Initial Probability Density Research Articles

Tests of human auditory temporal resolution: Simulations of Bayesian threshold estimation for auditory gap detection.

In an attempt to develop tests of auditory temporal resolution using gap detection, we conducted computer simulations of Zippy Estimation by Sequential Testing (ZEST), an adaptive Bayesian threshold estimation procedure, for measuring gap detection thresholds. The results showed that the measures of efficiency and precision of ZEST changed with the mean and standard deviation (SD) of the initial probability density function implemented in ZEST. Appropriate combinations of mean and SD values led to efficient ZEST performance; i.e., the threshold estimates converged to their true values after 10 to 15 trials.

Density Steering by Power Moments

This paper considers the problem of steering an arbitrary initial probability density function to an arbitrary terminal one, where the system dynamics is governed by a first-order linear stochastic difference equation. It is a generalization of the conventional stochastic control problem where the uncertainty of the system state is usually characterized by a Gaussian distribution. We propose to use the power moments to turn the infinite-dimensional problem into a finite-dimensional one and to present an empirical control scheme. By the designed control law, the moment sequence of the controls at each time step is positive, which ensures the existence of the control for the moment system. We then realize the control at each time step as a function in analytic form by a convex optimization scheme, for which the existence and uniqueness of the solution have been proved in our previous paper (Wu and Lindquist, 2022). Two numerical examples are given to validate our proposed algorithm.

Efficient Adaptive Monte Carlo Uncertainty Forecasting for High Dimensional Nonlinear Dynamic Systems

The problem of uncertainty forecasting for complex dynamical systems in the framework of particle methods can be effectively addressed through the solution methodology known as adaptive Monte Carlo (AMC). Monte Carlo (MC) methods involve discretizing the initial probability density function (pdf) followed by forward propagation of particles through system dynamics to obtain an approximate particle representation of the evolved state uncertainty. While simple to implement, MC faces questions surrounding transient statistical consistency and rate of convergence. AMC (adopted here) addresses these issues on-the-fly using defined bounds on estimation accuracy alongside ensemble enrichment routines. This paper presents several improvements in the ensemble enrichment module of AMC. The AMC platform is re-engineered to include the novel implementation of a parallel global stochastic optimization routine in conjunction with additional module enhancements that work together towards the goal of efficient forecasting. Moreover, the efficacy of algorithms utilized within every submodule of AMC are detailed and improved upon under the framework of arithmetic minimization and parallelization. Each submodule is profiled to determine computational bottlenecks, optimized or replaced with more efficient methods such as simulated annealing (SA), and parallelized ultimately leading to a clock time reduction of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$200 - 400\%$ </tex-math></inline-formula> for benchmark entry descent and landing, Lorenz-96, and Lotka-Volterra models.

Durability degradation simulation of RC structure based on gamma process considering two-dimensional chloride diffusion and life probabilistic prediction

In this paper, based on the gamma process, a new method for predicting the life of reinforced concrete (RC) structures eroded by chloride is presented considering two-dimensional (2D) chloride diffusion. First, according to the chloride diffusion theory, an analytical model and a finite difference numerical model considering 2D chloride diffusion under the action of multiple factors were established. Based on this, the Monte Carlo method and method proposed in this study were employed to predict the initial rust time of RC structures eroded by chloride. Histograms of the initial rust time and probability density function obtained using the two methods mentioned above were compared and analyzed to verify the validity of the proposed method. Finally, the life prediction model of the RC structure considering the initial rust time T1, rust expansion cracking time T2, and time when the concrete surface crack width reaches the limit crack width T3 was established, and its applicability was verified through an analysis of an engineering example. The research indicates that the gamma process-based probability models can approximately predict the time required for rebar corrosion of an RC structure eroded by chloride considering 2D diffusion.

