21february2012© 2012 The Royal Statistical Society

The expert has a mark from a crime-scene, and a print taken from a known individual, and is required to say either that the mark matches the print, and was therefore made by the accused; or that it does not match and was made by someone else. Traditionally, no room for doubt is allowed: a fingerprint expert will report that he or she is absolutely certain that an impression from the scene of a crime comes from the finger of the accused.

But in real life nothing is certain. Fingerprint ex-perts are exceedingly good at their jobs; nevertheless, different experts can and occasionally do give different opinions as to whether or not two impressions match. The crime-scene mark is unlikely to be perfect; it may be degraded, partial, distorted, or blurred. The expert may feel a small residue of doubt. If it is his belief that the mark ‘probably does’ or ‘almost certainly does’ or ‘is rather unlikely to’ match, he is forbidden to say so in court; in those cases, fingerprint evidence, for or against the accused, simply does not appear in the case and whatever supporting information content it may have is effectively wasted. (This happens in as many as 30% of the comparisons performed in a fingerprint bureau1). Strangely, DNA experts are required to give probabilities for their evidence of matching; fingerprint expert are forbidden to. This bizarre situation ought to be ended, in the interests of justice as well as of common sense.

Fingerprints at the crime-scene:

Statistically certain, or probable?

With dramatic suddenness [Inspector Lestrade] struck a match and by its light exposed a stain of blood upon the whitewashed wall. … It was the well-marked print of a thumb. …

“You are aware that no two thumb-marks are alike?”

“I have heard something of the kind,” re-plied Holmes.

Thus Sherlock Holmes in the case of ‘The Norwood Builder’, from The Return of Sherlock Holmes, first pub-lished in 1905. Since the dawn of modern detective work fingerprints have been regarded as the prime means of identifying an individual; their use as a means of personal identification goes back still further, to ancient China and even earlier (see box overleaf ). Nowadays, fingerprints are used by police forces and courts worldwide. It is commonly believed, and may very well be true, that no two people share an identical fingerprint. But forensic scientists do not deal with exact reproduction, but have to draw conclusions from imperfect prints found at crime-scenes. The comparison of fingerprints is not an objective science. Fingerprint experts rely on judgements and opin-ion. They give their testimony in court; and the testimony they give is an entirely subjective one. Yet at the same time that testimony is required to be definitive and absolute.

Fingerprints have been used for a century to identify criminals. But, astonishingly, fingerprint experts rely on subjective opinion, not on objective science. Yet they are required to claim absolute certainty for their judgements – a certainty that is mythical. Cedric Neumann brings probabilities and the hope of better justice to the courtroom; with Julian Champkin he explains the idea.

DNA experts are required to give probabilities. Fingerprint experts are forbidden. This bizarre situation should be ended

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If there is, say, a 75% probability of a match a jury should be told of that, to add or subtract from the balance of evidence against the accused, to weigh the scale on one side or the other. The present system, of a pretended certainty (one way or the other) that is not in fact there, can and should be improved upon.

What is needed is a way to give quanti-fied, numerical figures to the probability of a match.

How, then, can we introduce prob-ability to fingerprinting? The system itself has hardly changed since it was devised. The clas-sifications that Francis Galton and Sir Edward Henry introduced, of loops, arches and whorls, are still used today. For a more objective and probability-based approach we need ways to quantify these swirling shapes.

First, some definitions: a fingerprint, or print, is the record that is taken, at the police station or somewhere similar, from a known individual, using ink pad and roller, under con-trolled conditions and protocols. It is, if you like, the gold standard: the impression is as clear as possible, and we know for certain who it comes from. A mark, on the other hand, is the impres-sion that is found at the scene of the crime. It may be in blood, or grease, or powder; it may be smudged or smeared or distorted from a finger pressed at an angle; it may be incomplete; it may have other patterns – from wood grain, perhaps, if it is on a wooden surface – super-imposed on it; it may be distorted from being on a curved surface such as a bottle or a glass. It will probably have been photographed, or lifted by powder and tape by a scene-of-crime expert

using one of the techniques that crime-scene investigation shows portray rather fancifully on television. It will almost certainly not be as clear as our police-station fingerprint; and if you superimpose the two, almost certainly they will not exactly coincide.

So how can we establish whether police-station fingerprint and scene-of-crime mark comes from the same individual? And how can we give numerical probabilities for the certain-ty or uncertainty of our conclusion? The first problem is to find a set of features that define a fingerprint or mark. Then we can assign the probability that two similar but not identical sets of features come from the same source.

