Machine learning-based estimation of correlation dimension from short chaotic time series

Fractal dimension is an important tool for describing complex systems, while correlation dimension is a type of fractal dimension of time series. It can give a more accurate understanding of the characteristics of the system and plays an important role in practical applications. To address the challenge of accurately calculating the correlation dimension of short chaotic data, this paper explores a machine learning-based approach. This method leverages the universal approximation theory of neural networks to compare the Long Short-Term Memory (LSTM) network, the Transformer architecture, and the Backpropagation (BP) network in deep learning, and then selects the optimal approach. Ultimately, the method integrates the backpropagation (BP) neural network with a genetic programming (GP) algorithm for the calculation of the correlation dimension, thereby constructing a novel model. This method aims to expand the volume of chaotic short data and thereby better utilize data information. Initially, a BP neural network with ten hidden layer neurons is constructed, combined with the calculation of correlation dimension to form a new method. Subsequently, simulations are conducted on various commonly encountered chaotic systems, including Lorenz, Hénon, Chen, Logistic systems, solar activity time series and daily female births time series. And their mean squared error (MSE) values are recorded after data augmentation. Finally, the correlation dimensions of these expanded data sets are recalculated and compared with their original values. The findings reveal a remarkable proximity to the theoretical values, thereby validating the method's efficacy in augmenting chaotic short-term sequences. This approach not only enhances the utilization of original chaotic time series data but also paves the way for extracting deeper insights and greater value from short chaotic data.