Brownian non-Gaussian diffusion of self-avoiding walks

Three-dimensional Monte Carlo simulations provide a striking confirmation to a recent theoretical prediction: the Brownian non-Gaussian diffusion of critical self-avoiding walks. Although the mean square displacement of the polymer center of mass grows linearly with time (Brownian behavior), the initial probability density function is strongly non-Gaussian and crosses over to Gaussianity only at large time. Full agreement between theory and simulations is achieved without the employment of fitting parameters. We discuss simulation techniques potentially capable of addressing the study of anomalous diffusion under complex conditions like adsorption- or Theta-transition.

Adaptive polyhedral meshing for approximate dynamic programming in control

This work proposes a new criterion for adaptive meshing in polyhedral partitions which interpolate a value function in Approximate Dynamic Programming (ADP) in optimal control problems. The criterion adds new points to a simplicial mesh, based on: a user-defined initial condition probability density function which determines ‘influential’ regions of the state space, uncertainty (variance) propagation, and temporal-difference error. A collection of lemmas justifies the algorithmic proposal. Comparative analysis with other options in literature highlights the advantages of our proposal. The developed methods are applied to simulation examples and an experimental robotic setup.

Destination-Directed Trajectory Modeling, Filtering, and Prediction Using Conditionally Markov Sequences

In some problems, there is information about the destination of a moving object. An example is a flight from an origin to a destination. Such problems have three main components: an origin, a destination, and motion in between. We call such trajectories ending up at the destination destination-directed trajectories (DDTs). Described by an evolution law and an initial probability density, a Markov sequence is not flexible enough to model DDT well. The future (including destination) of a Markov sequence is completely determined probabilistically by its initial density and evolution law. One class of conditionally Markov (CM) sequences, called the CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> sequence (it includes the Markov sequence as a special case), has the following main components: a joint density of two endpoints and a Markov-like evolution law. This article proposes modeling DDT as CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> sequences. We study the CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L </sub> sequence, its dynamic model, and its realizations, all for DDT modeling. We demonstrate that CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> sequences naturally model DDT, enjoy several desirable properties for DDT modeling, and can be easily and systematically generalized if necessary. In addition, we study DDT filtering and trajectory prediction based on a CM <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</sub> model and compare them with those based on a Markov model. Several simulation examples are presented to illustrate DDT modeling and inference.

Probabilistic characteristics of nonlinear waves in nondispersive media of the hydrodynamic type.

The paper considers the probability distributions of the nonlinear wave characteristics in nondispersive media that satisfy the Riemann- and the Kardar-Parisi-Zhang-type equations. By using the Lagrangian and Euler relations of statistical descriptions, expressions are obtained for the probability density of the Riemann wave (displacement) integral through the initial probability density of displacement, velocity, and acceleration. The case of Gaussian initial statistics is considered when multivalued sections in nonlinear waves arise at arbitrarily small distances from the entrance. The expressions obtained in this case should be interpreted as the relative residence time of the process in a certain displacement range. It is shown that, due to the Riemann equation locality, the appearance of ambiguity in the wave profile, which occurs mainly at negative values, does not affect the probability density form at positive bias values.

Optimal Transport Based Tracking of Space Objects in Cylindrical Manifolds

In this paper, we examine the performance of satellite state estimation algorithms in the modified equinoctial coordinate system, defined on the cylindrical manifold of $\mathbb {R}^{5}\times \mathbb {S}$, where $\mathbb {R}$ is the space of reals and $\mathbb {S}$ denotes circular space. A comparison is made between an optimal transport based filter and ensemble Kalman filter algorithm in the context of satellite state estimation. The initial state joint probability density function is modeled in $\mathbb {R}^{5}\times \mathbb {S}$ using the Gauss von Mises distribution. The sensor noise for optimal transport filtering is modeled in the same manifold. The ensemble Kalman filter, by definition, requires the sensor noise to be Gaussian and is modeled in $\mathbb {R}^{6}$ for this problem. We observe that there is a clear advantage in using an optimal transport based filtering algorithm where we represent the initial condition uncertainty and sensor noise, in the cylindrical manifold. These two filtering algorithms are implemented on a simulated International Space Station orbit, with measurement at equal intervals. We compare two distinct scenarios in this simulation based study. In the first one, sensor noise characteristics are assumed to be known. For the second one, we present a new algorithm within the optimal transport framework with unknown sensor noise characteristics, a practical and relevant issue in the space-tracking problems. The optimal transport based algorithm provides more consistent and robust estimates compared to that of ensemble Kalman filter, in each of these scenarios.

