This applet produces a sample of Brownian Motion using midpoint displacement methods. This involves finding the midpoint of a line segment and then displacing it vertically up or down. Each midpoint value is represented as a random varianble of time in a probability space. Therefore the variance between samples is a factor in the vertical displacement. In the applet above, the initial variance of the space is entered and for each iteration (division of a line), and the variance for a particular midpoint displacement is calculated using the following formula:
As the initial variance gets larger, there would be more oscilations in the resulting curve. Also, as the initial variance gets smaller, the oscilations tend to decrease. To obtain a random vertical displacement, a Gaussian random number generator is used and the midpoint is then displaced with the following formula:
For the applet above, enter the number of iterations, and the initial variance. The number of iterations (n) indicates the number of sample of points to generate according to:
Similar to the applet above, this applet generates a sample of brownian motion, but this sample is derived using a generalization of the displacement variance. The generalized formula uses the Hurst exponent, H, which has values between 0 and 1. The Hurst exponent also allows the calculation of the fractal dimension of the Brownian motion sample:
Thus, this type of sample is called fractional Brownian motion. Similar to regular brownian motion , the displacement of the midpoint is dependent on variances and the random number generator. However, the variances are caluculated differently based on three arguments; the number of iterations, the initial variance, and the Hurst exponent.
In the fractional brownian motion applet, enter the number of iterations, the initial variance, and a value for the Hurst exponent (between 0 and 1 - eg. H=0.554). Click on the new sample button to generate a new sample with the field values filled.