Summary: We investigated 200 human skulls in order to define the variability of the piriform aperture. It was intended to demonstrate that this task can be managed by neural network. After recording the metric and non-metric data, descriptive analysis was applied, as well as hierachical cluster analysis and discriminant function. In addition to these conventional methods, a neural network of the Kohonen type (15 x 15 nodes) was established. The resulting classification does not contradict those of classic cluster analysis and discriminant analysis, but goes beyond them, due to the arrangement of the classes in two-dimensional dependencies, and furthermore by providing positions to intermediate forms not included in the original, net generating population. Therefore the neural network can be seen as a suitable method for the investigation of large collections of biological material and may be helpful in anatomy and anthropology as well as in forensic science for the identification of unknown material.
1.1. Variations of individual cases in a morphological collection bear important information of that object. This inherent information gives a more complete and general aspect of the object than any individual case from the collection taken by chance. An intuitive way to cope with this variability, and to demonstrate it, is to show instead of one example of several specific object types. Suitable methods to find types in sets are provided by modern information processing known as data mining. These methods take into account merely the measured data and omit any prerequisite. An armada of methods of so-called computational intelligence is aligned to data mining (Tsakonas et al. 2001). The method specified is determined by the kind of data (metric, metric in time-series, topological, or nominal), and whether a symbolic regression is needed, or heuristic rule-based schemes are applied (Berks and Graf von Keyserlingk 2000). It is common practice today to apply more than one computational intelligence approach either to ensure the results or to combine them in hybrid systems. The triple of cluster analysis, neural network, and factor analysis seems to be a suitable combination for anatomy and anthropology.
1.2. A neural net consists of a defined quantity of objects characterised by any number of variables described as a matrix of vectors. This matrix will then be folded so that every object or vector receives 6 neighbours in a hexagonal architecture or 8 neighbours in a quadrangular architecture instead of 2 in a linear arrangement. Of course the number of neighbours is lesser at the margins of the net.
1.3. The resulting structure, in the view from above, resembles a typical net, often also termed as "map" or "Kohonen map", where the vectors are no longer positioned in the level of representation. The objects are termed "knots" or "neurons", and the value of the vector is called the weight of the knot. The number of elements of vectors belonging to variables can be arbitrarily big.
1.4. If information is to be put into the net, this information must be transformed into vectors termed input vectors (Fig.1). These vectors are changing the net, which is called "training of the network". At the beginning, all elements of the vectors in the matrix of the net are combined with random numbers. Then an input vector is randomly chosen and compared with all other vectors in the net. The knot with the greatest similarity to the input vector will be searched. The specific knot with the greatest similarity is called the "winner neuron". If the winner neuron is found, an adaptational function will be processed with the winner neuron lying in the center.
1.5. Similarity can be defined mathematically in different ways: distance values from the Minkowski metric can be used as well as the scalar product. For our study we used the Minkowski metric. For adaptation of the surrounding, the Gaussian probability density function was used. The dynamic of the adaptation was controlled by the overall variance of the map. The adaptational function concerns two neighbouring knots and acts stronger if the knots are close together. The function determines the grade of adaptation between the input vector and the neighbouring vectors. Furthermore the working radius of the adaptational function will be determined. The adaptational function must be dynamic, because the circumstances of the net are changing during the training process. After the adaptation of the environment to the input vector, this vector has done its purpose. The next randomly chosen vector will then be put into the net and so on. The addition of vectors will be continued until a defined vector is positioned repeatedly at the same place. Now the information of the input vector is incorporated within the net. The vectors of the input matrix are lying disarranged together, but their position within the neural net is determined according to the "similarity" defined by the algorithms.
1.6. The aim of the investigation reported here, was to find prototypes and unique specimens of nose skeletons in a collection of recent human skulls and to demonstrate that the neural network yields a new method for analysing anatomical or anthropological material.
2.1. Twenty three anthropological parameters of the nose of 200 human skulls of the Aachen collection were used to train a neural net of the Kohonen type (Kohonen 1989). 98 skulls belong to male and 102 to female donors, all Central European persons. The median age was 76 years with a range between 36 and 100 years. At first, all specimens were characterised by their metric and non-metric parameters. As metric parameters, 23 anthropological parameters of the nose mainly according to Bräuer (1988) were used along with measurements of the area and the circularity. Non-metric parameters, such as the form of the nasal bones, were determined according to Martin & Saller (1959). Furthermore, the morphology of the caudal margin of the piriform aperture was classified according to Hovorka (1893) and the expression of the spina nasalis anterior was protocolled according to the scheme of Broca (1872).
