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

(2005). Time Series Analysis by State Space Methods. Technometrics: Vol. 47, No. 3, pp. 373-373.

Time Series Analysis by State Space Methods

Preface 1. Spatial voting models: the behavioural assumptions 2. The unidimensional spatial voting model 3. A two-dimensional spatial model 4. A general spatial model of candidate competition 5. The influence of candidate characteristics and abstention on election outcomes 6. Voting on budgets 7. Models of voter uncertainty 8. Institutions 9. Empirical testing of the spatial theory of elections 10. Concluding observations References Answers to selected problems Index.

The Spatial Theory of Voting: An Introduction

: Stable autoregressive (AR) and autoregressive moving average (ARMA) processes belong to the class of stationary linear time series. A linear time series is Gaussian if the distribution of the independent innovations (sigma (t)) is normal. Assuming that E sigma (t) = 0, some of the third order cumulants sub c xxx (m,n) = Ex(t)x(t+m)x(t+n) will be non-zero if the sigma (t) are not normal and E sigma cube(t)=0. If the relationship between x(t) and sigma (t) is non-linear, then (x(t)) is non-Gaussian even if the sigma (t) are normal. This paper presents a simple estimator of the bispectrum, the Fourier transform of (sub c xxx (m,n)). This sample bispectrum is used to construct a statistic to test whether the bispectrum of (x(t)) is non-zero. A rejection of the null hypothesis implies a rejection of the hypothesis that (x(t)) is Gaussian. A related test statistic is then presented for testing the hypothesis that (x(t)) is linear. The asymptotic properties of the sample bispectrum are incorporated in these test statistics. The tests are consistent as the sample size (N approaches infinity.

Testing for Gaussianity and Linearity of a Stationary Time Series.

The fundamental process of politics is the aggregation of citizens' preferences into a collective—a social—choice. We develop, interpret, and explain non-technically in this expository essay the definitions, assumptions, and theorems of a mathematical model of one aggregative mechanism—the electoral process. This mechanism is conceptualized here as a multidimensional model of spatial competition in which competition consists of candidates affecting turnout and the electorate's perception of each candidate's positions, and in which the social choice is a policy package which the victorious candidate advocates.This approach, inaugurated by Downs's An Economic Theory of Democracy, and falling under the general rubric “spatial models of party competition,” has been scrutinized, criticized, and reformulated. To clarify the accomplishments of this formulation we identify and discuss in section 2 the general democratic problem of ascertaining a social preference. We review critically in section 3 the definitions and assumptions of our model. We consider in sections 4 and 5 the logic of a competitive electoral equilibrium. We assume in section 4 that the electorate's preferences can be summarized and represented by a single function; the analysis in section 5 pertains to competition between two organizational structures or two opposed ideologies (i.e., when two functions are required to summarize and represent the electorate's preference). Finally, we suggest in section 6 a conceptualization of electoral processes which facilitates extending and empirically testing our model.

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An Expository Development of a Mathematical Model of the Electoral Process

There is no unified theory that can explain both voter choice and where choices come from. Hinich and Munger fill that gap with their model of political communication based on ideology.Rather than beginning with voters and diffuse, atomistic preferences, Hinich and Munger explore why large groups of voters share preference profiles, why they consider themselves "liberals" or "conservatives." The reasons, they argue, lie in the twin problems of communication and commitment that politicians face. Voters, overloaded with information, ignore specific platform positions. Parties and candidates therefore communicate through simple statements of goals, analogies, and by invoking political symbols. But politicians must also commit to pursuing the actions implied by these analogies and symbols. Commitment requires that ideologies be used consistently, particularly when it is not in the party's short-run interest.The model Hinich and Munger develop accounts for the choices of voters, the goals of politicians, and the interests of contributors. It is an important addition to political science and essential reading for all in that discipline."Hinich and Munger's study of ideology and the theory of political choice is a pioneering effort to integrate ideology into formal political theory. It is a major step in directing attention toward the way in which ideology influences the nature of political choices." --Douglass C. North." . . represents a significant contribution to the literature on elections, voting behavior, and social choice." --Policy CurrentsMelvin Hinich is Professor of Government, University of Texas. Michael C. Munger is Associate Professor of Political Science, University of North Carolina.

Melvin J. Hinich

Papers

Time Series Analysis by State Space Methods

The Spatial Theory of Voting: An Introduction

Testing for Gaussianity and Linearity of a Stationary Time Series.

An Expository Development of a Mathematical Model of the Electoral Process

Ideology and the theory of political choice