Maximum Likelihood Energy Based Acoustic Source Localization II

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Maximum Likelihood Energy Based Acoustic Source
Localization II

• -Xiaohong Sheng
• Feb. 24, 2003

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Energy Based Source Localization Model

• Assumptions
• Sound propagates in the free air,
• Target is pre-detected to be in a particular
  region of a sensor field.
• The region size is not very big so that targets
  are not far from the sensors.
• Sound source can still be treated as an
  omni-directional point.
• (We can assume the dimension of the engine of the
  vehicle is relative small compared with the
  distance between the sensor and the vehicle).
• The propagation medium (air) is roughly
  homogenous ( i.e. no gusty wind) and there is no
  sound reverberation.

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Energy Based Source Localization Model (Cont.)

• Assumption
• Noise is uncorrelated to signal,
• Signal is uncorrelated for different vehicle
• Acoustic Waveform is zero mean,
• Time window (T) is short enough so that acoustic
  waveform is stationary at this period, but long
  enough ( more than 400 sampling point) so that
  the average noise energy in this period can be
  assumed as Gaussian distributed.
• Model

• gi gain factor of the microphone
• Sk(t) energy emitted by the kth source
• ?k(t) Source Ks location during time interval
  t.
• ri sensor location of the ith sensor
• ?i(t) perturbation term that summarizes the net
  effects of background additive noise and the
  parameter modeling error.

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Normalized Energy Function

• Define

• Minimize Log-Likelihood Function

• Unknown Parameter

• Solution
• Projection Solution
• Set and insert the results into the
  Log-likelihood function to get the modified Cost
  function
• Expectation Maximization Solution
• Set to get ?j , S

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EM Solution for ML estimation

• Where

PH is projector of H
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EM Solution

• Initialization Initial estimates of ?j 1 j
  K
• Repeat until convergence
• Expectation Step.
• Estimate S
• Calculate Q based on estimated S
• Calculate Aj and Cj
• Maximization Step.
• Substitute QAj Cj into equation to get ?j ,

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Projection Solution

• Set

• Modified Likelihood Cost Function
• Insert the result to get the modified function

is the Reduced SVD of H
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Cramer-Rao Bounds

• CRB Definition
• Inverse of the Fisher Matrix
• lowest possible variance
• Purpose of CRB analysis
• indicate the performance bounds of a particular
  problem.
• facilitate analysis of factors that impact most
  on the performance of an algorithm.
• Fisher Matrix of our Energy Decay Model

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Cramer-Rao Bounds for the Energy Decay Model

• Fisher Matrix for our Model
• CRB for our Model
• Where

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Cramer-Rao Bounds for Single Target
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CRB Analysis (1)

• Assume
• gi and si are same for all sensors in the field..

then
Define
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CRB Analysis (2)-sensor deployment

• Way to reduce CRB
• i.e.
• When source is fixed, deploy the sensors
• symmetrically around this source, e.g.
• When source is moving, deploy the sensors
  uniformly distributed in the region, especially,
• When the source is along the road, deploy the
  sensors symmetrically along the two side of the
  road.
• Increase X and Y ,
• decrease the overall distance of the sensor to
  the target
• When the sensors are deployed close to the road,
  the weighted overall distance from the targets to
  the sensors is reduced. we can get smaller CRB.
• When the sensors are dense, there are more
  sensors close to the targets, therefore, smaller
  CRB we can approach

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Performance Analysis based on CRB

• Chebyshev's inequality
• For ML estimation, estimation variance
  asymptotically approach to its CRB, therefore
• Dense sensors that are close to the road gives
  smaller CRB, and therefore, smaller variance, so,
  it improves the performance in the sense that,
• is
  smaller.

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CRB Simulation sensor deployment

• Sensor Deployment, (a) dense sensor, (b) dense
  sensor close to road, (c) loose sensor located at
  one side, (d) loose sensor located at two side.

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Sensor deployment for CRB analysis-simulation
results

• Dense sensor deployed close to the road uniformly
  along the two side of the road gives the smallest
  CRB
• Dense sensor deployed uniformly along the road
  gives the second smallest CRB
• Loose sensor deployed uniformly along the road
  gives the third smallest CRB
• Loose sensor deployed one side along the road
  gives the largest CRB

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Sensor Deployment Based on CRB Analysis

• Closer to the road
• Dense Sensor
• Uniformly distributed in the region
• When targets are on the road, uniformly
  distributed along the two side of the road
• When targets are not necessary on the road
• Uniformly distributed on the plane

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Evaluation by Experiments and Simulation

