Stochastic Optimal Control

Recent analyses of serial correlations in cognitive tasks have provided preliminary evidence of the presence of a particular form of long-range serial dependence known as 1/f noise. It has been argued that long-range dependence has been largely ignored in mainstream cognitive psychology even though it accounts for a substantial proportion of variability in behavior (see, e.g., Gilden, 1997, 2001). In this article, we discuss the defining characteristics of long-range dependence and argue that claims about its presence need to be evaluated by testing against the alternative hypothesis of short-range dependence. For the data from three experiments, we accomplish such tests with autoregressive fractionally integrated moving-average time series modeling. We find that long-range serial dependence in these experiments can be explained by any of several mechanisms, including mixtures of a small number of short-range processes.

/pdf/estimation-and-interpretation-of-1-fa-noise-in-human-4g5t6b0fqv.pdf

Estimation and interpretation of 1/fα noise in human cognition

The 1/f family of fractal random processes model a truly extraordinary range of natural and man-made phenomena, many of which arise in a variety of signal processing scenarios. Yet despite their apparent importance, the lack of convenient representations for 1/f processes has, at least until recently, strongly limited their popularity. In this paper, we demonstrate that 1/f processes are, in a broad sense, optimally represented in terms of orthonormal wavelet bases. Specifically, via a useful frequency domain characterization for 1/f processes, we develop the wavelet expansion's role as a Karhunen-Loeve-type expansion for 1/f processes. As an illustration of potential, we show that wavelet based representations naturally lead to highly efficient solutions to some fundamental detection and estimation problems involving 1/f processes

Wavelet-based representations for the 1/f family of fractal processes : Fractals in electrical engineering

One of the most widely used method for the time-series analysis of continuous Global Navigation Satellite System (GNSS) observations is Maximum Likelihood Estimation (MLE) which in most implementations requires $$\mathcal{O }(n^3)$$
 operations for $$n$$
 observations. Previous research by the authors has shown that this amount of operations can be reduced to $$\mathcal{O }(n^2)$$
 for observations without missing data. In the current research we present a reformulation of the equations that preserves this low amount of operations, even in the common situation of having some missing data.Our reformulation assumes that the noise is stationary to ensure a Toeplitz covariance matrix. However, most GNSS time-series exhibit power-law noise which is weakly non-stationary. To overcome this problem, we present a Toeplitz covariance matrix that provides an approximation for power-law noise that is accurate for most GNSS time-series.Numerical results are given for a set of synthetic data and a set of International GNSS Service (IGS) stations, demonstrating a reduction in computation time of a factor of 10–100 compared to the standard MLE method, depending on the length of the time-series and the amount of missing data.

/pdf/fast-error-analysis-of-continuous-gnss-observations-with-4ldggnxcet.pdf

Fast error analysis of continuous GNSS observations with missing data

Spectrally resolved fast transient brain states in electrophysiological data

This paper discusses techniques for generating digital sequences of noise which simulate processes with certain known properties or describing equations. Part I of the paper presents a review of stochastic processes and spectral estimation (with some new results) and a tutorial on simulating continuous noise processes with a known autospectral density or autocorrelation function. In defining these techniques for computer generating sequences, it also defines the necessary accuracy criteria. These methods are compared to some of the common techniques for noise generation and the problems, or advantages, of each are discussed. Finally, Part I presents results on simulating stochastic differential equations. A Runge-Kutta (RK) method is presented for numerically solving these equations. Part II of the paper discusses power law, or 1/f/sup /spl alpha//, noises. Such noise processes occur frequently in nature and, in many cases, with nonintegral values for /spl alpha/. A review of 1/f noises in devices and systems is followed by a discussion of the most common continuous 1/f noise models. The paper then presents a new digital model for power law noises. This model allows for very accurate and efficient computer generation of 1/f/sup /spl alpha// noises for any /spl alpha/. Many of the statistical properties of this model are discussed and compared to the previous continuous models. Lastly, a number of approximate techniques for generating power law noises are presented for rapid or real time simulation. >

/pdf/discrete-simulation-of-colored-noise-and-stochastic-56wh4884s3.pdf

N.J. Kasdin

Papers

Discrete simulation of colored noise and stochastic processes and 1/f/sup /spl alpha// power law noise generation