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Register a new userOn robustness of discrete time optimal filters
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A new result on stability of an optimal nonlinear filter with respect to small perturbations on every step is established.
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The problem of extracting as much information as possible from a sequence of observations of a stationary stochastic process $X_0,X_1,...X_n$ has been considered by many authors from different points of view. It has long been known through the work of D. Bailey that no universal estimator for $textbf{P}(X_{n+1}|X_0,X_1,...X_n)$ can be found which converges to the true estimator almost surely. Despite this result, for restricted classes of processes, or for sequences of estimators along stopping times, universal estimators can be found. We present here a survey of some of the recent work that has been done along these lines.
Let ${X_n}$ be a stationary and ergodic time series taking values from a finite or countably infinite set ${cal X}$. Assume that the distribution of the process is otherwise unknown. We propose a sequence of stopping times $lambda_n$ along which we will be able to estimate the conditional probability $P(X_{lambda_n+1}=x|X_0,...,X_{lambda_n})$ from data segment $(X_0,...,X_{lambda_n})$ in a pointwise consistent way for a restricted class of stationary and ergodic finite or countably infinite alphabet time series which includes among others all stationary and ergodic finitarily Markovian processes. If the stationary and ergodic process turns out to be finitarily Markovian (among others, all stationary and ergodic Markov chains are included in this class) then $ lim_{nto infty} {nover lambda_n}>0$ almost surely. If the stationary and ergodic process turns out to possess finite entropy rate then $lambda_n$ is upperbounded by a polynomial, eventually almost surely.
We present a framework to interpret signal temporal logic (STL) formulas over discrete-time stochastic processes in terms of the induced risk. Each realization of a stochastic process either satisfies or violates an STL formula. In fact, we can assign a robustness value to each realization that indicates how robustly this realization satisfies an STL formula. We then define the risk of a stochastic process not satisfying an STL formula robustly, referred to as the STL robustness risk. In our definition, we permit general classes of risk measures such as, but not limited to, the conditional value-at-risk. While in general hard to compute, we propose an approximation of the STL robustness risk. This approximation has the desirable property of being an upper bound of the STL robustness risk when the chosen risk measure is monotone, a property satisfied by most risk measures. Motivated by the interest in data-driven approaches, we present a sampling-based method for estimating the approximate STL robustness risk from data for the value-at-risk. While we consider the value-at-risk, we highlight that such sampling-based methods are viable for other risk measures.
In a batch of synchronized queues, customers can only be serviced all at once or not at all, implying that service remains idle if at least one queue is empty. We propose that a batch of $n$ synchronized queues in a discrete-time setting is quasi-stable for $n in {2,3}$ and unstable for $n geq 4$. A correspondence between such systems and a random-walk-like discrete-time Markov chain (DTMC), which operates on a quotient space of the original state-space, is derived. Using this relation, we prove the proposition by showing that the DTMC is transient for $n geq 4$ and null-recurrent (hence quasi-stability) for $n in {2,3}$ via evaluating infinite power sums over skewed binomial coefficients. Ignoring the special structure of the quotient space, the proposition can be interpreted as a result of Polyas theorem on random walks, since the dimension of said space is $d-1$.
In this paper, we consider the optimal stopping problem on semi-Markov processes (SMPs) with finite horizon, and aim to establish the existence and computation of optimal stopping times. To achieve the goal, we first develop the main results of finite horizon semi-Markov decision processes (SMDPs) to the case with additional terminal costs, introduce an explicit construction of SMDPs, and prove the equivalence between the optimal stopping problems on SMPs and SMDPs. Then, using the equivalence and the results on SMDPs developed here, we not only show the existence of optimal stopping time of SMPs, but also provide an algorithm for computing optimal stopping time on SMPs. Moreover, we show that the optimal and -optimal stopping time can be characterized by the hitting time of some special sets, respectively.
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