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Time Series Modeling of Neuroscience Data

Tohru Ozaki
Champman & Hall/CRC
Publication Date: 
Number of Pages: 
Chapman & Hall/CRC Interdisciplinary Statistics Series
We do not plan to review this book.

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
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

Inference Problem (b) for State Space Models

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

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

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

Conclusion: The New and Old Problems


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