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Ballistics DNA Alain Beauchamp, PH.D. The path to a ballistic probability model PART I: Correlation score and probability PART II: Ballistic probability model PART III: How could we implement a probability model in a ballistic system? Conclusion and future work

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## Ballistics DNA

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**Ballistics DNA**Alain Beauchamp, PH.D.**The path to a ballistic probability model**• PART I: Correlation score and probability • PART II: Ballistic probability model • PART III: How could we implement a probability model in a ballistic system? • Conclusion and future work**Part I Correlation scores and probability**• Strengths and limitations of the current correlation score • Why are correlation scores hard to interpret? • Benefits of a probability “score”**Strength and limitations of the correlation score**• In the last 15 years, the correlation score has been in the core of FT’s ballistic systems • Strength of a correlation score: Useful as a ranking tool • Can compare score values computed with the same reference “A” (and same type of mark) • Score(A against B) > Score(A against C) means B looks more similar to A than C does**Strength and limitations of the correlation score (Cont’d)**• Limitations of a correlation score: • Correlation score is hard to interpret • Not useful as an intrinsic similarity measure • Examples: • Cannot compare score values computed with different references (same type of mark) • Score(A-B) > Score(C-D) DOES NOT mean that the A-B pair looks more similar than the C-D pair • Cannot compare score values computed from different marks • Score(A-B) for the Firing Pin > Score(A-B) for BreechFace DOES NOT mean B looks more similar to A on the FiringPin than on the BreechFace**The score is hard to interpret. Why?**5 reasons: • 1] Different algorithms for different marks • Characteristics of the correlatable features and the geometry are very different • FP/BF: circular contour and a wide variety of features • Ejector/Rimfire: polygonal contour • Bullets: stria only • 2] Algorithms change over time**The score is hard to interpret. Why? (Cont’d)**3] No unique cartridge or bullet score • More than 1 score per exhibit • Cartridge cases : • BF/FP/Ejector scores • Bullets (Land) • MaxPhase2D, PeakPhase2D, PeakScore2D • 3DScore • Number of score per exhibit expected to increase in the future • Cartridge cases: 3D scores • Bullets: • Added 3D Land score • GEA scores?**The score is hard to interpret. Why? (Cont’d)**• 4] Effect of the database size • As the database size increases, the probability to find non matches that look similar to a given reference increases • The probability to find a known match in the Top10 decreases even if the score does not change • The score value alone is not sufficient. The database size is an important factor as well. • “Universal law”, not only in ballistics systems**The score is hard to interpret. Individual Score Response**• 5] Each reference has its own “score response”. Example: • If two cartridges A and B are correlated against the same large database (with no match in it) Sometimes get two very different list of scores • For example, scores associated with A could be greater then scores associated with B**The score is hard to interpret. Individual Score Response**(Cont’d) • Experiment: Correlate 9LG bullets against the same large database (800 non matches) with BulletTRAX-3D • Compare their non match score distribution • Significant differences • high score region • position of the peak • Each bullet has its own statistical distribution of non match scores • No universal “score response” common to all bullets 9LG Bullet #A 9LG Bullet #B**Solution: Convert scores into probabilities**• Each of the previous problems can be solved using probabilities (in principle) • Different Algorithms: • Probability is a common concept for all score types • Algorithms change over time • Probability value may still change, but slightly • Distinct score response for each bullet/cartridge • Probability is a common concept for all exhibits • Effect of database size • Statistical models based on relevant data could quantify this effect • More than 1 score per bullet/cartridge • Compute a probability for each score and combine them to find a unique probability for the bullet/cartridge**How could we combine probabilities? Cartridge case**• Assume • we have a BF and a FP score for a pair of cartridge cases AND • the 2 following probabilities are known • P(FP): Confirmed match according to FP • P(BF): Confirmed match according to BF • 4 possible scenarios • Confirmed match according to BOTH FP and BF • Confirmed match according to FP ONLY • Confirmed match according to BF ONLY • Not a confirmed match**How could we combine probabilities? Cartridge case**(Cont’d) • FP/BF marks provide independent information • A combined probability is computed by assuming independent information • P Combined = 1 – (1-PBF)(1-PFP) • Results: • A mark with a low probability has no effect on the combined probability • As we add marks, the combined probability improves • Easy to generalize for 3 independent marks**How could we combine probabilities? Bullets**• The 4 bullets’ scores are not computed from independent information • Are computed from the same areas on the bullet • A combined probability cannot be computed by assuming independent information • Keep the highest probability only (conservative)**Conclusion: Part I**• The probability of being a match is a more meaningful concept than correlation score • Using probability solves all problems found with the interpretation of correlation scores • Probabilities of individual marks can be combined nicely • Challenge: Compute the probability of being a match for individual marks • Two main unknowns: • How to deal with the individual score response of each cartridge/bullet • How to predict the effect of database size**Part II Ballistic Probability Model**• Goal and constraints of the model • Hypothesis • Tests and results**Statistical model of scores: Goal & Constraints**• Project started in 2003 • Goal: Develop a model which • Converts the correlation score of a mark into a probability of being a match • Current constraints • We only have database of sister pairs • Tests with