A fingerprint is essentially a pattern. It is too complex for mathematics or for fingerprint

experts to analyse as a whole. Instead, experts select parts of it, which are deemed sufficient to distinguish it from any other fingerprint. Three basic overall patterns of fingerprint are arches, loops and whorls (see Figure 1), and these are good for general recognition and for classifying – as in the huge databases which police forces everywhere maintain; but for crime-scene comparisons the experts use finer details. Each of the few dozen ridges of skin that together form a fingerprint can have one of two things happen to it: it might come to an end; or it might divide into two (exception-ally three). These points of detail are called minutiae, or points, and it is by comparing the minutiae in a print and a trace that examiners form their opinions.

Quantitative observations of fingerprints

On the one hand, a single fingerprint pattern can contain dozens of these minutiae; there may be too many to examine them all. On the other hand, marks recovered from crime-scenes will only contain very few minutiae. When comparing marks and prints, experts will observe the minutiae present on the mark and search the print for correspondences and discordances. Historically, in most countries, 12 minutiae that matched each other in type, orientation and position have generally been considered sufficient to identify the source of the mark. Until 2001 the UK required 16 correspondences to establish proof of identity. Both these numbers arose through experience rather than statistical analysis.

The history of fingerprinting

By the year 246 bc Chinese officials were impressing their fingerprints into the clay seals used to seal documents. Some time before ad 851 an Arab merchant in China, Abu Zayd Hasan, witnessed Chinese merchants using fingerprints on silk or paper to authenticate loans. The Persian physician Rashid-al-Din Hamadani (1247–1318) refers to the Chinese practice of identifying people via their fingerprints and commented “Experience shows that no two individuals have fingers exactly alike.”

Modern use of fingerprinting began with Sir William Herschel. In India, from 1858, he used fingerprints to authenticate legal documents. Around 1880, Dr Henry Faulds, working in Tokyo, proposed a fingerprint scheme to the Metropolitan Police in London, who rejected it. Faulds wrote to Charles Darwin. Darwin, being too old and ill to work on it, passed the information to his cousin, Sir Francis Galton, who devised the classification system of whorls, loops and arches still in use today. Galton’s book Fingerprints (1892) encouraged its use in forensic science. He also performed one of the few statistical analyses of fingerprinting, calculating that the chance of two different individuals having the same fingerprints was about 1 in 64 billion. The first use of fingerprints to identify a murderer was in Argentina in 1892. Scotland Yard’s Fingerprint Bureau was established in 1901.

Main patterns of ridge flow ( , core (centre) of the patterns; , delta (point of convergence of the ridge flow) of the patterns): (a) whorl; (b) arch; (c) loop

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The reasoning that currently leads experts from minutiae to identification is essentially a psychological one that cannot be rationalized and rendered explicit. The method that my colleagues and I have presented2 also relies on those minutiae; but numbers are derived from them.

On any given finger impression, the most prominent minutiae – say six – can be selected and joined up, in a clockwise direction (see Figure 2). They will form a pattern – essen-tially a six-sided polygon around a centre. (The centre can be defined as the arithmetic mean of the Cartesian co-ordinates of our six points.) A polygon is a much simpler pattern than the whirling lines of a full print or mark. It is also much easier to analyse numerically. The basis of the method is to describe that polygon with a set of variables.

If the polygon has k corners – that is, was constructed from k minutiae - it contains 5k pieces of numerical information. They are, for each corner: its distance from the centre; its dis-tance from the next corner (going clockwise); the angle between the direction of the minutia and the radius line; the area of the triangle it forms with its neighbour and the centre; and the type of minutia it is – a ridge ending, a bifurcation,

or unknown (Table 1). These numbers can be recorded as a column, and the columns for each minutia can be set side by side. So, if k = 6, the information on the fingerprint is reduced to a 6 × 5 array of numbers.

Let us return to the scene of the crime, and the perhaps partial and degraded mark we have found there. This too can be reduced to a 6 × 5 array of numbers. Those numbers may be similar to, but not identical to, the ones in the previous fingerprint array. When we compare the two there will be similarities and also differences. How likely is it that they come from the same finger? And how can we assign numerical probabilities to that vague term “likely”?

