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Sep 21, 2013
09/13

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Jonathan M. Lilly

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The analysis of the fully three-dimensional and time-varying polarization characteristics of a modulated trivariate, or three-component, oscillation is addressed. The use of the analytic operator enables the instantaneous three-dimensional polarization state of any square-integrable trivariate signal to be uniquely defined. Straightforward expressions are given which permit the ellipse parameters to be recovered from data. The notions of instantaneous frequency and instantaneous bandwidth,...

Source: http://arxiv.org/abs/1104.2057v3

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Sep 18, 2013
09/13

by
Jonathan M. Lilly

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A modulated oscillation in two or three dimensions can be represented as the trajectory traced out in space by a particle orbiting an ellipse, the properties of which vary as a function of time. Generalizing ideas from signal analysis, the signal variability can be described in terms of kinematic quantities, the instantaneous moments, that formalize our intuitive notions of time-varying frequency and amplitude. On the other hand, if we observed an ellipse evolving in space we would seek to...

Source: http://arxiv.org/abs/1201.5916v1

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29

Sep 21, 2013
09/13

by
Jonathan M. Lilly; Sofia C. Olhede

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The generalizations of instantaneous frequency and instantaneous bandwidth to a bivariate signal are derived. These are uniquely defined whether the signal is represented as a pair of real-valued signals, or as one analytic and one anti-analytic signal. A nonstationary but oscillatory bivariate signal has a natural representation as an ellipse whose properties evolve in time, and this representation provides a simple geometric interpretation for the bivariate instantaneous moments. The...

Source: http://arxiv.org/abs/0902.4111v2

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Sep 17, 2013
09/13

by
Jonathan M. Lilly; Sofia C. Olhede

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An exact and general expression for the analytic wavelet transform of a real-valued signal is constructed, resolving the time-dependent effects of non-negligible amplitude and frequency modulation. The analytic signal is first locally represented as a modulated oscillation, demodulated by its own instantaneous frequency, and then Taylor-expanded at each point in time. The terms in this expansion, called the instantaneous modulation functions, are time-varying functions which quantify, at...

Source: http://arxiv.org/abs/0711.3834v3

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Jul 20, 2013
07/13

by
Jonathan M. Lilly; Sofia C. Olhede

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The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet {\em shape} with fixed...

Source: http://arxiv.org/abs/1203.3380v2

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Sep 21, 2013
09/13

by
Jonathan M. Lilly; Sofia C. Olhede

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The concept of a common modulated oscillation spanning multiple time series is formalized, a method for the recovery of such a signal from potentially noisy observations is proposed, and the time-varying bias properties of the recovery method are derived. The method, an extension of wavelet ridge analysis to the multivariate case, identifies the common oscillation by seeking, at each point in time, a frequency for which a bandpassed version of the signal obtains a local maximum in power. The...

Source: http://arxiv.org/abs/1104.2060v2

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Sep 23, 2013
09/13

by
Jonathan M. Lilly; Richard K. Scott; Sofia C. Olhede

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A method for extracting time-varying oscillatory motions from time series records is applied to Lagrangian trajectories from a numerical model of eddies generated by an unstable equivalent barotropic jet on a beta plane. An oscillation in a Lagrangian trajectory is represented mathematically as the signal traced out as a particle orbits a time-varying ellipse, a model which captures wavelike motions as well as the displacement signal of a particle trapped in an evolving vortex. Such oscillatory...

Source: http://arxiv.org/abs/1110.0140v2

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Jun 28, 2018
06/18

by
Adam M. Sykulski; Sofia C. Olhede; Jonathan M. Lilly

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We propose a simple stochastic process for modeling improper or noncircular complex-valued signals. The process is a natural extension of a complex-valued autoregressive process, extended to include a widely linear autoregressive term. This process can then capture elliptical, as opposed to circular, stochastic oscillations in a bivariate signal. The process is order one and is more parsimonious than alternative stochastic modeling approaches in the literature. We provide conditions for...

Topics: Statistics Theory, Methodology, Statistics, Mathematics

Source: http://arxiv.org/abs/1511.04128

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Jun 29, 2018
06/18

by
Adam M. Sykulski; Sofia C. Olhede; Jonathan M. Lilly

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The Whittle likelihood is a computationally efficient pseudo-maximum likelihood inference procedure which is known to produce biased parameter estimates for large classes of time series models. We propose a method for de-biasing Whittle likelihood parameter estimates for second-order stationary stochastic processes. We demonstrate how to compute the de-biased Whittle likelihood in the same $\mathcal{O}(n\log n)$ computational efficiency as standard Whittle likelihood. We prove that the method...

Topics: Methodology, Machine Learning, Mathematics, Statistics Theory, Statistics, Computation

Source: http://arxiv.org/abs/1605.06718

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Jun 28, 2018
06/18

by
Thomas E. Bartlett; Adam M. Sykulski; Sofia C. Olhede; Jonathan M. Lilly; Jeffrey J. Early

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We propose a novel algorithm for testing the hypothesis of nonstationarity in complex-valued signals. The implementation uses both the bootstrap and the Fast Fourier Transform such that the algorithm can be efficiently implemented in O(NlogN) time, where N is the length of the observed signal. The test procedure examines the second-order structure and contrasts the observed power variance - i.e. the variability of the instantaneous variance over time - with the expected characteristics of...

Topics: Methodology, Statistics

Source: http://arxiv.org/abs/1508.05593

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Jun 29, 2018
06/18

by
Arthur P. Guillaumin; Adam M. Sykulski; Sofia C. Olhede; Jeffrey J. Early; Jonathan M. Lilly

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We propose a new class of univariate nonstationary time series models, using the framework of modulated time series, which is appropriate for the analysis of rapidly-evolving time series as well as time series observations with missing data. We extend our techniques to a class of bivariate time series that are isotropic. Exact inference is often not computationally viable for time series analysis, and so we propose an estimation method based on the Whittle-likelihood, a commonly adopted...

Topics: Methodology, Fluid Dynamics, Atmospheric and Oceanic Physics, Applications, Physics, Statistics

Source: http://arxiv.org/abs/1605.09107