According to Mr. Ogden we do not use or apply mathematics in our senior project. So I am here to prove him wrong and apply it directly into Forensic Anthropology. One approach to this problem is to focus on the analysis cranial Cartesian coordinate points instead of the distances or angles they form. This requires some pre-processing of the data whcih are catalogged by a Forensic Anthropological Institution/Laboratory to construct proper shape variables. However, these vectors will retain all geometric information that could be collected from the cranium. That pre-processing step usually involves the registration of the configurations of landmarks for all specimens into a common coordinate system using a least-squares estimation of location and orientation parameters and a reasonable size standardization. This approach, in which data from individual specimens are fit to an iteratively computed mean configuration, is called Generalized Procrustes Analysis.
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| Traditional measurements of these skulls are based on anatomical landmarks I discussed on a later blog post. Landmarks included are prosthion, nasion, bregma, lambda, inion, opisthion, and basion. The measurements are the distances between the nasion and basion and the angle formed is from three points from the basion. For other noted refernces, let the record know that the anatomical landmark positions are encoded as Cartesian coordinates. |
3 Dimentional Identification implements this approach to help characterize human remains by using cranial remains. To do this, the user provides the program with three-dimensional coordinates of a subset of the landmarks described above. A reference database is then processed to extract appropriate reference samples. Then, the unknown is compared to the groups available in the reference sample to estimate group membership. Separate groupings are considered for each sex, but if sex can be determined by other means, the comparisons can be easily restricted to only female or male groups. The details of how this is accomplished follow:
The f function is the probability density function for the unknown and the group specified in the subscript. With unequal sample sizes and estimated means and covariance structures it leads to the second line of equation. On the third equation, v is the dimensionality of the space of the Anthropological data, ni is the size of the ith reference group. Again, the generalized inverse is used by the program to address singularities and avoid instabilities. The second diagnostic measure provided by the program is a “typicality” measure. This is simply the probability of an observation being as far or farther away from the mean as the unknown for a particular group. Typicality measures how likely it is that the unknown came from a particular population at all. For instance, an unknown will always be suggested as belonging to one of the available reference groups, but typicality measures how likely that is to be true for any given population. That is, the unknown could be suggested for membership in one (the closest) population, but still be so different that the probability of actually finding a specimen that different from the population is small. Again in such cases, the suggested assignment should be taken with an appropriate degree of skepticism. Typicality is computed by finding the probability of: for an F distribution with v and nf – v +1 degrees of freedom, F(v, nf – v + 1). In general, then, the program will suggest an assignment to the group whose mean is closest in the Mahalanobis sense to the unknown. Posterior probabilities can be used to assess how strong the evidence is this assignment versus that for assignment to other reference groups. The typicality can be used to assess how likely the unknown is to have come from a particular population regardless of how much closer it is to it than the other populations or how much that difference is similar to other such differences.
With this equation I received the result of:
rs:1904727279 through rs:184877381
H(1)=V(1)S(1)U(1)
H= 0.9948
G(1)=0.9848 G(2)= 0.9924
A=0.3485 G= 0.3505
A= 0.0606 G=0.9394
A=0.0076
G=0.9924
When entered these results onto the cranium software it depicted these images:
What this indicates from my observation is that even mathematical computations will result hand-in-hand with scientific analysis that Forensic Anthropologists come up with. The results I received from comparing cranial deformation and by solving the mathematical formulas I was able to answer the question of who the individual was. In this case a Mesoamerica (possibly of Mexican origin/ancestral background) between the age of his late 30s (39 years old) to early 40s (42 years old). This accuracy can be able to be matched with the missing person and create a scientific phenomena.
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