Measures of ruggedness using fuzzy-rough sets and fractals: Applications in medical time series

This paper attempts to characterize the medical time series by quantifying the ruggedness of the time series. The presence of two close data points on the time axis implies that these points are similar along the time axis. It creates the fuzzy similarity. Following the principle "similar causes create similar effects", we expect that the magnitudes corresponding to those two data points should also be similar. However, if other features are considered along with the time information, then those two apparently similar data points might look different. Consequently, when the other features are not considered, the magnitudes of those two similar points become different. It makes the time versus magnitude relationship one-to-many. Thus, the closeness creates the fuzziness, the one-to-many relationship creates the roughness, and together they form fuzzy-roughness. If the ruggedness is expressed as the fuzzy-roughness, then in some time series it is observed that the fuzzy-roughness of a part of the time series is similar to that of the whole time series. Specifically, the scaling up of the fuzzy-roughness follows the power law of fractal theory. Experiments on ICU data sets show that the ruggedness measure using the fuzzy-rough set based fractal dimension is more robust than the Hurst exponent, which is used frequently to measure the ruggedness of a fractal time series.