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# Contents

• Preface
• Notation
• Chapter 1: Introduction
• Examples of time series
• Dollar to Euro exchange rate
• Number of monthly airline passengers
• Heat dynamics of a building
• Prefator-prey relationship
• A first crash course
• Contents and scope of the book
• Chapter 2: Multivariate random variables
• Joint and marginal densities
• Conditional distributions
• Expectations and moments
• Moments of multivariate random variables
• Conditional expectation
• The multivariate normal distibution
• Distributions derived from the normal distribution
• Linear projections
• Problems
• Chapter 3: Regression-based methods
• The regression model
• The general linear model (GLM)
• Least squares (LS) estimates
• Maximum likelihood (ML) estimates
• Prediction
• Predictions in the general linear model
• Regression and exponential smoothing
• Predictions in the constant mean model
• Locally constant mean model and simple exponential smoothing
• Predictions in trend models
• Local trend and exponential smoothing
• Time series with seasonal variations
• The classic decomposition
• Holt-Winters procedure
• Global and local trend model - an example
• Problems
• Chapter 4: Linear dynamic systems
• Linear systems in the time domain
• Linear systems in the frequency domain
• Sampling
• The z-transform
• Frequently used operators
• The Laplace transform
• A comparison between transformations
• Problems
• Chapter 5: Stochastic processes
• Introduction
• Stochastic processes and their moments
• Characteristics for stochastic processes
• Covariance and correlation functions
• Linear processes
• Processes in discrete time
• Processes in continuous time
• Stationary processes in the frequency domain
• Commonly used linear processes
• The MA process
• The AR process
• The ARMA process
• Non-stationary models
• The ARIMA process
• Seasonal models
• Models with covariates
• Models with time-varying mean values
• Models with time-varying coefficients
• Optimal prediction of stochastic processes
• Prediction in the ARIMA process
• Problems
• Chapter 6: Identification, estimation, and model checking
• Introduction
• Estimation of covariance and correlation functions
• Autocovariance and autocorrelation function
• Cross-covariance and cross-correlation functions
• Identification
• Identification of the degree of differencing
• Identification of the ARMA part
• Cointegration
• Estimation of parameters in standard models
• Moment estimates
• The LS estimator for linear dynamic models
• The prediction error method
• The ML method for dynamic models
• Selection of the model order
• The autocorrelation functions
• Testing the model
• Information criteria
• Model checking
• Cross-validation
• Residual analyse
• Case study: Electricity consumption
• Problems
• Chapter 7: Spectral analysis
• The periodogram
• Harmonic analysis
• Properties of the periodogram
• Consistent estimates of the spectrum
• The truncated periodogram
• Lag- and spectral windows
• Approximative distributions for spectral estimates
• The cross-spectrum
• The co-spectrum and the quadrature spectrum
• Cross-amplitude spectrum, phase spectrum, coherence spectrum, gain spectrum
• Estimation of the cross-spectrum
• Problems
• Chapter 8: Linear systems and stochastic processes
• Relationship between input and output processes
• Moment relations
• Spectral relations
• Systems with measurement noise
• Input-output models
• Transfer function models
• Difference equation models
• Output error models
• Identification of transfer function models
• Multiple-input models
• Moment relations
• Spectral relations
• Identification of multiple-input models
• Estimation
• Moment estimates
• LS estimates
• Prediction error method
• ML estimates
• Output error method
• Model checking
• Prediction in transfer function models
• Minimum variance controller
• Intervention models
• Problems
• Chapter 9: Multivariate time series
• Stationary stochastic processes and their moments
• Linear processes
• The multivariate ARMA process
• Theoretical covariance matrix functions
• Partial correlation matrix
• q-conditioned partial correlation matrix
• VAR representation
• Non-stationary models
• The multivariate ARIMA process
• The multivariate seasonal model
• Time-varying models
• Prediction
• Missing values for some signals
• Identification of multivariate models
• Identification using pre-whitening
• Estimation of parameters
• Least squares estimation
• An extended LS method for multivariate ARMAX models (the Spliid method)
• ML estimates
• Model checking
• Problems
• Chapter 10: State space models of dynamic systems
• The linear stochastic state space model
• Transfer function and state space formulations
• Interpolation, reconstruction, and prediction
• The Kalman filter
• k-step predictions in state space models
• Empirical Bayesian description of the Kalman filter
• Some common models in state space form
• Signal extraction
• Time series with missing observations
• Estimation of autocorrelation functions
• ML estimates of space state models
• Problems
• Chapter 11: Recursive estimation
• Recursive LS
• Recursive LS with forgetting
• Recursive pseudo-linear regression (RPLR)
• Recursive prediction error methods (RPEM)
• Models with time-varying parameters
• The regression model with time-varying parameters
• Dynamic models with time-varying parameters
• Chapter 12: Real life inspired problems
• Prediction of wind power production
• Prediction of the consumption of medicine
• Effect of chewing gum
• Prediction of stock prices
• Wastewater treatent: Using root zone plants
• Scheduling system for oil delivery
• Warning system for slippery roads
• Statistical quality control
• Wastewater treatment: Modeling and control
• Sales numbers
• Modeling and prediction of stock prices
• Adaptive modeling of interest rates
• Appendix A: The solution to difference equations
• Appendix B: Partial autocorrelations
• Appendix C: Some results from trigonometry
• Appendix D: List of acronyms
• Appendix E: List of symbols
• Bibliography
• Index

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