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E-raamat: Multivariate Time Series With Linear State Space Structure

  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-May-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319285993
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Multivariate Time Series With Linear State Space StructureSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 09-May-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319285993
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This book presents a comprehensive study of multivariate time serieswith linear state space structure. The emphasis is put on both the clarity of thetheoretical concepts and on efficient algorithms for implementing the theory. In particular, it investigates the relationship between VARMA and statespace models, including canonical forms. It also highlights the relationshipbetween Wiener-Kolmogorov and Kalman filtering both with an infinite and afinite sample. The strength of the book also lies in the numerous algorithms includedfor state space models that take advantage of the recursive nature of themodels. Many of these algorithms can be made robust, fast, reliable andefficient. The book is accompanied by a MATLAB package called SSMMATLAB and awebpage presenting implemented algorithms with many examples and case studies. Thoughit lays a solid theoretical foundation, the book also focuses on practicalapplication, and includes exercises in each chapter. It is intended for

researchers and students working with linear state space models, and who arefamiliar with linear algebra and possess some knowledge of statistics.

Preface.- Computer Software.- Orthogonal Projection.- Linear Models.- Stationarity and Linear Time Series Models.- The State Space Model.- Time Invariant State Space Models.- Time Invariant State Space Models With Inputs.- Wiener-Kolmogorov Filtering and Smoothing.- SSMMATLAB.- Bibliography.- Author Index.- Subject Index.

Arvustused

The book under review is a mathematically solid and comprehensive text, covering in detail the main ingredients of linear estimation theory in state space models. Its emphasis is on the state estimation problems, rather than on statistical inference of the unknown parameters of the model, and from this point of view its scope and spirit is closer to the engineering literature, and to the standard reference . (Pavel Chigansky, Mathematical Reviews, May, 2017)

1 Orthogonal Projection
1(60)
1.1 Expectation and Covariance Matrix of Random Vectors
1(2)
1.2 Orthogonality
3(1)
1.3 Best Linear Predictor and Orthogonal Projection
4(4)
1.4 Orthogonalization of a Sequence of Random Vectors: Innovations
8(10)
1.5 The Modified Innovations Algorithm
18(3)
1.6 State Space Approach to the Innovations Algorithm
21(3)
1.7 Introduction to VARMA and State Space Models
24(12)
1.7.1 Innovations Algorithm for VARMA Models
27(3)
1.7.2 Covariance-Based Filter for State Space Models
30(6)
1.8 Further Topics Associated With Orthogonal Projection
36(8)
1.8.1 Sequential Update of an Orthogonal Projection
36(2)
1.8.2 The Law of Iterated Orthogonal Projection
38(1)
1.8.3 The Forward and Backward Prediction Problems
39(2)
1.8.4 Partial and Multiple Correlation Coefficients
41(3)
1.9 Introduction to the Kalman Filter
44(4)
1.10 Linear Regression and Ordinary Least Squares
48(3)
1.11 Historical Notes
51(1)
1.12 Problems
51(10)
Appendix
54(7)
2 Linear Models
61(52)
2.1 Linear Models and Generalized Least Squares
61(3)
2.2 Combined Linear Estimators
64(1)
2.3 Likelihood Function Definitions for Linear Models
65(5)
