Showing papers on "Cumulative distribution function published in 2004"
Book•
26 Mar 2004
TL;DR: This chapter discusses Random Variables, Linear Models and Linear Regression, and Statistical Inference, Parameter Estimation, and Model Verification, as well as some important Discrete Distributions.
Abstract: Preface1 IntroductionPart A: Probability and Random Variables2 Basic Probability Concepts3 Random Variables and Probability Distributions4 Expectations And Moments5 Functions of Random Variables6 Some Important Discrete Distributions7 Some Important Continuous DistributionsPart B: Statistical Inference, Parameter Estimation, and Model Verification8 Observed Data and Graphical Representation9 Parameter Estimation10 Model Verification11 Linear Models and Linear RegressionAppendix A: TablesAppendix B: Computer SoftwareAppendix C: Answers to Selected ProblemsSubject Index
370 citations
TL;DR: A novel approach is developed to derive the cumulative distribution functions (cdfs) of the selection-combining (SC) output signal-to-noise ratio (SNR) in equally correlated Rayleigh, Ricean, and Nakagami-m fading channels and shows that a set of equally correlated channel gains can be transformed into aSet of conditionally independent channel gains.
Abstract: We develop a novel approach to derive the cumulative distribution functions (cdfs) of the selection-combining (SC) output signal-to-noise ratio (SNR) in equally correlated Rayleigh, Ricean, and Nakagami-m fading channels. We show that a set of equally correlated channel gains can be transformed into a set of conditionally independent channel gains. Single-fold integral expressions are, therefore, derived for the cdfs of the SC output SNR. Infinite series representations of the output cdfs are also provided. New expressions are applied to analyze the average error rate, the outage probability, and the output statistics of SC. Numerical and simulation results that illustrate the effect of fading correlation on the performance of L-branch SC in equally correlated fading channels are provided.
271 citations
TL;DR: In this article, the acceleration component probability distribution function at Rλ =690 to probabilities of less than 10−7 was presented, which is an improvement of more than an order of magnitude over past measurements and allows us to conclude that the fourth moment converges and the flatness is approximately 55.
251 citations
TL;DR: In this paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability, and the PDEE is further reduced to a one-dimensional partial differential equation.
Abstract: Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.
231 citations
TL;DR: In this article, a rational method is proposed that defines the capacity of a building class by relating its deformation potential to its fundamental period of vibration at different limit states and comparing this with a displacement response spectrum.
Abstract: Earthquake loss estimation studies require predictions to be made of the propor- tion of a building class falling within discrete damage bands from a specified earthquake demand. These predictions should be made using methods that incorporate both computa- tional efficiency and accuracy such that studies on regional or national levels can be effec- tively carried out, even when the triggering of multiple earthquake scenarios, as opposed to the use of probabilistic hazard maps and uniform hazard spectra, is employed to real- istically assess seismic demand and its consequences on the built environment. Earthquake actions should be represented by a parameter that shows good correlation to damage and that accounts for the relationship between the frequency content of the ground motion and the fundamental period of the building; hence recent proposals to use displacement response spectra. A rational method is proposed herein that defines the capacity of a building class by relating its deformation potential to its fundamental period of vibration at different limit states and comparing this with a displacement response spectrum. The uncertainty in the geometrical, material and limit state properties of a building class is considered and the first- order reliability method, FORM, is used to produce an approximate joint probability density function (JPDF) of displacement capacity and period. The JPDF of capacity may be used in conjunction with the lognormal cumulative distribution function of demand in the classi- cal reliability formula to calculate the probability of failing a given limit state. Vulnerability curves may be produced which, although not directly used in the methodology, serve to illus- trate the conceptual soundness of the method and make comparisons with other methods.
215 citations
TL;DR: A method for determining the probability associated with any fence or observation is proposed based on the cumulative distribution function of the order statistics, which allows the statistician to easily assess the degree to which an observation is dissimilar to the majority of the observations.
201 citations
TL;DR: A variety of complex arithmetic problems can be solved using a single—and fairly simple—approach based on probability bounds analysis, which allows for dependencies other than independence, completely unknown dependence, and model uncertainty more generally.
