Fig. 1: Did you realize that all of the three figures show the same lattice model? If you have doubts, go and see for yourself. Click into the applet above and move your mouse. You are now able to rotate the lattice model as you like.
This is a striking example of how the gathering of information is often a matter of gaining the optimal vantage point. The statistical method of component analysis helps to determine this optimal vantage point, not only in two-dimensional (see figure below) and three-dimensional problems, but also in principle in data spaces of arbitrary dimensions.
Fig. 2: This diagram shows the same type of problem in a two-dimensional space. A cloud of data points is shown in two dimensions, and the density plot formed by projecting this cloud onto each of two axes, 1 and 2, are indicated. The projection onto axis 1 has maximum variance, and clearly shows the bimodal, or clustered character of the data.
Tiny software components are capable of finding such optimal representations of complex data spaces completely on their own. The mechanisms at work in this connection are not only very simple but are also more efficient than the calculation by means of traditional methods.
The upcoming article »Hebbian Neural Maximum Eigenfilters« is intended for mathematically trained readers and demonstrates the possibilities and efficiency of self-organizing neural structures.