• almost surely;
  • Bayesian estimation;
  • Brownian motion;
  • fractal;
  • fractal Brownian motion;
  • fractional Brownian motion;
  • fractal dimension;
  • fractal Gaussian noise;
  • fractional Gaussian noise;
  • functional;
  • Hausdorff-Besicovitch dimension;
  • Hurst parameter;
  • Kalman filter;
  • loss function;
  • measurable;
  • quadratic mean integral;
  • sample path integral;
  • self-similar;
  • wide-sense stationary


Fractal Brownian motion, also called fractional Brownian motion (fBm), is a class of stochastic processes characterized by a single parameter called the Hurst parameter, which is a real number between zero and one. fBm becomes ordinary standard Brownian motion when the parameter has the value of one-half. In this manner, it generalizes ordinary standard Brownian motion. Here, we precisely define fBm, compare it with Brownian motion, and describe its unique mathematical and statistical properties, including fractal behavior. Ideas of how such properties make these stochastic processes useful models of natural or man-made systems in life are described. We show how to use these processes as unique random noise representations in state equation models of some systems. We finally present statistical state equation estimation techniques where such processes replace traditional Gaussian white noises. WIREs Comp Stat 2011 3 149–162 DOI: 10.1002/wics.142

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