2.3.1 The Diffuse Likelihood
67(1)
2.3.2 The Transformation Approach and the Marginal Likelihood
68(1)
2.3.3 The Conditional Likelihood
68(1)
2.3.4 The Profile Likelihood
69(1)
2.4 Introduction to Signal Extraction
70(17)
2.4.1 Smoothing
70(2)
2.4.2 Smoothing with Incompletely Specified Initial Conditions
72(11)
2.4.3 Filtering
83(2)
2.4.4 Filtering with Incompletely Specified Initial Conditions
85(2)
2.4.5 Prediction
87(1)
2.5 Recursive Least Squares
87(10)
2.5.1 Square Root Form of RLS
91(2)
2.5.2 Fast Square Root Algorithms for RLS: The UD Filter
93(1)
2.5.3 Square Root Information Form of RLS
94(2)
2.5.4 Fast Square Root Information Algorithm for RLS
96(1)
2.6 Historical Notes
97(1)
2.7 Problems
97(16)
Appendix
102(11)
3 Stationarity and Linear Time Series Models
113(100)
3.1 Stochastic Processes
113(1)
3.2 Stationary Time Series
114(6)
3.2.1 Ergodicity
116(1)
3.2.2 The Autocovariance and Autocorrelation Functions and Their Properties
117(3)
3.3 Linear Time Invariant Filters
120(1)
3.4 Frequency Domain
121(7)
3.5 Linear Time Series Model Representation for a Stationary Process
128(3)
3.6 The Backshift Operator
131(1)
3.7 VARMA Models and Innovations State Space Models
132(5)
3.8 Minimality, Observability, and Controllability
137(16)
3.9 Finite Linear Time Series Models
153(4)
3.10 Covariance Generating Function and Spectrum
157(27)
3.10.1 Covariance Generating Function
158(5)
3.10.2 Spectrum
163(1)
3.10.3 Multivariate Processes
164(5)
3.10.4 Linear Operations on Stationary Processes
169(1)
3.10.5 Computation of the Autocovariance Function of a Stationary VARMA Model
170(7)
3.10.6 Algorithms for the Factorization of a Scalar Covariance Generating Function
177(6)
3.10.7 Algorithms for the Factorization of a Multivariate Covariance Generating Function
183(1)
3.11 Recursive Autoregressive Fitting for Stationary Processes: Partial Autocorrelations
184(14)
3.11.1 Univariate Processes
185(7)
3.11.2 Multivariate Processes
192(6)
3.12 Historical Notes
198(1)
3.13 Problems
198(15)
Appendix
203(10)
4 The State Space Model
213(110)
4.1 The State Space Model
213(1)
4.2 The Kalman Filter
214(5)
4.2.1 Innovations Model for the Output Process
215(1)
4.2.2 Triangular Factorizations of Var(Yt) and Var--1(Yt)
215(1)
4.2.3 Measurement and Time Updates
216(2)
4.2.4 Updating of the Filtered Estimator
218(1)
4.2.5 Sequential Processing
218(1)
4.3 Single Disturbance State Space Representation
219(1)
4.4 Square Root Covariance Filter
220(7)
4.4.1 Square Root Filter for the Single Disturbance State Space Model
220(1)
4.4.2 Fast Square Root Filter for the Single Disturbance State Space Model
221(1)
4.4.3 Square Root Filter for the Several Sources of Error State Space Model
221(1)
4.4.4 Measurement Update in Square Root Form
222(2)
4.4.5 Fast Square Root Algorithms for Measurement Update: The UD Filter
224(1)
4.4.6 Time Update in Square Root Form
225(1)
4.4.7 Fast Square Root Algorithms for Time Update
226(1)
4.5 A Transformation to get St = 0
227(3)
4.5.1 Another Expression for the Square Root Covariance Filter
228(1)
4.5.2 Measurement and Time Updates in Square Root Form When St ≠ 0
229(1)
4.6 Information Filter
230(3)
4.6.1 Measurement Update in Information Form
231(1)
4.6.2 Time Update in Information Form
231(1)
4.6.3 Further Results in Information Form
232(1)
4.7 Square Root Covariance and Information Filter
233(7)
4.7.1 Square Root Covariance and Information Form for Measurement Update
236(2)
4.7.2 Square Root Covariance and Information Form for Time Update
238(2)
4.8 Likelihood Evaluation
240(1)
4.9 Forecasting
241(1)
4.10 Smoothing
241(16)
4.10.1 Smoothing Based on the Bryson--Frazier Formulae
