Constraint local principal curve: Concept, algorithms and applications

• We present the concept of Constraint Local Principal Curve (CLPC) to reduce the computational complexity.
• We propose three CLPC algorithms that combine local optimization and adaptive radius to expand the range of applications and increase the solution quality.
• We present some numerical experiments with three simulation data sets and two measured GPS data sets in highway and railway.
• The CLPC algorithms can improve the accuracy and computational speed compared with the existing KPC algorithms.
• The features of the each CLPC algorithm are analyzed according to the comprehensive experiments.

Existing principal curve algorithms have some drawbacks such as time consuming and narrow application scope in practice, since these algorithms are mainly based on global optimization. In this paper, we present the concept of Constraint Local Principal Curve (CLPC), which uses local optimization methods and restricts the principal curve with two fixed endpoints to reduce the computational complexity. In addition, we propose three CLPC algorithms by Local Optimization and Adaptive Radius to expand the range of applications and increase the solution quality. The first algorithm, i.e., CLPCg is based on greedy thinking. The second algorithm, i.e., CLPCs uses one dimensional search and the last algorithm CLPCc combines the greedy thinking and one dimensional search. Then, we define six performance indices to evaluate the performance of the CLPC algorithms. Finally, we present some numerical experiments with three simulation data sets and two GPS measured data sets in both highway and railway. The results indicate that all of the three CLPC algorithms can obtain high-accuracy data from multiple low-accuracy data efficiently. The CLPC algorithms can improve the accuracy and computational speed compared with the existing K-segment principal curve (KPC) algorithm. In addition, CLPCc outperforms CLPCg and CLPCs according to the comprehensive experiments while CLPCg runs much faster than other ones.