Selected Areas in Data Fusion

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Selected Areas in Data Fusion

• Nalin Wickramarachchi and Saman Halgamuge

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Data Fusion Problem
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Mathematical Techniques

• Fusion process is a complex mathematical task and
  many issues need to be addressed
• Data in diverse formats, noisy and ambiguous
• analogue, digital, discrete, textual, imagery
• Data dimensionality and alignment
• coordinate systems, units, frequency, amplitude,
  timing
• Temporal alignment
• synchronisation of data,
• spatial distribution of sensors demands precise
  time measurements,
• data arrival at fusion node may not coincide due
  to variable propagation delays

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Fusion Technologies

• Fusion techniques are drawn from many different
  disciplines in mathematics and engineering.
• Probability and statistical estimation
  (Bayes,HMM..)
• Signal processing and information theory
• Image processing and pattern recognition
• Artificial intelligence (classical/modern)
• Information and communication technology
• Software engineering and networking
• Biological sciences
• control theory

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Bayes Theorem

• Assume we have a hypothesis space H, and dataset
  D. We can define three probabilities
• P(h) is the probability of h being the correct
  hypothesis before seeing any data. P(h) is called
  the prior probability of h.
• Example Chance of rain is 80 if we are close to
  the sea and in latitude X (no data has been
  seen).
• P(D) is the probability of seeing data D.
• P(Dh) is the probability of the data given h. It
  is called the likelihood of h with respect to D.

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Bayes Theorem

• Bayes theorem relates the posterior probability
  of a hypothesis given the data with the three
  probabilities mentioned before

Likelihood
Prior probability
Posterior probability
Evidence
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Maximum A Posteriori andMaximum Likelihood

• A method that looks for the hypothesis with
  maximum P(hD) is called a maximum a posteriori
  method or MAP.
• HMAP argmaxh P(hD)
• Sometimes we may assume that all a priori
  probabilities are equally likely in which case
  the method is called a maximum likelihood or ML
  method
• HML argmaxh P(Dh)

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Maximum A Posteriori andMaximum Likelihood

• Look among all hypotheses for the one with
  maximum a posteriori probability if using a MAP
  method, or the one with maximum likelihood If
  using an ML method.

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An example

• We do a test in the lab to check if a patient has
  cancer.
• We know only 0.008 of the population has cancer.
• The lab test is imperfect. It returns positive in
  98 of the cases where the disease is present
  (true positive rate) and it returns negative in
  97 of the cases where there is no disease (true
  negative rate).

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Example

• What is the probability that the patient has
  cancer given that lab result was positive?

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Bayesian Inference

• Suppose H1 , H2 , , Hn are mutually exclusive
  and exhaustive hypothesis that explains the
  observed data D.
• Then,

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Bayesian Inference

• Bayesian formulation can use subjective
  probabilities.
• It provides the probability of a hypothesis being
  true, given the evidence.
• It allows incorporation of priory knowledge of
  the likelihood of a hypothesis being true at all.
• In previous example, P() can be calculated as,
• P() (0.98)(0.008) (0.03)(0.992)
• It does not require knowledge of probability
  distribution functions.

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Hidden Markov Models
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Hidden Markov Models (HMM)

• Definition of HMM
• A stochastic process with an underlying
  stochastic state transition process that is not
  observable (hidden). The underlying process can
  only be inferred through a set of symbols emitted
  sequentially by the stochastic process.
• Example Dishonest Casino dealer States
  (hidden) F or L
• The set of symbols emitted 1,..,6

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HMM Problem

• Problem
• Given the set of symbols emitted sequentially,
  determine the state or sequence of states of the
  underlying hidden process.
• Solution
• Since this is a stochastic process, we can find
  most likely state or most likely state path
  (sequence of states) only.
• i.e., there is no perfect solution. But
  algorithms exist that find the best possible
  answer given the evidence.

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Applications

• Applications
• Signal processing (speech recognition, symbol
  identification in mobile communication, character
  recognition in images)
• Sensor fusion (target tracking)
• Bioinformatics (gene finding)
• Manufacturing (fault detection)
• Environment (weather prediction)

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Formulation

• Elements of an HMM
• The number of states.
• The number of distinct emission symbols in each
  state.
• The state transition probability matrix.
• The emission symbol probability matrix.
• The initial state distribution.

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Graphical Model

• Circles indicate states
• White arrows indicate probabilistic state
  transitions
• Yellow arrows indicate emission in each state

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Graphical Model

• Green circles are hidden states
• Dependent only on the previous state (Markov
  assumption)
• Next state depends only on the current state, and
  not on the history

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Graphical Model

• Purple circles are observed states
• Dependent only on the corresponding hidden state
• Observed state emitted symbol

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HMM Formulation
S
S
S
S
S
K
K
K
K
K

• S, K, P, A, B
• S s1sN are the hidden states
• K k1kM are the observations (emissions)

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HMM Formulation
A
A
A
A
A
S
S
S
S
S
B
B
B
K
K
K
K
K

• S, K, P, A, B
• P pi are the initial state probabilities
• A aij are the state transition probabilities
• B bik are the observation state probabilities

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Inference in an HMM

• Given an observation sequence, compute the most
  likely hidden state sequence (Casino problem)
• Given an observation sequence, compute the most
  likely current hidden state.
• What is the probability that a given observation
  sequence was generated by the HMM?
• Given an observation sequence and set of possible
  HMM models, which model most likely generated the
  observed sequence?

