Ant colony optimization for determining the optimal dimension and delays in phase space reconstruction

The selection of parameters in time-delay embedding for phase space reconstruction is crucial to chaotic time series analysis and forecasting. Although various methods have been developed for determining the parameters of embedding dimension and time delay for uniform embedding, the study of parameter selection for non-uniform embedding is progressed at a slow pace. In a non-uniform embedding which enables different dimensions in the phase space to have different time delays, the optimal selection of time delays presents a difficult optimization problem with combinatorial explosion. To solve this problem, this paper proposes an ant colony optimization (ACO) approach. The advantages of ACO for the embedding parameter selection problem are in two aspects. First, as ACO builds solution in an incremental way, it does not need to use a fixed embedding dimension as the encoding length of a solution. Instead, both the embedding dimension and the time delays can be optimized together. Second, ACO enables the use of problem-based heuristics. Therefore heuristics designed based on the original observed time series can be used to accelerate the search speed of ACO. Experimental results show that the proposed algorithm is promising.