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Keith D White, Jianbo Gao, Yihui Zhou; Fractal statistics of perceptual switching time series. Journal of Vision 2003;3(9):53. doi: 10.1167/3.9.53.
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© ARVO (1962-2015); The Authors (2016-present)
We examined perceptual switching during ambiguous depth perception, with a Necker cube (10 subjects); during ambiguous motion perception, with the Boneh et al rotating ball (56 subjects); and during binocular rivalry, with small moving gratings (56 subjects). Autocorrelation analysis of perceptual dominance time series showed that successive response durations are essentially independent, as has long been known, on a time scale of 20 sec to 1 min (10+ responses). However, variance-sample size analysis (VSS), data shuffling, and log-normal distributional properties all show that these time series behave like 1/f noise with long range correlations. VSS detects lack of response independence for a range from roughly 30 to 100 responses (time scale on the order of 2 to 10 min) as evidenced in three ways. First, the Hurst parameter (an index of fractal self-similarity derived from second-order statistics) was as large as 0.84. A stochastic process theoretically has this Hurst parameter value = 0.5 while a predictable process, such as a sine wave, has H(2) = 1.0. Secondly, shuffling the perceptual dominance periods of each time series into randomized order and then repeating the VSS analysis showed that variances in the naturally ordered time series were significantly larger than corresponding variances in the shuffled time series. This was expected given their Hurst correlations, and it was was shown empirically by F and binomial tests. Thirdly, the histograms of H(2) for shuffled time series, randomly reordered from 1000 to 9000 times each for a subset of the data, are sharply peaked near 0.50 with a standard deviation less than 0.10. This empirically estimated sampling distribution for H(2) shows that many of the natural time series have Hurst correlations unlikely to have happened by chance in random time series. Power spectral densities of these time series and the good fits of theoretically expected log-normal probability density functions to data histograms further support the method of multifractal analysis.
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