Math in forensics: Sherlock Holmes on steroids
Probability and statistics
The following are excerpts taken from “Essential Mathematics and Statistics for Forensic Science” by Craig Adam (Wiley-Blackwell)
Analysis and prediction using probability and statistics are becoming more a part of law enforcement investigation around the world. Fingerprints and DNA are only the tip of the iceberg. A forensic scientist’s work is different from other analytical scientists in that the results of the work is reported to a court of law, with the forensic scientist being the expert witness and subject to cross-examination.
Let’s look at the probabilistic basis for interpreting evidence. The probability of the evidence being at the crime scene (E) given the guilt of the accused may be written as P(E|G). We need to compare the alternative probability of the evidence given that the accused is innocent or not guilty (Ḡ), expressed as P(E|Ḡ). If the ratio of these alternatives is greater than one, then the proposition based on guilt is weighted more than that based on innocence and vice-versa.
Here, the likelihood ratio (LR) is created:
Now we need to consider each of these hypotheses to ensure that they are mutually exclusive and to facilitate the correct evaluation of each probability given specific data. So we now a more general for the LR where the mutually exclusive propositions H1 and H2 are to be defined:
Here H1 is the hypothesis that the prosecution would present and H2 for the defense, where H1 and H2 are mutually exclusive. Chapter 11.2.1 “Estimation and calculation of likelihood ratios” shows some concrete scenarios and examples. Then, the book moves on to Likelihood ratio, Bayes’ Rule and weight of evidence. Population data and interpretive databases in which frequency histograms are constructed from large amounts of data that describe the occurrence of some measurable characteristic within a certain specified population.
I will not go into the details here, but suffice it to say that most basic calculations are straightforward. The problem comes in when trying to understand how these methods should be applied and finding the meaning of the result. The referenced book explores these issues qualitatively using case studies as illustrations.
Angles and triangles
Blood spatter can be analyzed by trigonometry. Measuring the angles and distances and then calculating the third point of the triangle----the location of the tracker, is commonly done.
Ballistics trajectories like those from a shotgun can determine distance from the victim to the shooter, the shooter’s approximate height and information about the type of shotgun.
Blood stain formation
Math can determine, using fluid dynamics and equations, details about whether the blood is a droplet or a high velocity impact-based spatter. The direction of the impact can also be determined. Two key equations are used for the stain diameter and number of spines respectively:
D is the blood droplet diameter
v is velocity
ρ is density
η is viscosity
γ is surface tension
Methods for graphical analysis can be verified from these formulae to make calculations and predictions within the range of test data and to some extent outside of it.
A really good reference paper is “Deducing Drop Size and Impact Velocity
from Circular Bloodstains” from the Journal of Forensic Science
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