Location Estimation in
Sensor Networks
Moshe Mishali
(Wireless) Sensor Network
A wireless sensor network (WSN) is a wireless network consisting of spatially distributed autonomous devices using sensors to cooperatively monitor physical or environmental conditions, such as temperature, sound, vibration, pressure, motion or pollutants, at different locations.
Wikipedia
CodeBlue
Model
Fusion Center
Sensors
Maximum Likelihood Estimator
Given: are Gaussian i.i.d.
Then, the MLE is
Constrained Distributed Estimation
The communication to the fusion center is bandwidth-constrained.e.g. each sensor can send only 1 bit,
Variations
Deterministic or Bayesian Knowledge of noise structure
Known PDF (explicit) Known PDF with unknown parameters Unknown PDF (bounded or not)
Scalar or vector
Outline Known noise PDF Known noise PDF, but unknown parameters Unknown noise PDF (universal estimator) Advanced
Dynamic range considerations Detection in WSN Estimation under energy constraint (Compressive WSN)
Discussion
References
1. Z.-Q. Luo, "Universal decentralized estimation in a bandwidth constrained sensor network," IEEE Trans. on Inf. Th., June 2005
2. A. Ribeiro and G. B. Giannakis, "Bandwidth-constrained distributed estimation for wireless sensor Networks-part I: Gaussian case," IEEE Trans. on Sig. Proc., March 2006
3. A. Ribeiro and G. B. Giannakis, "Bandwidth-constrained distributed estimation for wireless sensor networks-part II: unknown probability density function," IEEE Trans. on Sig. Proc., July 2006
4. J.-J. Xiao and Z.-Q. Luo, “Universal decentralized detection in a bandwidth-constrained sensor network”, IEEE Trans. on Sig. Proc., August 2005
5. J.-J. Xiao, S. Cui, Z.-Q. Luo and A. J. Goldsmith, “Joint estimation in sensor networks under energy constraint”, IEEE Trans. on Sig. Proc., June 2005
6. W. U. Bajwa, J. D. Haupt, A. M. Sayyed and R. D. Nowak, “Joint source-channel communication for distributed estimation in sensor networks”, IEEE Trans. on Inf. Th., October 2007
Known Noise PDF – Case 1
Design:
CRLB for unbiased estimator based on the binary observations
Known Noise PDF – Case 1
min
Known Noise PDF – Case 2
Design:
Generalizing Case 2
Known Noise PDF
Example:
Known Noise PDF withUnknown Variance
Unknown Noise PDF
Setup
Binary observations:
Linear estimator:
1. Develop a universal linear -unbiased estimator for
2. Given such an estimator design the sensor network to achieve
Method
A Universal Linear -Unbiased Estimator
A necessary and sufficient condition
Construction (1)
Construction (2)
Fusion Center Estimator
To reduce MSE: Duplicate the whole system and average, OR Allocate sensor according to bit significance:
½ of the sensors for the 1st bit ¼ of the sensors for the 2nd bit, and so on…
Exact expressions can be found in [1] For small , it requires
Simulations
Simulations
Setup – Gaussian Noise PDF
The dynamic range of is large relativeto
Idea: Let each sensor use different quantization, so that some of the thresholds will be close to the real
Advanced I – Dynamic Range
Non-Identical Thresholds
Non-Identical Thresholds There is no close form for the log-
likelihood. However, there is a closed form for
the CRLB (for unbiased estimator):
Goal: minimize the CRLB instead of the MSE
Steps
1. Introduce “confidence” (i.e. prior) on 2. Derive lower-bound for the CRLB3. Derive upper-bound for the CRLB4. Implementation
Step 1/4 – “Confidence”
is the “confidence” (or prior) of The weighted Variance/CRLB:
The optimum:
Step 2/4 – Lower Bound
Derive:
+ necessary and sufficient condition for achievability
Numerically:
Step 3/4 – Upper Bound
For a uniform thresholds grid.
Select according [2, Th. 2] Then,
Step 4/4 - Implementation
1. Formulate an optimization problem for , which are the “closest” pair to the one of the condition of step 2.
2. Discretize the objective.
Advanced II – Detection
Fusion CenterConstraints:1. Each is a bit, 1 or 0.2. The noise PDF is unknown.
It is assumed that
Decentralized Detection
Suppose bounded noise Define Sensor decodes the th bit of ,
where The decision rule at the fusion center
is
Advanced III – Energy Constraint
FusionCenter
The BLUE estimator:
Setup
Advanced III – Energy Constraint
FusionCenter
Goal: Meet target MSE under quantization + total power constraints.
Probabilistic Quantization
Signalrange
Quant. Step
Bernoulli
The Quasi-BLUE estimator:
Power Scheduling
ConstConst
MSE due to BER: only a constant factor
Solution
1. Integer variable2. Non-Convex Transformation (Hidden convexity)
3. Analytic expression (KKT conditions)
Threshold strategy:1. The FC sends = threshold to all nodes (high power link).2. Each sensor observes his SNR (scaled by the path loss).3. If SNR> , send bits (otherwise inactive).
Simulations
Summary
Model Bandwidth-constrained estimation
Known Noise PDF Unknown Noise PDF
Extensions Detection Energy-constraint