Function Of State Probabilities Research Articles

Meta-state–space learning: An identification approach for stochastic dynamical systems

Available methods for identification of stochastic dynamical systems from input–output data generally impose restricting structural assumptions on either the noise structure in the data-generating system or the possible state probability distributions. In this paper, we introduce a novel identification method of such systems, which results in a dynamical model that is able to produce the time-varying output distribution accurately without taking restrictive assumptions on the data-generating process. The method is formulated by first deriving a novel and exact representation of a wide class of nonlinear stochastic systems in a so-called meta-state–space form, where the meta-state can be interpreted as a parameter vector of a state probability function space parameterization. As the resulting representation of the meta-state dynamics is deterministic, we can capture the stochastic system based on a deterministic model, which is highly attractive for identification. The meta-state–space representation often involves unknown and heavily nonlinear functions, hence, we propose an artificial neural network (ANN)-based identification method capable of efficiently learning nonlinear meta-state–space models. We demonstrate that the proposed identification method can obtain models with a log-likelihood close to the theoretical limit even for highly nonlinear, highly stochastic systems.

A Bayesian Estimation of Confidence Limits for Multi-state System Vulnerability Assessment With IEMI

A Bayesian approach based on the vulnerability distribution is proposed to estimate the confidence limits of the state probability and the threat level of multistate electronic systems interfered by intentional electromagnetic interference (IEMI). The vulnerability distribution is used to describe the state probability function of the multi-state system (MSS) for a given IEMI threat level. When a small number of test samples are under a specific threat level and prior MSS information is known, the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">posterior</i> estimation of the state probability function can be obtained based on Bayesian theory. Furthermore, to effectively apply the state probability function to assess the vulnerability of the MSS, the uncertainty of the state probability function is discussed. Considering state probabilities under the same threat level as random variables, all these state probabilities also follow the Dirichlet distribution. Thus, the confidence limits of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">posteriori</i> state probability and the threat level can be inferred by the marginal distribution of the state probability, which is induced by combining the Dirichlet and vulnerability distributions. Finally, a case study is given to demonstrate the details of the calculation process of the vulnerability distribution, the state probability function, and the confidence limits under a lognormal distribution.

A computational method for finding the availability of opportunistically maintained multi-state systems with non-exponential distributions

Availability is one of the most important performance measures of a repairable system. Among various mathematical methods, the method of supplementary variables is an effective way of modeling the steady-state availability of systems governed by non-exponential distributions. However, when all the underlying probability distributions are non-exponential (e.g., Weibull), the corresponding state equations are difficult to solve. To overcome this challenge, a new method is proposed in this article to determine the steady-state availability of a multi-state repairable system, where all the state sojourn times, as well as the maintenance times, are generally distributed. As an indispensable step, the well-posedness and stability of the system’s state equations are illustrated and proved using C0 operator semigroup theory. Afterwards, based on the generalized Integral Mean Value Theorem, the expression for system steady-state availability is derived as a function of state probabilities. Then, the original problem is transformed into a system of linear equations that can be easily solved. A simulation study and an instance studied in the literature are used to demonstrate the applications of the proposed method in practice. These numerical examples illustrate that the proposed method provides a new computational tool for effectively evaluating the availability of a repairable system without relying on simulation.

Multi-state balanced systems in a shock environment

Reliability analysis of balanced systems has become an important research topic. In previous studies, a balanced system only has two states, i.e., perfect functioning and complete failure. However, most practical systems have more than two states because of the complex system structure and complicated working environment. To fill in this research gap, a general multi-state balanced system is proposed by considering that the components in the system and the whole system both have multiple states. In this paper, component state transitions are assumed to be caused by external shocks. Based on the operating states of all components, the multiple states of the system are determined according to different system balance degrees, formulated by a balance function. Multi-state balanced systems based on state distance and symmetric state distance are constructed in detail based on two specific balance functions. The corresponding state probability functions and some other reliability indexes are derived by using a two-step finite Markov chain imbedding approach. Finally, numerical examples are presented and some future research topics are discussed.

