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Smoothness Priors Analysis of Time Series

Lecture Notes in Statistics Band 116

Genshiro Kitagawa, Will Gersch

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Smoothness Priors Analysis of Time Series addresses some of the problems of modeling stationary and nonstationary time series primarily from a Bayesian stochastic regression "smoothness priors" state space point of view. Prior distributions on model coefficients are parametrized by hyperparameters. Maximizing the likelihood of a small number of hyperparameters permits the robust modeling of a time series with relatively complex structure and a very large number of implicitly inferred parameters. The critical statistical ideas in smoothness priors are the likelihood of the Bayesian model and the use of likelihood as a measure of the goodness of fit of the model. The emphasis is on a general state space approach in which the recursive conditional distributions for prediction, filtering, and smoothing are realized using a variety of nonstandard methods including numerical integration, a Gaussian mixture distribution-two filter smoothing formula, and a Monte Carlo "particle-path tracing" method in which the distributions are approximated by many realizations. The methods are applicable for modeling time series with complex structures.


Einband Taschenbuch
Seitenzahl 280
Erscheinungsdatum 09.08.1996
Sprache Englisch
ISBN 978-0-387-94819-5
Verlag Springer US
Maße (L/B/H) 23,5/15,5/1,4 cm
Gewicht 880 g
Auflage 1996

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  • 1 Introduction.- 1.1 Background.- 1.2 What is in the Book.- 1.3 Time Series Examples.- 2 Modeling Concepts and Methods.- 2.1 Akaike’s AIC: Evaluating Parametric Models.- 2.1.1 The Kullback-Leibler Measure and the Akaike AIC.- 2.1.2 Some Applications of the AIC.- 2.1.3 A Theoretical Development of the AIC.- 2.1.4 Further Discussion of the AIC.- 2.2 Least Squares Regression by Householder Transformation.- 2.3 Maximum Likelihood Estimation and an Optimization Algorithm.- 2.4 State Space Methods.- 3 The Smoothness Priors Concept.- 3.1 Introduction.- 3.2 Background, History and Related Work.- 3.3 Smoothness Priors Bayesian Modeling.- 4 Scalar Least Squares Modeling.- 4.1 Estimating a Trend.- 4.2 The Long AR Model.- 4.3 Transfer Function Estimation.- 4.3.1 Analysis.- 4.3.2 A Transfer Function Analysis Example.- 5 Linear Gaussian State Space Modeling.- 5.1 Introduction.- 5.2 Standard State Space Modeling.- 5.3 Some State Space Models.- 5.4 Modeling With Missing Observations.- 5.5 Unequally Spaced Observations.- 5.6 An Information Square-Root Filter/Smoother.- 6 Contents General State Space Modeling.- 6.1 Introduction.- 6.2 The General State Space Model.- 6.2.1 General Filtering and Smoothing.- 6.2.2 Model Identification.- 6.3 Numerical Synthesis of the Algorithms.- 6.4 The Gaussian Sum-Two Filter Formula Approximation.- 6.4.1 The Gaussian Sum Approximation.- 6.4.2 The Two-filter Formula and Gaussian Sum Smoothing.- 6.4.3 Remarks on the Gaussian Mixture Approximation.- 6.5 A Monte Carlo Filtering and Smoothing Method.- 6.5.1 Introduction.- 6.5.2 Non-Gaussian Nonlinear State Space Model and Filtering.- 6.5.3 Smoothing.- 6.6 A Derivation of the Kalman filter.- 6.6.1 Preparations.- 6.6.2 Derivation of the Filter and Smoother.- 7 Applications of Linear Gaussian State Space Modeling.- 7.1 AR Time Series Modeling.- 7.2 Kullback-Leibler Computations.- 7.3 Smoothing Unequally Spaced Data.- 7.4 A Signal Extraction Problem.- 7.4.1 Estimation of the Time Varying Variance.- 7.4.2 Separating a Micro Earthquake From Noisy Data.- 7.4.3 A Second Example.- 8 Modeling Trends.- 8.1 State Space Trend Models.- 8.2 State Space Estimation of Smooth Trend.- 8.2.1 Estimation of a Smooth Trend.- 8.2.2 Smooth Trend Plus Autoregressive Model.- 8.3 Multiple Time Series Modeling: The Common Trend Plus Individual Component AR Model.- 8.3.1 Maximum Daily Temperatures 1971-1992.- 8.3.2 Tiao and Tsay Flour Price Data.- 8.4 Modeling Trends with Discontinuities.- 8.4.1 Pearson Family, Gaussian Mixture and Monte Carlo Filter Es-timation of an Abruptly Changing Trend.- 9 Seasonal Adjustment.- 9.1 Introduction.- 9.2 A State Space Seasonal Adjustment Model.- 9.3 Smooth Seasonal Adjustment Examples.- 9.4 Non-Gaussian Seasonal Adjustment.- 9.5 Modeling Outliers.- 9.6 Legends.- 10 Estimation of Time Varying Variance.- 10.1 Introduction and Background.- 10.2 Modeling Time-Varying Variance.- 10.3 The Seismic Data.- 10.4 Smoothing the Periodogram.- 10.5 The Maximum Daily Temperature Data.- 11 Modeling Scalar Nonstationary Covariance Time Series.- 11.1 Introduction.- 11.2 A Time Varying AR Coefficient Model.- 11.3 A State Space Model.- 11.3.1 Instantaneous Spectral Density.- 11.4 PARCOR Time Varying AR Modeling.- 11.5 Examples.- 12 Modeling Multivariate Nonstationary Covariance Time Series.- 12.1 Introduction.- 12.2 The Instantaneous Response-Orthogonal Innovations Model.- 12.3 State Space Modeling.- 12.4 Time Varying PARCOR VAR Modeling.- 12.4.1 Constant Coefficient PARCOR VAR Time Series Modeling.- 12.4.2 Time Varying PARCOR Coefficient VAR Modeling.- 12.5 Examples.- 13 Modeling Inhomogeneous Discrete Processes.- 13.1 Nonstationary Discrete Process.- 13.2 Nonstationary Binary Processes.- 13.3 Nonstationary Poisson Process.- 14 Quasi-Periodic Process Modeling.- 14.1 The Quasi-periodic Model.- 14.2 The Wolfer Sunspot Data.- 14.3 The Canadian Lynx Data.- 14.4 Other Examples.- 14.4.1 Phase-unwrapping.- 14.4.2 Quasi-periodicity in the Rainfall data.- 14.5 Predictive Properties of Quasi-periodic Process Modeling.- 15 Nonlinear Smoothing.- 15.1 Introduction.- 15.2 State Estimation.- 15.3 A One Dimensional Problem.- 15.4 A Two Dimensional Problem.- 16 Other Applications.- 16.1 A Large Scale Decomposition Problem.- 16.1.1 Data Preparation and a Strategy for the Data Analysis.- 16.1.2 The Data Analysis.- 16.2 Markov State Classification.- 16.2.1 Introduction.- 16.2.2 A Markov Switching Model.- 16.2.3 Analysis and Results.- 16.3 SPVAR Modeling for Spectrum Estimation.- 16.3.1 Background.- 16.3.2 The Approach and an Example.- References.- Author Index.