Kalman Filters and Dynamic Bayesian Networks

1. Kalman Filters andDynamic Bayesian Networks Markoviana Reading Group Srinivas Vadrevu Arizona State University

2. Markoviana Reading Group: Week3 Outline Introduction Gaussian Distribution Introduction Examples (Linear and Multivariate) Kalman Filters General Properties Updating Gaussian Distributions One-dimensional Example Notes about general case Applicability of Kalman Filtering Dynamic Bayesian Networks (DBNs) Introduction DBNs and HMMs DBNs and HMMs Constructing DBNs

3. Markoviana Reading Group: Week3 HMMs and Kalman Filters Hidden Markov Models (HMMs) Discrete State Variables Used to model sequence of events Kalman Filters Continuous State Variables, with Gaussian Distribution Used to model noisy continuous observations Examples Predict the motion of a bird through dense jungle foliage at dusk Predict the direction of the missile through intermittent radar movement observations

4. Markoviana Reading Group: Week3 Gaussian (Normal) Distribution Central Limit Theorem: The sum of n statistical independent random variables converges for n ? 8 towards the Gaussian distribution (Applet Illustration) Unlike the binomial and Poisson distribution, the Gaussian is a continuous distribution: ??= mean of distribution (also at the same place as mode and median) ?2 = variance of distribution y is a continuous variable (-8???y ??8? Gaussian distribution is fully defined by its mean and variance

5. Markoviana Reading Group: Week3 Gaussian Distribution: Examples Linear Gaussian Distribution Mean, ? and Variance, ? Multivariate Gaussian Distribution For 3 random variables Mean, ? = [m1 m2 m3] Covariance Matrix, Sigma = [ v11 v12 v13 v21 v22 v23 v31 v32 v33 ]

6. Markoviana Reading Group: Week3 Kalman Filters: General Properties Estimate the state and the covariance of the state at any time T, given observations, xT = {x1, �, xT} E.g., Estimate the state (location and velocity) of airplane and its uncertainty, given some measurements from an array of sensors The probability of interest is P(yt|xT) Filtering the state T = current time, t Predicting the state T < current time, t Smoothing the state T > current time, t

7. Markoviana Reading Group: Week3

8. Markoviana Reading Group: Week3

9. Markoviana Reading Group: Week3 Gaussian Noise & Example Next State is linear function of current state, plus some Gaussian noise Position Update: Gaussian Noise:

10. Markoviana Reading Group: Week3 Updating Gaussian Distributions Linear Gaussian family of distributions remains closed under standard Bayesian network operations One-step predicted distribution Current distribution P(Xt|e1:t) is Gaussian Transition model P(Xt+1|xt) is linear Gaussian The updated distribution Predicted distribution P(Xt+1|e1:t) is Gaussian Sensor model P(et+1|Xt+1) is linear Gaussian Filtering and Prediction (From 15.2):

11. Markoviana Reading Group: Week3

12. Markoviana Reading Group: Week3

13. Markoviana Reading Group: Week3

14. Markoviana Reading Group: Week3

15. Markoviana Reading Group: Week3

16. Markoviana Reading Group: Week3

17. Markoviana Reading Group: Week3

18. Markoviana Reading Group: Week3

19. Markoviana Reading Group: Week3

20. Markoviana Reading Group: Week3

21. Markoviana Reading Group: Week3 One-dimensional Example Update Rule (Derivations from Russel & Norvig) Compute new mean and covariance matrix from the previous mean and covariance matrix Variance update is independent of the observation Another variation of the update rule (from Max Welling, Caltech)

22. Markoviana Reading Group: Week3 The General Case Multivariate Gaussian Distribution Exponent is a quadratic function of the random variables xi in x Temporal model with Kalman filtering F: linear transition model H: linear sensor model Sigma_x: transition noise covariance Sigma_z: sensor noise covariance Update equations for mean and covariance Kt+1: the Kalman gain matrix F?t: predicted state at t+1 HF?t: the predicted observation Zt+1 � HFt: error in predicted observation

23. Markoviana Reading Group: Week3 Illustration

24. Markoviana Reading Group: Week3 Applicability of Kalman Filtering Popular applications Navigation, guidance, radar tracking, sonar ranging, satellite orbit computation, stock prize prediction, landing of Eagle on Moon, gyroscopes in airplanes, etc. Extended Kalman Filters (EKF) can handle Nonlinearities in Gaussian distributions Model the system as locally linear in xt in the region of xt = ?t Works well for smooth, well-behaved systems Switching Kalman Filters: multiple Kalman filters in parallel, each using different model of the system A weighted sum of predictions used

25. Markoviana Reading Group: Week3 Applicability of Kalman Filters

26. Markoviana Reading Group: Week3 Dynamic Bayesian Networks Directed graphical models of stochastic processes Extend HMMs by representing hidden (and observed) state in terms of state variables, with possible complex interdependencies Any number of state variables and evidence variables Dynamic or Temporal Bayesian Network??? Model structure does not change over time Parameters do not change over time Extra hidden nodes can be added (mixture of models) 2TBN Structure is replicated from slice to slice Stationary First-Order Markov process

27. Markoviana Reading Group: Week3 DBNs and HMMs HMM as a DBN Single state variable and single evidence variable Discrete variable DBN as an HMM Combine all state variables in DBN into a single state variable (with all possible values of individual state variables) Efficient Representation (with 20 boolean state variables, DBN needs 160 probabilities, whereas HMM needs roughly a trillion probabilities) Analogous to Ordinary Bayesian Networks vs Fully Tabulated Joint Distributions

28. Markoviana Reading Group: Week3 DBNs and Kalman Filters Kalman filter as a DBN Continuous variables and linear Gaussian conditional distributions DBN as a Kalman Filter Not possible DBN allows any arbitrary distributions Lost keys example

29. Markoviana Reading Group: Week3 Constructing DBNs Required information Prior distributions over state variables P(X0) The transition model P(Xt+1|Xt) The sensor model P(Et|Xt) Intra-Slice topology Inter-Slice topology (2TBN assumption)