Michael A. Kouritzin

Rates for branching particle approximations of continuous-discrete filters

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that t → Xt is a Markov process and we wish to calculate the measure-valued process t →µt(∙) ≐ P(Xt ∉∙|σ{Ytk , tk ≤ t}), where tk = ke and Ytk is a distorted, corrupted, partial observation of Xtk. Then, one constructs a particle system with observation-dependent branching and n initial particles whose empirical measure at time t;µnt , closely approximates µt. Each particle evolves independently of the other particles according to the law of the signal between observation times tk, and branches with small probability at an observation time. For filtering problems where e is very small, using the algorithm considered in this paper requires far fewer computations than other algorithms that branch or interact all particles regardless of the value of e. We analyze the algorithm on Levy-stable signals and give rates of convergence for E1/2[||µnt - µt||2√], ||∙ ||√ is a Sobolev norm, as well as related convergence results.

On the convergence of linear stochastic approximation procedures

Many stochastic approximation procedures result in a stochastic algorithm of the form h/sub k+1/=h/sub k/+1/k(b/sub k/-A/sub k/h/sub k/), for all k=1,2,3,... . Here, {b/sub k/,k=1,2,3...} is a R/sup d/-valued process, {A/sub k/,k=1,2,3,...} is a symmetric, positive semidefinite Re/sup d/spl times/d/-valued process, and {h/sub k/,k=1,2,3,...} is a sequence of stochastic estimates which hopefully converges to h/sup /spl Delta//=[lim/sub N/spl rarr//spl infin//1/N/spl Sigma//sub k=1//sup N/EA/sub k/]/sup -1/ {lim/sub N/spl rarr//spl infin//1/N/spl Sigma//sub k=1//sup N/Eb/sub k/} (assuming everything here is well defined). We give an elementary proof which relates the almost sure convergence of {h/sub k/,k=1,2,3,...} to strong laws of large numbers for {b/sub k/,k=1,2,3,...} and {A/sub k/,k=1,2,3,...}.

BAYESIAN MODEL SELECTION VIA FILTERING FOR A CLASS OF MICRO-MOVEMENT MODELS OF ASSET PRICE

This paper develops the Bayesian model selection based on Bayes factor for a rich class of partially-observed micro-movement models of asset price. We focus on one recursive algorithm to calculate the Bayes factors, first deriving the system of SDEs for them and then applying the Markov chain approximation method to yield a recursive algorithm. We prove the consistency (or robustness) of the recursive algorithm. To illustrate the construction of such a recursive algorithm, we consider a model selection problem for two micro-movement models with and without stochastic volatility, and provide simulation and real-data examples to demonstrate the effectiveness of the Bayes factor in the model selection for this class of models.

On exact filters for continuous signals with discrete observations

Many filtering applications have continuous state dynamics X/sub t/=/spl int//sub 0//sup t/m(X/sub s/)ds+/spl sigma/W/sub t/+/spl rho/, discrete observations Y/sub j/=Y(t/sub j/), and nonadditive or non-Gaussian observation noise. One wants to calculate the conditional probability Pr{Xt/spl isin/dz|Y/sub j/, 0/spl les/t/sub j//spl les/t} economically. In this paper we show that a combination of convolution, scaling, and substitutions efficiently solves this problem under certain conditions. Our method is easy to use and assumes nothing about the observations other than the ability to construct p(Y/sub j/)|X(t/sub j/), the conditional density of the jth observation given the current state.

Detecting network portscans through anomaly detection

In this note, we consider the problem of detecting network portscans through the use of anomaly detection. First, we introduce some static tests for analyzing traffic rates. Then, we make use of two dynamic chi-square tests to detect anomalous packets. Further, we model network traffic as a marked point process and introduce a general portscan model. Simulation results for correct detects and false alarms are presented using this portscan model and the statistical tests.

On Random Field Completely Automated Public Turing Test to Tell Computers and Humans Apart Generation

Herein, we propose generating CAPTCHAs through random field simulation and give a novel, effective and efficient algorithm to do so. Indeed, we demonstrate that sufficient information about word tests for easy human recognition is contained in the site marginal probabilities and the site-to-nearby-site covariances and that these quantities can be embedded directly into certain conditional probabilities, designed for effective simulation. The CAPTCHAs are then partial random realizations of the random CAPTCHA word. We start with an initial random field (e.g., randomly scattered letter pieces) and use Gibbs resampling to re-simulate portions of the field repeatedly using these conditional probabilities until the word becomes human-readable. The residual randomness from the initial random field together with the random implementation of the CAPTCHA word provide significant resistance to attack. This results in a CAPTCHA, which is unrecognizable to modern optical character recognition but is recognized about 95% of the time in a human readability study.

On the interrelation of almost sure invariance principles for certain stochastic adaptive algorithms and for partial sums of random variables

AbstractSince the novel work of Berkes and Philipp(3) much effort has been focused on establishing almost sure invariance principles of the form(1)

Rates of convergence in a central limit theorem for stochastic processes defined by differential equations with a small parameter

\left| {\sum\limits_{i = 1}^{|\_t\_|} {x_1 - X_t } } \right| \ll t^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - \gamma }