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Decoding Most Probable State Path
ot
ot-1
ot1

• Given an observation sequence and a model,
  compute the probability of the observation
  sequence

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Decoding
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Decoding
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Forward Algorithm
Define
?i(t) probability of getting length t
observation o1,,ot with final state being xt.
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Forward Algorithm
Then
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Backward Algorithm
Similarly
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Decoding Solution
Forward Procedure
Backward Procedure
Combination
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Best State Sequence

• Find the state sequence that best explains the
  observations
• Viterbi algorithm

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Viterbi Algorithm
x1
xt-1
j
oT
o1
ot
ot-1
ot1
The state sequence which maximizes the
probability of seeing the observations to time
t-1, landing in state j, and seeing the
observation at time t
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Viterbi Algorithm
x1
xt-1
xt
xt1
Recursive Computation
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Parameter Estimation
A
A
A
A
B
B
B
B
B

• Given an observation sequence, find the model
  that is most likely to produce that sequence.
• No analytical method.
• Given a model and observation sequence, update
  the model parameters to better fit the
  observations.

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Parameter Estimation
A
A
A
A
B
B
B
B
B
Probability of traversing an arc
Probability of being in state i
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Parameter Estimation
A
A
A
A
B
B
B
B
B
recursively update the estimates of the model
parameters.
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Example Tracking

• Track a single ground target via HMM approach
  that permit the exploitation of a priori
  knowledge about road-network features.
• Transition probability matrices exploit road
  knowledge.

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The Tracking Problem
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Exploiting Road Knowledge
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HMM-based tracker

• States geographic locations of the target and
  sensor observation.
• Probability transition matrix exploits a priori
  road information.
• Sensor performance model-based.
• Initial state distribution measurement-based.
• Estimation Approach Viterbi algorithm for best
  state sequence determination.

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HMM-based tracker

• Technique
• Partition the road segments into HMM block-matrix
  sections.
• Each block is associated with an HMM.
• Each HMM estimates its own subtrack.
• Join all subtracks to form the entire track.

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HMM-based tracker

• Transition Matrix
• Non-road states move toward the spatially-
  nearest road states.
• Road states move onto the immediately adjacent
  road states with equal probability.
• Initial state distribution
• First HMM probability one on the initial
  measurement state.
• Other HMMs probability one on the state closest
  to the last estimated state.

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Performance
-- target trajectory measurements
estimation regional HMMs
y (meters)
x (meters)
Source Chih-Chung Ke, Jesus Garcia Herrero,
James Llinas, Center for Multisource Information
Fusion, Stae Univ of NY at Buffalo, Calspan-UB
Research Center, Buffalo NY
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Sensor Scheduling
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Need for Sensor Management
Defense of Systems and Personnel
Achieve Assigned Missions
Signature Maintenance
Battlespace Awareness
Use of sensors necessary for offensive functions
Use of some sensors makes ship detectable
Environmental Sensors
Battlespace Sensors
Maneuvering Systems
Moving may disambiguate sensor data
Knowledge of environmental conditions can impact
choice of sensor settings
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Waveform Control for Target Tracking
Helicopter
Airliner
Fighter jet
Receiver
Transmitter
For best detection accuracy, transmitter
parameters must be adjusted for different types
of targets
Detector/ Tracker
Waveform control
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Optimum Sensor Scheduling

• Optimise choice of sensor/sensors for next
  observation,
• using current state of target
• using knowledge of environment
• to minimise cost of sensor operation (some
  measurements are more costly than others)
• to optimise accuracy of tracking/locating target

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Solution Strategy

• Markov decision process (MDP) and
• Partially observable Markov decision process
  (POMDP)
• similar to HMM problem, but in addition, there is
  a cost in each state transition
• each state path therefore has a cost associated
  with it, in addition to probability of taking
  that path.
• goal maximise the likelihood of reaching
  expected state with optimum cost.

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POMDP

• Underlying state is Markov chain and time
  varying.
• Only noisy measurements of state available.
• Original goal optimise cost function of expected
  state over time.
• Equivalent problem optimise equivalent function
  of information state over time.
• solution methods are available when state and
  measurement process are discrete and some states
  are more valuable than others.

Vikram Krishnamurthy, Algorithm for optimum
scheduling and management of HMM sensors, IEEE
Tran. on Sig. Proc., vol. 50, No. 6, 2002.
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References

• David Hall and Sonya McMullen, Mathematical
  Techniques in Multisensor Data Fusion, Artech
  House, 2004.
• Ng, G.W., Intelligent Systems Fusion, tracking
  and Control, Research Studies Press, 2003.
• Vikram Krishnamurthy, Algorithm for optimum
  scheduling and management of HMM sensors, IEEE
  Tran. on Sig. Proc., vol. 50, No. 6, 2002.
• Saman Halgamuge and Manfred Glesner, Fuzzy
  Neural Networks Between Functional Equivalence
  and Applicability, Int. Jrn. of Neural Systems,
  vol. 6, no. 2, 1995.
• THANK YOU
• My contact saman_at_unimelb.edu.au