Probabilistic Dynamical Systems Research Articles

Developing a detection model for a COVID-19 infected person based on a probabilistic dynamical system.

This paper presents a novel model to detect the COVID‐19 infected person from a Markovian feedback persons in a limited department capacity. The persons arrive one by one to the department and the balking and the retention of reneged person approaches are considered. There exists one server presents the service to these persons according to first‐come, first‐served (FCFS) discipline. An efficient and novel algorithm is presented to get the exact value of the probability of n persons in the department at any time interval. This algorithm depends on the Laplace transform to solve a probabilistic dynamical system of differential equations. By considering the exponential detection function and if the probability of the infected person in the department is equal to the probability of each one, then this algorithm is useful to obtain the detection probability of the infected one. Under steady state, the detection probability of the infected person is described. The usefulness of this model is illustrated for different capacities by using a numerical example to describe the behavior of probabilities of the persons in the department, the detection probabilities of the infected person as functions in time, and the mean time to detection.

Verifying Asymptotic Temporal Properties of Continuous-State Probabilistic Systems

We describe a theory of probabilistic Morse decomposition for continuous-state, probabilistic dynamical systems. Morse decompositions, studied in the topological theory of dynamical systems, allow reasoning about asymptotic behaviors of dynamical systems. We generalize notions of attractors, repellers, and invariant sets to the probabilistic context, and show how these can be used to describe the topological structure of the probabilistic dynamics. Additionally, we show a Lyapunov function characterization for probabilistic Morse decompositions. Our probabilistic Morse decompositions enable an abstraction-based verification methodology for asymptotic specifications such as “the trajectories asymptotically converge to a set with positive probability,” which are not expressible in usual linear temporal logics. We describe computational approaches to computing probabilistic Morse decompositions, using state-space gridding as well as Lyapunov functions. Interestingly, the construction of Morse decompositions is crucial: we show that existing abstraction-based techniques based on gridding the state space are not sufficiently powerful to verify asymptotic specifications.

Dynamical systems on weighted lattices: general theory

In this work a theory is developed for unifying large classes of nonlinear discrete-time dynamical systems obeying a superposition of a weighted maximum or minimum type. The state vectors and input-output signals evolve on nonlinear spaces which we call complete weighted lattices and include as special cases the nonlinear vector spaces of minimax algebra. Their algebraic structure has a polygonal geometry. Some of the special cases unified include max-plus, max-product, and probabilistic dynamical systems. We study problems of representation in state and input-output spaces using lattice monotone operators, state and output responses using nonlinear convolutions, solving nonlinear matrix equations using lattice adjunctions, stability and controllability. We outline applications in state-space modeling of nonlinear filtering; dynamic programming (Viterbi algorithm) and shortest paths (distance maps); fuzzy Markov chains; and tracking audio-visual salient events in multimodal information streams using generalized hidden Markov models with control inputs.

Decidability of Approximate Skolem Problem and Applications to Logical Verification of Dynamical Properties of Markov Chains

When studying probabilistic dynamical systems, temporal logic has typically been used to analyze path properties. Recently, there has been some interest in analyzing the dynamical evolution of state probabilities of these systems. In this article, we show that verifying linear temporal properties concerning the state evolution induced by a Markov chain is equivalent to the decidability of the Skolem problem -- a long-standing open problem in Number Theory. However, from a practical point of view, usually it is enough to check properties up to some acceptable error bound ϵ. We show that an approximate version of the Skolem problem is decidable, and that it can be applied to verify, up to arbitrarily small ϵ, linear temporal properties of the state evolution induced by a Markov chain.

Learning by imitation with the STIFF-FLOP surgical robot: a biomimetic approach inspired by octopus movements

Transferring skills from a biological organism to a hyper-redundant system is a challenging task, especially when the two agents have very different structure/embodiment and evolve in different environments. In this article, we propose to address this problem by designing motion primitives in the form of probabilistic dynamical systems. We take inspiration from invertebrate systems in nature to seek for versatile representations of motion/behavior primitives in continuum robots. We take the perspective that the incredibly varied skills achieved by the octopus can guide roboticists toward the design of robust motor skill encoding schemes and present our ongoing work that aims at combining statistical machine learning, dynamical systems, and stochastic optimization to study the problem of transferring movement patterns from an octopus arm to a flexible surgical robot (STIFF-FLOP) composed of two modules with constant curvatures. The approach is tested in simulation by imitation and self-refinement of an octopus reaching motion.

