modeoccurs popularcolours.The largecentral Each ofthemultiple thatthestandarddeviations increase at theneutral point.Figs 5(b) and 5(c) confirm show morevariability as we move away fromthe neutral withradius-the clusters thisand displaysthe priorforp(G, Ir) fortwo particular point. Fig. 6(a) confirms

95%/o of the mode.

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470 p(ral

=

WAKEFIELD, SKENE, SMITH AND EVETT .02)

(a,rI r.10)

0.00 Fig. 6.

O.01 (a)

0.02

0.03

1T

-50

0 (b)

50

0

distributionp (a) p (u,Ir) and (b) marginal (0)

nowtoFig. 5(d),themodeoccurring andonelarge. Referring ofr,onesmall choices towards alltend topoint which X= 3000 corresponds totheyellows, atapproximately ofthecolour to corresponds space,thisorientation theneutral region point.In this the blues, which again are prek= 500. The other mode at X= 1300 represents sides ofthe the andyellows neutral blues lieonopposite they point andso even though at X= 2400, havea flatter reflectof k.Greens, distribution, values lying havesimilar of two different often ariseas mixtures colours;see the ingthefactthatgreens tended to point towards theneutral we in Section 2.1. If thegreens point discussion distribution themarginal of bluesand yellows. Fig. 6(b) shows combinations p(O). to the was obtainedfrom approximations p(k, X) via discrete This distribution 500 reflects Themodeat approximately theprevious discussion of integrals. required andthegreens. theyellows theblues, 4. Implementation

4.1. Evaluation of LikelihoodRatio would expect the mode of p(k IX= 2400) to lie at k= - 600. The mode of this liesat X = 500 indicating are often thatthegreens distribution conditional actually towardsthe neutral between- 900 and 900 point. kis defined oriented dominantly

in equation thejointdistribution of a setofbivariate Each term (1.2) represents be from a single source. Reference to C and C can therefore colourmeasurements measurements, a collection and,ifz denotes dropped (zl,..*, Zk) ofk bivariate p(zI)=

k _

p(ziI) p(O I) dO,

(4.1)

discussed in Section 2 andassumed to be with byoneoftheforms p (ziI0) modelled of the form in 3. and a discussed Section of prior p (O II) specifying I, independent stilldepends on I, theinformation additional to theactual Notethattheintegral fibre type. colour measurements, including

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EVALUATION OF FIBRE TRANSFER EVIDENCE

471

4.2. Computation

the For anychoiceofp (zi IO) and p (O I), theevaluationof equation(4.1) requires numerical integral. evaluationof a five-dimensional severalnovel et al. (1985, 1987)describe Naylorand Smith(1982, 1983)and Smith Among of Bayesian methods. for the implementation strategies integration numerical very we outline which methods Gauss-Hermite Cartesian product theseare adaptive here. briefly if that,in one dimension, noting byfirst is motivated strategy The Gauss-Hermite the then x normal density a by polynomial be well approximated can an integrand an rule(withan exactanswerfrom willbe wellestimated bya Gauss-Hermite integral is actuallyof polynomial x normal form,withthe n-pointrule if the integrand polynomialcomponentof degreeup to 2n - 1). We thennote that,possiblyafter many of the marginalposteriordensitiesarisingin suitable reparameterization, can plausiblybe regardedas of polynomial x normalform.A Bayesianinference however,is likelyto exhibitseveral joint posteriordensityfor many parameters, whichwould make the applicationof Gauss-Hermiteproductrules dependencies, rulein each of theGauss-Hermite inefficient. Moreover,theimplementation highly for that and scale of location a the direction specification requires parameter values fortheseare not known. direction, and correct the gridat which a product forfinding iterative strategy thefollowing This suggests providethe basis for the accurate evaluationsof the likelihood x priorfunction of thejoint and henceany aspectsof interest constant evaluationof thenormalizing density. posterior Reparameterize individual parameters so that the resulting working by a polynomialx normalform.Often can be well represented parameters to take anyvalue on willallow theworking parameters thisparameterization thereal line. (b) Using initialestimatesof the joint posteriormean vectorand covariance scaled, transform further to a centred, parameters, matrixforthe working set of parameters. moreorthogonal forthese orthogonal (c) Using the derivedinitiallocation and scale estimates of interest of functions Cartesianproductintegration perform parameters, dimensioned grids. usingsuitably untilstable themeanand covarianceestimates, successively updating (d) Iterate, dimension. and between gridsof specified results are obtainedboth within (a) here, adopted fortheproblemdescribed The parameterization

