Time-varying Probability Distribution Research Articles

A Nonstationary Daily and Hourly Analysis of the Extreme Rainfall Frequency Considering Climate Teleconnection in Coastal Cities of the United States

The nonstationarity of extreme precipitation is now well established in the presence of climate change and low-frequency variability. Consequently, the implications for urban flooding, for which there are not long flooding records, need to be understood better. The vulnerability is especially high in coastal cities, where the flat terrain and impervious cover present an additional challenge. In this paper, we estimate the time-varying probability distributions for hourly and daily extreme precipitation using the Generalized Additive Model for Location Scale and Shape (GAMLSS), employing different climate indices, such as Atlantic Multi-Decadal Oscillation (AMO), the El Niño 3.4 SST Index (ENSO), Pacific Decadal Oscillation (PDO), the Western Hemisphere Warm Pool (WHWP) and other covariates. Applications to selected coastal cities in the USA are considered. Overall, the AMO, PDO and WHWP are the dominant factors influencing the extreme rainfall. The nonstationary model outperforms the stationary model in 92% of cases during the fitting period. However, in terms of its predictive performance over the next 5 years, the ST model achieves a higher log-likelihood in 86% of cases. The implications for the time-varying design rainfall in coastal areas are considered, whether this corresponds to a structural design or the duration of a contract for a financial instrument for risk securitization. The opportunity to use these time-varying probabilistic models for adaptive flood risk management in a coastal city context is discussed.

A unified transform method for general robust property of probabilistic Boolean control networks

This article proposes a unified transform method for general robust property of probabilistic Boolean control networks (PBCNs). Here, the general robust property of PBCNs means any given robust property of any given PBCN with probability one. Specifically, the method is that the investigation on any given robust property of any given PBCN with probability one can be equivalently transformed into the investigation on the given robust property of the corresponding Boolean control network (BCN). Note that the PBCN has time-varying probability distribution and the property can be set according to the needs of specific problems. As an application of the method, we study the robust set stabilization with probability one of PBCNs with positive time-varying probability distribution. We derive the results through a discussion of the robust set stabilization of the corresponding BCNs. Also, we characterize all of the time-optimal robust set stabilizers of the PBCNs. A reduced disturbed PBCN model of the lac operon in the Escherichia coli is provided to show the effectiveness of the achieved results.

Asynchronous Control of Stochastic Switched Boolean Control Networks With Piecewise-Homogeneous Dwell Time.

In this article, the l₁-induced performance of the stochastic switched Boolean control network (BCN) is investigated. The switched signal is considered to follow a time-varying probability distribution, the switching of which is considered to have a random dwell time. The asynchronous state feedback control (SFC) is studied to achieve the control objective. This kind of control can avoid the failure of the control due to the inconsistency between the system mode and the control mode, so the results obtained are more general. Using the semitensor product of matrices, the algebraic form of the considered BCN is represented. Under this framework, sufficient conditions are obtained to ensure that the closed-loop system is stochastic stabilized with a prescribed l₁-induced performance level ɣ. Parameters can be solved by inequalities. In addition, when the dwell time converges to infinity, the probability distribution of the switched signal becomes fixed. Necessary and sufficient conditions are presented to ensure the stabilization of the closed system under asynchronous SFC as well as the design of the asynchronous SFC. Then, sufficient condition is obtained for the prescribed l₁-induced performance level. Examples are presented to show the effectiveness of the obtained results.

Experimental methods and mathematical modeling quantify and predict DUSP1 RNA expression dynamics in response to dexamethasone.