2.3. A Kohonen map with 15 ×15 nodes was established. Measures of similarity in the Kohonen maps were done in the Minkowski metric. For adaptation of the surrounding the Gaussian probability density function was used. The dynamic of the adaptation was controlled by the overall variance of the map. The program was written in VBA for Microsoft Excel by ourselves. In addition, the data were exposed to a hierarchical cluster analysis, to evaluate the classification performed by the neural net (Everitt 1974, Kaufman and Rousseeuw 1990). Discriminant analysis was achieved to appreciate sex dimorphism. Cluster analysis, Discriminant analysis, and Factor Analysis were performed with SPSS for Windows Version 10.1 (Brühl and Zöfel 1998).
3.1. Figure 2 represents a Kohonen map with 15 × 15 nodes. The nodes marked black are those which were relatively frequently activated by nose skeleton vectors, namely as often as the number written at that place from 1000 trials taken by chance out of the whole population. If the number 55 is written this means that 5.5 % of the population activates this node. These activating skeletons may be considered as very similar to each other and therefore belong to the same type of nose skeleton. Only about 25% of the piriform aperture could be affiliated to distinct types if the threshold for a sound cluster is set to 20 activations per 1000.
3.2. Additionally nodes marked with grey background were very seldom activated, only one, two, or three times per 1000. Those nose skeletons which had rarely activated a node, must be considered as unique specimens; specimen 27/93, marked x in figure 2, is such an example, being very asymmetric (Fig. 3).
3.3. Figures 4 and 5 show the Kohonen maps during the learning phase so that a continuous spectrum of the activation is marked. Figure 4 demonstrates that the hierarchical cluster analysis and the Kohonen map classification are in accordance to a high degree. Class number two of the hierarchical cluster analysis corresponds to the nodes mainly activated by female skeletons, signed with number one in figure 4. The cluster 4 corresponds to nodes mainly activated by male skeletons in a female surrounding in the left upper corner of the Kohonen map.
3.4. Figure 5 represents the Kohonen map in the learning phase. Here the relation between activation of the nodes from male ( 2 ) or female skeletons ( 1 ) is demonstrated. ( 1 ) means activation only by female, ( 2 ) means activation by from male skeletons. The value in between means more or less from male or female specimens. If nodes are not marked in this map, they are not addressed by any of the nose skeletons of the applied population of 200 cases. The map in figure 2 points to the precise localisation of nodes activated by the specimens of skulls presented in figure 6 and the nose skeletons in figure 7. Notice the continuous change of the appearance of the piriform aperture from upper left to lower right: from a balanced relation of length and breadth, to a more pronounced breadth and then to a more pronounced length. The overall size of the nose skeletons is increased in the same way but this is not so evident in the pictures of figure 7 because of perspective distortion and in figure 6 because of the not standardised detail view.
4.1. In order to make the application of neural nets more general and foremost usable at multiple locations, the attribute "self-organising regarding orientation of the map" is a severe disadvantage, because maps from different sources could not be quantitatively compared, until now. To overcome this fault, we extended our software by two supplements: by map rotation and by map mirroring. If two maps are to be compared, one is rotated step by step and the discrepancy between the two is calculated as the sum of differences at each node. The best fit of the maps is in the position where this sum is minimal. In order to decide whether it is the absolute best correspondence or not, one has to apply the same procedure after mirroring one of the maps. The absolute minimum will be taken as indication for best fit.
4.2. To demonstrate the procedure, we took the sample of the measured nose skeletons (all 200 cases) and established a map from them (Fig. 8) and did the same for a random sample from them with only half the number of cases (Fig. 9).
4.3. It is obvious the second map has an individual topographical organisation, quite different from the first one. A rotation program is started (Fig. 10).
4.5. This example was only to illustrate the mechanism of the procedure. Until now, a non-conformity appears because simply calculating the differences between the types of the noses is a methodological simplificaton. In this way, the difference between type 1 and 2 appears to be of less importance than the difference between type 1 and 6. This seems by no means justified. Correspondence and non-correspondence should be better reduced to a yes or no answer. We will implement this in the next version of our program
4.6. The advantage of this quantitative approach is its suitability for comparisons with classical descriptive statistics. In future significance limits may be defined for maps to be considered as corresponding to each other or not. But at the moment more priority should be given to combine large collections from anatomical institutes to get a sound basis for anatomical and anthropological statements for what ever kind of material.
5.1. Classification by neural networks is suitable to characterise in great details the internal structure of a large population of a morphological structure, demonstrated here with the piriform aperture. The resulting classification does not contradict that of classic cluster analysis, but goes beyond that because of arrangement of the classes in two-dimensional dependencies and furthermore providing positions to intermediate forms, which were not included in the original, net generating population. Regarding the human nose skeleton studied here, the results points to the great individuality of the human nose, in contrast to the large bones of the human skeleton (for instance), as will be shown in a further study.
5.2. When presenting this kind of classification, frequently the question of validity of the results arises. The whole statistical methodology may be applied to the individual classes, of course, but the number of cases must be extremely increased to get reliable statements. This new approach may be helpful in legal and forensic medicine in order to determine unknown skeletal remains. It may be used in retrospective as well as in prospective ways. It also may be used for international standardisation.
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