V1 = 1si 2 = A2,

13

=

log

/

U2

U2 /

2

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472

WAKEFIELD, SKENE, SMITH AND EVETT

However,further powerand roofdensities. provedadquate fornormal,exponential byzl, . . ., Zk whenthebivariate intherangeofIt, A2,i and u2areimplied constraints is considered.This pointis discussedin more detailin Wakefield beta distribution The mean parameters pointis worthmentioning. et al. (1989). A further i, and A2 the horseshoe-shaped regionof Fig. 1 and so could be constrained mustlie within In practice,however,theyneverfall close to the boundary.Fig. 5(a) accordingly. butagain,inpractice, constrained bearsthisout. The variancesU2 and q2 are similarly thisis not a problem.

Studies 4.3. Sensitivity

of combinaElementsof the likelihoodratio (1.2) werecalculatedwitha variety roofand model(normal,double exponential, variability tionsof thewithin-garment in our uncertainty the choice of window (reflecting beta) and priorspecifications of the sensitivity are not givenhere. The resultsshowed, width).Detailed findings the values of individual wereclearly influenced that components although however, theresulting likelihood ratioswerewithin of a factor bythechoiceofmodeland prior, 2 in mostcases. feasible,the computational are computationally Althoughall such combinations with being bivariate normality costis influenced bythechoiceofmodelto someextent theuse oftheiterative Gauss-Hermite strategy. Furthermore, themostefficient given a factor of 10are currently to viewedas sufficient likelihood ratiosaccurateto within content.Thus themodelling and sensitivity exerciseshowsthat evidential represent The as thedefaultmodelis adequate at present. theadoptionof bivariate normality Impleexampleswhichfollowin Section6 havebeenanalysedunderthisassumption. is possibleon thecurrent rangeof personal of theintegration procedures mentation computer systemsalthough more powerful single-userworkstationsare more appropriate. Furtherdevelopmentsin computertechnologywill minimizethe overa morerealistic form. forchoosingbivariate distributional normality argument choice. are our preferred Bivariatebeta distributions and Refinements 5. Extensions eventI appearingin We now considerin a littlemore detail the conditioning all additional information availableaboutthefibres apart equation(1. 1). Irepresents co-ordinate measurements. theactual complementary from chromaticity of a typicaleventI, considerthe situation whereforeign As a simpleillustration of thefibres revealsthat fibres have been foundon a victim.An initialexamination they type(e.g. wool or nylon), (a) are of a particular such as delustrant etc. and (b) mayhave characteristics can be colour(e.g. redorblue) and possibly (c) to thenakedeyeare ofa particular as lightor dark etc. described fibres containing Then I is the eventthata suspecthas been foundwitha garment whichmatchthisdescription. of thecolourof bothrecovered fibres is then and control The precisemeasurement undertaken and y and x are obtained.

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EVALUATION OF FIBRE TRANSFER EVIDENCE

