Lecture Particle Filters

Transcription

1 FMS161/MASM18 Financial Statistics November 29, 2010

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)p K (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 K f (x k ) LLN E[f (X)]. k=1 We also know that: 1 K K f (x k ) E[f (X)] k=1 V[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]. as K. sup P K (x) P(x) 0 x [Donsker theorem] K (PK (x) P(x)) BB(P(x)), as K where BB is a Brownian Bridge and where P is X:s distribution function.

5 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 )

6 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 x0 k ), k=1 where x k 0 are particle samples from p(x 0)dx 0.

7 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.

8 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 xn+1 k ) p(y n+1 y 1:n ) w k w k = p(y n+1 xn+1 k ) as "weights", and then 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 k=1 w k δ(x n+1 x k n+1 ). This is the SIS - Sequential Importance Sampling Filter.

9 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.

10 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 indeces 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.

11 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 c Doucet, de Freitas & Gordon (2001)

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

13 OMXS30 logreturns

14 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 =

15 Normalised OMXS30 logreturns

16 Normplot: normalised OMXS30 logreturns Normal Probability Plot Probability Data

17 Value at risk Assume that we have a stochastic volatility model y t = v t η t log(vt 2 ) = α + β log(vt 1 2 t 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,vt 2). But p(y t y 1:t 1 ) = p(y t,v t y 1:t 1 )dv t = p(y t v t )p(v t y 1:t 1 )dv t. Easy to compute as p(v t y 1:t 1 ) K k=1 w kδ(v t v k t ).

18 Value at risk 5% logreturns Red dots are observations below value at risk. There are 61 red dots out of According to theory there should be about 50 (this is fair enough)