## Topic outline

• ### General

• Lecture 1, Sep 5, 2023: Course Arrangements, Statistics and Stochastics Folder
Not available unless: You belong to H01 (SISU)
##### Statistics and Stochastics

• Gradient, Jacobian and Hessian; Eigenvalues,
• Gaussian Random Variables; pdf, Mean and Covariance;
Joint and Conditional Random Variables
• Fundamental Equations of Linear Estimation
• Stochastic processes; Correlation; White noise process
• Random Sequences, Markov Processes; Markov
Sequences and Markov Chains

##### Equation collection, can be used in Exam

• Lecture 2, Sep 14, 2023: Basic concepts in estimation Folder
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##### Basic concepts in estimation

Random and nonrandom parameters
Definitions of estimates
• ML Maximum Likelihood
• MAP Maximum A Posteriori
• LS Least Squares
• MMSE Minimum Mean square Error
Measures of quality of estimates
• Unbiased, variance, consistency,the Cramer-Rao
lower bound, Fisher information, Efficiency

• Materials for homework 1. The deadline is 29-09-2023.

• Lecture 3, Sep 19, 2023: Linear Estimation in Static System Folder
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##### Linear Estimation in Static System

• Minimum Mean Square Error (MMSE)
• MMSE estimation of Gaussian random vectors
• Linear MMSE estimator for arbitrarily distributed random vectors
• LS estimation of unknown constant vectors from linear observations, batch form, recursive form.
• Apply the LS technique

• Lectrure 4, Sep 26, 2023: LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS, STATE ESTIMATION IN DISCRETE TIME LINEAR DYNAMIC SYSTEMS, KALMAN FILTER Folder
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##### LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS

• Gaussian pdf, mean and covariance
• Stochastic sequences, Markov property
• Discrete-time linear stochastic dynamic systems,  Prediction, propagation of mean and covariance
• Continuous-time linear stochastic dynamic systems, Propagation of mean and covariance

##### STATE ESTIMATION IN DISCRETE TIME LINEAR DYNAMIC SYSTEMS
• The estimation of the state vector of a stochastic linear dynamic system is considered.
• The state estimator for discrete-time linear dynamic systems driven by white noise — the (discrete-time) Kalman filter — is introduced.
• Estimation of Gaussian random vectors.
##### KALMAN FILTER

• For linear systems, white noise Gaussian processes
• Linear equations used for state prediction,  prediction of the measurement and for measurement update.
• Exact propagation and measurement  update equations for a priori and a posteriori covariances
• All pdfs stay exactly Gaussian, no need for approximations

• Lecture 5, Oct 3 2023: COMPUTATIONAL ASPECTS OF ESTIMATION, Implementation of Linear Estimation, Information Filter, THE CONTINUOUS-TIME LINEAR STATE ESTIMATION FILTER Folder
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##### Information Filter

• carries out the recursive computation of the inverse of the covariance matrix.
• an alternative to the “standard“ Kalman filter formulation and is less demanding computationally for systems with dimension of the measurement vector larger than that of the state.
• has the advantage that it allows the start-up of the estimation without an initial estimate,  i.e., with a noniformative prior
• Suitable for distributed estimation for example in swarm robotics

##### THE CONTINUOUS-TIME LINEAR STATE ESTIMATION FILTER

The linear minimum mean square error (LMMSE) filter for this continuous time problem, known as the Kalman-Bucy filter
The duality of the LMMSE estimation with the linear-quadratic (LQ)
control problem is discussed. These two problems have their
solutions determined by the same Riccati equation

• Materials for Homework 2. The deadline is 02.11.2023.

• Lecture 6, Oct 24, 2023: STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS; EXTENDED KALMAN FILTER Folder
Not available unless: You belong to H01 (SISU)
##### STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS

• Present the optimal discrete-time estimator.
• Discuss the difficulty in implementing it in practice due to memory requirements and computational requirements.
• Discuss numerical implementation of the optimal discrete-time estimator.
• Derive the suboptimal filter known as the extended Kalman Filter

##### EXTENDED KALMAN FILTER

• Very limited feasibility of the implementation of the optimal filter, the functional recursion, suboptimal algorithms are of interest.
• The recursive calculation of the sufficient statistic consisting of the conditional mean and variance in the linear-Gaussian case is the
simplest possible state estimation filter.
• As indicated earlier, in the case of a linear system with non-Gaussian random variables the same simple recursion yields an
approximate mean and variance.
• A framework similar is desirable for a nonlinear system. Such an estimator, called the extended Kalman filter (EKF), can be obtained
by a series expansion of the nonlinear dynamics and of the measurement equations.

• Lecture 7, Nov 7, 2023: Particle Filter; Example for EKF (Honfa ATV) Folder
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##### Particle Filter

▪ Represent belief by random samples
▪ Estimation of non-Gaussian, nonlinear processes
▪ Monte Carlo filter, Survival of the fittest, Condensation, Bootstrap filter, Particle filter
▪ Filtering: [Rubin, 88], [Gordon et al., 93], [Kitagawa 96]
▪ Computer vision: [Isard and Blake 96, 98]
▪ Dynamic Bayesian Networks: [Kanazawa et al., 95]d

• Lecture 8, Nov 14: MULTIPLE MODELS AND ADAPTIVE ESTIMATION Folder
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The multiple model (MM) algorithms; Hybrid systems — the system behaves according to one of a finite number of models, it is in one of several modes (operating regimes),
both discrete (structure/parameters) and continuous uncertainties

1. The static MM algorithm —for fixed (nonswitching) models
2. The optimal dynamic MM algorithm — for switching models —Markov chain, two
suboptimal approaches: Generalized pseudo-Bayesian (GPB) ; Interacting multiple model (IMM)

• Lecture 9, Nov 21, 2023: DUALITY BETWEEN ESTIMATION AND CONTROL; WIENER-NN MODEL Folder
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##### DUALITY BETWEEN ESTIMATION AND CONTROL

• Rudolf Kalman invented optimal state feedback control
• He invented the Kalman Filter, too.
• In Kalman’s paper with Bucy, it is shown that the problem of the optimal estimation is dual to the optimal control problem
• Here, the duality between estimation and control is shown using discrete formulations

##### WIENER -NN MODEL AND ITS USE IN MODELING OF BIOTECHNICAL PROCESSES
• Linear system: Orthogonal expansion of impulse response
• Reduced Wiener representation
• ’Feedforward’ Wiener-NN-model
• Wiener-NN with feedback model
• Comparison with nonlinear ‘regression’ models
• Identification of NOE-type models
• Case: Tricoderma fungi process producing enzyme