Abstract
In this paper we propose a numerical method to estimate the fractal dimension of stationary Gaussian stochastic processes using the random Euler numerical scheme and based on an analytical formulation of the fractal dimension for filtered stochastic signals. The discretization of continuous time processes through this random scheme allows us to find, numerically, the expected value, variance and correlation functions at any point of time. This alternative method for estimating the fractal dimension is easy to implement and requires no sophisticated routines. We use simulated data sets for stationary processes of the type Random Ornstein Uhlenbeck to graphically illustrate the results and compare them with those obtained whit the box counting theorem.
Highlights
A common practice when trying to measure irregular shapes, such as the perimeter of an island or the length of a coastline, is to use Euclidean geometry, ignoring the fact that these shapes do not correspond to the ones of ideal objects, such as polygons and circles, whose dimension is an integer
This section graphically illustrates some experimental numerical results related to the calculation of the fractal dimension for stationary Gaussian stochastic processes associated with random initial value problems (21) and (24)
This paper made a slight extension of the approach proposed in [14] for the numerical estimation of the fractal dimension of ergodic stationary Gaussian stochastic processes associated with the random Ornstein Uhlenbeck process
Summary
A common practice when trying to measure irregular shapes, such as the perimeter of an island or the length of a coastline, is to use Euclidean geometry, ignoring the fact that these shapes do not correspond to the ones of ideal objects, such as polygons and circles, whose dimension is an integer. The dimension of irregular shapes is a non-integer dimension known as the Hausdorff-Besicovitch dimension. In the early twentieth century, Lewis Fry Richardson found a relationship that allows determining the value of a constant that indicates the degree of roughness of a coast or a geographical border, and used it to calculate the border length between several countries [1].
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