Scalings and fractals in information geometry: Ornstein–Uhlenbeck processes

We propose a new methodology to understand a stochastic process from the perspective of information geometry by investigating power-law scaling and fractals in the evolution of information. Specifically, we employ the Ornstein–Uhlenbeck process where an initial probability density function (PDF) with a given width and mean value y0 relaxes into a stationary PDF with a width ϵ, set by the strength of a stochastic noise. By utilizing the information length which quantifies the accumulative information change, we investigate the scaling of with ϵ. When , the movement of a PDF leads to a robust power-law scaling with the fractal dimension . In general when , is possible in the limit of a large time when the movement of a PDF is a main process for information change (e.g. ). We discuss the physical meaning of different scalings due to PDF movement, diffusion and entropy change as well as implications of our finding for understanding a main process responsible for the evolution of information.

A class of individual-based models

We discuss a class of mathematical models of biological systemsВ at microscopic level - i.e. at the level of interacting individuals of a population. The class leads to partially integral stochastic semigroups- [5]. We state general conditions guaranteeing the asymptotic stability.В In particular under some ratherВ restrictive assumptions we observe that any, even non-factorized, initial probability density tends in the evolution to a factorized equilibrium probability density - [4]. We discuss possible applications of the general theory such as redistribution of individuals - [2], thermal denaturation of DNA [1], and tendon healing process - [3].В [1] M. Debowski, M. Lachowicz, and Z. Szymanska,В Microscopic description of DNA thermal denaturation, to appear.В [2] M. Dolfin, M. Lachowicz, and A. Schadschneider, A microscopic model of redistribution of individuals inside an 'elevator', In Modern Problems in Applied Analysis, P. Drygas and S. Rogosin (Eds.), Bikhauser, Basel (2018), 77--86; DOI: 10.1007/978--3--319--72640-3.[3] G. Dudziuk, M. Lachowicz, H. Leszczynski, and Z. Szymanska, A simple model ofВ collagen remodeling, to appear.[4] M. Lachowicz, A class of microscopic individual models corresponding to the macroscopic logistic growth, Math. Methods Appl. Sci., 2017, on--line, DOI: 10.1002/mma.4680[5] K. Pichor and R. Rudnicki, Continuous Markov semigroups and stability of transportВ equations, J. Math. Analysis Appl. 249, 2000, 668--685, DOI: 10.1006/jmaa.2000.6968

On the Transfer of a Number of Concepts of Statistical Radiophysics to the Theory of One-dimensional Point Mappings

In the article, the possibility of using a bispectrum under the investigation of regular and chaotic behaviour of one-dimensional point mappings is discussed. The effectiveness of the transfer of this concept to nonlinear dynamics was demonstrated by an example of the Feigenbaum mapping. Also in the work, the application of the Kullback-Leibler entropy in the theory of point mappings is considered. It has been shown that this information-like value is able to describe the behaviour of statistical ensembles of one-dimensional mappings. In the framework of this theory some general properties of its behaviour were found out. Constructivity of the Kullback-Leibler entropy in the theory of point mappings was shown by means of its direct calculation for the ”saw tooth” mapping with linear initial probability density. Moreover, for this mapping the denumerable set of initial probability densities hitting into its stationary probability density after a finite number of steps was pointed out.