Overactivation of mitogen-activated protein kinase (MAPK) signaling pathways is key to multiple inflammatory responses, and synthetic glucocorticoids (GC), such as Dexamethasone (Dex), have long been used to treat inflammatory pathologies. Upon cell entry, Dex binds to the glucocorticoid receptor (GR), which initiates translocation to the nucleus. Nuclear GR then interacts with Glucocorticoid Regulatory Elements (GRE's) to promote transcription of anti-inflammatory genes including dual-specificity phosphatase 1 (DUSP1). In turn, DUSP1 encodes for the protein mitogen-activated protein kinase phosphatase 1 (MKP-1), which regulates the cell's anti-inflammatory response through dephosphorylation of the P38 and JUN N-terminal kinase (JNK) MAPK pathways. In this presentation, we combine single-cell measurements and discrete stochastic models to elucidate the spatiotemporal mechanisms by which Dex modulates GR localization and DUSP1 transcription. Using single-molecule inexpensive fluorescent in situ hybridization, we quantify the nascent transcription, spatial localization, and cellular heterogeneity of DUSP1 mRNA for hundreds of individual HeLa cells at 20 different time points following application of continuous 100 nM Dex stimulation. We observe a rapid increase in DUSP1 transcription sites (TS's) as early as 10-20 minutes, followed by a four-fold increase in expression of mature DUSP1 mRNA that peaks at 75-90 minutes and a temporary decrease in DUSP1 expression at 120-150 minutes post stimulation. We found that expression levels stay elevated during longer time course experiments, and high cellular heterogeneity is observed throughout the post-stimuli response. We used these experimental data to infer parameters and mechanisms for a discrete stochastic model that quantitatively reproduces the time-varying probability distributions for single-cell DUSP1 expression at all time points following Dex stimulation. Finally, we use this model inferred from DUSP1 expression to predict GR translocation dynamics and compare to direct measurement of GR localization dynamic using single-cell immunocytochemistry.

Proposal and evaluation of nonstationary versions of SPEI and SDDI based on climate covariates for regional drought analysis

Multivariate standardized drought indices (MSDIs) are helpful in effectively characterizing, monitoring, and forecasting drought events over various spatial–temporal scales. Currently, there is a need to develop nonstationary versions of such indices for use in climate change scenarios to account for nonstationarities in different hydrometeorological variables (e.g., precipitation, evapotranspiration) affecting droughts. This article contributes nonstationary versions of two drought indices: SPEI (Standardized Precipitation-Evapotranspiration Index) and SDDI (Standardized Deficit Distance Index). They consider time-varying probability distribution whose location parameter is determined using climate covariates. It is suggested to delineate homogeneous drought regions using a fuzzy dynamic clustering approach to characterize regional droughts. Furthermore, a multi-site block bootstrap-based modified seasonal Mann-Kendall test is proposed to detect regional trends in drought analysis. For the first time, the performances and agreement of different pairs of stationary and nonstationary versions of three MSDIs, namely SPEI, SDDI, RDI (Reconnaissance Drought Index), are analyzed at at-site and regional scales through a case study on Karnataka state, India. The strength of agreement between different pairs of MSDIs is moderate to fair, and climate indices chosen for their estimation differed across regions and months. In analysis with nonstationary MSDIs, (i) at-site mean intensity and frequency of moderate/severe/extreme drought events were higher, whereas (ii) at-site mean duration and mean/maximum severity, and regional drought severity and intensity (discerned using severity-area-frequency and intensity-area-duration relationships) were generally lower relative to their stationary counterparts. Furthermore, nonstationary and stationary versions of RDI yielded higher regional drought intensity values for higher accumulation periods.

A reliability analysis method of dynamic irregularity for track–viaduct system with low stiffness