Vulnerability Assessment of a Multistate Component for IEMI Based on a Bayesian Method

A method based on the Bayesian approach is proposed in this paper to assess the vulnerability of a multistate component when the available experimental data are insufficient. According to the proposed definition of hierarchical vulnerability distributions, each state probability function of components is represented by adjacent level distributions, and the restriction of vulnerability distributions is analyzed. Depending on the credibility interval of the parameters of vulnerability distributions, the joint prior probability density functions (PDFs) of these parameters can be obtained in different scenarios. Therefore, the joint posterior PDF of these parameters can be obtained by combining the joint prior PDF and the likelihood function based on a Bayesian formula. Then, the posterior probability function of each state can be obtained by the proposed method. In realistic situations, the estimation of the state probability functions will be more practical than single estimation of the state probability under a specific threat level of intentional electromagnetic interference (IEMI). The specific process of the method is presented, while the hierarchical vulnerability distributions are represented by lognormal cumulative distribution functions. Finally, a case study demonstrates the effectiveness of the proposed method.

Analyzing an infinite buffer batch arrival and batch service queue under batch-size-dependent service policy

In this paper, we investigate an infinite-buffer queue with batch-arrival and batch-service wherein a single server operates under random serving capacity rule with service time dependent on the size of the batch under the service. First, we derive the probability generating function of state probabilities at service completion epoch, from which an entire spectrum regarding queue-length at various epochs is extracted. Using the departure epoch probabilities, we establish a stable relationship between departure and random epochs probabilities based on ‘rate in = rate out’ approach. Further, random epoch probabilities are used to obtain pre-arrival epoch probabilities. Finally, we illustrate our analytical results by means of numerical computation which includes the case of multiple roots.

Chloride Flux in Prestin-Expressing Cells

In prestin-expressing cells, intracellular Cl- ([Cl-]i) flux plays a preeminent role in promoting prestin activity, especially since the resulting Vh shift in prestin's state-probability function (inferred from nonlinear capacitance - NLC) along the Vm axis will effect a motile response. For example, with perforated patch clamp and local perfusion, changing extracellular Cl concentration from 1 mM to 140 mM using prestin's NLC as a measure of intracellular Cl indicates a several mM increase in intracellular Cl concentration. Nevertheless, the mechanism underlying Cl- flux in prestin-expressing cells is not clear.To better define Cl- flux, the CFP-YFP-based ratiometric Cl indicator (Cl-sensor, J. Neurosci. Meth., 2008, 170, 67) was modified to reduce pH sensitivity by shifting the pKa away from the physiological pH range. Using excitation ratiometric imaging of fluorescence, [Cl]i was measured in either induced- or non-induced prestin-expressing HEK cell-lines transfected with our new Cl-sensor construct. Upon changing local extracellular perfusion from 0.2 mM to 140 mM Cl buffer, fluorescence measures indicate a significant difference between induced- and non-induced prestin cell-lines.We also evaluated [Cl]i flux using a Cl-sensor fusion product of prestin at the C-terminal to better gauge flux near prestin's intracellular binding site. Comparison of [Cl]i flux in HEK-293T cells transfected with a normal prestin-Cl-sensor or a prestin construct with a C-terminal deletion that eliminated NLC (P709-Cl-sensor) showed no statistical difference in flux. These data suggest that prestin's NLC is separable from Cl- movement induced by prestin, as we previously suggested (Bai et al., BJ, 3179, 2009). Further experiments are underway, including voltage dependence of flux by simultaneously monitoring fluorescence ratio signals and NLC using perforated patch.(Supported by NIDCD DC00273 and NIDCD DC 008130)

Reliability assessment for fuzzy multi-state systems

Fuzzy multi-state system (FMSS) is defined as a multi-state system (MSS) consisting of multi-state elements (MSE) whose performance rates and transition intensities are presented as fuzzy values. Due to the lack, inaccuracy or fluctuation of data, it is oftentimes impossible to evaluate the performance rates and transition intensities of MSE with precise values. This is true especially in continuously degrading elements that are usually simplified to MSE for computation convenience. To overcome these challenges in evaluating the behaviour of MSS, fuzzy theory is employed to facilitate MSS reliability assessment. Given the fuzzy transition intensities and performance rates, the state probabilities of MSE and MSS are also fuzzy values. A fuzzy continuous-time Markov model with finite discrete states is proposed to assess the fuzzy state probability of MSE at any time instant. The universal generating function with fuzzy state probability function and performance rate is applied to evaluate fuzzy state probability of MSS in accordance with the system structure. A modified FMSS availability assessment approach is introduced to compute the system availability under the fuzzy user demand. In order to obtain the membership functions of the indices of interest, parametric programming technique is employed according to Zadeh's extension principle. The effectiveness of the proposed method is illustrated and verified via reliability assessment of a multi-state power generation system.

