Schedule

1 Essentials of Signals, Systems and Stochastic Processes 1.1 Probability Theory .1.1.1 Random Variables 1.1.2 Probability Distribution, Density and Events Bayesian Perspective Joint and Marginal Probability Conditional Probability and Bayes’ rule Operations on Random Variables 1.1.3 Expectation Variance, Covariance and Correlation Moments and the Moment Generating Function1.1.4 Common Probability Density Functions Uniform Density Gaussian Density Multivariable Gaussian Density Chi-Squared Density1.2 Stochastic Processes1.2.1 Stationary Processes and the (Auto) Covariance and (Auto) Correlation functions.1.2.2 Cross-Covariance and Cross-Correlation Functions1.2.3 Power Spectral Density1.2.4 Linear Systems subject to stochastic input 1.3 Quasi-stationary signals 1.4 Stochastic Convergence 1.4.1 Convergence in Mean1.4.2 Convergence in Probability 1.4.3 Convergence with Probability 1

1.4.4 Convergence in Distribution

2 Estimation Methods2.1 Minimum Mean Square Error Estimation2.2 Maximum A Posteriori Estimation 2.3 Unbiased Parameter Estimation3 Minimum Mean Square Error Parameter Estimation3.1 The Bias-Variance Error Trade-Off3.2 Risk and Average Risk. The Bayes Estimator Risk estimation methods¶

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SURE, Empirical Bayes, Variational Bayes¶

3.43 Linear in the Parameters Models

4 Linear in the Parameters Models 475 Dynamical Models 495.1 Model Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2 Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.1 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . 495.2.2 The Extended Invariance Principle . . . . . . . . . . . . . . . . . . . . . . . 495.2.3 The Prediction Error Method . . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.4 Multi-Step Least-Squares Methods . . . . . . . . . . . . . . . . . . . . . . . 495.2.5 Instrumental Variable Methods . . . . . . . . . . . . . . . . . . . . . . . . . 495.2.6 Indirect Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.3.1 Maximum Likelihood Estimation . . . . . . . . . . . . . . . . . . . . . . . 495.3.2 The Prediction Error Method . . . . . . . . . . . . . . . . . . . . . . . . . . 49Multi-Step Least-Squares Methods . . . . . . . . . . . . . . . . . . . . . . . 49Subspace Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Instrumental Variable Methods . . . . . . . . . . . . . . . . . . . . . . . . . 49Bayesian Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Time versus Frequency Domain Identification . . . . . . . . . . . . . . . . . 49Continuous Time Model Identification . . . . . . . . . . . . . . . . . . . . . 495.4 Software Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49and Probabilistic Models5.2 Estimation Methods5.2.1 Maximum Likelihood Estimation5.2.2 The Extended Invariance Principle 5.2.3 The Prediction Error Method5.2.4 Multi-Step Least-Squares Methods5.2.5 Instrumental Variable Methods5.2.6 Indirect Inference5.3 Linear Models5.3.1 Maximum Likelihood Estimation5.3.2 The Prediction Error Method5.4 Multi-Step Least-Squares Methods5.5 Subspace Identification5.6 Instrumental Variable Methods5.7 Bayesian Methods5.8 Time versus Frequency Domain Identification5.9 Continuous Time Model Identification6 Model Quality 516.1 Variance Quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.1.1 Fundamental Geometric Principles . . . . . . . . . . . . . . . . . . . . . . . 516.1.2 Fundamental Structural Results . . . . . . . . . . . . . . . . . . . . . . . . 516.1.3 Variability of Estimated Frequency Response . . . . . . . . . . . . . . . . . 51A General Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Details for FIR Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Details for ARMAX Models . . . . . . . . . . . . . . . . . . . . . . . . . . 51Details OE and BJ Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 516.1.4 Variability of Nonlinear System Estimates . . . . . . . . . . . . . . . . . . . 516.1.5 Bootstrap Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51CONTENTS 56.1.1 Fundamental Geometric Principles 6.1.2 Fundamental Structural Results 6.1.3 Variability of Estimated Frequency Response6.1.4 Variability of Nonlinear System Estimates6.1.5 Bootstrap Methods7 Experiment Design 537.1 Identifiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.2 Persistence of Exciation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.3 Input Signal Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537.3.1 Common Input Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53PRBS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Sums of Sine-Waves and Crest Factor Correction . . . . . . . . . . . . . . . 537.4 Application Oriented Experiment Design . . . . . . . . . . . . . . . . . . . . . . . 537.5 Adaptive Experiment Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 ModelValidation 558.1 Residual whiteness Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.2 Input to residual correlation tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558.3 Model Error Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 Applications 579.1 Closed Loop Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.2 Network Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.3 Errors-in-Variables Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579.4 Block-structured Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . .