Continuous Estimation in Continuous Estimation in WLAN PositioningWLAN Positioning

ByBy

Tilen MaTilen Ma

Clarence FungClarence Fung

ObjectiveObjective

Area-Based Probability (ABP)Area-Based Probability (ABP)

Continuous Space Estimation(CSE)Continuous Space Estimation(CSE)– Center of MassCenter of Mass– Time-AveragingTime-Averaging

Point MappingPoint Mapping

ConclusionConclusion

Applying Area-based ApproachApplying Area-based Approach

Area-Based Probability (ABP)Area-Based Probability (ABP)

Advantages:Advantages:

Presents the user an understanding of the Presents the user an understanding of the system in a more natural and intuitive system in a more natural and intuitive mannermanner

High accuracyHigh accuracy

More mathematical approachMore mathematical approach

Steps in using ABPSteps in using ABP

Decide the Areas

Measure Signals at Different Areas

Create a Training Set

Measure Signals at Current Position

Create a Testing Set

Find out the Probability of Being at Different Areas

Calculate Probability Density

Return the Area with Highest Probability

Area Based ProbabilityArea Based Probability

We compute P(SWe compute P(St t |A|Aii) for every area A) for every area Aii ,i=1…m, using the ,i=1…m, using the Gaussian assumptionGaussian assumption

Finding Probability DensityFinding Probability Density– the object must be at one of the 12 areas the object must be at one of the 12 areas – ΣP(Ai | St) =1 for all i ΣP(Ai | St) =1 for all i

Max{P(AMax{P(Ai i |S|Stt) } = Max{c*P(S) } = Max{c*P(St t |A|Aii) } ) } = Max{P(S= Max{P(St t |A|Aii) }) }

Return the area AReturn the area Aii with top probability with top probability

Applying Area-based ApproachApplying Area-based Approach

There are two approach to estimate There are two approach to estimate position:position:– Discrete Space EstimationDiscrete Space Estimation– Continuous Space EstimationContinuous Space Estimation

Discrete Space EstimationDiscrete Space Estimation

LimitationLimitation

Only one of the discrete locations in the traOnly one of the discrete locations in the training set is returnedining set is returned

Cannot return the intermediate locationsCannot return the intermediate locations

Low accuracyLow accuracy

Not desirable for location-based applicatioNot desirable for location-based application. Eg. Tour guiden. Eg. Tour guide

Introduction to CSEIntroduction to CSE

Continuous Space EstimationContinuous Space EstimationAdvantage:Advantage:– Return locations may or may not be in the Return locations may or may not be in the

training settraining set– Higher accuracyHigher accuracy– Suitable for mobile applicationSuitable for mobile application

Two techniques:Two techniques:– Center of MassCenter of Mass– Time-AveragingTime-Averaging

Center of MassCenter of Mass

Assume n locationsAssume n locations

Treat each location in the training set as Treat each location in the training set as an objectan object

Each object has a weight equals to its Each object has a weight equals to its probability densityprobability density

Obtain Center of Mass of n objects using Obtain Center of Mass of n objects using their weighted positionstheir weighted positions

Center of MassCenter of Mass

Let p(i) be the probability of a location xLet p(i) be the probability of a location x ii, i=1,2 …, i=1,2 …

nn

Let Y be the set of locations in 2D space and Y(i) Let Y be the set of locations in 2D space and Y(i) is the corresponding position of xis the corresponding position of x ii

The Center of Mass is given by:The Center of Mass is given by:

Center of MassCenter of Mass

ExampleExample

Time-AveragingTime-Averaging

Use a time-average window to smooth the Use a time-average window to smooth the resulting location estimatedresulting location estimated

Obtain the result location by averaging the Obtain the result location by averaging the last W locations estimated by discrete-last W locations estimated by discrete-space estimatorspace estimator

Time-AveragingTime-Averaging

Given a stream of location estimates xGiven a stream of location estimates x11,x,x22,…,x,…,xtt

The current location xThe current location xcc is estimated by is estimated by

Time-AveragingTime-Averaging

Problem with CSEProblem with CSE

Locations in training setLocations in training set

Estimated positionEstimated position

Point MappingPoint Mapping

Goal : Map the result to the closest point in Goal : Map the result to the closest point in the set of all possible locationsthe set of all possible locations

Step 1: Divide the corridor into several line Step 1: Divide the corridor into several line segments L segments L ii

Step 2: We define each line segment L Step 2: We define each line segment L ii by by

an equation:an equation:

PP = = PPi1i1 + u + uii ( (PPi2i2 – – PPi1i1) )

– PPi1i1 (x (xi1i1,y,yi1i1) is the starting point of L ) is the starting point of L ii

– PPi2i2 (x (xi2i2,y,yi2i2) is the end point of L ) is the end point of L ii

Let the point Let the point EE(x(xee,y,yee) be the estimated point) be the estimated point

Let ILet Iii be the point of intersection between L be the point of intersection between L ii(P(Pi1i1PPii

22) and the line at the tangent to L) and the line at the tangent to L ii passing throug passing throug

h h EE

Step 3: Finding the distances DStep 3: Finding the distances D ii between the est between the est

imated point E and L imated point E and L i i for all ifor all i

the dot product of the tangent and Lthe dot product of the tangent and L ii is 0, thus is 0, thus

(E - I(E - Iii) dot () dot (PPi2i2 – – PPi1i1) = 0 ) = 0

Solving this we have,Solving this we have,

Substituting uSubstituting uii into the equation of L into the equation of Lii gives gives

the point of intersection Ithe point of intersection Iii as as

x = xx = xi1i1 + u + uii (x (xi2i2 – x – xi1i1))

y = yy = yi1i1 + u + uii (y (yi2i2 – y – yi1i1) )

DDii is equal the distance between I is equal the distance between Iii and E and E

Step 4:check if IStep 4:check if Iii lies in the line segment L lies in the line segment Lii

i.e. ui.e. uii lies between 0 and 1 lies between 0 and 1

Step 5:Step 5:

Return IReturn Iii lying in L lying in Lii and with smallest D and with smallest Dii

ConclusionConclusion

Continuous Space Estimation solves the Continuous Space Estimation solves the limitation in Discrete Space Estimationlimitation in Discrete Space Estimation

Continuous Space Estimation improves Continuous Space Estimation improves the accuracy in determining position the accuracy in determining position

Point Mapping overcome out of bound Point Mapping overcome out of bound problem in CSE problem in CSE

THE ENDTHE END