- Provides an overview of time series models for the study of neuroscience data
- Illustrates the methods using real examples
- Supplies computational guidance on the application of the methods
- Bridges the gap between complex mathematical models and applied science

Recent advances in brain science measurement technology have given researchers access to very large-scale time series data such as EEG/MEG data (20 to 100 dimensional) and fMRI (140,000 dimensional) data. To analyze such massive data, efficient computational and statistical methods are required.

**Time Series Modeling of Neuroscience Data** shows how to efficiently analyze neuroscience data by the Wiener-Kalman-Akaike approach, in which dynamic models of all kinds, such as linear/nonlinear differential equation models and time series models, are used for whitening the temporally dependent time series in the framework of linear/nonlinear state space models. Using as little mathematics as possible, this book explores some of its basic concepts and their derivatives as useful tools for time series analysis. Unique features include:

- A statistical identification method of highly nonlinear dynamical systems such as the Hodgkin-Huxley model, Lorenz chaos model, Zetterberg Model, and more
- Methods and applications for Dynamic Causality Analysis developed by Wiener, Granger, and Akaike
- A state space modeling method for dynamicization of solutions for the Inverse Problems
- A heteroscedastic state space modeling method for dynamic non-stationary signal decomposition for applications to signal detection problems in EEG data analysis
- An innovation-based method for the characterization of nonlinear and/or non-Gaussian time series
- An innovation-based method for spatial time series modeling for fMRI data analysis

The main point of interest in this book is to show that the same data can be treated using both a dynamical system and time series approach so that the neural and physiological information can be extracted more efficiently. Of course, time series modeling is valid not only in neuroscience data analysis but also in many other sciences and engineering fields where the statistical inference from the observed time series data plays an important role.

**Introduction**Time-Series Modeling

Continuous-Time Models and Discrete-Time Models

Unobserved Variables and State Space Modeling

Dynamic Models for Time Series Prediction

Time Series Prediction and the Power Spectrum

Fantasy and Reality of Prediction Errors

Power Spectrum of Time Series

Discrete-Time Dynamic Models

Linear Time Series Models

Parametric Characterization of Power Spectra

Tank Model and Introduction of Structural State Space Representation

Akaike’s Theory of Predictor Space

Dynamic Models with Exogenous Input Variables

Multivariate Dynamic Models

Multivariate AR Models

Multivariate AR Models and Feedback Systems

Multivariate ARMA Models

Multivariate State Space Models and Akaike’s Canonical Realization

Multivariate and Spatial Dynamic Models with Inputs

Continuous-Time Dynamic Models

Linear Oscillation Models

Power Spectrum

Continuous-Time Structural Modeling

Nonlinear Differential Equation Models

Some More Models

Nonlinear AR Models

Neural Network Models

RBF-AR Models

Characterization of Nonlinearities

Hammerstein Model and RBF-ARX Model

Discussion on Nonlinear Predictors

Heteroscedastic Time Series Models

Related Theories and Tools

Prediction and Doob Decomposition

Looking at the Time Series from Prediction Errors

Innovations and Doob Decompositions

Innovations and Doob Decomposition in Continuous Time

Dynamics and Stationary Distributions

Time Series and Stationary Distributions

Pearson System of Distributions and Stochastic Processes

Examples

Different Dynamics Can Arise from the Same Distribution.

