Time Series Modeling of Neuroscience Data (Hardcover)

IntroductionTime-Series ModelingContinuous-Time Models and Discrete-Time ModelsUnobserved Variables and State Space ModelingDynamic Models for Time Series PredictionTime Series Prediction and the Power SpectrumFantasy and Reality of Prediction ErrorsPower Spectrum of Time SeriesDiscrete-Time Dynamic ModelsLinear Time Series ModelsParametric Characterization of Power SpectraTank Model and Introduction of Structural State Space RepresentationAkaike’s Theory of Predictor SpaceDynamic Models with Exogenous Input VariablesMultivariate Dynamic ModelsMultivariate AR ModelsMultivariate AR Models and Feedback Systems Multivariate ARMA ModelsMultivariate State Space Models and Akaike’s Canonical Realization Multivariate and Spatial Dynamic Models with InputsContinuous-Time Dynamic ModelsLinear Oscillation ModelsPower SpectrumContinuous-Time Structural ModelingNonlinear Differential Equation ModelsSome More ModelsNonlinear AR ModelsNeural Network ModelsRBF-AR ModelsCharacterization of NonlinearitiesHammerstein Model and RBF-ARX ModelDiscussion on Nonlinear PredictorsHeteroscedastic Time Series ModelsRelated Theories and ToolsPrediction and Doob DecompositionLooking at the Time Series from Prediction ErrorsInnovations and Doob DecompositionsInnovations and Doob Decomposition in Continuous TimeDynamics and Stationary DistributionsTime Series and Stationary DistributionsPearson System of Distributions and Stochastic ProcessesExamplesDifferent Dynamics Can Arise from the Same Distribution.Bridge between Continuous-Time Models and Discrete-Time ModelsFour Types of Dynamic ModelsLocal Linearization BridgeLL Bridges for the Higher Order Linear/Nonlinear Processes LL Bridges for the Processes from the Pearson SystemLL Bridge as a Numerical Integration SchemeLikelihood of Dynamic ModelsInnovation ApproachLikelihood for Continuous-Time ModelsLikelihood of Discrete-Time ModelsComputationally Efficient Methods and AlgorithmsLog-Likelihood and the Boltzmann EntropyState Space ModelingInference Problem (a) for State Space ModelsState Space Models and InnovationsSolutions by the Kalman FilterNonlinear Kalman FiltersOther SolutionsDiscussionsInference Problem (b) for State Space ModelsIntroductionLog-Likelihood of State Space Models in Continuous TimeLog-Likelihood of State Space Models in Discrete TimeRegularization Approach and Type II LikelihoodIdentifiability ProblemsArt of Likelihood MaximizationIntroductionInitial Value Effects and the Innovation LikelihoodSlow Convergence ProblemInnovation-Based Approach versus Innovation-Free .Approach Innovation-Based Approach and the Local Levy State Space Models Heteroscedastic State Space ModelingCausality AnalysisIntroductionGranger Causality and LimitationsAkaike CausalityHow to Define Pair-Wise Causality with Akaike MethodIdentifying Power Spectrum for Causality AnalysisInstantaneous CausalityApplication to fMRI DataDiscussionsConclusion: The New and Old ProblemsReferencesIndex