Kalman Filters and Dynamic Bayesian Networks

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Kalman Filters andDynamic Bayesian Networks

• Markoviana Reading Group
• Srinivas Vadrevu
• Arizona State University

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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

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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

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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

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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

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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(ytxT)
• Filtering the state
• T current time, t
• Predicting the state
• T lt current time, t
• Smoothing the state
• T gt current time, t

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Gaussian Noise Example

• Next State is linear function of current state,
  plus some Gaussian noise
• Position Update
• Gaussian Noise

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Updating Gaussian Distributions

• Linear Gaussian family of distributions remains
  closed under standard Bayesian network operations
• One-step predicted distribution
• Current distribution P(Xte1t) is Gaussian
• Transition model P(Xt1xt) is linear Gaussian
• The updated distribution
• Predicted distribution P(Xt1e1t) is Gaussian
• Sensor model P(et1Xt1) is linear Gaussian
• Filtering and Prediction (From 15.2)

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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)

• - ?t1 is weighted mean of new observation Zt1
  and the old mean ?t
• Observation unreliable ? ?2z is large (more
  attention to old mean)
• Old mean unreliable ? ?2t is large (more
  attention to observation)

• - ?2 is variance or uncertainty, K is the Kalman
  gain
• K 0 ? no attention to measurement
• - K 1 ? complete attention to measurement

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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

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Illustration
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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

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Applicability of Kalman Filters
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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

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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

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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

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Constructing DBNs

• Required information
• Prior distributions over state variables P(X0)
• The transition model P(Xt1Xt)
• The sensor model P(EtXt)
• Intra-Slice topology
• Inter-Slice topology (2TBN assumption)