Sampling Methods: Particle Filtering

Transcription

1 Penn State Sampling Methods: Particle Filtering CSE586 Computer Vision II CSE Dept, Penn State Univ

2 Penn State Recall: Importance Sampling Procedure to estimate E P (f(x)): 1) Generate N samples x i from Q(x) 2) form importance weights 3) compute empirical estimate of E P (f(x)), the expected value of f(x) under distribution P(x), as

3 Penn State Resampling Note: We thus have a set of weighted samples (xi, wi i=1,,n) If we really need random samples from P, we can generate them by resampling such that the likelihood of choosing value x i is proportional to its weight w i This would now involve now sampling from a discrete distribution of N possible values (the N values of x i ) Therefore, regardless of the dimensionality of vector x, we are resampling from a 1D distribution (we are essentially sampling from the indices 1...N, in proportion to the importance weights w i ). So we can using the inverse transform sampling method we discussed earlier.

4 Penn State Sequential Monte Carlo Methods Sequential Importance Sampling (SIS) and the closely related algorithm Sampling Importance Sampling (SIR) are known by various names in the literature: - bootstrap filtering - particle filtering - Condensation algorithm - survival of the fittest General idea: Importance sampling on time series data, with samples and weights updated as each new data term is observed. Well-suited for simulating recursive Bayes filtering!

5 Penn State Recall: Bayes Filtering Two-step Iteration at Each Time t: Motion Prediction Step: Data Correction Step (Bayes rule):

6 Penn State Recall: Bayes Filtering Problem: in general we get intractible integrals Motion Prediction Step: Data Correction Step (Bayes rule):

7 Penn State Sequential Monte Carlo Methods Intuition: Represent probability distributions by samples (called particles). Each particle is a guess at the true state. For each one, simulate it s motion update and add noise to get a motion prediction. Measure the likelihood of this prediction, and weight the resulting particles proportional to their likelihoods.

8 Penn State Back to Bayes Filtering This integral in the denominator of Bayes rule disappears as a consequence of representing distributions by a weighted set of samples. Since we have only a finite number of samples, the normalization constant will be the sum of the weights! Data Correction Step (Bayes rule):

9 Penn State Back to Bayes Filtering Now let s write the Bayes filter by combining motion prediction and data correction steps into one equation. new posterior data term motion term old posterior

10 Penn State Monte Carlo Bayes Filtering Assume the posterior at time t-1 (which is the prior at time t) has been approximated as a set of N weighted particles: So that Where is the delta dirac function Useful property:

11 Penn State Monte Carlo Bayes Filtering Then the motion prediction integral simplifies to a summation Motion prediction integral The prior had been approximated by N particles Exchange order of summation and integration Property of Dirac delta function

12 Penn State Monte Carlo Bayes Filtering Our Bayes filtering equation thus simplifies as well Plugging in result from previous page Bringing term that doesn t depend on i into the summation

13 Penn State Monte Carlo Bayes Filtering Our new posterior is therefore but this is still not amenable to computation in closed-form for arbitrary motion models and likelihood functions (e.g. we would have to integrate it to compute the normalization constant c) Idea : Let s approximate the posterior as a set of N samples! Idea 2 : Hey wait a minute, the prior was already represented as a set of N samples! Why don t we just update each of those?

14 Penn State Monte Carlo Bayes Filtering Approach: for each sample x i t-1, generate a new sample x i t from convenient proposal distribution by importance sampling using some So, generate a sample and compute its importance weight

15 Penn State Monte Carlo Bayes Filtering We then can approximate our posterior as where

16 Penn State SIS Algorithm

17 w w w Robert Collins Penn State SIS Degeneracy Unfortunately, pure SIS suffers from degeneracy. In many cases, after a few iterations, all but one particle will have negligible weight. Illustration of degeneracy: Time 1 Time 10 Time 19

18 Penn State Resampling to Combat Degeneracy Sampling with replacement to get N new samples, each having equal weight 1/N Samples with high weight get replicated Samples with low weight die off Concentrates particles in areas of higher probability

