controlling uncertainty in personal positioning at minimal measurement cost
DESCRIPTION
Controlling Uncertainty in Personal Positioning at Minimal Measurement Cost. * Hui Fang, * Wen-Jing Hsu, and ' Larry Rudolph * Singapore-MIT Alliance Nanyang Technological University ' MIT. Outline. Background & introduction Problem statement Location inference model - PowerPoint PPT PresentationTRANSCRIPT
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Controlling Uncertainty in Personal
Positioning at Minimal
Measurement Cost
*Hui Fang, *Wen-Jing Hsu, and ' Larry Rudolph*Singapore-MIT Alliance
Nanyang Technological University' MIT
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Outline
Background & introduction
Problem statement
Location inference model
Energy-saving strategies
Conclusion
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Personal mobile positioning devices
GPS Cell tower
Mobile device
2008-01-23 19:51:46
CellID=(525, 5, 12, 51153)
GPS=(1.34690833333333, 103.678925)2008-01-23 19:53:06
CellID=(525, 5, 12, 13901)
GPS=(1.34690833333333, 103.678925)2008-01-23 19:55:46
CellID=(525, 5, 12, 51683)
GPS=None2008-01-23 19:55:51
CellID=(525, 5, 12, 51683)
GPS=(1.34690833333333, 103.678925)2008-01-23 19:55:56
CellID=(525, 5, 12, 51683)
GPS=(1.34690833333333, 103.678925)
GPS/Cell-ID records
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Uncertainty in Positioning Measurement
Certainty of position estimate
A cell tower ID covers a larger area
GPS, error 3-5 meters
GPS > Cell-ID
Cost. Battery-energy consumption per probe.
GPS > Cell-ID
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Problem statement
Goal
Estimate a mobile user's actual position at a given point
of time by individualized means
Requirements
Sufficient certainty (i.e. above threshold) on estimates
Efficient on energy consumption
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Representation of a positional estimate
A positional estimate is a 2-D
Gaussian random variable z.
σ is a standard deviation reflecting uncertainty.
u is the best estimate of the actual location z.
(u,σ)
0
68.3% chance staying in circle (radius=σ)
x
y
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Combining two estimates Given two estimates
(u, V1) and (v,V2)
New estimate
The new estimate is closer to the one with greater certainty;
The certainty of new estimate increases.
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Movement model
Position xk at time tk
Measurement zk
Involved noises:
wk velocity noise
rk measurement noise
A journey with 5 measurements over time [t0,t3]
We assume:*user follows a given journey
*velocity noise is also Gaussian
t0 t1 t2
t3
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Estimate new position over time
Given:xk, last positionτk , time elapse
vk Vw , velocity mixed with noise
New estimatexk+1 advance to k+1 time step
* Estimate becomes more uncertain over time;* Uncertainty curve is non-linear.
time
unce
rtai
nty
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Complete location inference model
Measurement update
Prediction update
More details, refer to:[1]An introduction to the Kalman filter. G.Welch et al. 2001[2]Simultaneous GPS and Cell-ID Records for Personal Positioning and Location Inferences. H. Fang et al., 2007
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Energy saving versus certainty Estimate is safe when its uncertainty (denoted by
standard deviation σ) is smaller than a threshold Each positioning probe costs a portion of battery-
energy
Problem: minimize the overall cost while keeping all the estimates safe
Trade-off?
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Measurement strategies
Question: For one type of device, how does
certainty change when carrying out probes
periodically?
time
Probe A Probe A Probe A
ΔT = constant
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Periodic probing
Given a fixed probe time
interval Δt , we infer the
new uncertainty curve σk+1
Δt
σk+1
Note: σw velocity noise
σrk+1 measurement noise
σk uncertainty at time step k
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Periodic probing (cont’d)
We further showed that when periodic probing Certainty will converge to a fixed value with any initial
estimate
In order to keep certainty non-decreasing, either measurement must be sufficiently precise,
or the time interval small. *
* Minimum probe frequency f,
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Probe strategy with 2 types of devices
Two types of positioning devices
available, probes A and B with diff. costs
and certainty
A
B
Decision problem Which probe type to use,
A or B?When to carry out
probes?How many probes each
time?
time
t0 t1 t2
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Extending the safe duration of a journey
Safe duration: the portion of time while estimate uncertainty is below the threshold, in a journey of time T.
Connected probes: when probe A is carried out at the end of probe B’s safe duration, they are called connected.
Overlapped probes: when probe A is carried out in the middle (endpoint excl.) of probe B’s safe duration, they are overlapped.
uncertainty
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Lazy probing strategy Question: When 2 probes available,
what timing strategy leads to the total longest safe duration?
We showed that lazy probing strategy is best
the non-overlapped and connected
probing sequence achieves the
longest safe duration.
Each probe starts only after the
previous one runs out its safe
duration.
threshold
lazy
overlapped
T1 > T2
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Choosing right probe timing & type
Probe timing Lazy probing strategy (for multi probes) is best . When the
estimate is still safe, it is better to keep lazy.
Probe type We showed that: When initial uncertainty equals to threshold,
two lazy probes can change the order without reducing safe
duration.
Safe duration
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Choosing the right probe type
We showed:
All min-cost safe probe sequences are equivalent to one of
A...AB...B
B...BA...A
A
B
ABAB AABB
Cost(S) = ΣN1 C(A) + ΣN2 C(B)
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Algorithm of computing an optimal strategy
Compute safe duration of probe A and B, ta, tb;
N1 = T/ ta, seq* = None, cost* = inf
for i = 1 to N1:
j = [(T- i*ta )/ tb]
seq = {A1…Ai B1…Bj}If cost (seq) < cost*:
seq* = seq
cost* = Σi cost(A) + Σj cost(B) return seq*
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Computing an optimal strategy
We showed:
A min-cost probe sequence can be obtained by
comparing at most N1+N
2 candidates.
Further, the min-cost solution can be found in both
N1 set and N
2 set.
Time complexity. min(N1 , N
2)
is needed
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Conclusion
Based on our position inference model, we present:
Optimal probing strategies for Integrating measurements from multiple positioning devices
minimizing energy while maintaining the estimate certainty