Two-dimensional space

The Kohonen network or self-organizing map (SOM) was developed by Teuvo Kohonen [11]. It can be used to classify a set of input vectors according to their similarity. The result of such a network is usually a two-dimensional map. Thus, the Kohonen network is a method for projecting objects from a multidimensional space into a two-dimensional space. This projection keeps the topology of the multidimensional space, i.e., points which are close to one another in the multidimensional space are neighbors in the two-dimensional space as well. An advantage of this method is that the results of such a mapping can easily be visualized. 

Consider a vector as an arrow in two-dimensional space. Now superimpose x-y coordinates on the 2-space, arbitrarily placing the origin on the taiP of the arrow. 

This kind of matr ix is called a Hessian matrix. The derivatives give the cmvatme of V(x[,X2) in a two-dimensional space because there are two masses, even though both masses are constrained to move on the -axis. As we have already seen, these derivatives are pari of the Taylor series expansion... 

If this can be done in a two-dimensional space, it can (in principle) be done in an n-space. 

These six matrices can be verified to multiply just as the symmetry operations do thus they form another three-dimensional representation of the group. We see that in the Ti basis the matrices are block diagonal. This means that the space spanned by the Tj functions, which is the same space as the Sj span, forms a reducible representation that can be decomposed into a one dimensional space and a two dimensional space (via formation of the Ti functions). Note that the characters (traces) of the matrices are not changed by the change in bases. 

Multidimensional planar chromatographic separations, as we have seen, require not only a multiplicity of separation stages, but also that the integrity of separation achieved in one stage be transferred to the others. The process of separation on a two-dimensional plane is the clearest example of multidimensional separations. The greatest strength of MD-PC, when properly applied, is that compounds are distributed widely over two-dimensional space of high zone (peak) capacity. Another... 

The important aspect of this problem is that while the penultimate model involves a four dimensional parameter space, the model discrimination problem can be reduced to a two dimensional space by dealing with functions of the original parameters. This approach requires that probabilities for array locations in the four dimensional r, r/, rj, rj ) space be mapped to array locations in the ((ri-r/), (rj-rj )) space. 

For instance, the first row of the matrix X defines a point with the coordinates (x, y,) in the space defined by the two orthogonal axes = (I 0) and = (01). Factor rotation means that one rotates the original axes = (1 0) and = (0 1) over a certain angle 9. With orthogonal rotation in two-dimensional space both axes are rotated over the same angle. The distance between the points remains unchanged. 

The line L divides the two-dimensional space in two regions (corresponding to the two classes). All points or objects (combinations of x, and X2) below the line yield a value for the function, which we will call NET(xj,X2) (eq. (44.2)), larger (smaller) than 0 the points above the line yield a value of NET(xi,X2) smaller (larger) than 0 the points on the line satisfy eq. (44.1) and thus yield a value, 0. 

A scatter plot provides a quick visual summary of where numerous data points exist in two-dimensional space. Scatter plots are commonly used to investigate positive or negative correlation between two variables plotted on the X and Y axes. 

Fig. 8.6. Representation of 14 multivariately characterized objects in a two-dimensional space of variables where the clusters are connected into four groups (a) and classified into two differently chosen groups (b,c), respectively d shows a nested clustering of B within A... 

A one-dimensional SOM is less effective at filling the space defined by input data that cover a two-dimensional space (Figure 3.22) and is rather vulnerable to entanglement, where the ribbon of nodes crosses itself. It does, however, make a reasonable attempt to cover the sample dataset. 

As we will see in more detail in Chapter 2, three vectors in a two-dimensional space cannot be independent and there exists an infinity of relationships such that... 

Rotation matrices can be defined for an arbitrary number of dimensions. They are particularly useful to examine compositional data in three-dimensional spaces in search for regularities unsuspected in two-dimensional spaces. Commercial software (e.g., Systat ) exists that produces geometric transformations in a convenient way. 

Figures 8a and 8b present the simulated current transients obtained from the self-affine fractal interfaces of r/ = 0.1 0.3 0.5 and r] = 1.0 2.0 4.0, respectively, embedded by the Euclidean two-dimensional space. It is well known that the current-time relation during the current transient experiment is expressed as the generalized Cottrell equation of Eqs. (16) and (24).154 So, the power exponent -a should have the value of - 0.75 for all the above self-affine fractal interfaces.

Figure 9. Dependence of the scaled length SL on the projected segment size SS on a logarithmic scale obtained from the self-affine fractal profiles in Figure 7 by using the triangulation method for Euclidean two-dimensional space. Flere, S means d In SL / d In SS. Reprinted from H. -C. Shin et al., A study on the simulated diffusion-limited current transient of a self-affine fractal electrode based upon the scaling property, J. Electroanal. Chem. 531, p. 101, Copyright 2002, with permission from Elsevier Science.

The above operation is iterated at various segment sizes. Finally, the self-similar fractal dimension of the profile embedded by the two-dimensional space is given by... 

FIGURE 2.17 Projection of the object points from a two-dimensional variable space on to a direction b45 giving a latent variable with a high variance of the scores, and therefore a good preservation of the distances in the two-dimensional space. 

Then he came to extending the division of continuous two-dimensional space into the third dimension. He restricted his examinations to polyhedra and found one of the five space-filling parallelohedra, which were discovered by E. S. Fedorov as capable of filling the space in parallel orientation without gaps or overlaps. Fedorov was one of the three scientists who determined the number (230) of three-dimensional space groups. The other two were Arthur Schoenflies and the amateur William Barlow. 

A two-dimensional space-filling structure is obviously present but is not easily discussed in terms of "strands and "crosslinks." Continued on next page. 

---