Learning by imitation with the STIFF-FLOP surgical robot: a biomimetic approach inspired by octopus movements

Transferring skills from a biological organism to a hyper-redundant system is a challenging task, especially when the two agents have very different structure/embodiment and evolve in different environments. In this article, we propose to address this problem by designing motion primitives in the form of probabilistic dynamical systems. We take inspiration from invertebrate systems in nature to seek for versatile representations of motion/behavior primitives in continuum robots. We take the perspective that the incredibly varied skills achieved by the octopus can guide roboticists toward the design of robust motor skill encoding schemes and present our ongoing work that aims at combining statistical machine learning, dynamical systems, and stochastic optimization to study the problem of transferring movement patterns from an octopus arm to a flexible surgical robot (STIFF-FLOP) composed of two modules with constant curvatures. The approach is tested in simulation by imitation and self-refinement of an octopus reaching motion.

Probabilistic polynomial dynamical systems for reverse engineering of gene regulatory networks.

Elucidating the structure and/or dynamics of gene regulatory networks from experimental data is a major goal of systems biology. Stochastic models have the potential to absorb noise, account for un-certainty, and help avoid data overfitting. Within the frame work of probabilistic polynomial dynamical systems, we present an algorithm for the reverse engineering of any gene regulatory network as a discrete, probabilistic polynomial dynamical system. The resulting stochastic model is assembled from all minimal models in the model space and the probability assignment is based on partitioning the model space according to the likeliness with which a minimal model explains the observed data. We used this method to identify stochastic models for two published synthetic network models. In both cases, the generated model retains the key features of the original model and compares favorably to the resulting models from other algorithms.

Empirical comparison of Markov and quantum models of decision making

There are at least two general theories for building probabilistic–dynamical systems: one is Markov theory and another is quantum theory. These two mathematical frameworks share many fundamental ideas, but they also differ in some key properties. On the one hand, Markov theory obeys the law of total probability, but quantum theory does not; on the other hand, quantum theory obeys the doubly stochastic law, but Markov theory does not. Therefore, the decision about whether to use a Markov or a quantum system depends on which of these laws are empirically obeyed in an application. This article derives two general methods for testing these theories that are parameter free, and presents a new experimental test. The article concludes with a review of experimental findings from cognitive psychology that evaluate these two properties.

Probabilistic control of chaos through small perturbations

We study the method of probabilistic control of chaotic dynamics (I. Antoniou, V. Basios, F. Bosco, Int. J. Bif., Chaos 6 (1996) 1563; I. Antoniou, V. Basios, F. Bosco, Absolute controllability condition for probablistic control of Chaos, Int. J. Bif., Chaos 8(2) (1998) 409) in connection with small random perturbations of the map parameter. By constructing a probabilistic dynamical system equivalent to the perturbed chaotic map we calculate a characteristic exponent which gives controllability conditions in terms of the applied perturbations.

Spectral decomposition of expanding probabilistic dynamical systems

Abstract We study probabilistic combinations of expanding dynamical systems, which we call expanding probabilistic dynamical systems, in one dimension. If the system is composed by exact endomorphisms we prove that the probabilistic dynamical system is an exact Markov semigroup, and we determine a generalized spectral decomposition of the associated Markov operator on densities for an example of the tent map coupled with the 2-Renyi map.

Spectral decomposition of contracting probabilistic dynamical systems

We study probabilistic couplings of a finite class of linear contractions of the unit interval. For this reason we call these systems ‘contracting probabilistic dynamical systems’. The resulting system is a stationary Markov process with deterministic outcomes. Each outcome results from the application of one of the contractions. The selection of each contraction is conditioned probabilistically. We study these systems through the spectral properties of their corresponding sthocastic and Markov operators.

Absolute Controllability Condition for Probabilistic Control of Chaos

We derive an absolute condition for probabilistic control [Antoniou et al., 1996] of the unstable dynamics of the β-adic Renyi map. The controllability condition over the parameters of the probabilistic dynamical system is valid for a wide class of observables and is obtained from the fixed point solution of the associated Markov operator for measures.

ANALYSIS OF CATEGORICAL TIME-SERIES TEXT RECALL DATA USING A CONNECTIONIST MODEL

Categorical time-series are generated by discrete-time probabilistic dynamical systems which can only be in one of a small number of finite states at any given instant in time. A novel statistical methodology based upon log-linear modelling is proposed for analyzing categorical time-series data which allows one to incorporate a considerable amount of prior knowledge directly into the data analysis. The statistical model can be shown to be formally equivalent to a connectionist (i.e., artificial neural network) model, Methods for model selection and hypothesis testing using the new statistical model for samples with large numbers of observations are then developed using asymptotic statistical theory. To illustrate this new method of categorical time-series data analysis, the model is applied to the analysis of text free recall data from children and adults. These analyses indicated that the model can successfully use the order of recalled text propositions to discriminate among alternative theories of prior knowledge and alternative treatment conditions. The reliability of the large sample statistical tests were also checked using a boot-strap methodology and found to be acceptable.