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

Using a modification of the Hinich, J Time Ser Anal 3(3):169–176, (1982) bispectrum test for nonlinearity and Gaussianity, the residuals of the Tiao and Box, J Am Stat Assoc 76:802–816, (1981) constrained and unconstrained VAR models for the gas furnace data reject the assumption of Gaussianity and linearity over a grid of bandwidths for estimating the bispectrum. These findings call into question the specification of the linear VAR and VARMA models assumed by Tiao and Box, J Am Stat Assoc 76:802–816, (1981). Utilizing the alternative Hinich J Nonparametr Stat 6:205–221, (1996) nonlinearity test, the residuals of the VAR model were shown to exhibit episodic nonlinearity. The sensitivity of the findings to outliers is investigated by estimating and testing the residuals of L1 and MINIMAX models from 1–6 lags. Building on the linear dynamic specification, a multivariate adaptive regression splines (MARS) model is estimated, using two software implementations, and shown to remove the nonlinearity in the residuals. Leverage plots were used to illustrate the “cost” of imposing a linearity assumption. Out-of-sample forecasting tests from 1–6 periods ahead found that using the sum-of-squared errors criteria, the MARS model out performed ACE, GAM and projection pursuit models.

Melvin J. Hinich

Papers

Detecting and modeling nonlinearity in the gas furnace data

Discussion of "The positive theory of legislative institutions"

Introduction to continuous probability theory

Importance of Centrist Ideologies in Democratic Elections

Estimating linear filters with errors in variables using the Hilbert transform