Lecture Particle Filters. Magnus Wiktorsson

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1 Lecture Particle Filters Magnus Wiktorsson

2 Monte Carlo filters The filter recursions could only be solved for HMMs and for linear, Gaussian models. Idea: Approximate any model with a HMM. Replace p(x) with p K (x) = K k=1 λ kδ(x x k ), s.t. K k=1 λ k = 1. The by far most common choice for λ k = 1/K. p K (x) is called the empirical density.

3 Convergence results We know that f(x)pk (x)dx f(x)p(x)dx = E[f(X)] for all functions f(x) such that E[f(X)] <. Proof: f(x)p K (x)dx = 1 K We also know that: 1 K K k=1 K k=1 f(x k ) E[f(X)] V[f(x)] f(x k ) LLN E[f(X)]. CLT Z, Z N(0, 1).

4 Convergence results II Introduce the empirical distribution function P K (x) = 1 K K k=1 1 {X k x} We can show that P K (x) a.s. P(x). Other convergence results include: [Glivenko-Cantelli theorem]. sup P K (x) P(x) 0 x as K. [Donsker theorem] K (PK (x) P(x)) BB(P(x)), as K where BB is a Brownian Bridge and where P is the distribution function of X.

5 Applications The Monte Carlo filter can thus approximate any functions E [ϕ(x n ) y 1:n ] (1) by using the empirical measure K p K (x n y 1:n ) = λ k δ(x n x k n) (2) k=1 leading to K E [ϕ(x n ) y 1:n ] λ k ϕ(x k n) (3) k=1

6 Filter Recursions We need to compute the following equations: Initialization p(x 0 ) Prediction p(x n+1 y 1:n ) = p(x n+1 x n )p(x n y 1:n )dx n. Filter update p(x n+1 y 1:n+1 ) = p(y n+1 x n+1 )p(x n+1 y 1:n ). p(y n+1 y 1:n )

7 Initialization Sample particle using your favorite method (builtin, inverse method, acceptance rejection, MCMC, importance sampling etc.) from p(x 0 ). Thus, we get p K (x 0 ) = 1 K K δ(x 0 x k 0), k=1 where x k 0 are particle samples from p(x 0)dx 0. Hint: It may be a good idea to have many particles in this step...

8 Prediction Prediction p(x n+1 y 1:n ) = p(x n+1 x n )p(x n y 1:n )dx n We need to generate a sample from p K (x n+1 y 1:n ) Solution: Simulate particles x k n+1 p(x n+1 x k n)dx n.

9 Prediction Prediction p(x n+1 y 1:n ) = p(x n+1 x n )p(x n y 1:n )dx n We need to generate a sample from p K (x n+1 y 1:n ) Solution: Simulate particles x k n+1 p(x n+1 x k n)dx n. Or possibly K p K (x n+1 y 1:n ) = λ k p(x n+1 x k n) k=1

10 Filter Recursions I Filter update p(x n+1 y 1:n+1 ) = p(y n+1 x n+1 )p(x n+1 y 1:n ). p(y n+1 y 1:n ) Can be approximated by p K (x n+1 y 1:n ) 1 K K k=1 δ(x n+1 x k n+1 ), Interpret w k = p(y n+1 x k n+1 ) p(y n+1 y 1:n ) as weights, and then w k w k = p(y n+1 x k n+1 ) Thus p K (x n+1 y 1:n+1 ) K k=1 w kδ(x n+1 x k n+1 ). Normalize w k = w k / K k=1 w k such that p K (x n+1 y 1:n+1 ) = K w k δ(x n+1 x k n+1). k=1 This is the SIS - Sequential Importance Sampling Filter.

11 Filter Recursions II Filter update p(x n+1 y 1:n+1 ) = p(y n+1 x n+1 )p(x n+1 y 1:n ). p(y n+1 y 1:n ) The SIS filter breaks down after only a few iterations! Try this during the computer exercise?! Why? The weighting causes uneven weights. The particle sample is often reduced to a few significant particles after a few iterations.

12 Filter Recursions III Filter update p(x n+1 y 1:n+1 ) = p(y n+1 x n+1 )p(x n+1 y 1:n ). p(y n+1 y 1:n ) Solution: Find a method that gives the particles even weights. I) Resample from {x k n+1, w k}. Let I k be a vector of indexes generated by sampling from w k. Then {x I k n+1, 1/K} is an alternative representation of p K (x n+1 y 1:n+1 ). II) Use an importance sampler to generate more particles in relevant areas. III) Use two-stage (auxilliary variable) samplers etc. Resampling is easy and is needed even if a fancy sampler is used.

13 Propagation and resampling for the Bootstrap filter i=1,...,n=10 particles {x ~,N } (i) -1 t-1 {x ~ (i) (i),w ~ } t-1 t-1 (i) -1 {x t-1,n } {x ~ -1,N } (i) t {x ~ ~,w } (i) t (i) t Doucet, de Freitas & Gordon (2001)

14 Likelihood computation The likelihood is computed through direct calculations p(y t y 1:t 1 ) = p(y t, x t y 1:t 1 )dx t = p(y t x t )p(x t y 1:t 1 )dx t. The resulting expression for the likelihood is p(y t y 1:t 1 ) K w k p(y t x k t ) k=1 BUT the likelihood is discontinuous in the parameters space! This means that the likelihood is difficult to optimize (stoch. approximation etc. is needed)

15 OMXS30 logreturns

16 Estimated log Taylor82vol x(t) = v t η t, η t N(0, 1) log(v 2 t ) = α + β log(v 2 t 1) + σe t, e t N(0, 1) ˆα = , ˆβ = , ˆσ =

17 Normalised OMXS30 logreturns

18 Normplot: normalised OMXS30 logreturns Normal Probability Plot Probability Data

19 Value at risk Assume that we have a stochastic volatility model y t = v t η t (4) log(v 2 t ) = α + β log(v 2 t 1) + σe t (5) Compute VAR γ = inf{ξ R : ξ p(y t y 1:t 1 )dy t = γ} We know the density for y t v t, e.g. N(0, v 2 t ).

20 Value at risk Assume that we have a stochastic volatility model y t = v t η t (4) log(v 2 t ) = α + β log(v 2 t 1) + σe t (5) Compute VAR γ = inf{ξ R : ξ p(y t y 1:t 1 )dy t = γ} We know the density for y t v t, e.g. N(0, v 2 t ). But p(y t y 1:t 1 ) = = p(y t, v t y 1:t 1 )dv t (6) p(y t v t )p(v t y 1:t 1 )dv t. (7) Easy to compute as p(v t y 1:t 1 ) K k=1 w kδ(v t v k t ).

21 Value at risk 1% logreturns Red x:s are observations below value at risk. There are 61 red dots out of According to theory there should be about

22 Parameter estimation Not easy as the log-likelihood is discontinuous. A few alternatives are The EM (or MCEM or ideally SAEM) algorithm Particle MCMC (PMCMC) Gradient free direct maximization Iterated filtering

Lecture Particle Filters

Lecture Particle Filters FMS161/MASM18 Financial Statistics November 29, 2010 Monte Carlo filters The filter recursions could only be solved for HMMs and for linear, Gaussian models. Idea: Approximate any model with a HMM. Replace