Dynamic Probabilistic Approach for Long-Term Pavement Restoration Program with Added User Cost

A dynamic probabilistic-based approach has been developed for generating a long-term pavement restoration program for a given pavement system. The probabilistic approach applies the basic principles of stochastic processes to predict pavement conditions. Initial state probabilities and transition probabilities are the two main parameters required to develop the future state probability functions used in formulating an effective optimum decision policy. The future state probabilities are only functions of the restoration variables representing potential restoration actions. An optimum decision policy is deployed to yield a pavement restoration program comprising ( n) restoration plans corresponding to ( n) transitions. The derived pavement restoration program takes into consideration the long-term transitional performance and budget requirements. The applied decision policy is based on either maximizing the expected system condition ratings subjected to budget constraints or minimizing the net system costs subjected to desired expected system condition ratings. The net system cost may include restoration cost, deteriorated pavement added user cost, and work-zone added user cost. The two decision policy options are subjected to other constraints placing limits on the state probability functions and restoration variables. The restoration variables as applied to state probability functions result in optimum models that are sequentially solved to maintain their linearity. Sample results from a case study have indicated the usefulness of the developed dynamic probabilistic approach in yielding potential long-term pavement restoration programs.

Impacts of surface mine valley fills on headwater floods in eastern Kentucky

The potential impacts of valley fills associated with mountaintop removal/valley fill (MTR/VF) coal mining on downstream flooding in the coalfields of eastern Kentucky and adjacent states are a subject of public debate and scientific uncertainty. This study explored two aspects of this issue. First, hydrologic indices of relative runoff production and surface and subsurface flow detention were applied to conditions typical of headwater and low-order drainage basins in eastern Kentucky. Results show that there is a clear risk of increased flooding (greater runoff production and less surface flow detention) following MTR/VF operations, and suggest that, on balance, valley fills are more likely to increase rather than decrease flood potential. However, there is a wide range of outcomes, qualitatively and quantitatively. Flood risks can be increased or decreased, and the degree of either may vary markedly. The effects of MTR/VF mining on downstream peak flows are highly contingent on local pre- and post-mining conditions, and it would be unwise to apply generalizations to specific sites. Second, the occurrence of flash floods downstream of MTR/VF operations when nearby unmined areas did not flood or had less severe floods has frequently been explained (without supporting data) in terms of locally greater precipitation. The likelihood of such short-range variability of storm precipitation is evaluated by applying the state probability function to NEXRAD radar estimates of precipitation for two 2001 storms which produced flash floods in eastern Kentucky. The spatial structure of the storm precipitation indicates that at the scale of the analysis (pixel size of approximately 2 km) large local variations in storm precipitation are unlikely—that is, the probability of nearby hollows or low-order drainage basins receiving substantially different storm precipitation totals is low.

Cl- flux through a non-selective, stretch-sensitive conductance influences the outer hair cell motor of the guinea-pig.

Outer hair cells underlie high frequency cochlear amplification in mammals. Fast somatic motility can be driven by voltage-dependent conformational changes in the motor protein, prestin, which resides exclusively within lateral plasma membrane of the cell. Yet, how a voltage-driven motor could contribute to high frequency amplification, despite the low-pass membrane filter of the cell, remains an enigma. The recent identification of prestin's Cl- sensitivity revealed an alternative mechanism in which intracellular Cl- fluctuations near prestin could influence the motor. We report the existence of a stretch-sensitive conductance within the lateral membrane that passes anions and cations and is gated at acoustic rates. The resultant intracellular Cl- oscillations near prestin may drive motor protein transitions, as evidenced by pronounced shifts in prestin's state-probability function along the voltage axis. The sensitivity of prestin's state probability to intracellular Cl- levels betokens a more complicated role for Cl- than a simple extrinsic voltage sensor. Instead, we suggest an allosteric modulation of prestin by Cl- and other anions. Finally, we hypothesize that prestin sensitivity to anion flux through the mechanically activated lateral membrane can provide a driving force that circumvents the membrane's low-pass filter, thus permitting amplification at high acoustic frequencies.

