Characterization Of Probability Distributions Research Articles

PaperObjective classificationof empirical probability distributionsand the issue of event detection

The paper concentrates on the issue of classifica-tion of empirical probability distributions (histograms), whichis useful both in management situations and in event detectionor event mining. While existing approaches to event detectionconcentrate on the use of selected moments or other character-istics of empirical probability distributions, we postulate thatfull empirical distribution preserves more of needed informa-tion then selected moments of this distribution, thus multiplecriteria classification of distributions can be most effective inevent detection.

Objective classification of empirical probability distributions and the issue of event detection

The paper concentrates on the issue of classification of empirical probability distributions (histograms), which is useful both in management situations and in event detection or event mining. While existing approaches to event detection concentrate on the use of selected moments or other characteristics of empirical probability distributions, we postulate that full empirical distribution preserves more of needed information then selected moments of this distribution, thus multiple criteria classification of distributions can be most effective in event detection.

Characterizations of Probability Distribution by Some Sequential Relative Reliability Measures: An Application of Completeness in a Hilbert Space

In this paper, by means of the concept of completeness in functional analysis, we characterize probability distributions using several relative measures of residual life of an item at independent sequences of random times. These sequences are the order statistics and the record statistics of a random sequence of times. The reliability quantities of the mean residual life, the vitality function, and the survival function are considered to construct the main characteristics of distributions.

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

AbstractUncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling‐based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time‐consuming if the simulation‐based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. An algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail.This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Statistical and Graphical Methods of Data Analysis > Sampling

Uncertainty-Aware Principal Component Analysis.

We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertainty-aware PCA. In addition to the accuracy and performance gained by our approach over sampling-based strategies, our formulation allows us to perform sensitivity analysis with regard to the uncertainty in the data. For this, we propose factor traces as a novel visualization that enables to better understand the influence of uncertainty on the chosen principal components. We provide multiple examples of our technique using real-world datasets. As a special case, we show how to propagate multivariate normal distributions through PCA in closed form. Furthermore, we discuss extensions and limitations of our approach.

Halfspace depth does not characterize probability distributions

We give examples of different multivariate probability distributions whose halfspace depths coincide at all points of the sample space.

Non-parametric estimation of reciprocal coordinate subtangent for right censored dependent scheme

The concept of reciprocal coordinate subtangent (RCST) has been used as a useful tool to study the monotone behavior of a continuous density function and for characterizing probability distributions. In this paper, we propose a non-parametric estimator for RCST based on the censored dependent data. Asymptotic properties of the estimator are established under suitable regularity conditions. A simulation study is carried out to examine the performance of the estimator. The usefulness of the estimator is also examined through a real data.

Option-implied objective measures of market risk

Foster and Hart (2009) introduce an objective measure of the riskiness of an asset that implies a bound on how much of one’s wealth is ‘safe’ to invest in the asset while (a.s.) guaranteeing no-bankruptcy in the long run. In this study, we translate the Foster-Hart measure from static and abstract gambles to dynamic and applied finance using nonparametric estimation of risk-neutral densities from S&P 500 call and put option prices covering 2003 to 2013. This exercise results in an option-implied market view of objective riskiness. The dynamics of the resulting ‘option-implied Foster-Hart bound’ are analyzed and assessed in light of well-known risk measures including value at risk, expected shortfall and risk-neutral volatility. The new measure is shown to be a significant predictor of ahead-return downturns. Furthermore, it is able to grasp more characteristics of the risk-neutral probability distributions than other measures, furthermore exhibiting predictive consistency. The robustness of the risk-neutral density estimation method is analyzed via a bootstrap.

Moscow smoke haze in October 2014: Variations in gaseous air pollutants

The composition of the Moscow atmosphere during the joint smoke screening by near fires in the Moscow region and far forest fires in October 2014 is studied. According to ecological monitoring data, the ratio of mass concentrations of nitrogen oxides and carbon monoxide in “fresh” smokes of near fires reached 0.35; in “old” smokes of far fires, decreased to 0.025. The ratio of the nitrogen oxide concentration to the sum of oxide and dioxide concentrations reached 0.87 in “fresh” smokes” and decreased to zero in “old” ones. It has been shown that statistical characteristics of variations and empirical probability distributions of carbon monoxide and nitrogen oxide concentrations in the smoky Moscow atmosphere in October 2014 radically differ from the corresponding characteristics and empirical distributions in a smokeless atmosphere and during the smoke events in 2002 and 2010. The empirical concentration distribution function for carbon monoxide can be approximated by piecewise exponential functions; for nitrogen oxide, by piecewise power functions. During the smoke event in Moscow in October 2014, a close correlation between the concentration of carbon monoxide and total concentration of nitrogen oxides was observed in a wide range of time scales, which testifies to the joint formation of the abovementioned pollutants during forest fires. The coefficient of the spectral correlation between ozone and nitrogen dioxide concentrations on different time scales varied from +1 to–1, which indicates the complexity of relationships between ozone and nitrogen oxides in a smoky atmosphere.

