A new principal curve algorithm and standard deviation clouds for non-parametric ordered data analysis

Principal curves are a study of the underlying structure of a data cloud. We modify Kegl´s [2] polygonal line algorithm by assuming that data points are vertices on different continuous curves which implies data ordering. We also develop a representation of curve deviation from the polygonal path by creating a deviation cloud based on computing a measure of the variance of the curves from the polygonal path. For the purposes of this paper, we consider the input curves to be vertex representations of independent polygonal paths. Comparisons of the presented algorithm on various data sets with that of Verbeek et al. [3] are given to illustrate differences when using ordered data represented as multiple continuous curves. We further consider applications of this algorithm to the evaluation of multiobjective optimization algorithm convergence for biobjective optimization. We present preliminary results for NSGA-II on ZDT1, ZDT2, and ZDT3 in order to show how this methodology could be used.