19 Penn State Generic Particle Filter

20 Penn State Sample Importance Resample (SIR) SIR is a special case of the generic particle filter where: - the prior density is used as the proposal density - resampling is done every iteration therefore and thus cancellation the old weights are all equal due to resampling

21 Penn State SIR Algorithm

22 Penn State Drawing from the Prior Density note, when we use the prior as the importance density, we only need to sample from the process noise distribution (typically uniform or Gaussian). Why? Recall: x k = f k (x k-1, v k-1 ) v is process noise Thus we can sample from the prior P(x k x k-1 ) by starting with sample x i k-1, generating a noise vector v i k-1 from the noise process, and forming the noisy sample x i k = f k (x i k-1, v i k-1) If the noise is additive, this leads to a very simple interpretation: move each particle using motion prediction, then add noise.

23 Penn State Robert Collins SIR Filtering Illustration M m m k M x 1 ) ( 1 1, x M m m k m k w x 1 ) ( ) (, M m m k M x 1 ) ( ~ 1, M m m k M x 1 ) ( 1 1, M m m k m k w x 1 ) ( 1 ) ( 1, M m m k M x 1 ) ( 1 ~ 1, M m m k M x 1 ) ( 2 1,

24 Penn State Problems with SIS/SIR Degeneracy: in SIS, after several iterations all samples except one tend to have negligible weight. Thus a lot of computational effort is spent on particles that make no contribution. Resampling is supposed to fix this, but also causes a problem... Sample Impoverishment: in SIR, after several iterations all samples tend to collapse into a single state. The ability to representation multimodal distributions is thus short-lived.

25 Particle Filter Failure Analysis References King and Forsyth, How Does CONDENSATION Behave with a Finite Number of Samples? ECCV 2000, Karlin and Taylor, A First Course in Stochastic Processes, 2nd edition, Academic Press, 1975.

26 Particle Filter Failure Analysis Summary Condensation/SIR is aymptotically correct as the number of samples tends towards infinity. However, as a practical issue, it has to be run with a finite number of samples. Iterations of Condensation form a Markov chain whose state space is quantized representations of a density. This Markov chain has some undesirable properties high variance - different runs can lead to very different answers low apparent variance within each individual run (appears stable) state can collapse to single peak in time roughly linear in number of samples tracker may appear to follow peaks in the posterior even in the absence of any meaningful measurements. These properties generally known as sample impoverishment

27 Stationary Analysis For simplicity, we focus on tracking problems with stationary distributions (posterior should be the same at any time step). [because it is hard to really focus on what is going on when the posterior modes are deterministically moving around. Any movement of modes in our analysis will be due to behavior of the particle filter]

28 A Simple PMF State Space Consider 10 particles representing a probability mass function over 2 locations. PMF state space: {(0,10)(1,9)(2,8)(3,7)(4,6)(5,5) (6,4)(7,3)(8,2)(9,1)(10,0)} 1 2 (4,6) We will now instantiate a particular two-state filtering model that we can analyze in closed-form, and explore the Markov chain process (on the PMF state space above) that describes how particle filtering performs on that process.

29 Discrete, Stationary, No Noise Assume a stationary process model with no-noise process model: X k+1 = F X k + v k I Identity 0 no noise process model: X k+1 = X k

30 Perfect Two-State Ambiguity Let our two filtering states be {a,b}. We define both prior distribution and observation model to be ambiguous (equal belief in a and b). P(X 0 ) =.5 X 0 = a.5 X 0 = b P(Z X k ) =.5 X 0 = a.5 X 0 = b from process model: a P(X k+1 X k ) = b a 1 0 b 0 1

31 These are exact propagation equations. Robert Collins Prediction: Recall: Recursive Filtering predicted current state state transition previous estimated state Update: estimated current state measurement predicted current state normalization term

32 Analytic Filter Analysis Predict =.5 =.5 Update =.25/( ) =.5 =.25/( ) =.5

33 Analytic Filter Analysis Therefore, for all k, the posterior distribution is P(X k z 1:k ) =.5 X k = a.5 X k = b which agrees with our intuition in regards to the stationarity and ambiguity of our two-state model. Now let s see how a particle filter behaves...