• Purposes
• Compare the performance of ML estimation and NLS
  estimation
• Compare the performance of Projection solution
  with Exhaustive Search and MR search and EM
  Solution
• Compare the performance when the source have
  different (S2/S1)
• Experiment
• Single Target Localization for AAV and DW Data
• Simulation
• Exhaustive and MR search of projection solution
  for ML estimation
• two targets moving in opposite direction for the
  cases when the energy source of the two targets
  are similar (simulated data only).
• two targets moving in opposite direction for the
  cases when the energy source of the two targets
  have significant deference (simulated data only).
• EM solution for ML estimation
• two targets moving in opposite direction for the
  cases when the energy source of the two targets
  are similar (simulated data only).
• Two-targets moving in opposite direction for the
  cases when the energy source of the two targets
  are similar or have significant difference
  (simulated data only)

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Experiments Sensor Deployment

• Sensor deployment, road coordinate and region
  specification for experiments

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Experiments (AAV)

• AAV ground truth, localization estimation results
  and estimation error histogram based on ML
  algorithm with projection solution and NLS
  algorithm (MR search is used, grid size is 44,
  22 and 11. Estimation results look bias from
  the ground-truth, see discussion for reasoning)

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Experiments (DW)

• AAV ground truth, localization estimation results
  and estimation error histogram based on ML
  algorithm with projection solution and NLS
  algorithm (MR search is used, grid size is 44,
  22 and 11. Estimation results look bias from
  the ground-truth, see discussion for reasoning)

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Discussion

• Both ML and NLS algorithms perform well
  estimations of target location.
• ML algorithm with projection solution outperforms
  to NLS algorithm
• Less estimation error,
• smaller maximum error
• NLS algorithm needs less bandwidth.
• NLS doesn't use noise variance for its estimation
  while ML algorithm does need it.
• NLS algorithm save about 1/4 bandwidth.
• Localization estimation results look bias from
  the real ground-truth.
• Inaccurate GPS measurement, sharp background
  noise or sensor faults
• Ground truth also looks bias from the road

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Simulation (Deployment and Ground truth)

• (a) sensor deployment and road coordinate for
  simulations

• (b) Ground truth for two targets moving in the
  opposite direction

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Comparison of ML estimation -Projection Solution
with ES and MRS and Direct solution with EM
algorithm
(b)
(a)
(Distance Estimation Error)

• Estimation distance error comparison for
  projection solution using MR search and
  exhaustive search and Direct solution with EM
  algorithm
• (a) target 1, (b) target 2
• two targets moving in opposite direction
• Noise is uniformly distributed from 0.01ymax to
  0.04ymax,
• s1s, s21.2s
• Projection solution has much better performance
  than Direct solution with EM algorithm

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Simulation for comparison of ML estimation
-Projection Solution with ES and MRS and Direct
solution with EM algorithm
(Variance Estimation Error)
(a)
(b)

• Estimation distance error comparison for
  projection solution using MR search and
  exhaustive search and Direct solution with EM
  algorithm
• (a) target 1, (b) target 2
• two targets moving in opposite direction
• Noise is uniformly distributed from 0.01ymax to
  0.04ymax,
• s1s, s21.2s
• Estimation Variance of Projection Solution
  approaches to its CRB ( reach its performance
  bounds

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Comparison for two targets having different
energy sourceprojection solution with MR search
only
Estimation error difference when two targets have
different s2/s1
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Discussion

• Algorithm with best performance
• ML estimation with projection solution
• Its variance approaches to its CRB.
• Disadvantage of Projection Solution
• Heavy computation burden
• Algorithm with fastest computation
• EM Algorithm
• convergence within 5 or 6 times of iteration
• Disadvantage of EM algorithm
• Easy to track into local minimum
• Previous Energy ratio method ( Nonlinear Least
  Square)
• Advantage
• No need to use the variance estimation
• Disadvantage of NLS
• Single Target localization
• Performance is not as good as Projection solution
• When two targets have significant different
  source energy
• The target with low energy is more ambiguous

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Discussion

• Projection solution with MR search
• Reduce the computer burden a lot
• Performance is still outstanding
• Using previous estimated location
• Reduce the search region
• Improves the efficiency
• Conclusion
• Projection Solution with MR search and using the
  previous estimation location gives outstanding
  performance and reasonable computation burden
• The above estimation is based on the assumptions
• Noise i.i.d., Gaussian Distributed
• No sharp background noise
• Sensor fault doesnt happen
• The number of targets in the region has been
  estimated ( need to use sequential analysis)

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Conclusion

• Robust
• averaging of instantaneous power over a
  pre-defined time interval
• Accuracy,
• Estimation error is within 15 meters for
  simulated data ( projection solution)
• More than 80 of estimation error is within 20
  meters for real data
• Efficiency,
• For Projection solution with MR search and
  reduced search region based on previous location
  estimation
• when there is k targets, we need 16k4k search
  times
• When k2, need 272 search for every location
  operation
• For Direct solution with EM algorithm
• Need only 5 or 6 times of iteration
• Low communication bandwidth
• each sensor is to report only the energy reading
  during each detection/localization time interval.
  