A class of microscopic individual models corresponding to the macroscopic logistic growth

We study a class of microscopic models corresponding to the standard macroscopic logistic equation. The models at microscale refer to a number of interacting individuals and are in terms of linear evolution equations related to Markov jump processes. The asymptotic large time behavior for the microscopic models is obtained. Moreover, it is shown that any, even nonfactorized, initial probability density tends in the evolution to a factorized equilibrium probability density.

Optimal Transport Over a Linear Dynamical System

We consider the problem of steering an initial probability density for the state vector of a linear system to a final one, in finite time, using minimum energy control. In the case where the dynamics correspond to an integrator ( $\dot{x}(t) = u(t)$ ) this amounts to a Monge-Kantorovich Optimal Mass Transport (OMT) problem. In general, we show that the problem can again be reduced to solving an OMT problem and that it has a unique solution. In parallel, we study the optimal steering of the state-density of a linear stochastic system with white noise disturbance; this is known to correspond to a Schrodinger bridge . As the white noise intensity tends to zero, the flow of densities converges to that of the deterministic dynamics and can serve as a way to compute the solution of its deterministic counterpart. The solution can be expressed in closed-form for Gaussian initial and final state densities in both cases.

Kramers problem for a dimer: Effect of noise correlations.

The Kramers problem for a dimer in a bistable piecewise linear potential is studied in the presence of correlated noise processes. The effect of such a correlation is to redistribute the thermal power between the dynamical degrees of freedom, and this leads to significant deviations in the dynamics of the system from the case of independent noise processes. The distribution of first passage times from one minima to the basin of attraction of the other minima is found to have exponentially decaying tails with the parameter dependent on the amount of correlation and the coupling between the particles. The strong coupling limit of the problem is analyzed using adiabatic elimination, where it is found that the initial probability density relaxes towards a stationary value on the same time scale as the mean escape time when the noise intensity of the system is low. For higher noise fluctuations, the relaxation towards the stationary state is slower in comparison to escape times. In the extreme limit of perfect anticorrelation, the random dynamical system behaves as a deterministic system in a steady state in which the center of mass starting from the unstable maxima moves down the hill and gets trapped in the potential minima. The implications for polymer dynamics in a potential are discussed.

Polymer quantization of a self-gravitating thin shell

We study the quantum mechanics of self-gravitating thin shell collapse by solving the polymerized Wheeler-DeWitt equation. We obtain the energy spectrum and solve the time dependent equation using numerics. In contradistinction to the continuum theory, we are able to consistently quantize the theory for super-Planckian black holes, and find two choices of boundary conditions which conserve energy and probability, as opposed to one in the continuum theory. Another feature unique to the polymer theory is the existence of negative energy stationary states that disappear from the spectrum as the polymer scale goes to zero. In both theories the probability density is positive semi-definite only for the space of positive energy stationary states. Dynamically, we find that an initial Gaussian probability density develops regions of negative probability as the wavepacket approaches $R=0$ and bounces. This implies that the bouncing state is a sum of both positive and negative eigenstates.

Novel mapping in non-equilibrium stochastic processes

We investigate the time-evolution of a non-equilibrium system in view of the change in information and provide a novel mapping relation which quantifies the change in information far from equilibrium and the proximity of a non-equilibrium state to the attractor. Specifically, we utilize a nonlinear stochastic model where the stochastic noise plays the role of incoherent regulation of the dynamical variable x and analytically compute the rate of change in information (information velocity) from the time-dependent probability distribution function. From this, we quantify the total change in information in terms of information length and the associated action , where represents the distance that the system travels in the fluctuation-based, statistical metric space parameterized by time. As the initial probability density function’s mean position (μ) is decreased from the final equilibrium value (the carrying capacity), and increase monotonically with interesting power-law mapping relations. In comparison, as μ is increased from and increase slowly until they level off to a constant value. This manifests the proximity of the state to the attractor caused by a strong correlation for large μ through large fluctuations. Our proposed mapping relation provides a new way of understanding the progression of the complexity in non-equilibrium system in view of information change and the structure of underlying attractor.

