Analysis of Local Space/Time Statistics and Dimensions of Attractors Using Singular Value Decomposition and Information Theoretic Criteria

Abstract

An algorithm to estimate the average local intrinsic dimension (<LID>) of an attractor using information theoretic criteria is explored in this work. Using noisy sample data the <LID> is computed from an eigenanalysis of local attractor regions, indicating the local orthogonal directions along which the data is clustered. Singular value decomposition (SVD) is used to calculate the eigenvalues as well as to determine the rank of the local phase space data matrix. The <LID> algorithm requires the separation of signal eigenvalues, i.e. the dominant eigenvalues, from the noise eigenvalues for which thresholding mechanisms based on two information theoretic criteria are used. The two information theoretic criteria which we consider for signal/noise separation are the Akaike Information Criterion (AIC) and the Minimum Description Length (MDL) of Rissanen and Schwarz. Several test cases are presented. Results are then compared to the correlation dimension, or fractal dimension, as computed by the Grassberger-Procaccia method. The MDL separation technique produces results in good agreement with the correlation dimension in the range of signal-to-noise ratio (SNR) between 5 to 12 dB. Also, <LID> is calculated in two different ways, as either a spatial average, <LID>_s, or as a temporal average, <LID>_t. <LID>_s is computed by averaging LID values over randomly selected local regions on the attractor. <LID>_t is computed by first generating consecutive data sets and then averaging LID values over local regions which are restrained to be in approximately the same local vicinity of the attractor. In this way we may compare LID statistics over time and space separately. In addition, we consider the effects of the sampling time, correlation time, window size, and resolution on the <LID> results.