Bridge between Continuous-Time Models and Discrete-Time Models

Four Types of Dynamic Models

Local Linearization Bridge

LL Bridges for the Higher Order Linear/Nonlinear Processes

LL Bridges for the Processes from the Pearson System

LL Bridge as a Numerical Integration Scheme

Likelihood of Dynamic Models

Innovation Approach

Likelihood for Continuous-Time Models

Likelihood of Discrete-Time Models

Computationally Efficient Methods and Algorithms

Log-Likelihood and the Boltzmann Entropy

State Space Modeling

Inference Problem (a) for State Space Models

State Space Models and Innovations

Solutions by the Kalman Filter

Nonlinear Kalman Filters

Other Solutions

Discussions

Inference Problem (b) for State Space Models

Introduction

Log-Likelihood of State Space Models in Continuous Time

Log-Likelihood of State Space Models in Discrete Time

Regularization Approach and Type II Likelihood

Identifiability Problems

Art of Likelihood Maximization

Introduction

Initial Value Effects and the Innovation Likelihood

Slow Convergence Problem

Innovation-Based Approach versus Innovation-Free .Approach

Innovation-Based Approach and the Local Levy State Space Models

Heteroscedastic State Space Modeling

Causality Analysis

Introduction

Granger Causality and Limitations

Akaike Causality

How to Define Pair-Wise Causality with Akaike Method

Identifying Power Spectrum for Causality Analysis

Instantaneous Causality

Application to fMRI Data

Discussions

Conclusion: The New and Old Problems

**Tohru Ozaki** is a mathematician and statistician. He received his BSc in mathematics from the University of Tokyo in 1969. He then joined the Institute of Statistical Mathematics (ISM), Tokyo, in 1970 and study and worked with Hirotugu Akaike. He received his DSc from Tokyo Institute of Technology in 1981 under the supervision of Akaike. From 1987 to 2008, he was a professor at ISM and, after Akaike’s retirement, served as the director of the prediction and control group. His major research areas include time series analysis, nonlinear stochastic dynamic modeling, predictive control, signal processing and their applications in neurosciences, control engineering, and financial engineering.

While he was at ISM, Ozaki was engaged in various projects in applied time series analysis in science and engineering: EEG dynamic inverse problems, spatial time series modeling of fMRI data, causality analysis in behavioral science, modeling nonlinear dynamics in ship engineering, predictive control design in fossil power plant control, seasonal adjustment in official statistics, heteroscedastic modeling and risk-sensitive control in financial engineering, nonlinear dynamic modeling in macroeconomics, spectral analysis of seismology data, point process modeling of earthquake occurrence data, river-flow prediction in stochastic hydrology, etc.

Ozaki retired from ISM in 2008. Since then he has been a visiting professor at Tohoku University, Sendai, Japan, and at Queensland University of Technology, Brisbane, Australia. He has been involved in supporting several research projects (in dynamic modeling of neuroscience data, fossil power plant control design, and risk-sensitive control in financial engineering) in universities and industry. He has also led, through his international research network, a time series research group called Akaike Innovation School from his office in Mount Fuji and organizes seminars every summer.

"This is a very unusual book on time series, with much that is new, innovative , and usually not found in other books on time series, for example multivariate AR models, multivariate dynamic models, causal analysis and the Doob decomposition, and so on. Among the major pleasures of browsing through the book are the acquaintance with ‘Laplace’s Demon’, seeing Pearsonian and multimodal distributions as stationary distributions for dynamic models, Einstein’s inductive use of Boltzmann entropy—to mention just a few of the novelties. But the hard core of the book is about state space modeling and its application to neuroscience data. The pages 331 through 351 are a richly textured but precise and detailed introduction to state space modeling. Here is a lovely summary by Ozaki that I have not seen elsewhere—it deals with time series dynamics …"

—Jayanta K. Ghosh, *International Statistical Review* (2013), 81

"This book is essential for every quantitative scientist who is interested in developing rigorous statistical models for analyzing brain signals. It is written by an expert statistician who has made significant contributions to the area of time series and stochastic processes. … His expertise on this subject and interest on the deep issues of statistical modeling of brain signals are clearly reflected in the character of this book. This book builds an important foundation for neurostatistics … it is truly unique in its treatment of the topic because it has an eye towards modeling brain signals, such as electroencephalograms and functional magnetic resonance images, and thus builds on the specifics that are directly relevant to these particular data. … At the University of California, Irvine, researchers have used this book recently and found it to be very helpful. Moreover, I intend to use this book as the primary text for a special topic course on neurostatistics in the Department of Statistics."

—Hernando Ombao, *Journal of Time Series Analysis*, 2013