Quarter Century of a Paradigm Shift in a Statistical Modeling Technology

I entered the Institute of Statistical Mathematics in April 1989, the beginning year of the Heisei period. Just before the end of the Heisei period, in March, 2019, I stepped down from the director and at the same time retired from the institute. Therefore, the ``Heisei 30 years'' are the life of a researcher for me. In this article, I summarize the major advances in statistical modeling technologies that form the basis of statistical thinking such as data-based prediction and discrimination from my point of view.

Key words: Bayesian modeling, particle filter, kernel methods, deep learning.

Grammar of Statistical Science and Logical Reasoning

Although the standardized research process of science should be a statistical and quantitative exploration process in order to give appropriate laws to the Nature or societies, the main research process of science with program as an artificial and weak order may be logical and qualitative, as suggested by Goertz and Mahoney (2012). From a statistical scientific point of view, I illustrate some situations in which the difference between quantitative and logical processes is noticeable. I also discusses the possibility of unifying the two scientific approaches along the Grammar of Science by Pearson (1892).

Key words: Grammar of science, program, law, quantitative research, qualitative research.

Use of a State Space Model in Time Series Analysis

In time series analysis, the state space model was initially used as means for realizing the maximum likelihood estimation of the ARMA model and for obtaining an optimum gain in statistical control. Since 1980, it has played a role as a platform for handling various kinds of time series modeling, such as nonstationary time series modeling, nonlinear modeling, signal extraction, missing-value processing, self-organizing state-space modeling and data assimilation. In pararell to the development of the use of the state-space model, various algorithms extending the conventional Kalman filter, such as Gaussian-sum filter, non-Gaussian filter, particle filter, etc. have been developed for state estimation of the generalized state-space model. In this article, we focus on the research of the Institute of Statistical Mathematics, and outline the development of state-space models and related calculational methods and their applications.

Key words: State estimation, nonstationary model, nonlinear model, Kalman filter, non-Gaussian filter, particle filter.

The Role of Information Criterion AIC in Statistical Science

The Akaike information criterion (AIC) provides a useful tool for evaluating models estimated by the maximum likelihood method, and a number of successful applications of AIC have been reported in diverse fields of the natural and social sciences. AIC was essentially derived as an estimator of the Kullback-Leibler information from the predictive point of view, and it provided a new paradigm for model selection and evaluation problems in statistical science. The first objective of this paper is to provide a brief explanation of the concept and derivation of the AIC and related criteria.
With the development of modeling techniques such as regularization, sparse modeling, and Bayes modeling, it is necessary to present criteria that enable us to evaluate models constructed by various estimation procedures. The second objective of this paper is to review the AIC type of information criteria for evaluating models estimated by various techniques, with emphasis on the choice of the adjusted parameters, including a smoothing parameter. We review some advances in Bayesian information-theoretic criteria, where criteria were constructed, using the concept of the degrees of freedom as a bias-corrected adjustment. We also describe information-theoretic criteria for evaluating a Bayesian predictive distribution, derived from the fundamental principle behind AIC.

Key words: AIC, ABIC, BIC, DIC, GIC, PIC, TIC, WAIC.

The Development of Statistical Seismology: A Personal Experience and View

I provide an overview of the development of statistical seismology in Japan and my research experience. Some focuses are placed on prediction of seismic activity by point process models, and statistical diagnostic analysis of anomalous seismic activities searching relations to physical phenomena.

Key words: Seismic activity, point processes, ETAS model, hierarchical Bayesian method, probability forecasts, foreshock discrimination.

The Tube Method: Theory and Applications

The tube method is an integral geometric method that is used to estimate the upper tail probabilities of the maxima of Gaussian random fields. The Euler characteristic method is used for the same purpose. In this paper, the concepts underlying the tube method and Euler characteristic method are investigated, and their associated applications in statistics and recent technical developments are discussed.

Key words: Euler characteristic method, simultaneous confidence band, look-elsewhere effect, projection pursuit, singular model, VBM data analysis.

Particle Filter and Data Assimilation

State estimation by the particle filter requires many realizations of the state vector, called particles, for representing non-Gaussian distributions. On the other hand, data assimilation based on a numerical simulation model cannot adopt an assimilation algorithm that is computationally expensive because computational resources need to be assigned to time integration of the state vector by the simulation model. Therefore, the methodology for data assimilation by the particle filter is considered to still be in the research stage. One important issue is how to avoid so-called the filter degeneracy, in which weights for resampling concentrate to a single particle. This article reviews particle filter algorithms on the basis of importance sampling, specifically the implicit particle filter and the equivalent-weights particle filter.

Key words: Particle filter, data assimilation, importance sampling, implicit particle filter, equivalent-weights particle filter.

A Historic Review of Research on Optimization at the Institute of Statistical Mathematics

Optimization is an important discipline in statistical science. Broadening the frontier of tractable optimization problems directly leads to innovation of statistical science. This is a main reason why optimization research has been conducted at the Institute of Statistical Mathematics. In this paper, we review historic development of optimization research at the institute, and its contributaion to statistical sciences. The topics include duality theory of infinite-dimensional linear programming and its application to statistics, convergence analysis of the steepest descendent algorithm and Kaczmarz method, differential-geometric approach to nonlinear programming, interior-point algorithms and information geometry for linear programming, second-order cone programming and semidefinite programming, and various applications.

Key words: Infinite-dimensional optimization, steepest descent, kaczmarz method, nonlinear optimization, interior-point methods, information geometry.

The Design Matrices of an Age-Period-Cohort Modelfor a Standard Cohort Table

This paper gives the explicit expressions of the design matrices of an age-period-cohort model for standard cohort table data classified by age group and survey period, obtained from repeated cross-sectional surveys, to separate the effects of age, period, and cohort factors. The matrices are derived from imposing equality or zero-sum constraints on the cell parameters in a cohort table as well as from visual inspection.

Key words: APC model, age, period and cohort effects, cell parameters, equality or zero-sum constraints, repeated cross-sectional surveys.