• This is a very favorable feature as wireless
  sensor network typically has only narrow band
  communication capability

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Sensitivity Analysis

• Parameters
• Sensor Gain gi,
• Measured Energy yi
• Estimated noise mean and variance µi, si
• Sensor Location ri
• Noise Character N(µi, si )? , i.i.d.?
• ?2, by Central Limit theorem -gti.i.d. Gaussian
• Decay a2?
• Location estimation based on ML estimation
  (Project Sol)

Wont discuss here

• looking for the target locations whose
  constructed subspace HGD has the maximum
  projection energy for normalized energy Z

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Normalized Energy ZContributed by yi, µi, si

• Parameters that cause the change of Z
• Measured Energy yi
• Estimated noise mean and variance µi, si
• When Perturbation from Z, ?Z is parallel to Z,
  i.e. ?Z Z,
• the required subspace HH ?H almost the same as
  H .
• when Z?Z is orthogonal to Z, - Z
• Get largest change of the required subspace, HH
  ?H.
• When ?Z is not parallel to Z,
• ?(?ZZ)Z f(?Zi, Z, ?Zi), for i1,2,n
• the degree of changing the required subspace is
  affected by f(?Zi, Z, ?Zi),

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Change Subspace
Since ?Z0, Z0, ?ZZ90 Max Change Space is
?ZZ90gt(?Zi ? Zi 0)
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Sensitivity Analysis Simulation Results ( for Z)

• Effects of Z perturbation with different
  direction and size ( Target 2 is similar)
• when ?Z - Z, estimation error increases a lot
• When ?Z Z, estimation error most keep same
• When ?Z - Z and ?Z bigger, estimation error
  increase mostly
• Estimation error incensement also relates the D
  MATRIX

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Sensitivity of Z - contributed by noise mean,
variance and energy measurement

• Sensitivity of µi, si, yi
• When there is no sharp background noise, i.e.,
  background noise is quasi-stable,
• µi and si estimation error are small compared to
  acoustic energy, so, the effects of µi and si
  estimation error to the location estimation is
  small
• When there is peak background noise,
• sharp background noise can be treated as yi by
  our CFAR detector, and therefore, the direction
  of Z changes a lot, i.e.,
• ?(?ZZ)Z f(?Zi, Z, ?Zi), can be large
• When sensor works well,
• yi measurement error is normally small,
  ?(Z?Z)Z, is small since ?Zi is small
• When sensor fault occurs,
• ?yi is big, ?(Z?Z)Z and ?Zi can be large,
• ?(Z?Z)Z, also depend on the density of the
  sensors, ?(?ZZ)Z f(?Zi, Z, ?Zi),

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Sensor Gain Perturbation

• Reduced SVD of H

• Sensor Gain Perturbation changes the space of H
• Subspace Composed by
• Subspace Composed by H

Assume is less than smallest
nonzero singular value of H.
Location Estimation based on ML estimation
changes to
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Comparison of sensor gain perturbation with other
parameter perturbations

• Sensitive to sensor gain
• Assume there is no sharp background noise and no
  sensor fault
• Sensitivity is related to sensor deployment (D)

Sensitivity Simulation for different parameters (
Gain, noise mean and variance, sensor location, )
Assume there is no sharp background noise and
sensor works well
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Sensitivity Conclusion

• Current algorithm is sensible to sharp background
  noise, sensor fault and gain estimation error.
• When there is no sharp background noise and
  sensor works well, sensitivity order of current
  algorithm is
• gain estimation error(-10)
• Noise variance estimation error(-10)
• Sensor location error ( within 10 meters)
• Noise mean estimation error ( -10)
• Dense sensor can reduce the sensitivity of these
  parameters
• Proper sensor deployment reduces the effects of
  gain estimation error

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Performance Enhancement

• Purpose
• Robust to sharp noise
• Robust to sensor fault
• Method
• Sub-band Analysis
• Sub-band Detection, Sub-band Localization
• Sequential Analysis
• Using Tracking Results
• Modeling the energy transition by Markov Modeling
• Filter out the sharp noise
• Combine them together
• Robust Test