Measures of thermodynamic irreversibility in deterministic and stochastic dynamics

It is generally observed that if a dynamical system is sufficiently complex, then as time progresses it will share out energy and other properties amongst its component parts to eliminate any initial imbalances, retaining only fluctuations. This is known as energy dissipation and it is closely associated with the concept of thermodynamic irreversibility, measured by the increase in entropy according to the second law. It is of interest to quantify such behaviour from a dynamical rather than a thermodynamic perspective and to this end stochastic entropy production and the time-integrated dissipation function have been introduced as analogous measures of irreversibility, principally for stochastic and deterministic dynamics, respectively. We seek to compare these measures. First we modify the dissipation function to allow it to measure irreversibility in situations where the initial probability density function (pdf) of the system is asymmetric as well as symmetric in velocity. We propose that it tests for failure of what we call the obversibility of the system, to be contrasted with reversibility, the failure of which is assessed by stochastic entropy production. We note that the essential difference between stochastic entropy production and the time-integrated modified dissipation function lies in the sequence of procedures undertaken in the associated tests of irreversibility. We argue that an assumed symmetry of the initial pdf with respect to velocity inversion (within a framework of deterministic dynamics) can be incompatible with the Past Hypothesis, according to which there should be a statistical distinction between the behaviour of certain properties of an isolated system as it evolves into the far future and the remote past. Imposing symmetry on a velocity distribution is acceptable for many applications of statistical physics, but can introduce difficulties when discussing irreversible behaviour.

Excited State Intramolecular Proton Transfer Dynamics of Oxadiazole‐based Dyes#

Excited state intramolecular proton transfer (ESIPT) dynamics of two oxadiazole‐based model compounds, 2,5‐bis‐[5‐(4‐tert‐butyl‐phenyl)‐[1,3,4]oxadiazol‐2‐yl]‐phenol (SOX) and 2,5‐bis‐[5‐(4‐tert‐butyl‐phenyl)‐[1,3,4]oxadiazol‐2‐yl]‐benzene‐1,4‐diol (DOX) have been investigated by time‐resolved fluorescence. SOX and DOX have one and two intramolecular hydrogen bond moieties, respectively, where ESIPT may occur. Time‐resolved fluorescence fully resolves the ESIPT dynamics and establishes the two‐state conversion between the enol and keto isomers in the S1 potential energy surface. The apparent inconsistency between the ESIPT rate and the steady‐state spectrum is settled by invoking the conformational inhomogeneity including the intermolecular hydrogen bonding with protic solvents. The ESIPT rates of SOX and DOX are not as fast as those of typical ESIPT molecules, which indicates a finite barrier for the reaction. For DOX, ESIPT is enabled for only one hydroxyl moiety, and the ESIIPT rate is twice as fast as that of SOX. We propose a symmetric potential energy surface along the two proton transfer coordinates, and bifurcation of the initial probability density of the enol isomer into the two potential minima representing the keto isomer.

Stochastic tool wear assessment in milling difficult to machine alloys

In the machining industry, maximising profit is intuitively a primary goal; therefore continuously increasing machining process uptime and consequently productivity and efficiency is crucial. Tool wear plays an important factor in both machining uptime and quality, and since tool failure is related to the surface quality and the dimensional accuracy of the end product, it is essential to quantify and predict this phenomenon with the best possible certainty. One of the most common ways of tool wear prediction is through the use of low cost spindle current sensing technology which is used to measure spindle power consumption in CNC machines and relate power increase to tool wear. In this work, two methods of stochastic filtering (i.e. Kalman and particle filter) were used in predicting tool flank wear in machining difficult-to-machine materials through spindle power consumption measurements. Results show a maximum of 15% average error in estimation, which indicates the good potential of using stochastic filtering techniques in estimating tool flank wear. In addition, the particle filter was used for online estimation of a spindle power model parameter with uniform and Gaussian mixture models as the initial probability density functions, and the evolution of this parameter to the true posterior density function over time was investigated.