0902 regular meeting

Outline

Inferring High-Level Behavior from Low-LevelSensors

Donald J. Patterson, Lin Liao, Dieter Fox, and Henry KautzUBICOMP 2003

Speaker: Cheng-Chang Hsieh (a.k.a. Don)

Database LabNCU CSIE, Taiwan

September 2, 2008

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Outline

Outline

1 Introduction2 Tracking on a Graph3 Parameter Learning4 Experiments5 Conclusions and Future Work

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Introduction

Part I

Introduction

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IntroductionThe ProblemIntroduction

The Problem

Transportation RoutinesHow to predict the transportation mode of a user?

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IntroductionThe ProblemIntroduction

Introduction

A method of learning a Bayesian model of a travelermoving through an urban environment.The model is

1 implemented using particle filters2 learned using Expectation-Maximization.

Particle filters are variants of Bayes filters for estimatingthe state of a dynamic system.Apply Expectation-Maximization (EM) to learn typicalmotion patterns in a completely unsupervised manner.

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Tracking on a Graph

Part II

Tracking on a Graph

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Tracking on a GraphBayesian Filtering on a GraphParticle Filter Based Implementation

Tracking on a Graph

The model is a graph G = (V,E).Edges correspond to straight sections of roads and footpaths.Vertices represent either an intersection, or to model acurved road as a set of short straight edges.

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Tracking on a GraphBayesian Filtering on a GraphParticle Filter Based Implementation

Bayesian Filtering on a Graph

The key idea of Bayes lters is to recursively estimate theposterior probability density over the state spaceconditionedon the data collected so far.

p(xt|z1:t) ∝ p(zt|xt)∫p(xt|xt−1)p(xt−1|z1:t−1)dxt−1

z1:t: a sequence of observations.xt: a state. (the position and velocity of the object)p(zt|xt): the likelihood of making observation zt given thelocation xt.mt ∈ {BUS,FOOT,CAR}

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Tracking on a GraphBayesian Filtering on a GraphParticle Filter Based Implementation

Fig.: Two-slice Dynamic Bayes Net Model

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Tracking on a GraphBayesian Filtering on a GraphParticle Filter Based Implementation

SISR: Sequential Importance Sampling with Re-sampling

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Parameter Learning

Part III

Parameter Learning

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Parameter LearningE-stepM-stepImplementation Details

Parameter Learning

Learning means adjusting the model parameters to betterfit the training data.Learn the motion model based solely on

1 a map and2 a stream of non-continuous and noisy GPS sensor data.

Location and transportation mode are hidden variables.they cannot be observed directlythey have to be inferred from the raw GPS measurements.

EM solves this problem by iterating between anExpectation step (E-step) and a Maximization step(M-step).

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Parameter LearningE-stepM-stepImplementation Details

E-step

The E-step estimates p(x1:t|z1:t,Θ(i−1)).x1:t: states.z1:t: observations.Θ(i−1): parameters we want to estimate at the i− 1-thiteration.

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Parameter LearningE-stepM-stepImplementation Details

M-step

The goal of the M-step is to maximize the expectation ofp(z1:t, x1:t|Θ) over the distribution of x1:t obtained in theE-step by updating the parameter estimations.

Θ(i) = arg maxΘ

n∑j=1

log p(z1:t, x(j)1:t |Θ)

Θ(i) = arg maxΘ

n∑j=1

(log p(z1:t|x(j)

1:t ) + log p(x(j)1:t |Θ)

)Θ(i) = arg max

Θ

n∑j=1

(log p(x(j)

1:t |Θ))

n: the number of particles.x

(j)1:t : the state history of the j-th particle.

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Parameter LearningE-stepM-stepImplementation Details

M-step (Cont.)

p(z1:t, x(j)1:t |Θ) =

p(z1:t, x(j)1:t ,Θ)

p(Θ)=p(z1:t, x

(j)1:t ,Θ)

p(x(j)1:t ,Θ)

p(x(j)1:t ,Θ)p(Θ)

p(z1:t, x(j)1:t |Θ) = p(z1:t|x(j)

1:t ,Θ) · p(x(j)1:t |Θ)

p(z1:t, x(j)1:t |Θ) = p(z1:t|x(j)

1:t ) · p(x(j)1:t |Θ)

log p(z1:t, x(j)1:t |Θ) = log p(z1:t|x(j)

1:t ) + log p(x(j)1:t |Θ)

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Parameter LearningE-stepM-stepImplementation Details

Implementation Details

p(et|et−1,mt−1): the edge transition probability.p(mt|et−1,mt−1): the mode transition probability.

Define:1 αt(et,mt): the number of particles on edge et and in modemt at time t in the forward pass of particle filtering.

2 βt(et,mt): the number of particles on edge et and in modemt at time t in the backward pass of particle filtering.

3 ξt−1(et, et−1,mt−1): the probability of transiting from edgeet−1 to et in mode mt−1 at time t− 1.

4 ψt−1(mt, et−1,mt−1): the probability of transiting frommode mt−1 to mt in edge et−1 at time t− 1.

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Parameter LearningE-stepM-stepImplementation Details

ξt−1 and ψt−1

ξt−1(et, et−1,mt−1) ∝αt−1(et−1,mt−1)p(et|et−1,mt−1)βt(et,mt−1)

ψt−1(mt, et−1,mt−1) ∝αt−1(et−1,mt−1)p(mt|et−1,mt−1)βt(et−1,mt)

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Parameter LearningE-stepM-stepImplementation Details

Update The Parameters

After we have ξt−1 and ψt−1 for all the t from 2 to T , weupdate the parameters as:

p(et|et−1,mt−1)

=expected number of transitions from et−1 to et in mode mt−1

expected number of transitions from et−1 in mode mt−1

=∑T

t=2 ξt−1(et, et−1,mt−1)∑Tt=2

∑et∈Neighbors of et−1

ξt−1(et, et−1,mt−1)andp(mt|et−1,mt−1) =expected number of transitions from mt−1 to mt on edge et−1

expected number of transitions from mt−1 on edge et−1

=∑T

t=2 ψt−1(mt, et−1,mt−1)∑Tt=2

∑mt∈{BUS,FOOT,CAR} ψt−1(mt, et−1,mt−1)

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Parameter LearningE-stepM-stepImplementation Details

Algorithm: EM-based Parameter Learning

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Parameter LearningE-stepM-stepImplementation Details

Algorithm: EM-based Parameter Learning (Cont.)

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Experiments

Part IV

Experiments

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ExperimentsMode Estimation and PredictionLocation Prediction

Experiments

The test data set consists of logs of GPS data.The data contains position and velocity informationcollected at 2-10 second intervals.

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ExperimentsMode Estimation and PredictionLocation Prediction

Prediction Accuracy of Mode Transition Changes

Model Precision RecallDecision Tree with Speed and Variance 2% 83%

Prior Graph Model, w/o bus stops and bus routes 6% 63%Prior Graph Model, w/ bus stops and bus routes 10% 80%

Learned Graph Model 40% 80%

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ExperimentsMode Estimation and PredictionLocation Prediction

Fig.: Location Prediction

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Conclusions and Future WorkThe END

Part V

Conclusions and Future Work

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Conclusions and Future WorkThe END

Conclusions and Future Work

The good predictive user-specic models can be learned inan unsupervised fashion.The key idea is to apply a graph-based Bayes lter to track aperson’s location and transportation mode on a street map.

1 Making positive use of negative information.2 Learning daily and weekly patterns.3 Modeling trip destination and purpose.4 Using relational models to make predictions about novel

events.

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Conclusions and Future WorkThe END

Q & A

Thanks for your attention!

Q & A

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