Fractional Brownian Series Research Articles

Self-similarity of higher-order moving averages

In this work, higher-order moving average polynomials are defined by straightforward generalization of the standard moving average. The self-similarity of the polynomials is analyzed for fractional Brownian series and quantified in terms of the Hurst exponent H by using the detrending moving average method. We prove that the exponent H of the fractional Brownian series and of the detrending moving average variance asymptotically agree for the first-order polynomial. Such asymptotic values are compared with the results obtained by the simulations. The higher-order polynomials correspond to trend estimates at shorter time scales as the degree of the polynomial increases. Importantly, the increase of polynomial degree does not require to change the moving average window. Thus trends at different time scales can be obtained on data sets with the same size. These polynomials could be interesting for those applications relying on trend estimates over different time horizons (financial markets) or on filtering at different frequencies (image analysis).

Sequences of predictive eye movements form a fractional Brownian series – implications for self-organized criticality in the oculomotor system

When human subjects are presented with a pair of visual targets that alternate periodically, they track the targets with rapid eye movements known as saccades. In previous work we demonstrated that at low pacing rates (<0.5 Hz), saccades have a latency of about 180 ms, and the latencies are uncorrelated from trial to trial. At high pacing rates (>0.6 Hz), latencies are much shorter: subjects make predictive saccades that anticipate target motion. The predictive latencies are correlated and appear to form a fractional Brownian motion. Here we confirm this finding by examining the rate of decay of nonlinear forecasting of predictive latencies. We further characterize the nature of predictive saccade latencies through the use of detrended fluctuation analysis and surrogate data. These results lead us to conclude that predictive saccades may exhibit a form of self-organized criticality, which enables rapid response to changes in stimulus timing. We provide an experimental demonstration of this.

Comparison of human performance with algorithms for estimating fractal dimension of fractional Brownian statistics

Five standard methods for estimating fractal dimension were compared by means of one-dimensional fractional Brownian series generated by four different algorithms yielding series with different statistics. The same one-dimensional series were also displayed as jagged lines and as one-dimensional luminance patterns for judgments by human observers. Only the algorithm implementing the maximum-likelihood method, which required that the generation statistic for the fractal series be known, gave better performance than human observers in estimating fractal dimension. However, when the method of generation is not known, one of the four other standard methods for estimating fractal dimension must be used, and these performed significantly worse than human observers.

Multiscale sources of spatial variation in soil. I. The application of fractal concepts to nested levels of soil variation

SummarySoil variation has often been considered to be composed of‘functional’ or ‘systematic’ variation that can be explained, and random variation (‘noise’) that is unresolved. The distinction between systematic variation and noise is entirely scale dependent because increasing the scale of observation almost always reveals structure in the noise. The white noise concept of a normally distributed random function must be replaced to take into account the nested, autocorrelated and scale‐dependent nature of unresolved variations. Fractals are a means of studying these phenomena. The Hausdorff‐Besicovitch dimension D is introduced as a measure of the relative balance between long‐ and short‐range sources of variation; D can be estimated from the slope of a double logarithmic plot of the semivariogram. The family of Brownian linear fractals is introduced as the model of ideal stochastic fractals.Data from published and unpublished soil studies are examined and compared with other environmental data and simulated fractional Brownian series. The soil data are fractals because increasing the scale of observation continues to reveal more and more detail. But soil does not vary exactly as a Brownian fractal because its variation is controlled by many independent processes that can cause abrupt transitions or local second order stationarity. Estimates of D values show that soil data usually have a much higher proportion of short‐range variation than landform or ground water surfaces. The practical implication is that interpolation of soil property values based on observations from single 10 cm auger observations will be unsatisfactory and that some method of bulking or block kriging should be used whenever longrange variations need to be mapped.