To evaluate the reliability of the dynamic irregularity of a track structure in satisfying the safety requirements, a stochastic virtual track inspection (SVTI) system that can be used to predict the time-varying probability distribution of dynamic irregularity was developed by combining virtual track inspection (VTI) system and the generalized probability density evolution method (GPDEM) in this study. A train–track–bridge dynamic interaction model (TTBDIM) was built using the Universal Mechanism platform. The inertial reference method (IRM) was simulated by finding the frequency-domain quadratic integration of the axle box acceleration and filtering it, based on which the deterministic VTI system was established. For the sake of probability analysis, the track geometry irregularities (TGI) was considered as a stochastic variable. A number-theory-based uniform experimental design was used to determine the representative points in the high-dimensional probability space. SVTI was established by coupling the deterministic VTI system and the generalized probability density evolution equation. The probability density evolution function of the dynamic irregularity was solved for using the finite difference method with the total variation diminishing (TVD) format. In this study, floating slab tracks on a U-shaped beam viaduct were taken as an example to analyze the influence of track stiffness irregularity (TSI) on the basic signal characteristics of the dynamic irregularity of the tracks using the deterministic VTI system. Furthermore, the SVTI system was used to analyze the influence of TSI on the time-varying probability distribution and determine whether the dynamic inspection requirements of the dynamic irregularity were satisfied. The research findings suggested that the TSI of the track–viaduct system with a low stiffness had a significant influence on the time-varying characteristics of the dynamic irregularity probability distribution. As the stiffness of the track–viaduct system decreased, the mean probability distribution of the dynamic irregularity extrema shifted to the higher-level area with an increasing standard deviation. When rubber spring floating slab tracks (RSFSTs) and steel spring floating slab tracks (SSFSTs) were laid on the 35-m-span U-shaped beam, the probabilities of dynamic irregularity exceeding the limit were 16.13% and 30.02%, respectively. The deterministic VTI system might not be able to detect irregularities exceeding the limit within the limited simulation mileage, which could lead to misjudgments and track regularity overestimation. In conclusion, the proposed SVTI system will provide direction for the design of track–viaduct systems with low stiffnesses in urban rail transit construction.

Modeling Time-Varying Random Objects and Dynamic Networks

Samples of dynamic or time-varying networks and other random object data such as time-varying probability distributions are increasingly encountered in modern data analysis. Common methods for time-varying data such as functional data analysis are infeasible when observations are time courses of networks or other complex non-Euclidean random objects that are elements of general metric spaces. In such spaces, only pairwise distances between the data objects are available and a strong limitation is that one cannot carry out arithmetic operations due to the lack of an algebraic structure. We combat this complexity by a generalized notion of mean trajectory taking values in the object space. For this, we adopt pointwise Fréchet means and then construct pointwise distance trajectories between the individual time courses and the estimated Fréchet mean trajectory, thus representing the time-varying objects and networks by functional data. Functional principal component analysis of these distance trajectories can reveal interesting features of dynamic networks and object time courses and is useful for downstream analysis. Our approach also makes it possible to study the empirical dynamics of time-varying objects, including dynamic regression to the mean or explosive behavior over time. We demonstrate desirable asymptotic properties of sample based estimators for suitable population targets under mild assumptions. The utility of the proposed methodology is illustrated with dynamic networks, time-varying distribution data and longitudinal growth data.

Functional Models for Time-Varying Random Objects

SummaryFunctional data analysis provides a popular toolbox of functional models for the analysis of samples of random functions that are real valued. In recent years, samples of time-varying object data such as time-varying networks that are not in a vector space have been increasingly collected. These data can be viewed as elements of a general metric space that lacks local or global linear structure and therefore common approaches that have been used with great success for the analysis of functional data, such as functional principal component analysis, cannot be applied. We propose metric covariance, a novel association measure for paired object data lying in a metric space (Ω, d) that we use to define a metric autocovariance function for a sample of random Ω-valued curves, where Ω generally will not have a vector space or manifold structure. The proposed metric autocovariance function is non-negative definite when the squared semimetric d2 is of negative type. Then the eigenfunctions of the linear operator with the autocovariance function as kernel can be used as building blocks for an object functional principal component analysis for Ω-valued functional data, including time-varying probability distributions, covariance matrices and time dynamic networks. Analogues of functional principal components for time-varying objects are obtained by applying Fréchet means and projections of distance functions of the random object trajectories in the directions of the eigenfunctions, leading to real-valued Fréchet scores. Using the notion of generalized Fréchet integrals, we construct object functional principal components that lie in the metric space Ω. We establish asymptotic consistency of the sample-based estimators for the corresponding population targets under mild metric entropy conditions on Ω and continuity of the Ω-valued random curves. These concepts are illustrated with samples of time-varying probability distributions for human mortality, time-varying covariance matrices derived from trading patterns and time-varying networks that arise from New York taxi trips.

