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    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 1, Sep 5, 2023: Course Arrangements, Statistics and Stochastics Mapp
       Course Arrangements
      Statistics and Stochastics

      • Gradient, Jacobian and Hessian; Eigenvalues,
      Eigenvectors, and Quadratic Forms
      • 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



    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 2, Sep 14, 2023: Basic concepts in estimation Mapp
      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

    • Inlämningsuppgift icon

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

    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 3, Sep 19, 2023: Linear Estimation in Static System Mapp
      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

    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lectrure 4, Sep 26, 2023: LINEAR DYNAMIC SYSTEMS WITH RANDOM INPUTS, STATE ESTIMATION IN DISCRETE TIME LINEAR DYNAMIC SYSTEMS, KALMAN FILTER Mapp
      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

    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 5, Oct 3 2023: COMPUTATIONAL ASPECTS OF ESTIMATION, Implementation of Linear Estimation, Information Filter, THE CONTINUOUS-TIME LINEAR STATE ESTIMATION FILTER Mapp
      COMPUTATIONAL ASPECTS OF ESTIMATION
      Implementation of Linear Estimation
      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

    • Inlämningsuppgift icon

      Materials for Homework 2. The deadline is 02.11.2023.

    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 6, Oct 24, 2023: STATE ESTIMATION FOR NONLINEAR DYNAMIC SYSTEMS; EXTENDED KALMAN FILTER Mapp
      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.

    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 7, Nov 7, 2023: Particle Filter; Example for EKF (Honfa ATV) Mapp
      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

    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 8, Nov 14: MULTIPLE MODELS AND ADAPTIVE ESTIMATION Mapp

      MULTIPLE MODELS AND ADAPTIVE ESTIMATION

      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)

    • Tillgänglig om: Du tillhör gruppen H01 (SISU)
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      Lecture 9, Nov 21, 2023: DUALITY BETWEEN ESTIMATION AND CONTROL; WIENER-NN MODEL Mapp
      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