Characterization of medical time series using fuzzy similarity-based fractal dimensions

This paper attempts to characterize medical time series using fractal dimensions. Existing fractal dimensions like box, information and correlation dimensions characterize the time series by measuring the rate at which the distribution of the time series changes when the length (or radius) of the box (or hypersphere) is changed. However, the measured dimensions significantly vary when the box (or hypersphere) position is changed slightly. It happens because the data points just outside the box (or hypersphere) are not accounted for, and all the data points inside the box or hypersphere are treated equally. To overcome these problems, the hypersphere is converted to a Gaussian, and thus the hard boundary becomes soft. The Gaussian represents the fuzzy similarity between the neighbors and the point around which the Gaussian is constructed. This concept of similarity is exploited to propose a fuzzy similarity-based fractal dimension. The proposed dimension aims to capture the regularity of the time series in terms of how the fuzzy similarity scales up/down when the resolution of the time series is decreased/increased. Experiments on intensive care unit (ICU) data sets show that the proposed dimension characterizes the time series better than the correlation dimension.