The effect of round-off error on long memory processes

We study how the round-off (or discretization) error changes the statistical properties of a Gaussian long memory process. We show that the autocovariance and the spectral density of the discretized process are asymptotically rescaled by a factor smaller than one, and we compute exactly this scaling factor. Consequently, we find that the discretized process is also long memory with the same Hurst exponent as the original process. We consider the properties of two estimators of the Hurst exponent, namely the local Whittle (LW) estimator and the Detrended Fluctuation Analysis (DFA). By using analytical considerations and numerical simulations we show that, in presence of round-off error, both estimators are severely negatively biased in finite samples. Under regularity conditions we prove that the LW estimator applied to discretized processes is consistent and asymptotically normal. Moreover, we compute the asymptotic properties of the DFA for a generic (i.e. non Gaussian) long memory process and we apply the result to discretized processes.Comment: 44 pages, 4 figures, 4 table

Analysis of Round Off Errors with Reversibility Test as a Dynamical Indicator

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analysed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.Comment: International Journal of Bifurcation and Chaos, 201

Wireless Software Synchronization of Multiple Distributed Cameras

We present a method for precisely time-synchronizing the capture of image sequences from a collection of smartphone cameras connected over WiFi. Our method is entirely software-based, has only modest hardware requirements, and achieves an accuracy of less than 250 microseconds on unmodified commodity hardware. It does not use image content and synchronizes cameras prior to capture. The algorithm operates in two stages. In the first stage, we designate one device as the leader and synchronize each client device's clock to it by estimating network delay. Once clocks are synchronized, the second stage initiates continuous image streaming, estimates the relative phase of image timestamps between each client and the leader, and shifts the streams into alignment. We quantitatively validate our results on a multi-camera rig imaging a high-precision LED array and qualitatively demonstrate significant improvements to multi-view stereo depth estimation and stitching of dynamic scenes. We release as open source 'libsoftwaresync', an Android implementation of our system, to inspire new types of collective capture applications.Comment: Main: 9 pages, 10 figures. Supplemental: 3 pages, 5 figure

Revealing Relationships among Relevant Climate Variables with Information Theory

A primary objective of the NASA Earth-Sun Exploration Technology Office is to understand the observed Earth climate variability, thus enabling the determination and prediction of the climate's response to both natural and human-induced forcing. We are currently developing a suite of computational tools that will allow researchers to calculate, from data, a variety of information-theoretic quantities such as mutual information, which can be used to identify relationships among climate variables, and transfer entropy, which indicates the possibility of causal interactions. Our tools estimate these quantities along with their associated error bars, the latter of which is critical for describing the degree of uncertainty in the estimates. This work is based upon optimal binning techniques that we have developed for piecewise-constant, histogram-style models of the underlying density functions. Two useful side benefits have already been discovered. The first allows a researcher to determine whether there exist sufficient data to estimate the underlying probability density. The second permits one to determine an acceptable degree of round-off when compressing data for efficient transfer and storage. We also demonstrate how mutual information and transfer entropy can be applied so as to allow researchers not only to identify relations among climate variables, but also to characterize and quantify their possible causal interactions.Comment: 14 pages, 5 figures, Proceedings of the Earth-Sun System Technology Conference (ESTC 2005), Adelphi, M

Efficient Distributed Estimation of Inverse Covariance Matrices

In distributed systems, communication is a major concern due to issues such as its vulnerability or efficiency. In this paper, we are interested in estimating sparse inverse covariance matrices when samples are distributed into different machines. We address communication efficiency by proposing a method where, in a single round of communication, each machine transfers a small subset of the entries of the inverse covariance matrix. We show that, with this efficient distributed method, the error rates can be comparable with estimation in a non-distributed setting, and correct model selection is still possible. Practical performance is shown through simulations

On the Complexity of Bandit Linear Optimization

We study the attainable regret for online linear optimization problems with bandit feedback, where unlike the full-information setting, the player can only observe its own loss rather than the full loss vector. We show that the price of bandit information in this setting can be as large as $d$, disproving the well-known conjecture that the regret for bandit linear optimization is at most $\sqrt{d}$ times the full-information regret. Surprisingly, this is shown using "trivial" modifications of standard domains, which have no effect in the full-information setting. This and other results we present highlight some interesting differences between full-information and bandit learning, which were not considered in previous literature