Fractal dimension and local intrinsic dimension

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Abstract
An algorithm to estimate the average local intrinsic dimension of an attractor is described. Of particular concern is the ability to estimate local attractor dimensionality at moderate values of signal-to-noise ratio (SNR). The algorithm is based on the estimation of the local intrinsic dimension (LID) of various randomly chosen regions around the attractor. The LID is determined by the number of significant singular values, or rank, of the data matrix for data points within the local region. The rank of this matrix is an integer and, in general, varies with position on the attractor. To obtain an overall estimate of the local attractor dimension a weighted average of all the local regions is calculated. Results are then compared to conventional estimates of fractal dimension. For high SNR results are in good agreement with the fractal dimension as calculated by the Grassberger-Procaccia algorithm. Limited test results indicate that the LID technique may be more robust when trying to estimate, or bound, the fractal dimension in the presence of moderate amounts of noise.

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