Reconstruction of noisy system correlation dimension

• Main Authors: Diudin, M. S., Kalaidin, E. N.
• Format: Article
• Language: English
• Published: Saratov State University 2020-04-01
• Series: Известия высших учебных заведений: Прикладная нелинейная динамика
• Subjects: correlation dimension; correlation integral; additive noise
• Online Access: https://andjournal.sgu.ru/sites/andjournal.sgu.ru/files/text-pdf/2020/04/diudin.pdf
• ISSN: 0869-6632; 2542-1905

Purpose. Measurement of correlation dimension is considered in a dynamic system with additive random noise. To determine the correlation dimension correctly, it is necessary to eliminate the shift of horizontal coordinate in the graph of the correlation integral caused by the increase in distance between the points due to addition of random noise. Methods. To calculate the correlation dimension of a dynamic system, it is proposed to use the Grassberger–Procaccia algorithm and then change the calculation results according to the properties of the dynamics’ random component. When an additive normally distributed random noise is added, the distances between attractor points (calculated using the Euclidean norm) become random values distributed over a non-central χ-distribution that has a mathematical expectation greater than the distance before the noise was added. Results. Eliminating the shift of horizontal coordinate allows to get a flat section on the plot of the local slope of correlation integral, which coincides in vertical coordinate with the same section for a system without added noise. Local slope value for this section is close to fractal dimension of the systems under study. Conclusion. Proposed algorithm allows to accurately measure the correlation dimension of dynamic systems with additive (observable) random noise, to refine the estimate of the standard deviation of random noise obtained by other methods.