241(7)
4.10.2 Smoothing With the Single Disturbance State Space Model
248(1)
4.10.3 The Rauch--Tung--Striebel Recursions
249(3)
4.10.4 Square Root Smoothing
252(3)
4.10.5 Square Root Information Smoothing
255(2)
4.11 Covariance-Based Filters
257(1)
4.12 Markov Processes
258(12)
4.12.1 Forwards Markovian Models
260(1)
4.12.2 Backwards Markovian Models
261(1)
4.12.3 Backwards Models From Forwards State Space Models
262(7)
4.12.4 Backwards State Space Model When the Πt are Nonsingular
269(1)
4.12.5 The Backwards Kalman Filter
270(1)
4.13 Application of Backwards State Space Models to Smoothing
270(4)
4.13.1 Two-Filter Formulae
271(1)
4.13.2 Backwards Model When Π-1 n+1 and the Ft are Nonsingular
272(2)
4.14 The State Space Model With Constant Bias and Incompletely Specified Initial Conditions
274(7)
4.14.1 Examples
275(2)
4.14.2 Initial Conditions in the Time Invariant Case
277(2)
4.14.3 The Diffuse Likelihood
279(1)
4.14.4 The Profile Likelihood
280(1)
4.14.5 The Marginal and Conditional Likelihoods
281(1)
4.15 The Augmented-State Kalman Filter and the Two-Stage Kalman Filter
281(9)
4.16 Information Form and Square Root Information Form of the Bias Filter
290(5)
4.17 Fast Square Root Information Form of the Bias Filter
295(1)
4.18 Evaluation of the Concentrated Diffuse Log-likelihood with the TSKF and the Information Form Bias Filter
295(1)
4.19 Square Root Information Form of the Modified Bias-free Filter
296(2)
4.20 The Two-stage Kalman Filter With Square Root Information Bias Filter
298(6)
4.20.1 Evaluation of the Concentrated Diffuse Log-likelihood with the TSKF-SRIBF
300(1)
4.20.2 The Diffuse Likelihood When the Square Root Information Form of the Modified Bias-free Filter is Used
301(1)
4.20.3 Forecasting With the TSKF--SRIBF
301(1)
4.20.4 Smoothing With the TSKF--SRIBF Without Collapsing
302(1)
4.20.5 Square Root Information Smoothing With the Modified Bias-Free Filter
303(1)
4.21 Collapsing in the TSKF--SRIBF to Get Rid of the Nuisance Random Variables
304(15)
4.21.1 Examples of Collapsing
306(3)
4.21.2 Evaluation of the Concentrated Diffuse Log-likelihood With the TSKF--SRIBF Under Collapsing
309(1)
4.21.3 Smoothing with the TSKF--SRIBF Under Collapsing
309(10)
4.22 Historical Notes
319(1)
4.23 Problems
320(3)
5 Time Invariant State Space Models
323(82)
5.1 Covariance Function of a Time Invariant Model
324(1)
5.2 Stationary State Space Models
324(2)
5.3 The Lyapunov Equation
326(4)
5.4 Covariance Generating Function
330(2)
5.5 Computation of the Covariance Function
332(1)
5.6 Factorization of the Covariance Generating Function
332(5)
5.7 Cointegrated VARMA Models
337(3)
5.7.1 Parametrizations and State Space Forms
340(1)
5.7.2 Forecasting
340(1)
5.8 The Likelihood of a Time Invariant State Space Model
340(1)
5.9 Canonical Forms for VARMA and State Space Models
341(13)
5.9.1 VARMA Echelon Form
341(3)
5.9.2 State Space Echelon Form
344(3)
5.9.3 Relation Between VARMA and State Space Echelon Forms
347(2)
5.9.4 Echelon and Overlapping Parametrizations
349(5)
5.10 Covariance Factorization for State Space Echelon Forms
354(2)
5.11 Observability and Controllability
356(5)
5.12 Limit Theorems for the Kalman Filter and the Smoothing Recursions
361(3)
5.12.1 Solutions of the DARE
361(2)
5.12.2 Convergence of the DARE
363(1)
5.13 Fast Kalman Filter Algorithm: The CKMS Recursions
364(5)
5.14 CKMS Recursions Given Covariance Data
369(2)
5.15 Fast Covariance Square Root Filter
371(6)
5.15.1 J-Orthogonal Householder Transformations
375(2)
5.16 The Likelihood of a Stationary State Space Model
377(1)
5.17 The Innovations Algorithm Approach for Likelihood Evaluation