Fast dynamic nonparametric distribution tracking in electron microscopic data

In situ transmission electron microscope (TEM) adds a promising instrument to the exploration of the nanoscale world, allowing motion pictures to be taken while nano objects are initiating, crystalizing and morphing into different sizes and shapes. To enable in-process control of nanocrystal production, this technology innovation hinges upon a solution addressing a statistical problem, which is the capability of online tracking a dynamic, time-varying probability distribution reflecting the nanocrystal growth. Because no known parametric density functions can adequately describe the evolving distribution, a nonparametric approach is inevitable. Towards this objective, we propose to incorporate the dynamic evolution of the normalized particle size distribution into a state space model, in which the density function is represented by a linear combination of B-splines and the spline coefficients are treated as states. The closed-form algorithm runs online updates faster than the frame rate of the in situ TEM video, making it suitable for in-process control purpose. Imposing the constraints of curve smoothness and temporal continuity improves the accuracy and robustness while tracking the probability distribution. We test our method on three published TEM videos. For all of them, the proposed method is able to outperform several alternative approaches.

Distributed Fictitious Play for Multiagent Systems in Uncertain Environments

A multiagent system operates in an uncertain environment about which agents have different and time-varying probability distributions on the environment—beliefs—that, as time progresses, converge to a common probability distribution, i.e., an asymptotic belief. A global utility function that depends on the realized state of the environment and actions of all the agents determines the system's optimal behavior. We define the asymptotically optimal action profile as a Nash equilibrium of the potential game defined by considering the expected utility with respect to the asymptotic belief. At finite time, however, agents have not entirely congruous beliefs about the state of the environment and may select conflicting actions. This paper proposes a variation of the fictitious play algorithm, which is proven to converge to equilibrium actions if the state beliefs converge to a common distribution at a rate that is at least linear. In conventional fictitious play, agents infer about others’ future behavior by computing histograms of past actions and best respond to their expected payoffs integrated with respect to these histograms. In the variations developed here, histograms are built using knowledge of actions taken by nearby nodes and best responses are further integrated with respect to the local beliefs on the state of the environment. We exemplify the use of the algorithm in coordination games.

Application of gamma process model to estimate the lifetime of photovoltaic modules

The analysis of long term data for degradation of PV modules suffers from volatility and uncertainty due to intrinsic and extrinsic factors. The low rate of degradation causes analysis complexity and ambiguity. In this study, methods used to estimate the PV module lifetimes were reviewed in terms of degradation of power output with time. Under the assumption that degradation is continuous, gradual, and monotonic, the gamma process model can explain the sampling and temporal uncertainties of lifetime data. Examples are provided to demonstrate the use of gamma process model for long term and accelerated lifetime test (ALT) data. Three types of lifetime estimation method were compared for long term operation data. Although they all gave similar estimated lifetime, the gamma process model gave the most applicable results to determine warranty life. The gamma process model can also express the condition variation at inspection and the lifetime variation at failure level as probability distributions. A method to determine warranty life is proposed using an age based replacement policy. For ALT data, we estimated the lifetime from degradation data using the Arrhenius equation for standard environmental conditions and applied the gamma process model to obtain time varying probability distributions for condition and lifetime. Service life was estimated as the median, while warranty life was estimated as the minimum rate of increase of optimal replacement time.

A Survey on Gain-Scheduled Control and Filtering for Parameter-Varying Systems

This paper presents an overview of the recent developments in the gain-scheduled control and filtering problems for the parameter-varying systems. First of all, we recall several important algorithms suitable for gain-scheduling method including gain-scheduled proportional-integral derivative (PID) control,H2,H∞and mixedH2/H∞gain-scheduling methods as well as fuzzy gain-scheduling techniques. Secondly, various important parameter-varying system models are reviewed, for which gain-scheduled control and filtering issues are usually dealt with. In particular, in view of the randomly occurring phenomena with time-varying probability distributions, some results of our recent work based on the probability-dependent gain-scheduling methods are reviewed. Furthermore, some latest progress in this area is discussed. Finally, conclusions are drawn and several potential future research directions are outlined.