Comparison of configurations and optimization

The comparison of the two arrays can be visualized by returning to the polygons that generated them. If we lay the centres of the two polygons on top of one another we can rotate them until the distance between correspond-ing corners is minimized. A little maths and geometry will find the best position. We can add up all the six corner-differences to give a single number, or score, that expresses the similarities and differences between the two patterns. This is very similar in principle to matching algorithms implemented in the UK national fingerprint database known as Ident1.

Basil Rathbone as Sherlock Holmes, considering a probable match? Rathbone played Holmes in more than a dozen Hollywood movies from 1939 onwards

Figure 2. Illustrations of the radial triangulation used to order minutiae. The variables listed in Table 1 are indicated in grey in (a). The arrows show the ‘directions’ of the minutiae. The differences between (a) and (b) are presented by using thinner lines

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The score is actually the differences be-tween the numbers in the two arrays when they have been collectively minimized. The total difference between any two minutiae (corners) is the weighted arithmetic sum of the squared differences between the five variables measured on every minutia. The weightings are simple multipliers whose values are chosen to optimise the algorithm. The optimiza-tion criterion is to end up with a measure of distance such that mark–print pairs from the same finger (within source) should tend to yield small scores and mark–print pairs from different fingers (between source) should tend to yield large scores. Those interested may find the mathematics, the geometry and the weightings in the paper by Neumann et al. cited earlier2.

Properly interpreting the similarities and differences observed between a crime-scene mark and the control print from a suspect requires considering two alternative hypoth-eses. First, the hypothesis that the prosecu-tion in a courtroom will try to prove: namely, that the trace and the source come from the same finger – that is, the differences can be explained by the partial and degraded nature of the mark. The method also needs to be fair to the defence, which would be trying to prove the opposite hypothesis, that the fingerprint and trace come from different fingers – that is, the similarities between the mark and the suspect’s print are coincidental – and here we are interested in knowing how likely it is to randomly select an individual who will exhibit such similarities.

For use in courtrooms, we need a method to translate the scores calculated between marks and prints into probabilities assigned to the two hypotheses mentioned above. The first hypothesis assumes that the suspect is the real source of the crime-scene mark; it requires studying the range of possible marks that this suspect can leave, and assessing whether the mark recovered on the crime-scene fits within this range. Under this hypothesis, scores are calculated between the crime-scene mark and ‘pseudo-marks’ – that is, a large number of marks, generated by computer from the suspect’s real fingerprint, each pseudo-mark varying from the real print according to al-gorithms that simulate possible distortions, pressures and angles of pressure and the like; the pseudo-marks represent the range of dif-ferent traces that the suspect’s actual finger could have made. They let us determine how

likely it is to observe the crime-scene mark if it was truly left by the suspect. By definition of the score used in this study, a large amount of small scores will result in a high probability supporting this hypothesis.

The second hypothesis assumes that somebody other than the suspect is the real source of the crime-scene mark; in other words, it assumes that the suspect is wrongfully accused and that the similarities observed between the mark and his finger are coincidental. This hypothesis requires studying the number of people who have fin-gers that can generate a mark with the same features that are observed on the crime-scene mark, and assessing how rare or common are these features. Under this hypothesis, scores are calculated between the crime-scene mark and pseudo-marks generated from fingers of randomly selected individuals to determine how likely it is to observe fingers exhibiting the features of the crime-scene mark by ran-dom chance. By definition of the score used in this study, a large amount of small scores will result in a high probability of observing the features on the crime-scene mark by chance (common features) and support the hypoth-esis that the association between the crime-scene mark and the suspect is coincidental; while a negligible amount of small scores will result in a very low probability of observing those features by chance (rare features) and support the hypothesis that the association between the mark and the suspect is evidence in the case.

The ratio between the two probabilities is called a likelihood ratio. The higher the likelihood ratio, the stronger the evidence in favour of the hypothesis that the suspect is the source of the crime-scene mark; the smaller

the likelihood ratio, the stronger the evidence in favour of the hypothesis that the suspect is not the source of the crime-scene mark.

Testing the model

The method needs to be tested and checked against reality. If print and trace come from the same fingers, the model must give a high probability supporting that hypothesis; if they come from different fingers, the model must reflect that fact equally strongly.

Ideally, we want neither that the method generates misleading evidence in favour of the prosecution, nor in favour of the defence. We want to be sure that if the numbers extracted from the mark and print polygons seem similar, this could not be accidental resemblance. To test this under the most difficult conditions, a validation experiment was performed: traces from the scenes of 364 crimes committed in North Wales were compared to fingerprints obtained from the national fingerprint data-base of the USA.