34 Particle Filter Consider 10 particles representing a probability mass function over our 2 locations {a,b}. In accordance with our ambiguous prior, we will initialize with 5 particles in each location P(X 0 ) = a b

35 Condensation (SIR) Particle Filter 1) Select N new samples with replacement, according to the sample weights (equal weights in this case) 2) Apply process model to each sample (deterministic motion + noise) (no-op in this case) 3) For each new position, set weight of particle in accordance to observation probability (all weights become.5 in this case) 4) Normalize weights so they sum to one (weights are still equal )

36 Condensation as Markov Chain (Key Step) Recall that 10 particles representing a probability mass function over 2 locations can be thought of as having a state space with 11 elements: {(0,10)(1,9)(2,8)(3,7)(4,6)(5,5)(6,4)(7,3)(8,2)(9,1)(10,0)} (5,5) a b

37 Condensation as Markov Chain (Key Step) We want to characterize the probability that the particle filter procedure will transition from the current configuration to a new configuration: {(0,10)(1,9)(2,8)(3,7)(4,6)(5,5)(6,4)(7,3)(8,2)(9,1)(10,0)} (5,5)? a b

38 Condensation as Markov Chain (Key Step) We want to characterize the probability that the particle filter procedure will transition from the current configuration to a new configuration: {(0,10)(1,9)(2,8)(3,7)(4,6)(5,5)(6,4)(7,3)(8,2)(9,1)(10,0)}? (5,5) Let P(j i) be prob of transitioning from (i,10-i) to (j,10-j) a b

39 Example N=10 samples (5,5) a b.2051 (4,6).1172 a b (3,7) a b (5,5) a b (6,4) a b.25 0 P( j 5) j 0 10

40 0 Full Transition Table P( j i) i.25 P( j 5) 0 0 j j 10

41 P(j 5) The Crux of the Problem from (5,5), there is a good chance we will jump to away from (5,5), say to (6,4) P(j 6) P(j 7) once we do that, we are no longer sampling from the transition distribution at (5,5), but from the one at (6,4). But this is biased off center from (5,5) and so on. The behavior will be similar to that of a random walk.

42 0 Another Problem P( j i) P(0 0) = 1 i (0,10) and (10,0) are absorbing states! 10 0 j 10 P(10 10) = 1

43 Observations The Markov chain has two absorbing states (0,10) and (10,0) Once the chain gets into either of these two states, it can never get out (all the particles have collapsed into a single bucket) There is a nonzero probability of getting into either absorbing state, starting from (5,5) These are the seeds of our destruction!

44 Simulation

45 Some sample runs with 10 particles

46 N=10 More Sample Runs N=20 N=100

47 average time to absorbtion Robert Collins Average Time to Absorbtion number of particles N Dots - from running simulator (100 trials at N=10,20,30...) Line - plot of 1.4 N, the asymptotic analytic estimate (King and Forsyth)

48 More Generally Implications of stationary process model with no noise, in a discrete state space. any time any bucket contains zero particles, it will forever after have zero particles (for that run). there is typically a nonzero probability of getting zero particles in a bucket sometime during the run. thus, over time, the particles will inevitably collapse into a single bucket.

49 Extending to Continuous Case A similar thing happens in more realistic cases. Consider a continuous case with two stationary modes in the likelihood, and where each mode has small variance with respect to distance between modes. mode1 mode2

50 Extending to Continuous Case The very low variance between modes is fatal to any particles that try to cross from one to the other via diffusion. mode1 mode2

51 Extending to Continuous Case Each mode thus becomes an isolated island, and we can reduce this case to our previous two-state analysis (each mode is one discrete state) mode1 mode2 a b