An online EEG-based brain–computer interface for controlling hand grasp using an adaptive probabilistic neural network

This paper presents a new online single-trial EEG-based brain–computer interface (BCI) for controlling hand holding and sequence of hand grasping and opening in an interactive virtual reality environment. The goal of this research is to develop an interaction technique that will allow the BCI to be effective in real-world scenarios for hand grasp control. One of the major challenges in the BCI research is the subject training. Currently, in most online BCI systems, the classifier was trained offline using the data obtained during the experiments without feedback, and used in the next sessions in which the subjects receive feedback. We investigated whether the subject could achieve satisfactory online performance without offline training while the subjects receive feedback from the beginning of the experiments during hand movement imagination. Another important issue in designing an online BCI system is the machine learning to classify the brain signal which is characterized by significant day-to-day and subject-to-subject variations and time-varying probability distributions. Due to these variabilities, we introduce the use of an adaptive probabilistic neural network (APNN) working in a time-varying environment for classification of EEG signals. The experimental evaluation on ten naïve subjects demonstrated that an average classification accuracy of 75.4% was obtained during the first experiment session (day) after about 3 min of online training without offline training, and 81.4% during the second session (day). The average rates during third and eighth sessions are 79.0% and 84.0%, respectively, using previously calculated classifier during the first sessions, without online training and without the need to calibrate. The results obtained from more than 5000 trials on ten subjects showed that the method could provide a robust performance over different experiment sessions and different subjects.

Optimal enumeration of state space of finitely buffered stochastic molecular networks and exact computation of steady state landscape probability.

BackgroundStochasticity plays important roles in many molecular networks when molecular concentrations are in the range of 0.1 μM to 10nM (about 100 to 10 copies in a cell). The chemical master equation provides a fundamental framework for studying these networks, and the time-varying landscape probability distribution over the full microstates, i.e., the combination of copy numbers of molecular species, provide a full characterization of the network dynamics. A complete characterization of the space of the microstates is a prerequisite for obtaining the full landscape probability distribution of a network. However, there are neither closed-form solutions nor algorithms fully describing all microstates for a given molecular network.ResultsWe have developed an algorithm that can exhaustively enumerate the microstates of a molecular network of small copy numbers under the condition that the net gain in newly synthesized molecules is smaller than a predefined limit. We also describe a simple method for computing the exact mean or steady state landscape probability distribution over microstates. We show how the full landscape probability for the gene networks of the self-regulating gene and the toggle-switch in the steady state can be fully characterized. We also give an example using the MAPK cascade network. Data and server will be available at URL: .ConclusionOur algorithm works for networks of small copy numbers buffered with a finite copy number of net molecules that can be synthesized, regardless of the reaction stoichiometry, and is optimal in both storage and time complexity. The algorithm can also be used to calculate the rates of all transitions between microstates from given reactions and reaction rates. The buffer size is limited by the available memory or disk storage. Our algorithm is applicable to a class of biological networks when the copy numbers of molecules are small and the network is closed, or the network is open but the net gain in newly synthesized molecules does not exceed a predefined buffer capacity. For these networks, our method allows full stochastic characterization of the mean landscape probability distribution, and the steady state when it exists.

Time-varying photon probability distribution of individual molecules at room temperature

The radiation field emitted from rhodamine molecules adsorbed on a glass surface is investigated at room temperature. Transitions between Poissonian and sub-Poissonian statistics are monitored by tracking the probability of detecting photon pairs after pulsed optical excitation. Fluctuations are analyzed by recording simultaneously the inter-photon time distribution, the emission maximum, and the fluorescence lifetime with a time-resolution in the ms–s time range. The presented technique enables us to determine unequivocally whether an observed chromophoric system behaves as a single quantum emitter at any time.

