There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This paper proposes a very tractable approach to modeling changes in regime. The parameters of an autoregression are viewed as the outcome of a discrete-state Markov process. For example, the mean growth rate of a nonstationary series may be subject to occasional, discrete shifts. The econometrician is presumed not to observe these shifts directly, but instead must draw probabilistic inference about whether and when they may have occurred based on the observed behavior of the series. The paper presents an algorithm for drawing such probabilistic inference in the form of a nonlinear iterative filter

A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle.

Today's Web-enabled deluge of electronic data calls for automated methods of data analysis. Machine learning provides these, developing methods that can automatically detect patterns in data and then use the uncovered patterns to predict future data. This textbook offers a comprehensive and self-contained introduction to the field of machine learning, based on a unified, probabilistic approach. The coverage combines breadth and depth, offering necessary background material on such topics as probability, optimization, and linear algebra as well as discussion of recent developments in the field, including conditional random fields, L1 regularization, and deep learning. The book is written in an informal, accessible style, complete with pseudo-code for the most important algorithms. All topics are copiously illustrated with color images and worked examples drawn from such application domains as biology, text processing, computer vision, and robotics. Rather than providing a cookbook of different heuristic methods, the book stresses a principled model-based approach, often using the language of graphical models to specify models in a concise and intuitive way. Almost all the models described have been implemented in a MATLAB software package--PMTK (probabilistic modeling toolkit)--that is freely available online. The book is suitable for upper-level undergraduates with an introductory-level college math background and beginning graduate students.

Machine Learning : A Probabilistic Perspective

Forecasting with artificial neural networks: the state of the art

Recurrence plots for the analysis of complex systems

This paper presents a non standard analysis of systematic nonstationarity in data on ionospheric motions. The methodology is applicable to any analysis of ambient back-ground noise in communication systems which use the atmosphere as a medium. The data used in this paper were obtained from an array of radar-like devices called phase-path sounders, which measure the velocity of the vertical motion of the neutral gases in the ionosphere. The model fits between 36% and 5%, of the variation of the log spectrum for frequencies less than 3 CPH and greater than 6 CPH using a period of 24 hours (diurnal) and the first harmonic of 12 hours (semidiurnal).

Fitting a Nonstationary Model of Ionospheric Motions

The purpose of this article is to investigate the ability of an assortment of frequency domain bandpass filters proposed in the economics literature to extract a known periodicity. The specific bandpass filters investigated include a conventional Discrete Fourier Transform filter, together with the filter recently proposed in Iacobucci-Noullez (2004, 2005). We employ simulation methods whereby the above-mentioned filters are applied to artificial data in order to investigate their cycle extraction properties. We also investigate the implications and complications that may arise from the Gibbs Effect in practical settings that typically confront applied macroeconomists.

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Discrete Fourier Transform Filters as Business Cycle Extraction Tools: An Investigation of Cycle Extraction Properties and Applicability of ‘Gibbs’ Effect

Interest has been growing in testing for nonlinearity or chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We designed and ran a single-blind controlled competition among five highly regarded tests for nonlinearity or chaos with ten simulated data series. The data generating mechanisms include linear processes, chaotic recursions, and nonchaotic stochastic processes; and both large and small samples were included in the experiment. The data series were produced in a single blind manner by the competition manager and sent by e-mail, without identifying information, to the experiment participants. Each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test. The results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos.

A Single-Blind Controlled Competition Among Tests For Nonlinearity And Chaos

Analyzing several musical instrument tones using the randomly modulated periodicity model

Consider a collection of m sensors, immersed in an isotropic noise field, that is designed to detect the arrival of an unknown time‐limited signal of length T seconds However, the power spectrum S(ω) of the signal is known Asssume that the direction of the signal propagation is known approximately, but due to local perturbations of the medium or motions of the sensors, the precise arrival time of the signal at each sensor is unknown, and the signals cannot be added coherently Time adjustments can be made for each sensor channel so that the signal is assumed to arrive during a sampling window of T seconds if some part or all of the signal is within the window for each sensor If the noise is white, the weak‐signal optimal detection procedure is to compute U = Δπ ∫ 0π/Δ Ŝ(ω) S(ω) dω, where S(ω) is the observed power spectrum averaged over all the sensors and Δ is the interval between discrete observations, and then to compare U with a threshold The signal is claimed to be present if U exceeds the thresh

Melvin J. Hinich

Papers

Fitting a Nonstationary Model of Ionospheric Motions

Discrete Fourier Transform Filters as Business Cycle Extraction Tools: An Investigation of Cycle Extraction Properties and Applicability of ‘Gibbs’ Effect

A Single-Blind Controlled Competition Among Tests For Nonlinearity And Chaos

Analyzing several musical instrument tones using the randomly modulated periodicity model

Detection of an Incoherent Finite Signal Using a Number of Sensors