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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

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Questions

Thank you!

Fang Hui

Nanyang Technological University

fang0025@ntu.edu.sg