377(2)
5.18 Finite Forecasting
379(1)
5.19 Finite Forecasting Using the Innovations Algorithm
380(2)
5.20 Inverse Process
382(3)
5.21 Method of Moments Estimation of VARMA Models
385(4)
5.22 Historical Notes
389(1)
5.23 Problems
390(15)
Appendix
392(13)
6 Time Invariant State Space Models with Inputs
405(44)
6.1 Stationary State Space Models with Inputs
407(1)
6.2 VARMAX and Finite Linear Models with Inputs
408(4)
6.3 Kalman Filter and Likelihood Evaluation for the State Space Model with Inputs
412(3)
6.4 The Case of Stochastic Inputs
415(4)
6.5 Canonical Forms for VARMAX and State Space Models with Inputs
419(9)
6.5.1 VARMAX Echelon Form
419(3)
6.5.2 State Space Echelon Form
422(3)
6.5.3 Relation Between the VARMAX and the State Space Echelon Forms
425(1)
6.5.4 Decoupled VARMAX Echelon Form
426(1)
6.5.5 Decoupled State Space Echelon Form
427(1)
6.6 Estimation of VARMAX Models Using the Hannan--Rissanen Method
428(6)
6.7 Estimation of State Space Models with Inputs Using Subspace Methods
434(5)
6.8 Fast Estimation of State Space Models with Inputs
439(4)
6.9 The Information Matrix
443(1)
6.10 Historical Notes
444(1)
6.11 Problems
444(5)
Appendix
447(2)
7 Wiener--Kolmogorov Filtering and Smoothing
449(72)
7.1 The Classical Wiener--Kolmogorov Formulae
449(16)
7.1.1 Wiener--Kolmogorov Smoothing
450(4)
7.1.2 Wiener--Kolmogorov Filtering
454(5)
7.1.3 Polynomial Methods
459(2)
7.1.4 Prediction Based on the Semi-infinite Sample
461(2)
7.1.5 Innovations Approach
463(2)
7.2 Wiener--Kolmogorov Filtering and Smoothing for Stationary State Space Models
465(20)
7.2.1 Recursive Wiener--Kolmogorov Filtering and Smoothing
468(4)
7.2.2 Covariance Generating Function of the Process
472(1)
7.2.3 Covariance Generating Functions of the State Errors
473(3)
7.2.4 Computing the Filter Weights
476(3)
7.2.5 Disturbance Smoothing and Interpolation
479(3)
7.2.6 Covariance Generating Functions of the Disturbance Errors
482(2)
7.2.7 Equivalence Between Wiener--Kolmogorov and Kalman Filtering
484(1)
7.2.8 Nonstationary Time Invariant State Space Models
484(1)
7.3 Wiener--Kolmogorov Filtering and Smoothing in Finite Samples
485(29)
7.3.1 Finite Generating Functions
485(4)
7.3.2 Innovations Representation
489(1)
7.3.3 Covariance Generating Function
490(1)
7.3.4 Inverse Process
490(1)
7.3.5 Finite Wiener--Kolmogorov Filtering and Smoothing
491(2)
7.3.6 Finite Wiener--Kolmogorov Filtering for Multivariate Processes with State Space Structure
493(21)
7.4 Historical Notes
514(1)
7.5 Problems
515(6)
8 SSMMATLAB
521(6)
8.1 Introduction
521(1)
8.2 Kalman Filter and Likelihood Evaluation
522(1)
8.3 Estimation and Residual Diagnostics
522(1)
8.4 Smoothing
523(1)
8.5 Forecasting
523(1)
8.6 Time Invariant State Space Models
523(1)
8.7 ARIMA and Transfer Function Models
524(1)
8.8 Structural Models
525(1)
8.9 VARMAX Models
525(1)
8.10 Cointegrated VARMA Models
526(1)
Bibliography 527(6)
Author Index 533(4)
Subject Index 537
Dr. Víctor Gómez is a statistician and technical advisor at the Spanish Ministry of Finance and Public Administrations in Madrid. His professional activity involves statistical, econometric and, above all, time series analysis of macroeconomic data, mostly in connection with short term economic analysis. More recently, he has focused on research in the field of time series analysis and the development of software for time series analysis. He has also taught numerous courses on time series analysis and related topics such as short-term forecasting, seasonal adjustment methods or time series filtering. 
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