This US database contains over 600 million prints. Its experts searched it for the nearest match to each Welsh crime-scene trace; in each case, checks were made that the owner of the US finger concerned could not possibly be the source of the crime committed in North Wales. This gave us a gold standard of certainty with which to test our method’s powers of discrimination.

In each case it was found that the US fin-gerprint which corresponded most closely – out of over 600 million of them – to the trace at the Welsh crime-scene was numerically different enough not to be giving misleading evidence that the US person was the source of the mark. In other words, a prosecution

Table 1. Data extracted from a minutiae configuation

Notation Description Units

δ Radius – the distance between a minutia and the centre of the polygon Pixels

σ Side length – the distance between a minutia and the next contiguous minutia in a clockwise direction

Pixels

θ Angle – the angle between the direction of a minutia and the line from the centre of the polygon to the minutia

Radians

α Area – the area of the triangle constituted by a minutia, the next contiguous minutia and the centre of the polygon

Pixels

τ Type – type of the minutia {ridge ending, bifurcation, unknown} [0, 1, 2]

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based on that fingerprint evidence would not have been brought based on the model. This was reassuring: it was a tough test, yet the method would in no case have helped to convict an innocent person. What of the reverse hypothesis? Fingerprint evidence is more usually used in court to help identify the guilty.

The North Wales crime-scene database was from cases that had been brought to court and successfully prosecuted; so the fingerprints from the true British-based perpetrators of the crimes were also available to compare to the traces. Here, too, the model performed satis-factorily. There is a clear separation between the likelihood ratios in the cases where print and mark came from the same person, and the cases where they came from different people. The more minutiae were used, the clearer was the separation.

The future

In the immediate future, we do not see that the current courtroom practice of presenting categorical opinions about fingerprints will change. But in the longer term we expect an evolution towards a framework that is similar to that which underpins DNA evidence. Already this work has formed the basis of training workshops in the UK, USA and in Europe. We have seen several years of courtroom battles in relation to DNA evidence. They have proved to be beneficial to the science. We must expect similar battles over this method. But the notion of a quantitative measure of the strength or weakness of evidence involves subtle issues which many lawyers fail to understand. It remains to be seen how future legal battles play out, but we see models such as this one as a powerful platform for change.

References1. Neumann, C., Mateos-Garcia, I.,

Langenburg, G., Kostroski, J., Skerrett, J.E. and Koolen, M. (2011) Operational benefits and challenges of the use of fingerprint statistical models: a field study. Forensic Science International, 212, 32–46.

2. Neumann, C., Evett, I.W. and Skerrett, J. (2012) Quantifying the weight of evidence assigned to a forensic fingerprint comparison: a new paradigm. Journal of the Royal Statistical Society, Series A, 175, 1–26.

Cedric Neumann is currently an Assistant Professor in Statistics at Pennsylvania State University. Cedric obtained his Ph.D. in Forensic Science from the oldest forensic programme in the world at the University of Lausanne, Switzerland. He has worked on forensic projects for the United States Secret Service and has managed a research team at the Forensic Science Service in the UK.

Playing the lottery with a little bit of stats know-how...Ian McHale dreams of winning the lottery. As a statistician, does he stand a better chance? With Rose D. Baker, he explains…

Lottery games around the world follow the same basic structure: players choose m numbers from a panel of M numbers. The most popular lottery in the UK is the National Lottery’s Lotto, where players choose 6 numbers from 49. Prizes are given for matching at least three of the prize numbers drawn; the more numbers a player matches, the bigger the prize. If no player matches all six prize numbers then the jackpot (the prize for matching all six numbers) is “rolled over” to the next draw.

There is roughly a 1 in 14 million chance of a single ticket winning the UK Lotto jackpot – see Table 1 for the chances of a lesser prize – and for every £1 ticket we purchase, we would expect over the long run

less than 50 pence back. On the face of it the lottery is a very poor investment and financially a player would be better off keeping the £1 in his or her pocket (or putting

Table 1. The probability of winning a UK Lotto prize

Number of ball matches Proportion of prize pool Probability of winning

6 52% (plus any rollover or bonus) 7.15 × 10–8

5 + bonus 16% 4.29 × 10–7

5 10% 1.80 × 10–5

4 22% 9.69 × 10–4

3 Fixed £10 prize 0.0177

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