On the quantification and efficient propagation of imprecise probabilities resulting from small datasets

Abstract This paper addresses the problem of uncertainty quantification and propagation when data for characterizing probability distributions are scarce. We propose a methodology wherein the full uncertainty associated with probability model form and parameter estimation are retained and efficiently propagated. This is achieved by applying the information-theoretic multimodel inference method to identify plausible candidate probability densities and associated probabilities that each method is the best model in the Kullback-Leibler sense. The joint parameter densities for each plausible model are then estimated using Bayes’ rule. We then propagate this full set of probability models by estimating an optimal importance sampling density that is representative of all plausible models, propagating this density, and reweighting the samples according to each of the candidate probability models. This is in contrast with conventional methods that try to identify a single probability model that encapsulates the full uncertainty caused by lack of data and consequently underestimate uncertainty. The result is a complete probabilistic description of both aleatory and epistemic uncertainty achieved with several orders of magnitude reduction in computational cost. It is shown how the model can be updated to adaptively accommodate added data and added candidate probability models. The method is applied for uncertainty analysis of plate buckling strength where it is demonstrated how dataset size affects the confidence (or lack thereof) we can place in statistical estimates of response when data are lacking.

PRODIGEN: visualizing the probability landscape of stochastic gene regulatory networks in state and time space

BackgroundVisualizing the complex probability landscape of stochastic gene regulatory networks can further biologists’ understanding of phenotypic behavior associated with specific genes.ResultsWe present PRODIGEN (PRObability DIstribution of GEne Networks), a web-based visual analysis tool for the systematic exploration of probability distributions over simulation time and state space in such networks. PRODIGEN was designed in collaboration with bioinformaticians who research stochastic gene networks. The analysis tool combines in a novel way existing, expanded, and new visual encodings to capture the time-varying characteristics of probability distributions: spaghetti plots over one dimensional projection, heatmaps of distributions over 2D projections, enhanced with overlaid time curves to display temporal changes, and novel individual glyphs of state information corresponding to particular peaks.ConclusionsWe demonstrate the effectiveness of the tool through two case studies on the computed probabilistic landscape of a gene regulatory network and of a toggle-switch network. Domain expert feedback indicates that our visual approach can help biologists: 1) visualize probabilities of stable states, 2) explore the temporal probability distributions, and 3) discover small peaks in the probability landscape that have potential relation to specific diseases.

Multi-objective optimization of multi-purpose multi-reservoir systems under high reliability constraints

The supply of municipal water in Germany is often governed by policies that warrant occurrence-based reliabilities of 99.5 % or more. Such reliabilities have to be considered in the optimization of reservoir operation. A reliability-based optimization requires sufficiently long simulation periods of several thousand years of time series of inflow and demand. However, long simulation periods lead, especially for multi-objective parameterization–simulation–optimization (MOPSO), to unacceptable computational burden. Therefore, techniques need to be developed that increase the computational efficiency of MOPSO for such optimization problems. In this paper, a novel Monte Carlo recombination method (MCR) is proposed. MCR reduces the length of inflow time series significantly while preserving critical statistics, i.e., characteristics of probability distributions and variability of wet and dry conditions. It could be shown that simulations based on these shortened time series allow for highly efficient MOPSO and yield comparable Pareto-fronts and reliabilities. For the demonstration of the capabilities of MCR, it is integrated into a MOPSO framework for the optimization of a multi-purpose multi-reservoir system in the Eastern Ore Mountains, Germany. For this real-world application, synthetic time series of a length of 10,000 years are generated and reduced to 882 years, which results in a reduction of the computational burden by a factor of eleven. A validation of the results shows that the MOPSO framework allows for optimization of operational policies that yield reliabilities over 99.95 % on a monthly scale and up to 99.7 % on an annual timescale.

On Characterizing Probability Distributions by Conditional Expectation of Two Order Statistics

On Characterizing Probability Distributions by Conditional Expectation of Two Order Statistics

How certain are we about the certainty-equivalent long term social discount rate?

Theoretical arguments for using a term structure of social discount rates (SDR) that declines with the time horizon have influenced government guidelines in the US and Europe. The certainty equivalent discount rate that often underpins this guidance embodies uncertainty in the primitives of the SDR, such as growth. For distant time horizons the probability distributions of these primitives are ambiguous and the certainty equivalent itself is uncertain. Yet, if a limited set of characteristics of the unknown probability distributions can be agreed upon, ‘sharp’ upper and lower bounds can be defined for the certainty-equivalent SDR. Unfortunately, even with considerable agreement on these features, these bounds are widely spread for horizons beyond 75 years. So while estimates of the present value of intergenerational impacts, including the social cost of carbon, can be bounded in the presence of this ambiguity, they typically remain so imprecise as to provide little practical guidance.