Analysis of a nonparametric blind equalizer for discrete-valued signals

An investigation is carried out to identify the variables that may affect the numerical properties of an inverse filtering method of blind equalization for linear channels with discrete input. The analysis is under a nonparametric framework in which all coefficients of the inverse filter (equalizer) can be freely chosen. It reveals, in particular, that the filter length plays two contradictory roles-increasing the length always helps improve the accuracy of inverse filtering, but when the filter is too long, the numerical properties of the method may deteriorate. Other influential variables include the constellation and (possibly time-varying) probability distribution of the input signals. The method is also shown to be highly efficient for nonparametric channel estimation, as was shown for estimating parametric channels. Simulations are carried out to verify the analytical findings concerning the numerical properties.

Minimizing the average cost of paging and registration: A timer-based method

Methods of balancing call registration and paging are developed in this paper. Given that the probability distribution on the user location as a function of time is either known or can be calculated, previous work shows the existence of lower bounds on the average cost of paging. Here these bounds are used in conjunction with a Poisson incoming-call arrival model to formulate the paging/registration optimization problem in terms of timeout parameters, $\tau _m$; the maximum amount of time to wait before registering given the last known location was $\tau _m$. Timer-based methods, as opposed to location-based methods, do not require the user to record and process location information during the time between location updates. This feature might be desirable for minimizing mobile transceiver use during idle periods. We then consider uniform motion processes where a spatial translation of starting location produces an identical spatial translation of the associated time-varying probability distribution. This leads to a universal timeout parameter $\tau $ which may be readily calculated. We study $\tau $ and the minimum cost of paging/registration for a simple model of user motion and compare our results to an earlier method of location-based paging/registration cost minimization.

Minimizing the average cost of paging under delay constraints

Efficient paging procedures help minimize the amount of bandwidth expended in locating a mobile unit. Given a probability distribution on user location, it is shown that the optimal paging strategy which minimizes the expected number of locations polled $E[L]$ is to query each location sequentially in order of decreasing probability. However, since sequential search over many locations may impose unacceptable polling delay, D, optimal paging subject to delay constraints is considered. It is shown that substantial reductions in E[L] can be had even after moderate constraints are imposed on acceptable D (i.e., D \leq 3). Since all methods of mobility management eventually reduce to considering a time-varying probability distribution on user location, this work should be applicable to a wide range of problems in the area. most notably those with additive cost structures.

Numerical methods for modeling computer networks under nonstationary conditions

Numerical techniques for modeling computer networks under nonstationary conditions are discussed, and two distinct approaches are presented. The first approach uses a queuing theory formulation to develop differential equation models which describe the behavior of the network by time-varying probability distributions. In the second approach, a nonlinear differential equation model is developed for representing the dynamics of the network in terms of time-varying mean quantities. This approach allows multiple classes of traffic to be modeled and establishes a framework for the use of optimal control techniques in the design of network control strategies. Numerical techniques for determining the queue behavior as a function of time for both approaches are discussed and their computational advantages are contrasted with simulation. >

Probability and stochastic modelling of water quality parameters in the Thames River

Baseline information on water quality parameters is essential to the development of policies to manage and control con taminants in streams. Investigations on the probability behaviour of monthly statistical estimates reveal that the governing probability distributions are not fixed but change on a monthly basis; and they are non-Gaussian. Probability density functions are developed using a five-parameter polynomial density function, which allows successful preservation of the key moments of the baseline water quality sequence. Stochastic models for transformed series of concentration of total phosphorus and suspended solids are determined using the ARIMA process. Since the ARIMA family of processes requires that the underlying distributions for monthly events be Gaussian and time-invariant, appropriate transformations are made for mapping the monthly water quality data from its parent distribution to a N(0,1) distribution. The model constructed for the transformed monthly phosphorus variable is an ARIMA [1, 1, 0], while the transformed suspended solids series of monthly time step is adequately described using a white noise ARIMA model. The methodology developed provides a framework for modelling baseline water quality data at various tributaries. The transformations guarantee that the probability distribution of the observed series is incorporated into the model structure. Therefore, synthetic water quality series generated using these models reproduce the non-Gaussian time-varying probability distributions and maintain the serial relationship between consecutive monthly events. Key words: ARIMA process, non-Gaussian probability distribution, water quality time series, suspended solids data, phosphorus concentration series.