Spiral structures are one of the most difficult patterns to classify. Spiral time series data has a helical movement with time that is both difficult to predict as well as classify. This paper focusses on how structural information about spirals can be useful in providing critical information to a neural network for their recognition. Results are presented on neural network solutions to the classical two-spiral problem by extracting structural and rotational information from the spiral training data. The results show that in both two and three dimension, the spirals can be easily recognised by neural networks if they are trained on the temporal structural changes.
1. SPIRAL STRUCTURES
In our previous paper in Pattern Recognition Letters it was highlighted that spiral classification is a difficult problem which can be reasonably well solved using a fuzzy nearest neighbour classifier. Our earlier reservation for not using a neural network was that neural networks have convergence problems with raw coordinate data as inputs for recognising spirals. The two spiral benchmark from Carnegie Mellon repository in particular has been highly researched and it has been found that sophisticated neural networks have difficulty in classifying spirals with reasonable accuracy. In this paper we summarise the problem in short again for new readers and develop an input selection method for neural networks that allows them to recognise spirals in two or more dimensions with relative ease.
Spiral data is found in several natural and physical domains. The classic double helix DNA, the motion of particles in cyclotrons, spiral feed in manufacturing, spiral galaxies and spiral movement of financial stocks are some of the well-known examples. Spirals are particularly intriguing because of their high levels of non-linearity and resistance to shape transformation under rotation, translation or other scalar operations. Spirals structures are also attractive for their temporal properties and are found to be particularly hard to classify. For recognition purposes, spiral recognition problems are specially attractive since we can manipulate their complexity with relative ease for testing classifier efficiency.
The Carnegie Mellon benchmark can generate a spiral structure with the parameters: user defined density, maximum radius allowed and the total number of points needed. If a total of N data points are to be generated, then the spiral shape parameters change as follows, 1≤ i ≤ N: angle = ( i.π) / (16.density ) radius = maxRadius.((104.density) - i) / (104.density)) ...(1) ...(2)
x = radius.cos(angle) y = radius.sine(angle)
where density and maxRadius parameters are pre-defined, i is time, and π= 3.14.
The two spiral classification problem is shown in Figure 1. The two spirals coil around the origin and are symmetrical in nature. The two spirals are governed by two parameters: density ϕ and radius σ. The density variable defines the total number of points generated within an envelope defined by the radius. Data belonging to two different classes lie on these two different spirals (represented as a sequence of white and black circles in Figure 1). By manipulating spiral parameters, it is possible to generate different spirals with varying radius and length. The main advantage of using equations 1-4 for simulating spiral behaviour is very important for managing complexity of the problem. This artificial problem can be made complex (by either decreasing the maximum radius for the same number of points or increasing density for the same maximum radius), or made easy by doing the reverse. By manipulating the problem complexity, a more general set of tools can be explored. Figure 1. 2D Spiral data scatterplot....