SIS Sequential Importance Sampling Advanced Methods In Simulation Winter 2009 Presented by: Chen Bukay, Ella Pemov, Amit Dvash

Talk Layout  SIS – Overview and algorithm  Random walk – SIS simulation  Nonlinear Filtering – Overview & Added value  Nonlinear Filtering – Simulation

Importance Sampling - General Overview  Importance Sampling –  The most fundamental variance reduction technique  Leads to a dramatic variance reduction – particularly when estimating rare event probabilities  Target – Expected performance of- Likelihood Ratio Estimator - the sample performance importance sampling density probability density of X

SIS - Overview  Sequential Importance Sampling  Also known as “Dynamic Importance Sampling”.  Simply means importance sampling that carried out in sequential manner.  Why Sequential?  Problematic to sample from multi-dimensional vector  Dependency between the variables  It is difficult to sample from

SIS - Overview  Assumptions –  X is decomposable  can present g(x) –  Easy to sample from g(x) sequentially

SIS – Overview (cont’)  It is easy to generate sequentially from  Generate from  ….. We get – Due to the product rule of probability we can write - The Likelihood function -

SIS – Overview (cont’) Likelihood till time t Likelihood till time t-1

SIS – Overview  In order to update the likelihood ratio recursively, we need to know how to calculate  We know  In order to calculate it requires integrating over  There options to solve this –  Use auxiliary pdfs that can be easy evaluated and each is a good approximation to considered hard integral Easy to calculate Where

SIS – Algorithm  SIS algorithm (Sequential) 1.For each finite t = 1,…,n, Sample from 2.Compute where and 3.Repeat N times and estimate via  SIS algorithm (Dynamic) 1.At time t, arrival of t th sample 2.Sample x t N times according to 3.Calculate 4.estimate according to the existing samples (1,…,t) t = 1,…,n Parallel computing

SIS Algorithm - Sequential 1 st sample: 2 nd sample:. N th sample: Calculate Estimate by Computing

SIS Algorithm - Dynamic 1 st sample: 2 nd sample:. N th sample: At Time t =1 Calculate recalculate Calculate recalculate At time t =2 At time t=n Estimate by Computing With the existing samples

Random Walk  Problem statement  Reminder -  Go forward  Probability p  Go backward  probability q  p < q (has drift to - )  Goal – estimating the rare event probability of reaching state K (large number) before 0 (zero) starting at k. 0K12…k start

Random Walk – Simulation Result

SIS Application: Non Linear Filtering

State Space Models x T x1x1 x2x2 x3x3 y2y2 y3y3 y1y1

Dynamic Model Measurement Equation State Equation Observation Equation HMM

State Space Models cont’ Known pdf - P w Known pdf - P v Markov Property

Linear Models Kalman Filter  Linear Dynamic models  Linear Measurement Equations  v, w, x 0 – Gaussian & independent  Kalman Filter is the optimal estimator (MSE)

 Assuming models  Motion models - Linear/Non-Linear State Dynamic  Linear/Non-Linear Measurement Equations  v, u, x 0 – independent, not necessarily Gaussian General Models

Problem Description θaθa θbθb θcθc LOP – Line Of Position Observers – Known exact location (x a,,y a ) (x b,y b ) (x c,,y c ) Target – Unknown location (x e,,y e )

θaθa θbθb θcθc Bearing Only Measurements (x e,,y e ) (x a,,y a ) (x b,y b ) (x c,,y c )

Bearing Only Measurements

Non-Linear Filtering  Motivation  Non linear dynamic/measurement equations  Noise distribution not Gauss  Kalman Filter:  No longer the optimal estimator (MSE)  EKF – Linearization of the state space Equations  Suboptimal estimator  Convergence is not guaranteed 

The Bootstrap Filter  Represent the pdf as a set of rv (and not as a function)  The Bootstrap Filter – Recursive algorithm for propagating and updating these rv samples  Samples are naturally concentrated in regions of high probability “Novel Approach to nonlinear/non Gaussian Bayesian state estimation” N.J. Gordon, D.J. Salmond & A.F.M Smith

Motivation For having P (X(k)|Y(1:k)) MSE ML

The Bootstrap Filter Recursive Calculation of P (X(k)|Y(1:k)) Assume we know Bayes & y t |x t independent of y 1:t-1

The importance Sampling

The Bootstrap Filter Algorithm 1. Initialization: k = 0, Generate x 0 i ~Px 0 i = 1…N 2. Measurement Update: Given y k calculate likelihood for each current sample

Algorithm (cont’) 3. Re-Sampling - Sample N samples from {x k *i } i=1:N, with replacement, where the probability to choose the i-th particle is q k i at stage k 4. Prediction: Pass the new samples through the system Equation 5. Set k = k+1 and return to 2 The Bootstrap Filter

V x = -0.1 [km/sec] Vy= 0.01 [km/sec] dt = 300 [sec] Measurement Variance ~ 1 o (141,,141) [km] (100,120) [km]

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Simulations

150 time steps

Backup

Markov Chain  Markov property –  Given the present state, future states are independent of the past states.  The present state fully captures all the information that could influence the future evolution of the process.  The changes of state are called transitions, and the probabilities associated with various state-changes are called transition probabilities P=

F(X) - calculations

Where to put Markov PropertyBayes Rule Normalization Constant

Where to put 2