Data Mining - Mehmed Kantardzic [256]
15.4 RADIAL VISUALIZATION
Radial visualization is a technique for representation of multidimensional data where the number of dimensions are significantly greater then three. Data dimensions are laid out as points equally spaced around the perimeter of a circle. For example, in the case of an 8-D space, the distribution of dimensions will be given as in Figure 15.7.
Figure 15.7. Radial visualization for an 8-dimensional space.
A model of springs is used for point representation. One end of n springs (one spring for each of n dimensions) is attached to n perimeter points. The other end of the springs is attached to a data point. Spring constants can be used to represent values of dimensions for a given point. The spring constant Ki equals the value of the ith coordinate of the given n-dimensional point where i = 1, … , n. Values for all dimensions are normalized to the interval between 0 and 1. Each data point is then displayed in 2-D under condition that the sum of the spring forces is equal to 0. The radial visualization of a 4-D point P(K1, K2, K3, K4) with the corresponding spring force is given in Figure 15.8.
Figure 15.8. Sum of the spring forces for the given point P is equal to 0.
Using basic laws from physics, we can establish a relation between coordinates in an n-dimensional space and in 2-D presentation. For our example of 4-D representation given in Figure 15.8, point P is under the influence of four forces, F1, F2, F3, and F4. Knowing that every one of these forces can be expressed as a product of a spring constant and a distance, or in a vector form
it is possible to calculate this force for a given point. For example, force F1 in Figure 15.8 is a product of a spring constant K1 and a distance vector between points P(x, y) and D1(1,0):
The same analysis will give expressions for F2, F3, and F4. Using the basic relation between forces
we will obtain
Both the i and j components of the previous vector have to be equal to 0, and therefore:
or
These are the basic relations for representing a 4-D point P*(K1,K2,K3,K4) in a 2-D space P(x, y) using the radial-visualization technique. Similar procedures may be performed to get transformations for other n-dimensional spaces.
We can analyze the behavior of n-dimensional points after transformation and representation with two dimensions. For example, if all n coordinates have the same value, the data point will lie exactly in the center of the circle. In our 4-D space, if the initial point is P1*(0.6, 0.6, 0.6, 0.6), then using relations for x and y its presentation will be P1(0, 0). If the n-dimensional point is a unit vector for one dimension, then the projected point will lie exactly at the fixed point on the edge of the circle (where the spring for that dimension is fixed). Point P2*(0, 0, 1, 0) will be represented as P2(−1, 0). Radial visualization represents a nonlinear transformation of the data, which preserves certain symmetries. This technique emphasizes the relations between dimensional values, not between separate, absolute values. Some additional features of radial visualization include:
1. Points with approximately equal coordinate values will lie close to the center of the representational circle. For example, P3*(0.5, 0.6, 0.4, 0.5) will have 2-D coordinates P3(0.05, −0.05).
2. Points that have one or two coordinate values greater than the others lie closer to the origins of those dimensions. For example, P4*(0.1, 0.8, 0.6, −0.1) will have a 2-D representation P4(−0.36, −0.64). The point is in a third quadrant closer to D2 and D3, points where the spring is fixed for the second and third dimensions.
3. An n-dimensional line will map to the line or in a special case to the point. For example,