Adam M. Sykulski

A global surface drifter data set at hourly resolution

The surface drifting buoys, or drifters, of the Global Drifter Program (GDP) are predominantly tracked by the Argos positioning system, providing drifter locations with O(100 m) errors at nonuniform temporal intervals, with an average interval of 1.2 h since January 2005. This data set is thus a rich and global source of information on high-frequency and small-scale oceanic processes, yet is still relatively understudied because of the challenges associated with its large size and sampling characteristics. A methodology is described to produce a new high-resolution global data set since 2005, consisting of drifter locations and velocities estimated at hourly intervals, along with their respective errors. Locations and velocities are obtained by modeling locally in time trajectories as a first-order polynomial with coefficients obtained by maximizing a likelihood function. This function is derived by modeling the Argos location errors with t location-scale probability distribution functions. The methodology is motivated by analyzing 82 drifters tracked contemporaneously by Argos and by the Global Positioning System, where the latter is assumed to provide true locations. A global spectral analysis of the velocity variance from the new data set reveals a sharply defined ridge of energy closely following the inertial frequency as a function of latitude, distinct energy peaks near diurnal and semidiurnal frequencies, as well as higher-frequency peaks located near tidal harmonics as well as near replicates of the inertial frequency. Compared to the spectra that can be obtained using the standard 6-hourly GDP product, the new data set contains up to 100% more spectral energy at some latitudes.

Frequency domain estimation of integrated volatility for Ito processes in the presence of market-microstructure noise

This paper proposes a novel multiscale estimator for the integrated volatility of an Ito process in the presence of market microstructure noise (observation error). The multiscale structure of the observed process is represented frequency by frequency, and the concept of the multiscale ratio is introduced to quantify the bias in the realized integrated volatility due to the observation error. The multiscale ratio is estimated from a single sample path, and a frequency-by-frequency bias correction procedure is proposed, which simultaneously reduces variance. We extend the method to include correlated observation errors and provide the implied time-domain form of the estimation procedure. The new method is implemented to estimate the integrated volatility for the Heston and other models, and the improved performance of our method over existing methods is illustrated by simulation studies.

Fractional Brownian Motion, the Matérn Process, and Stochastic Modeling of Turbulent Dispersion

Stochastic process exhibiting power-law slopes in the frequency domain are frequently well modeled by fractional Brownian motion (fBm). In particular, the spectral slope at high frequencies is associated with the degree of small-scale roughness or fractal dimension. However, a broad class of real-world signals have a high-frequency slope, like fBm, but a plateau in the vicinity of zero frequency. This low-frequency plateau, it is shown, implies that the temporal integral of the process exhibits diffusive behavior, dispersing from its initial location at a constant rate. Such processes are not well modeled by fBm, which has a singularity at zero frequency corresponding to an unbounded rate of dispersion. A more appropriate stochastic model is a much lesser-known random process called the Matern process, which is shown herein to be a damped version of fractional Brownian motion. This article first provides a thorough introduction to fractional Brownian motion, then examines the details of the Matern process and its relationship to fBm. An algorithm for the simulation of the Matern process in O(N log N) operations is given. Unlike fBm, the Matern process is found to provide an excellent match to modeling velocities from particle trajectories in an application to two-dimensional fluid turbulence.

A Widely Linear Complex Autoregressive Process of Order One

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 stationarity, and derive the form of the covariance and relation sequence of this model. We describe how parameter estimation can be efficiently performed both in the time and frequency domain. We demonstrate the practical utility of the process in capturing elliptical oscillations that are naturally present in seismic signals.

On-Line Adaptation of Exploration in the One-Armed Bandit with Covariates Problem

Many sequential decision making problems require an agent to balance exploration and exploitation to maximise long-term reward. Existing policies that address this tradeoff typically have parameters that are set a priori to control the amount of exploration. In finite-time problems, the optimal values of these parameters are highly dependent on the problem faced. In this paper, we propose adapting the amount of exploration performed on-line, as information is gathered by the agent. To this end we introduce a novel algorithm, e-ADAPT, which has no free parameters. The algorithm adapts as it plays and sequentially chooses whether to explore or exploit, driven by the amount of uncertainty in the system. We provide simulation results for the one armed bandit with covariates problem, which demonstrate the effectiveness of e-ADAPT to correctly control the amount of exploration in finite-time problems and yield rewards that are close to optimally tuned off-line policies. Furthermore, we show that e-ADAPT is robust to a high-dimensional covariate, as well as misspecified models. Finally, we describe how our methods could be extended to other sequential decision making problems, such as dynamic bandit problems with changing reward structures.

Exact simulation of noncircular or improper complex-valued stationary Gaussian processes using circulant embedding

This paper provides an algorithm for simulating improper (or noncircular) complex-valued stationary Gaussian processes. The technique utilizes recently developed methods for multi-variate Gaussian processes from the circulant embedding literature. The method can be performed in O(n log2 n) operations, where n is the length of the desired sequence. The method is exact, except when eigenvalues of prescribed circulant matrices are negative. We evaluate the performance of the algorithm empirically, and provide a practical example where the method is guaranteed to be exact for all n, with an improper fractional Gaussian noise process.

Trading strategies for markets: a design framework and its application

In this paper, we present a novel multi-layered framework for designing strategies for trading agents. The objective of this work is to provide a framework that will assist strategy designers with the different aspects involved in designing a strategy. At present, such strategies are typically designed in an ad-hoc and intuitive manner with little regard for discerning best practice or attaining re-usability in the design process. Given this, our aim is to put such developments on a more systematic engineering footing. After we describe our framework, we then go on to illustrate how it can be used to design strategies for a particular type of market mechanism (namely the Continuous Double Auction), and how it was used to design a novel strategy for the Travel Game of the International Trading Agent Competition.

Analysis of Non-Stationary Modulated Time Series with Applications to Oceanographic Surface Flow Measurements

We extend the concept of a modulated nonstationary process to account for rapidly time-evolving correlation structure. This correlation varies sufficiently fast to make existing theory for nonstationary processes not applicable. The rapid variation in the correlations challenges state-of-the-art methods to make inferences. Even for stationary processes, exact inference in the time domain is often not computa- tionally viable. A well-established and fast approximation, known as the Whittle likelihood, is used as a pseudo-likelihood approach. We discuss how the Whittle likelihood can be extended to our given class of nonstationary modulated processes. Simulation studies reveal the power of our proposed methodology. We demonstrate the performance of our method on the analysis of ocean surface currents measured by freely-drifting instruments, a dataset which is pivotal to understanding global climate patterns.

A Power Variance Test for Nonstationarity in Complex-Valued Signals

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 stationary signals generated via the bootstrap method. Our algorithmic procedure is capable of learning different types of nonstationarity, such as jumps or strong sinusoidal components. We illustrate the utility of our test and algorithm through application to turbulent flow data from fluid dynamics.