Stochastic models and statistical analysis for clock noise

Abstract

In this primer we first give an overview of stochastic models that can be used to interpret clock noise. Because of their statistical tractability, we concentrate on fractionally differenced (FD) processes, which we relate to fractional Brownian motion, discrete fractional Brownian motion, fractional Gaussian noise and discrete pure power-law processes. We discuss several useful extensions to FD processes, namely, composite FD processes, autoregressive fractionally integrated moving average (ARFIMA) processes and time-varying FD processes. We then consider the statistical analysis of clock noise in terms of how these models are manifested via the spectral density function (SDF) and the wavelet variance (WV), the latter being a generalization of the well-known Allan variance. Both the SDF and the WV decompose the process variance with respect to an independent variable (Fourier frequency for the SDF, and scale for the WV); similarly, judiciously chosen estimators of the SDF and WV decompose the sample variance of clock noise. We give an overview of the statistical properties of estimators of the SDF and the WV and show how these estimators can be used to deduce the parameters of stochastic models for clock noise.

Wavelet-Tsallis Entropy Detection and Location of Mean Level-Shifts in Long-Memory fGn Signals

“…F-ARIMA time series are LRD as long as 0 < d < 1/2 and provide the additional ability of modeling short-memory and long-memory in the same signal structure. For further references on the properties of fGn and f-ARIMA processes, the reader is referred to [17,[23][24][25] …”

Wavelet q-Fisher Information for Scaling Signal Analysis

“…Various definitions of the scaling property have been proposed in the scientific literature, some based on their characteristics such as self-similarity or long-memory, others based on the behaviour of their power spectral density (PSD). In this article, a scaling process is a random process for which the associated PSD behaves as a power-law in a range of frequencies [3,21], i.e.,…”

“…The persistence of scaling processes can also be quantified by the index α and within this framework, scaling processes possess negative persistence as long as α < 0, positive weak long-persistence when 0 < α < 1 and positive strong long-persistence whenever α > 1. Scaling signals encompasses a large family of well-known random signals, e.g., fBms, fGns [22], pure power-law processes [21], multifractal processes [3], etc. FBm, B H (t), comprises a family of Gaussian, self-similar processes with stationary increments and because of the Gaussianity, it is completely characterized by its autocovariance sequence (ACVS), which is given by,…”

Quantum probes for the spectral properties of a classical environment

“…In particular, in the case of phase damping, i.e. pure dephasing, it has been shown that the interaction of a quantum system with a quantum bath can be written in terms of a random unitary evolution driven by a classical stochastic process [4,5].The characterization of classical noise is often performed by collecting a series of measurements to estimate the autocorrelation function and the spectral properties [6][7][8][9][10][11]. This procedure is generally time consuming and may require the control of a complex system.…”

“…The characterization of classical noise is often performed by collecting a series of measurements to estimate the autocorrelation function and the spectral properties [6][7][8][9][10][11]. This procedure is generally time consuming and may require the control of a complex system.…”