473

oddsonthe totheposterior contribution (1.1),theoverall equation As weseefrom of the garment is a combination thesuspect's comefrom fibres having recovered onI. just conditioned (1.2),basedony andx andI, andan oddsratio ratio likelihood on the depends term, p(CI I)/p(CI I), clearly of thelatter Detailedquantification For ofa suspect anda garment. theidentification surrounding circumstances specific or was the becauseof thegarment, identified primarily was thesuspect example, We on other grounds? ofthe suspect tothe identification found subsequently garment the the effect of concentrate on but instead shall not pursuesuch issueshere, in (a)-(c) onp (OII). information though intheory, (b) isstraightforward ofanIbased on (a) and/or account Taking inpractice. ofthe Suchinformation thesubset simply identifies someissues itraises of 3 which the kernel Section is to be applied. on methodology database informano additional is usually there suchas woolandcotton, fibres Fornatural good to obtain and thus the database is sufficient available present tionoftype (b) of rare fibre or where a combination For estimates. situations types density kernel leadsto a cross-sectional and delustrant shapeand radiusof fibres type, synthetic is notyet toyield database precise thepresent sufficiently large initial match, precise ratiocan stillbe In suchcases thelikelihood of colourdistribution. descriptions obtained datapoints available. thenear uniform from thefew priors using calculated willthusbe smallbuttheinitial valueof thecolourmeasurements The evidential willusually havehigh evidential content. ofthekind Information matching process colour that an apparent match has beenmadeis not described by(c) andbythefact which In effect, it imposes a further to incorporate. conditioning, straightforward for theappropriate ofthedatabase thekernel, by(a) and(b), implied that part implies entire horseshoeofthe colour rather than over the a subset space, isdefined over only is precisely to quantify earlier described (seeFig. 1). Whatis difficult region shaped in (c) andthefact ofthecolourspaceis defined bythedescription which subregion has beenobtained. a prima that facievisualmatch istodefine theappropriate ofthecolour subset space tothis Oursolution problem Thesizeofthis circle location ofthe recovered fibres. centred atthe mean tobea circle science caseofficers. Theregion ofa match forensic does, after waschosen consulting butouranalyses tobereasonably onthemean location proved tosome depend extent, modified The implied form ofp(OII) is then to theexactvaluechosen. insensitive in Section3 by replacing estimate of the the kernel density from thatdiscussed circle use of a The to the circular one constrained by region. p(Al ', A2) component

has negligible impacton thevalues of thansome othershapedneighbourhood rather case the likelihoodratioobtainedand thisapproach has been used in the following studies. 6. Illustrative Analyses

I 6.1. Example Thevictim intoa garden andwasrobbed ofherhandbag. wasdragged Thevictim

fibre jumper. The assailantwas wearinga was wearinga greenwool and synthetic Threegreen werefoundon thesuspect's wool fibres checkedcottonshirt. grey-green werefoundon thevictim's cottonfibres jumper. 10 control shirt.Seven grey-green and thevictim's weretakenfromboththe suspect'sshirt jumper. fibres