Properties of the elasticity of a continuous random variable. A special look at its behavior and speed of change

ABSTRACTBelzunce et al. (1995) define the elasticity for non negative random variables as the reversed proportional failure rate (RPFR). Veres-Ferrer and Pavía (2012, 2014b) interpret it in economic terms, extending its definition to variables that can also take negative values, and briefly present the role of elasticity in characterizing probability distributions. This paper highlights a set of properties demonstrated by elasticity, which shows many similar properties to the reverse hazard function. This paper pays particular attention to studying the increase/decrease and the speed of change of the elasticity function. These are important properties because of the characterizing role of elasticity, which makes it possible to introduce our hypotheses and knowledge about the random process in a more meaningful and intuitive way. As a general rule, it is observed the need for distinguishing between positive and negative areas of the support.

Uncertainty and Sensitivity Analysis of Cost Assumptions for Global Longterm Poliovirus Risk Management

The world will stop the use of oral poliovirus vaccines (OPV) after certification of global wild poliovirus eradication, but investments must continue in long-term poliovirus risk management, including some level of use of the inactivated poliovirus vaccine (IPV). The health economic justification of these activities depends on their assumed costs and savings associated with prevented polio cases. We characterize probability distributions for economic inputs of an existing global model of long-term poliovirus risk management. Using a fixed set of 120 realizations of the stochastic model, we estimate the corresponding expected incremental net benefits (INBs) of OPV cessation compared to continued OPV use over a 40-year time period for a large sample of the uncertain cost inputs. We also explore the impact of some specific assumptions about future IPV costs. Cost-related uncertainty substantially influences the INBs for OPV cessation compared to continued OPV use, although nearly all simulations resulted in positive expected global INBs. IPV cost, OPV administration costs, and average treatment costs emerged as the most influential uncertainties. A potential drop in the IPV cost starting in the 2020’s may result in an expected economic benefit of $1.5-4.5 billion mainly due cost savings in higher-income countries, depending on the timing and magnitude of the cost decrease and on whether they apply to combination vaccine products. Cost-related uncertainties may lead to significantly higher or lower expected long-term benefits of polio eradication and OPV cessation, and efforts to further reduce costs, particularly associated with IPV vaccine cost and delivery, will likely yield significant benefits.

Option-Implied Objective Measures of Market Risk

Foster and Hart (2009) introduce an objective measure of the riskiness of an asset that implies a bound on how much of one’s wealth is ‘safe’ to invest in the asset while (a.s.) guaranteeing no-bankruptcy. In this study, we translate the Foster-Hart measure from static and abstract gambles to dynamic and applied finance using nonparametric estimation of risk-neutral densities from S&P 500 call and put option prices covering 2003 to 2013. The dynamics of the resulting ‘option-implied Foster-Hart bound’ are assessed in light of other well-known option-implied risk measures including value at risk, expected shortfall and risk-neutral volatility, as well as high moments of the densities and several industry measures. Rigorous variable selection reveals that the new measure is a significant predictor of (large) ahead-return downturns. Furthermore, it grasps more characteristics of the risk-neutral probability distributions in terms of moments than other measures and exhibits predictive consistency. The robustness of the risk-neutral density estimation is analyzed via Monte Carlo methods.

Pressure drop and pressure fluctuations in spouted beds with binary mixtures of particles

When handling fine particles in spouted beds, addition of coarse particles has shown improved spouting stability than single particle systems. However, segregation may still occur because of insufficient mixing in binary mixtures, which will adversely influence the process performance. Therefore, in this study, analysis of pressure drop and its fluctuation signals were for the first time used to understand mechanisms of flow regime transitions in spouted beds with binary mixtures. The results showed that the typical varying sequence of pressure drop can be observed for spouted bed with binary mixtures and the peak pressure drop is related to the mixing degree of particles, which is mainly influenced by the inter-particle forces between fine particles and their counterpart coarse ones. The statistic characteristics of pressure drop time series, i.e., average value, standard deviation and probability distributions, were found to vary for different flow regimes. Therefore, they could be used for the characterization of these flow regimes. The spouting stability of binary mixtures can be reflected by power spectrum analysis; the influences of particles size and density difference on spouting stability were discussed through power spectral analysis.

A Discrete Analogue of Reciprocal Coordinate Subtangent and its Role in Characterization Problems

Abtsrcat The concept of reciprocal coordinate subtangent (RCST) has been used in statistical literature as a useful tool to study the monotone behaviours of a continuous density function and for characterizing probability distributions through its functional forms. Motivated by this, in the present paper, we consider a discrete analogue of RCST introduced by Gupta et al. (1997) and study its usefulness in modeling probability mass functions. Characterization results are proved for models viz. geometric, discrete Burr and modified power series. The concept is also extended to the weighted models and proves some useful results as models such as logarithmic series, residual lifetime and partial sum (renewal) distributions. Finally, a new definition for discrete analogue of RCST is introduced and examined its usefulness in modeling proportional hazard models in discrete time.

On Some Properties of a Heterogeneous Transfer Function Involving Symmetric Saturated Linear (SATLINS) with Hyperbolic Tangent (TANH) Transfer Functions

For transfer functions to map the input layer of the statistical neural network model to the output layer perfectly, they must lie within bounds that characterize probability distributions. The heterogeneous transfer function, SATLINS_TANH, is established as a Probability Distribution Function (p.d.f), and its mean and variance are shown.