Spatial Structures and Scale in Categorical Maps

Spatial variability is typically represented by categorical maps or by samples taken at specific locations (point data). The regions or patches represented in categorical maps are intuitive and consistent with much geographical (and ecological, pedological, etc.) theory, but point data are more amenable to geostatistics and other approaches which allow examination of scales of variability and of spatial structure. The state probability function (SPF) is introduced as a tool for evaluating the spatial structure of landscapes represented as categorical maps. The SPF is based on the degree of similarity between mapped units or classificatory categories a given distance apart. A plot of the SPF vs. distance produces a graph which can be interpreted similarly to a variogram. The method is tested/illustrated using detailed soil map data from eastern North Carolina. The SPF is shown to be appropriately sensitive to major landscape boundaries across which the suite of soil types changes, and to the degree of clustering of similar units.

Inherited vs. acquired complexity in east Texas weathering profiles

Complexity in weathering profiles may reflect variability in the parent material and/or complex feedback relationships within regolith weathering systems. Because regolith formation modifies or destroys the parent material, it is difficult to determine the extent to which variability in the weathering profile is inherited, as opposed to created by the weathering processes. The state probability function (SPF), designed to model the spatial pattern of transitions in system states, is applied to complex weathering profiles formed in glauconitic materials in east Texas to address this issue. The weathering profile is remarkably complex, with units ranging from unweathered parent material to highly modified weathering products similar to those found in the Bt horizons of the overlying Rhodic Paleudalf. There is a very general vertical sequence with more weathered material near the top and less weathered material near the bottom; however, the pattern is quite irregular. Highly weathered units are found everywhere except at the base of the study section, and much less developed units are found throughout. Though the SPF indicates a high degree of spatial complexity, the value never approaches or exceeds that which would be expected in a random vertical sequence of weathering horizons. Variation inherited from the parent material would produce SPF values greater than that for a random sequence. The complex patterns in the east Texas regolith are thus attributable primarily to feedbacks within the weathering system and indicate that the complexity is dominantly acquired. Whatever anisotropy there may have been in the parent material has been magnified and overprinted by complex interactions within the weathering system.

Divergent evolution and the spatial structure of soil landscape variability

The spatial structure of soil variability at the landscape scale was examined on adjacent geomorphic surfaces dating from 80 to 200 ka in eastern North Carolina. The purpose was to determine whether there is evidence at broader scales (distances of 10 2–10 4 m) for the divergent evolution observed in the field at very detailed scales (distances of 10 0–10 2 m). The state probability function (SPF), which measures spatial dependence for categorical environmental data along a transect, was applied to soil series mapped at a 1:24,000 scale. The older Talbot Terrace and younger Pamlico Terrace surfaces showed distinctly different patterns of spatial variability. The range of spatial dependence was shorter on the older surface (about 200 vs. 300 m), and the SPF was higher at any given distance, indicating more variability. The SPF for the Pamlico surface also indicates a periodicity related to fluvial dissection of the landscape, which is not readily detectable on the Talbot transect despite its greater degree of dissection. The results confirm earlier field studies which suggest that pedogenesis is marked by divergence, whereby differences in initial conditions or local perturbations persist and increase to produce a more variable soil cover.

Statistical Theory of Subcritically-Excited Strong Turbulence in Inhomogeneous Plasmas. II

A statistical description is developed for a self-sustained subcritical turbulence in inhomogeneous plasmas. Interchange mode in the presence of inhomogeneous magnetic field, which reveals a submarginal strong turbulence, is considered. Nonlinear dispersion relation is extended to a Langevin equation for a dressed test mode, in which nonlinear interactions are kept as renormalized drag and random self noise. Based upon the assumption that the random noise has a faster time scale, the solutions are obtained for the fluctuation level, decorrelation rate, auto- and cross-correlation functions and spectrum. They are expressed as nonlinear functions of non-equilibrium parameters like gradient. Extended fluctuation-dissipation theorem (Einstein relation) is described as statistical relations. Then the Langevin equation is reformulated into a Fokker-Planck equation of the probability distribution function. The steady state probability function is solved. Imposing the constraint of the self-noise, the power-law d...

Statistical Theory of Subcritically-Excited Strong Turbulence in Inhomogeneous Plasmas. I

A statistical description is developed for a self-sustained subcritical turbulence in inhomogeneous plasmas. Interchange mode in the presence of inhomogeneous magnetic field, which reveals a submarginal strong turbulence, is considered. Nonlinear dispersion relation is extended to a Langevin equation for a dressed test mode, in which nonlinear interactions are kept as renormalized drag and random self noise. Based upon the assumption that the random noise has a faster time scale, the solutions are obtained for the fluctuation level, decorrelation rate, auto- and cross-correlation functions and spectrum. They are expressed as nonlinear functions of non-equilibrium parameters like gradient. Extended fluctuation-dissipation theorem (Einstein relation) is described as statistical relations. Then the Langevin equation is reformulated into a Fokker-Planck equation of the probability distribution function. The steady state probability function is solved. Imposing the constraint of the self-noise, the power-law distribution with respect to the fluctuation amplitude is obtained in the tail of the distribution function.

