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

The problem of detecting a non-Gaussian time series in the presence of additive Gaussian or non-Gaussian noise is cast into a classical hypothesis testing framework, using the sample bispectrum as the test statistic The power of the test is demonstrated as a function of signal-to-noise ratio, the degree of skewness of the signal, and processing parameters The results are compared to the power of a classical energy detection test It is concluded that the bispectrum can be used effectively to detect non-Gaussian signals in the presence of interfering noise and that it may perform better, depending on the degree of non-Gaussianity, than energy detection >

Detection of non-Gaussian signals in non-Gaussian noise using the bispectrum

Aliasing is a signal-confounding problem that arises when a continuous-time signal is sampled at a rate slower than twice the highest frequency component of a Fourier series representation of the signal. Aliasing can be especially serious for social-science time series applications, since the sampling designs used to construct most social-science data bases are fixed by considerations other than the nature of the underlying continuous-time mechanisms. After collecting sampled data, it is of value to test the observations for the presence of aliasing. It is shown that the nature of the support set of a sampled band-limited stationary signal can be used to motivate an amended version of the Hinich bispectrum test for Gaussianity (Hinich 1982) as a test for aliasing.

A Test for Aliasing Using Bispectral Analysis

The aggregation of individual preferences into a social preference ordering is one of the most fundamental of the problems in social science. The general problem is non-trivial even for a small group of individuals. Given plausible conditions which a social preference ordering should satsify, Arrow demonstrates in his classic work [1] that even when individual preferences satisfy reasonable conditions there need not exist a general social welfare function or preference ordering if the number of involved persons and alternatives under consideration is greater than or equal to three. Only in the special case of two alternatives is the device of decision by majority rule proven to provide a generally satisfactory social preference ordering for an arbitrary number of individuals.

On the power and importance of the mean preference in a mathematical model of democratic choice

This paper presents a theory which rationalizes voting in terms of the marginal utility a citizen derives from contributing a small amount of effort in the political process when the cost of voting is small. Citizens abstain when the marginal cost of voting exceeds the marginal perceived benefit. A simple choice rule for voting in a two candidate race is derived from the theory. This rule depends on the voter's subjective belief about the election outcome as well as his preferences for the candidates. The key assumption is that the voter's utility increases if he votes for a winner, or decreases if he votes for a loser. This assumption is no less plausible than the assumption that voters believe they can be pivotal.

Voting as an act of contribution

We examine a method recently proposed by Hinich and Patterson (mimeo, University of Texas at Austin, 1995) for testing the validity of specifying a GARCH error structure for financial time series data in the context of a set of ten daily Sterling exchange rates. The results demonstrate that there are statistical structures present in the data that cannot be captured by a GARCH model, or any of its variants. This result has important implications for the interpretation of the recent voluminous literature which attempts to model financial asset returns using this family of models.

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Melvin J. Hinich

Papers

Detection of non-Gaussian signals in non-Gaussian noise using the bispectrum

A Test for Aliasing Using Bispectral Analysis

On the power and importance of the mean preference in a mathematical model of democratic choice

Voting as an act of contribution

Episodic nonstationarity in exchange rates