Sampling Methods Henrik I. Christensen Robotics & Intelligent - - PowerPoint PPT Presentation

Introduction Basic Algorithms Example Summary

Sampling Methods

Henrik I. Christensen

Robotics & Intelligent Machines @ GT Georgia Institute of Technology, Atlanta, GA 30332-0280 hic@cc.gatech.edu

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Introduction Basic Algorithms Example Summary

Outline

1

Introduction

2

Basic Algorithms

3

Small Robot Example

4

Summary

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Introduction Basic Algorithms Example Summary

Introduction

Last time we talked about approximation of distributions using deterministic approximation Not clear that it is always possible to generate such solutions. Might not be possible to generate an estimate of the distribution How can we find the distribution / expectation of a function wrt a probability distribution p(z)? I.e. can we evaluate E[f ] =

• f (z)p(z)dz

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Introduction Basic Algorithms Example Summary

Introduction

The general idea with sampling

Collect a set of samples z(l) from p(z) Evaluate ˆ f = 1 L

L

• l=1

f (z(l))

ˆ f is a good estimate of the mode/expectation and the variance of the estimator can be computed Main challenge - how can we generates samples from p(z)?

Henrik I. Christensen (RIM@GT) Sampling Methods 4 / 23

Introduction Basic Algorithms Example Summary

Introduction

In some cases a graphical model is available From earlier (8.1.2) we have p(z) =

M

• i=1

p(zi|pai) where pai is the set of parents to a node i We can then sample the conditional distributions Do a pass through the tree and compute the distribution

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Introduction Basic Algorithms Example Summary

Outline

1

Introduction

2

Basic Algorithms

3

Small Robot Example

4

Summary

Henrik I. Christensen (RIM@GT) Sampling Methods 6 / 23

Introduction Basic Algorithms Example Summary

Standard distributions

It is common to generate pseudo random sequences using a standard random number generator If we want to transform the z according to a function f(.) so that y=f(z) we have p(y) = p(z)

• dz

dy

• Integrating we have

z = h(y) = y

−∞

p(ˆ y)dˆ y So we are after y = h−1(z)

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Introduction Basic Algorithms Example Summary

Standard Distributions

Consider the standard exponential distribution p(y) = λ exp(−λy) Then h(y) = 1 − exp(λy) ⇒ y = −λ−1 ln(1 − z) Now the uniform z will have an exponential distribution

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Introduction Basic Algorithms Example Summary

Uniform Gaussian (in 2D)

Consider a mapping f1(z) = 2z − 1 Reject all values where z2

1 + z2 2 > 1

Assign p(z1, z2) = 1/π This is a uniform distribution inside the unit circle Generate the mapping yi = zi −2 ln zi r2 2 The joint distribution is then p(y1, y2) =

• 1

√ 2π exp(−y2

1

2 ) 1 √ 2π exp(−y2

2

2 )

• Henrik I. Christensen (RIM@GT)

Sampling Methods 9 / 23

Introduction Basic Algorithms Example Summary

Standard Distributions

Transformation depends upon the ability to calculate and then invert the indefinite integral Sometimes more general approaches are required for sampling

1

Rejection Sampling

2

Important Sampling

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Introduction Basic Algorithms Example Summary

Rejection Sampling

Assume we have a distribution p(z) which is difficult to access, but we have a scaled version ˜ p(z) which is the same up to a scaling p(z) = 1 Zp ˜ p(z) We can propose a distribution q(z) We can then consider kq(z) ≥ ˜ p(z)

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Introduction Basic Algorithms Example Summary

Rejection Sampling - Idea z0 z u0 kq(z0) kq(z)

• p(z)

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Introduction Basic Algorithms Example Summary

Rejection Sampling - Example

z p(z) 10 20 30 0.05 0.1 0.15

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Introduction Basic Algorithms Example Summary

Rejection Sampling - Example

z p(z) −5 5 0.25 0.5

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Introduction Basic Algorithms Example Summary

Importance Sampling

Assume we could generate a uniform tessellation of space We can now evaluate E[f ] ≈

• l

p(z(l))f (z(l)) Assume we have a simpler distribution q(z) that is easier to sample E[f ] =

• f (z)p(z)dz

=

• f (z)p(z)

q(z)q(z)dz ≈ 1 L

• l

p(z(l)) q(z(l))f (z(l)) the ratio r = p(z)/q(z) is known as the importance weight We can normalize to have weights wi = ri/ rl

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Introduction Basic Algorithms Example Summary

Sampling Importance Resampling

The sampling can be re-evaluated over time to avoid a uniform distribution of sampling up-front. I.e. we should sample more densely close major probability masses.

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Introduction Basic Algorithms Example Summary

Outline

1

Introduction

2

Basic Algorithms

3

Small Robot Example

4

Summary

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Introduction Basic Algorithms Example Summary

Monte-Carlo Based Localization

Monte-Carlo based methods is using a sample model for approximation of the pose estimate Using a grid model as presented earlier Assume we have a number of particles in a collection St =

• (s(i)

t , π(i) t )|i = 1..N

• each particle is a hypothesis for the position of the robot, and π(i)

t

is an associated weight We can now approximate p(st|z0, z1, ...zt) for any distribution of the pose hypotheses

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Introduction Basic Algorithms Example Summary

Monte-Carlo Strategy

1 Draw N samples from an initial PDF. Typically a uniform distribution.

Give each sample a weight of 1

N

2 Propagate the motion information and draw a new sample from the

distribution p(s(i)

t+1|s(i) t , ot)

3 Set the weight of the sample to π(i)

t+1 = p(zt+1|s(i) t+1) ∗ π(i) t

based on sensory input

4 Generate a new sample set by drawing samples from the current set

and a basis distribution (typically uniform). Normalize the weights

5 Go back to step 2 Henrik I. Christensen (RIM@GT) Sampling Methods 19 / 23

Introduction Basic Algorithms Example Summary

Monte-Carlo Example

Example of particle distribution around estimate of position Sonar readings for update

• f the position

Video of system in

• peration

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Introduction Basic Algorithms Example Summary

Monte-Carlo Example

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Introduction Basic Algorithms Example Summary

Outline

1

Introduction

2

Basic Algorithms

3

Small Robot Example

4

Summary

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Introduction Basic Algorithms Example Summary

Summary

Sampling based methods are efficient for many applications The key is typically in the sampling/re-sampling strategy There are quite a few ways sampling can be applied Next time we will cover the popular MCMC strategy

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