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Sub-band Characteristics
Energy
Energy Ratio
Time

• Noisy Node 1

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Sub-band Characteristics
Energy
Energy Ratio

• Good Node 42

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Sub-band Characteristics
Energy
Energy Ratio

• Node 47

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Sub-band Characteristics
Energy
Energy Ratio

• Noisy Node 49

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Sub-band Characteristics
Energy
Energy Ratio

• Noisy Node 61

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Sub-band Analysis

• Energy at some bandwidth is dominated
• Energy ratio of different bandwidth is an
  important parameter, Filter out some background
  noise and reduce the fault alarm by this
  parameter
• Use sub-band energy can improve performance of
  detection and localization
• Detection
• Node detection
• Sub-band detection, filter out the detection
  results using the energy sub-band ratio
• Region detection
• Use sub-band node detection results to detect the
  target based on that band
• Fuse sub-band detection results

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Sub-band Analysis (Detection)
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Sub-band Analysis (Detection)
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Sub-band Analysis-Localization

• Localization Based on sub-band energy
• -ML estimation with projection solution

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Sub-band Analysis-Localization

• There always exist some sub-band localization
  results with good performance at particular time

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Sub-band Localization ( Conclusion)

• There always exists some sub-band localization
  with good performance
• Noise ( or other vehicle) might cover some
  sub-band of the vehicle we are interested, but
  not necessary to cover all the sub-band of that
  particular vehicle
• If we can always identify which sub-band
  localization results are the correct results, we
  can get the good performance most of the time
• How?
• Using tracking results
• Robust Test
• MAP estimation

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Performance Enhancement

• Tracking Results
• For each sub-band, use MAP instead of ML
• Cost Function
• Ignore the prior source distribution
• EM Solution -Projection Solution
• ( find the sub-subspace
  constraint on certain subspace
• with Gaussian distribution)
• could be complicated
• Another way to do this is to use ML estimation
  again and Use the ML estimation results for each
  sub-band and then use the tracking results to do
  some robust test

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Performance Enhancement

• ML estimation for each sub-band
• Use Tracking Results to test the sub-band
  localization results
• Other Test
• Bayes Test
• Digitize the Localization Estimation by Gaussian
  Distribution Function, get the prior probability
  for each possible localization, and do bayes test
• MinMax Test
• There is a probability that the vehicle could
  stop suddenly or change direction, use minMax
  test to identify which case for the recent
  motion, and then use that case distribution
  function to test the results according the
  sub-band localization resluts
• Balanced Test
• Equalize the false alarm and miss detection

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Performance EnhancementFault Sub-band
Identification (Noise Cancellation)

• Suppose there is l sub-band ratio for sensor
  node, and the classifier identify that the signal
  is from ith vehicle
• For jth Sub-band for each sensor Node, the
  Matching Score is (Xj is the sub-band ratio
  vector for (jth sub-band energy) /(total energy)
  for all nodes)
• MI ? Euclidean distance
• MR-1?Mahalanobis (N is the number of trials)
• The T2 test statistics is

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Performance EnhancementSensor Fault
Identification- sub-band analysis-noise
cancellation

• State Transition Matrix
• Suppose vehicle has four states
• ( A,B,C,D), each state has different sub-band
  ratio pattern,
• HMM modeling
• Use Observation X to estimate current state,
• Use that state character to do T2 test for each
  sensor node

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Localization Enhancement-HMM, Bayes Test,
Balance Test, Robust Test

• Vehicle can stop or change direction suddenly
• But the maximum speed wont change
• Define states as follow
• For localization results at each sub-band XX1,
  X2,X18, estimate the possible current state
• HMM Model
• Bayes Test, Balance Test ( using the tracking
  results)
• Robust Test, (MinMax Test)
• Use that state, test the localization results for
  each sub-band localization

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Node Detection
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Region Detection
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Localization
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Future Work

• Use sequential energy reading (Space-Time
  Analysis)
• estimate the number of the targets in the region
  (?)
• filter out the abnormal sensor data or sharp
  background noise (done with sub-band and HMM
  model)
• Markov Random Field Modeling ?
• Use tracker estimation as the pre-prior
  probability (done)
• MAP? ( ML is the special issue on MAP with
  equally prior probability)
• Sensor deployment analysis for performance
  enhancement
• From CRB analysis (done),
• From Parameter sensitivity,
• From Region Detection,
• Noise and interference cancellation
• Energy computation restrict to a sub-band
  ?(done)
• Other Method?
• Combine other localization method?
• Time-Delayed estimation?
• PIR? Others?