Lacunarity of the zero crossings of Gaussian processes

A lacunarity analysis of the zero crossings derived from Gaussian stochastic processes with oscillatory autocorrelation functions is evaluated and reveals distinct multiscaling signatures depending on the smoothness and degree of anticorrelation of the process. These bear qualitative similarities and quantitative distinctions from an oscillatory deterministic signal and a Poisson random process both possessing the same mean interval size between crossings. At very small and large scales compared with the correlation length of the random processes, the lacunarity is similar to the Poisson but exhibits significant departures from Poisson behavior if there is a zero-frequency component to the process's power spectrum. A comparison of exact results with the gliding box technique that is frequently used to determine lacunarity demonstrates its inherent bias.

Figure 1

A comparison between the exact and simulation estimates of $\mathrm{var}\left[n\right(r\left)\right]$ as a function of normalized box-size for the $g_1$ correlation function.

Figure 2

Lacunarity and its dimensionless slope (inset) for processes with autocorrelation functions $g_i$ together with the Poisson case for reference.

Figure 3

Plots of the interval variance as the periodicity parameter $a$ increases for processes with autocorrelation functions (a) $g_1\left(\tau \right)cos\left(a\tau \right)$ and (b) $g_1\left(\tau \right){cos}^2(a\tau /\sqrt{2})$. Dashed red lines indicate reference scalings of the simulation variance at large $a$.

Figure 4

Lacunarity curves for processes with oscillatory autocorrelation functions (a) $g_i\left(\tau \right)cos\left(a\tau \right)$ and the deterministic signal with the same period and (b) $g_i\left(\tau \right){cos}^2(a\tau /\sqrt{2})$ with the nonoscillatory Poisson curve, when $a=10$. Insets show the periodicities that occur when box sizes match multiples of the mean interval length. Within the main plot in (a), for $\overline{R}r\ge 7/2$ the peak-to-peak lacunarity values for the deterministic signal are plotted as a dashed gray line. The inset in (a) shows that peaks of the full lacunaity curve occur when $r$ is an odd multiple of ${\overline{R}}^{-1}$.

Figure 5

Lacunarity and dimensionless slope curves for processes with oscillatory autocorrelation functions $g_i\left(\tau \right)cos\left(a\tau \right)$ [(a) and (c)], and $g_i\left(\tau \right){cos}^2(a\tau /\sqrt{2})$ [(b) and (d)], when $a=100$, compared with the deterministic signal with the same period and the nonoscillatory Poisson process. Within the main plot in (a), for $\overline{R}r>7/2$ the peak-to-peak lacunarity values for the deterministic signal are plotted as a dashed gray line, and in (c) the corresponding lacunarity slope as a solid gray line. The inset in (a) shows that peaks of the full lacunaity curve occur when $r$ is an odd multiple of ${\overline{R}}^{-1}$.

Figure 6

Sequences [(a) and (c)] illustrating the correlation of zero-crossing intervals and sample functions [(b) and (d)] showing the different types of periodicity, when $a=100$. Plots (a) and (b) are for the process ${\rho }_{11}$, and horizontal lines in (a) denote 1.5 standard deviations above (orange) and below (yellow) the mean. Plots (c) and (d) are for the process ${\rho }_{12}$, and the crossings flip between the two branches that are symmetrically disposed about the interval mode (horizontal black line), excursions greater than $\sqrt{2}\langle T\rangle$ occurring off-axis. Vertical dashed lines in plots (a) and (c) indicate the sections of the corresponding sample functions shown in (b) and (d), and the two red circles in (d) represent an interval of size $\approx 25\langle T\rangle /\sqrt{2}$.