Abstract
Multifractal processes are key to modeling complex nonlinear systems. MITRE has applied fractal theory to agent-based combat simulations to understand complex behavior on the battlefield. The outstanding features of general fractal processes are long-range correlation and intermittency. If B is the lower band edge frequency of the high-pass signal component, the flatness function F(B), defined as the ratio of the fourth-order moment to the square of the second-order moment of a stationary process, is a measure of the intermittency or burstiness of a random process at small scales. If F(B) increases with no upper bound as B increases, then the process is intertmittent. In this work, we have derived an expression for the fourth-order structure function of the increments of a fractional Brownian motion (fBm) process through the use of integrals over the generalized multispectrum. It was concluded that the flatness function is independent of the lower edge of the high-pass signal component B, as expected of an fBm.
Approved for Public Release: 12-1203. This work was supported by MITRE internally funded research.
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Notes
- 1.
\(\delta (\frac{\alpha } {\tau } ) = \tau \delta (\alpha )\).
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Christou, C.T., Jacyna, G.M. (2013). On the Fourth-Order Structure Function of a Fractal Process. In: Andrews, T., Balan, R., Benedetto, J., Czaja, W., Okoudjou, K. (eds) Excursions in Harmonic Analysis, Volume 1. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston. https://doi.org/10.1007/978-0-8176-8376-4_14
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DOI: https://doi.org/10.1007/978-0-8176-8376-4_14
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