A population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models

This paper addresses the problem of Monte Carlo approximation of posterior probability distributions. In particular, we have considered a recently proposed technique known as population Monte Carlo (PMC), which is based on an iterative importance sampling approach. An important drawback of this methodology is the degeneracy of the importance weights when the dimension of either the observations or the variables of interest is high. To alleviate this difficulty, we propose a novel method that performs a nonlinear transformation on the importance weights. This operation reduces the weight variation, hence it avoids their degeneracy and increases the efficiency of the importance sampling scheme, specially when drawing from a proposal functions which are poorly adapted to the true posterior. For the sake of illustration, we have applied the proposed algorithm to the estimation of the parameters of a Gaussian mixture model. This is a very simple problem that enables us to clearly show and discuss the main features of the proposed technique. As a practical application, we have also considered the popular (and challenging) problem of estimating the rate parameters of stochastic kinetic models (SKM). SKMs are highly multivariate systems that model molecular interactions in biological and chemical problems. We introduce a particularization of the proposed algorithm to SKMs and present numerical results.Comment: 35 pages, 8 figure

Graph-based particle filter in indoor positioning

Tässä diplomityössä käsitellään sisätilapaikannusta ja siihen tarkoitettua suodatinta WLAN-tukiasemista sekä BLE-lähettimistä saatujen mittausten avulla. Työssä esitellään tarkemmin sisätilapaikannusmenetelmä, joka käyttää mittausten lisäksi tehokkaasti myös rakennuksen pohjapiirustukseen perustuvaa karttainformaatiota käyttäjän sijainnin määrittämisessä. Suodatuksen taustalla olevan teorian lisäksi työssä esitellään suodattimen toiminta ja saadut tulokset todellisia mittauksia sisältävien testireittien avulla. Työssä esiteltävä graafipohjainen partikkelisuodatin on graafipohjaiselle karttarakenteelle luotu partikkelisuodatin, joka rajoittaa käyttäjän sijaintia ja liikkumista rakennuksen rakenteiden mukaisesti. Graafipohjainen karttarakenteessa käytäviä sekä pieniä huoneita mallinnetaan solmupisteiden välisten linkkien avulla ja suurempia avoimia alueita mallinnetaan avoimina tiloina, jossa käyttäjän liikkuminen on mahdollista kahdessa ulottuvuudessa. Tässä työssä esitellään myös graafipohjainen tilamalli, joka mallintaa käyttäjän liikkeen mahdollisimman todenmukaisesti käyttäjän päämäärätietoisuuden huomioiden. Graafipohjaista partikkelisuodattimen suorituskykyä testataan työssä esitetyllä karttarakenteella Tampereen teknillisen yliopiston Tietotalosta kerättyjen testireittien avulla kahdella eri mittaustiheydellä. Suodattimen antamia paikannustuloksia verrataan eri tilamallia käyttävän graafipohjaisen partikkelisuodattimen sekä vakionopeusja vakiopaikkamallia käyttävien Kalmanin suodattimien tuloksiin. Saatujen tulosten ja niistä tehtyjen analyysien pohjalta, esitellyn suodattimen paikannustarkkuus havaitaan vertailumenetelmiä paremmaksi lähes jokaisessa testitilanteessa

Markov Chain Monte Carlo Estimation of Stochastic Volatility Models with Finite and Infinite Activity Lévy Jumps: Evidence for Efficient Models and Algorithms

A financial model plays a key role in the valuation and risk management of financial derivatives, and it serves as an important tool for investors to measure the risk exposure of their portfolios and make predictions and decisions. However, the popular affine stochastic volatility models without jumps, such as the Heston model, have been questioned in the finance literature in terms of their appropriateness for modelling stock prices and pricing derivatives. Many alternative model specifications have been proposed in recent decades, including the specification of non-affine variance dynamics and the inclusion of Lévy jumps. However, the complexity introduced by further model specifications leads to poor probabilistic properties, and hence most popular estimation methods are not applicable. The Bayesian estimation method is among the few that work. In this thesis, I discuss the role of new model specifications and investigate the performance of Bayesian estimation methods. First, I use an extensive empirical data set to study how the use of infinite-activity Lévy jumps in stock returns and variance improves model performance. The stock returns and variance are driven by diffusions and different Lévy jumps, including the finite-activity compound Poisson jump and infinite-activity Variance Gamma and Normal Inverse Gaussian (NIG) jumps. Moreover, the non-affine linear variance process is compared to the affine square-root stochastic process. With the conventional Markov Chain Monte Carlo (MCMC) algorithms, including the Gibbs sampler and Metropolis-Hastings (MH) methods, and the Damien-Wakefield-Walker method to cope with complicated posteriors, eighteen different model specifications are estimated using the joint information of the S&P 500 index and the VIX index for 1996 – 2009. There is clear evidence that in terms of the goodness of fit and option pricing performance, a relatively parsimonious model with infinite-activity NIG jumps in returns and non-affine variance dynamics is particularly competitive. In the second part of the thesis, I examine the performance of advanced MCMC algorithms. The efficiency of the MH algorithm has been questioned because of its slow mixing speed, especially in the presence of high dimensions and a strong dependence between model parameters and state variables. Generally, a class of algorithms seeks to improve the MH by constructing more effective proposals, and another combines the MCMC with the Sequential Monte Carlo algorithms. To investigate, I first conduct simulation studies to compare the estimation performance of seven advanced Bayesian estimation methods against the MH. Specifically, I use the affine Heston model, the affine Bates model, and an affine model with NIG return jumps, and examine whether the different jump structures affect the estimation results. Second, I estimate the non-a ffine model with NIG return jumps using the joint information of the S&P 500 index and the VIX for 2002–2005 with selected algorithms that perform well in the simulation studies. The results of the simulation and empirical studies are mixed about the performance of the algorithms. The Fast Universal Self-tuned Sampler algorithms are particularly competitive in generating virtually independent samples and achieving the fastest mixing with a fixed number of MCMC runs, and their performance is stable regardless of the model specifications. However, they are computationally expensive. The computational costs of the Particle Markov Chain Monte Carlo (PMCMC) methods are much cheaper and also efficient in mixing, and they perform best when estimating the models without jumps/with NIG jumps in the simulation studies, as well as in the fit to the VIX in the empirical studies. However, the PMCMC methods are more vulnerable to model specifications than the other algorithms; in particular, the rare large compound Poisson jumps in the Bates model significantly reduce the acceptance rate and worsen the estimation performance of the PMCMC methods