BulletTRAX-3D scores • The model should find the same performance as the large database study • As the database size increases, the probability to find a known match in the first position should decrease**Ballistic Statistical Model: Hypothesis**• Any mathematical or physical model starts with a small number of hypotheses/laws/axioms • Need hypotheses for the (3D bullet) ballistic model • Need to find something common to all bullet score distributions • However, each bullet has its own score response**Hypothesis (Cont’d)**• Non Match Statistical distribution • Experiment already discussed: Correlate 9LG bullets against the same large database (800 non matches) • Compare their non match score distribution (3D) • Differences • in the high score region • in the position of the peak • Similarity: • The distributions have a similar shape 9LG Bullet #1 9LG Bullet #2**Hypothesis (Cont’d)**• Core Hypothesis: The non match score distribution of all bullets • Has the same universal “shape” (up to a shift and stretch factor) • This shape is independent of calibre, material and quality of the marks Can be broken into two hypotheses • Hypothesis I: • The non match score distribution of each bullet is fully characterized by only two parameters: • its mean (position of the peak) • its width • Hypothesis II: • If we remove the effect of these 2 parameters, the non match score distributions of bullets are strictly identical • The effect of the 2 parameters is removed as follows • Shift the overall distribution at the same peak position for every bullet • Shrink or expand the overall distribution to get the same width for every bullet**Hypothesis (Cont’d)**• The effect of the 2 parameters is removed as follow • Shift the mean to 0 • Shrink to unit width • Get very similar distributions! • Small variations due to limited data 9LG Bullet #1 9LG Bullet #2**Ballistic Statistical Model: Testing the model**• 4 steps: • Compute 3D correlation scores from a large database study with BulletTRAX-3D • 4 calibers, 2 materials/compositions • Compute the individual parameters for each bullet (Hypothesis I) • mean and width of its non match score distribution • Determine a Universal Non Match score distribution (Hypothesis II) • By simulations, predict the performance of the correlation algorithm as a function of database size**Testing the model :Database General Information***Pittsburgh bullets database (Allegheny County Coroner’s Office Forensic Laboratory Division)**Testing the model :Compute individual parameters**• For each bullet • get an approximation of the universal distribution (Hypothesis II) • The scores are normalized by this process • For each bullet: • Mean and width are computed • The distribution is • Shifted the mean to 0 • Rescaled to unit width **Add up the “approximated” universal distributions found**for all bullets Smooth shape even in high score region Universal Normalized distribution for non match scores Testing the model :Define a “universal” non match distribution**Testing the model :Simulations**• The simulation reproduces the operations done in a real large database study • Real study (with sister pairs) • For each reference bullet • Introduce its known match in the database of size N • Compute all correlation scores between the reference and (N+1) bullets in the database • Find the rank of the known match • Compute the performance of the correlation algorithm (number of known matches at the first position)**Testing the model :Simulations (Cont’d)**• Simulation: • For each reference bullet • Select randomly N non match (normalized) correlation scores from the universal score distribution • Normalize the (known) score of its known match by using • the reference’s individual parameters (mean and width of its non match score distribution) • Introduce the normalized score of its known match in the (generated) non-match score list • Find the rank of the known match • Compute the simulated performance of the correlation algorithm • Repeat the same process for several databases sizes N**Testing the model Simulations (Cont’d)**Probability that the sister is at the first position as a function of its “normalized” score S • Dark circles: experimental data • Dark curve: Result from the model • Gray curves: Prediction for other database sizes 8**Testing the model Simulations (Cont’d)**Summary of the figure • If the sister has a “normalized score” = 8 • The probability to be in first position is • 90% for N = 500 • 70% for N = 2K • 20% for N = 10K • If we want the sister to be at the first position with a 95% probability, • its score must be • 9 for N = 500 • 10 for N = 2K • 12 for N = 10K**Part II: Summary**• A statistical model of non match scores was built • a database of 2000 bullets, 4 calibers, 2 compositions/materials • 3D correlation on BulletTRAX-3D • Hypothesis: • The non match score distribution has the same shape for all bullets (except for a shift and stretch factor) • The model computes the probability that the sister with a given score is in first position • The prediction agrees with the actual performance in the large database study • Performance decreases as the database size increases**Part III**How could we implement a probabilistic model in a ballistic system?**How could we implement a probabilistic model in a ballistic**system? • Correlate a given bullet against a large database • From the (large) list of scores, compute the two characteristic parameters of the reference bullet • mean and width of its non match score distribution • Compute the probability that the bullet in the first position is a match by using • The universal non match score distributions • Two characteristic parameters computed previously • Actual score of the bullet at the first position • Information about match score distributions (unknown yet)**How could we implement a probabilistic model in a ballistic**system? (Cont’d) • Repeat the same process for all score types • MaxPhase2D, PeakPhase2D, PeakScore2D • 3DScore • Combine the 4 probabilities into a unique probability for the bullet**Future work**• Improving the model with new large database studies (new calibers) • Test on cartridges • Get a better knowledge of sister score distributions • The current study was done with sister pairs only • Use the model to improve correlation algorithms

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