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 References Index