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474

V

WAKEFIELD, SKENE, SMITH AND EVETT

V

0.32 -

x

0.335 0.3350.332** x

x

0.31-~~~~

0.31-

x

X

0.3-

~~~x

0.3250.32 -

x I 0.36 I 0.38 I 0.4 (a) l 0.42 0.44

u

0.315 I

x 0.315 l 0.32 (b) I 0.325

u

Fig. 7. Control (x) cotton grey-green

and recovered (X) fibre measurements for (a) the green wool and (b) the

thegreenwool. Fig. 7(a) givesa scatterplot measurements showing Considerfirst and controlfibres.The value of the likelihoodratiobased on fromboth recovered thesedata alone is 700. ratiois 30. The thevalue of thelikelihood cotton, Considering onlythegrey-green fibre are shownin Fig. 7(b). measurements controland recovered ratiofortwoon I, thelikelihood In mostcases, it can be arguedthat,conditional thelikelihood is the productof the separateratios.Overall,therefore, way transfer ratioin thiscase is 21000. 6.2. Example 2 theforensic was asked to scientist Followinga complexseriesof armedrobberies, ofgetaway linksbetween three and a number it cars. In particular, suspects investigate a knife to investigate thelinkbetween foundin thepossessionof one of was required cashbagswhich and several had beensplitopen. We shall thesuspects purplenylon-66 consider from thecash bags and the knife and (a) purplefibres foundinone ofthecarsand matched with a suitownedbyone (b) bluewool fibres of the suspects. 6.2.1. Purple cash bags had werecontacted and onlytwo batchesof thistypeof fibres The manufacturers werefoundon theknife and chromaticity were co-ordinates everbeenmade. 73 fibres all thebags werevisually a randomly selected15ofthese.Fibresfrom calculatedfrom butthey could be splitintotwosetsbyvisiblespectroscopy and thin indistinguishable 13 fibres matched the first batch and theremaining twofibres layerchromatography. for each of the matchedthe second batch. There were fivecontrolmeasurements of thecontroland recovered fibresforthe second batch is batches.The scatterplot there are onlytworecovered ratio thelikelihood fibres, shownin Fig. 8(a). Although moments of modelparameters indicated that,in thisexample,both is 217. Posterior

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EVALUATION OF FIBRE TRANSFER EVIDENCE

V V

475

0.41 -

x x 0.3750.376035 0.36 x x . X~~~~~~~ x X x

0.4 -

x

x

x

0.38 I 0.35 I 0.355 (a) I 0.36 0.365

u

0.355 I

0.34

I

0.345

I

l 0.35 (b)

I 0.355

I 0.36

u

for(a) thepurplecash bags and (b) theblue measurements Fig. 8. Control( x ) and recovered (-) fibre wool suit

and thedispersion theorientation oftherecovered and control data werevery similar. The highvalue is not a consequenceof therarity of thepurplecolour as theratiois on I. The factthata knife conditional was recovered with fibres thatmatched therare colourof thecash bags would normally increasethestrength of evidence.However, thisaspectof theevidencemustbe modelledviap (CI I)/p(Cl I). 6.2.2. Blue wool suit 15 blue wool fibreswerefoundand thesematchedfibres takenfroma blue suit In thiscase 20 control ownedbyone ofthesuspects. measurements weretakenand the subsequentmeasurements, along withthe recovered co-ordinates, are displayedin Fig. 8(b). In this case a likelihoodratio of 0.8 was obtained,the data being sufficiently in dispersionbetweenthe recoveredand control numerousto suggestdifferences samples. Acknowledgements herewas madepossiblebya Home Office The workreported grant supporting J.C. Wakefield. We also acknowledgeinvaluable assistance fromP. E. Cage, A. W. Hartshorneand D. K. Laing of the Home OfficeCentral Research and Support and R. Cook oftheMetropolitan Establishment Police ForensicScienceLaboratory. References

Box, G. E. P. and Tiao, G. C. (1973) Bayesian Inferencein Statistical Analysis,p. 157. New York: Addison-Wesley. Evett, I. W. (1984) A quantitative theoryfor interpreting transfer evidencein criminalcases. Appl. Statist.,33, 25-32. and forensicscience: problemsand perspectives. (1987) Bayesian inference Statistician, 36, 99-106. C. G. G. (1987) Evaluationofthelikelihood Evett,I. W., Cage, P. E. and Aitken, ratioforfibre transfer cases. Appl. Statist.,36, 174-180. evidencein criminal

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WAKEFIELD, SKENE, SMITH AND EVETT

A. W., Cook, R. and Robinson,G. (1987)A fibre Laing,D. K., Hartshorne, data collection forforensic and examination methods.J. Forens. Sci., 32, 364-369. scientists-collection Laing, D. K., Hartshorne, A. W. and Harwood, R. J. (1986) Colour measurements on singletextile fibres. Forens. Sci. Int., 30, 65-77. Lindley,D. V. (1977) A problemin forensic science.Biometrika, 64, 207-213. Naylor,J. C. and Smith,A. F. M. (1982) Applicationsof a methodforthe efficient computation of distributions. posterior Appl. Statist.,31, 214-225. modelin clinicalchemistry: (1983) A contamination an illustration of a methodfortheefficient of posterior distributions. computation Statistician, 32, 82-87. B. W. (1986) DensityEstimation and Data Analysis.London: Chapman and Silverman, for Statistics Hall. and Smith,A. F. M., Skene,A. M., Shaw, J. E. H. and Naylor,J. C. (1987) Progresswithnumerical graphicalmethodsforpracticalBayesianstatistics. Statistician, 36, 75-82. Smith, A. F. M., Skene, A. M., Shaw, J. E. H., Naylor, J. C. and Dransfield, M. (1985) The of theBayesianparadigm.CommunsStatist.Theory implementation Meth., 14, 1079-1102. A. F. M. and Evett,I. W. (1989)A case study in forensic J. C., Skene,A. M., Smith, science: Wakefield, transfer evidence.ResearchReport.Nottingham theevaluationof fibre Statistics Group,University of Nottingham.

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