Connected and degraded text recognition using hidden Markov model

We have applied a Hidden Markov Model (HMM) and level-building dynamic programming algorithm to the problem of robust machine recognition of connected and degraded characters forming words in a poorly printed text. The recognition system consists of preprocessing, subcharacter segmentation and feature extraction, followed by supervised learning or recognition. A structural analysis algorithm is used to segment a word into subcharacter segments irrespective of the character boundaries, and to identify the primitive features in each segment such as strokes and arcs. The states of the HMM for each character are statistically represented by the subcharacter segments, and the state characteristics are obtained by determining the state probability functions based on the training samples. In order to recognize an unknown word, subcharacter segmentation and feature extraction are performed and the transition probabilities between character models are used for the transition between characters in the string. A level-building dynamic programming algorithm combines segmentation and recognition of the word in one operation and chooses the best probable grouping of characters for recognition of an unknown word. The computer experiments demonstrate the robustness and effectiveness of the new system for recognizing words formed by degraded and connected characters.

The concentration equation of polymer solution in nonhomogeneous flow field

We have derived formally exact equations for the concentration of polymer in an externally imposed field using a time-dependent projection operator formalism. Two different bases are chosen as projectors; the initial state basis (equilibrium) at t=0 and the final state basis (steady state) at t→∞. The initial state basis is natural for studying transient behavior at short time and the final state basis is good for long time asymptotic behavior. Several methods have been suggested to obtain overall time dependences of the concentration, such as an interpolation model for the approximate initial and final state formulae. The general framework will be useful in studying the macroscopic characteristics of polymer solutions exposed to arbitrary flow fields in confined geometries using the molecular inputs. A simple illustration of polymer migration caused by the flow geometry and hydrodynamic interactions is provided. Calculation of the steady state probability function of polymer chains is also given.

Wigner’s inequality, quantum-mechanical probability functions and hidden-variable theories

We investigate the algebraic properties of the quantum-mechanical singlet state probability functions and the algebraic conditions which these functions satisfy. We, then, argue that violation of Wigner’s inequality by quantum mechanics is not significant in deriving conclusions about locally realistic hidden-variable theories, in particular, and local realism, in general.

Multiple-criteria robust interactive decision analysis (MCRID) for optimizing public policies

Important public policy decisions are often very complex, involving many factors, large uncertainties, and multiple conflicting objectives. Decision analysis, which is a formal framework for dealing with such issues, has been applied to a variety of public-sector problems over the last two decades. A limitation of using decision analysis is the often onerous assessment burden it places on the decision maker (DM). We have developed a novel interactive procedure and Decision Support System (DSS) for performing decision analysis on the personal computer, called robust interactive decision analysis (RID), which avoids the difficult problems of precisely assessing a DM's utility and state probability functions associated with traditional decision treee analysis. The RID method permits a DM to voluntarily and interactively express strong (viz, sure) binary preferences for actions, partial decision functions, and full decision functions, and only imprecise probability and utility function assessments. The inputs are used with various dominance operators via a mathematical programming framework to prune the state probability, utility, and decision space until there is convergence on a relatively small subset of efficient strategies or an optimal choice strategy. Conceptually, the operation of the RID method can be regarded as a state-pruning system and its computer implementation as a DSS. In this paper, we extended the RID concept and DSS to deal with multiple-criteria decisions, obviating the need to precisely and completely assess attribute weights (tradeoffs), consequences, and/or value/utility functions. The approach, which we call multiple-criteria robust interactive decision analysis, MCRID, is developed, illustrated, and shown to be effective in eliminating many inefficient alternatives. However, since the aggregate utility of each alternative is interval-valued, it then becomes difficult to intuitively choose an optimal alternative from among the remaining efficient ones since the intervals are not defined by any specific probability distributions. We have tentatively shown that aggregate utility intervals ultimately converge toward a normal density function, and can be so approximated with relatively few attributes. Hence, stochastic dominance is employed to further filter out alternatives. An interactive goal programming (GP) technique is then developed to assist the DM in choosing an optimal alternative.