183 citations
TL;DR: In this article, the first-order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis, which reduces the chance of an increase in the nonlinearity of the limit-state function.
Abstract: In the approximation methods of reliability analysis, nonnormal random variables are transformed into equivalent standard normal random variables. This transformation tends to increase the nonlinearity of a limit-state function and, hence, results in less accurate reliability approximation. The first-order saddlepoint approximation for reliability analysis is proposed to improve the accuracy of reliability analysis. By the approximation of a limit-state function at the most likelihood point in the original random space and employment of the accurate saddlepoint approximation, the proposed method reduces the chance of an increase in the nonlinearity of the limit-state function. This approach generates more accurate reliability approximation than the first-order reliability method without an increase in the computational effort. The effectiveness of the proposed method is demonstrated with two examples and is compared with the first- and second-order reliability methods.
172 citations
TL;DR: The basic theoretical and numerical properties of clouds are discussed, and they are related to histograms, cumulative distribution functions, and likelihood ratios, and to consistent possibility and necessity measures of Jamison and Lodwick.
Abstract: Clouds are a concept for uncertainty mediating between the concept of a fuzzy set and that of a probability distribution. A cloud is to a random variable more or less what an interval is to a number. We discuss the basic theoretical and numerical properties of clouds, and relate them to histograms, cumulative distribution functions, and likelihood ratios. We show how to compute nonlinear transformations of clouds, using global optimization and constraint satisfaction techniques. We also show how to compute rigorous enclosures for the expectation of arbitrary functions of random variables, and for probabilities of arbitrary statements involving random variables, even for problems involving more than a few variables. Finally, we relate clouds to concepts from fuzzy set theory, in particular to the consistent possibility and necessity measures of Jamison and Lodwick.
146 citations
TL;DR: Quantile and conditional quantile statistical thinking, as I have innovated it in my research since 1976, is outlined in this comprehensive survey and introductory course in quantile data analysis.
Abstract: Quantile and conditional quantile statistical thinking, as I have innovated it in my research since 1976, is outlined in this comprehensive survey and introductory course in quantile data analysis. We propose that a unification of the theory and practice of statistical methods of data modeling may be possible by a quantile perspective. Our broad range of topics of univariate and bivariate probability and statistics are best summarized by the key words. Two fascinating practical examples are given that involve positive mean and negative median investment returns, and the relationship between radon concentration and cancer.
132 citations
07 Jun 2004
TL;DR: An efficient statistical timing analysis algorithm that can handle arbitrary (spatial and structural) causes of delay correlation is described and derives the entire cumulative distribution function of the circuit delay using a new mathematical formulation.
Abstract: An efficient statistical timing analysis algorithm that can handle arbitrary (spatial and structural) causes of delay correlation is described. The algorithm derives the entire cumulative distribution function of the circuit delay using a new mathematical formulation. Spatial as well as structural correlations between gate and wire delays can be taken into account. The algorithm can handle node delays described by non-Gaussian distributions. Because the analytical computation of an exact cumulative distribution function for a probabilistic graph with arbitrary distributions is infeasible, we find tight upper and lower bounds on the true cumulative distribution. An efficient algorithm to compute the bounds is based on a PERT-like single traversal of the sub-graph containing the set of N deterministically longest paths. The efficiency and accuracy of the algorithm is demonstrated on a set of ISCAS'85 benchmarks. Across all the benchmarks, the average rms error between the exact distribution and lower bound is 0.7%, and the average maximum error at 95th percentile is 0.6%. The computation of bounds for the largest benchmark takes 39 seconds.
TL;DR: This work studies the performance of a dual SC receiver over correlated Weibull fading channels with arbitrary parameters and derives novel closed-form analytical expressions for the probability density function, the cumulative distribution function, and the moments of the output signal-to-noise ratio (SNR).
Abstract: Ascertaining the importance of the dual selection combining (SC) receivers and the suitability of the Weibull model to describe mobile fading channels, we study the performance of a dual SC receiver over correlated Weibull fading channels with arbitrary parameters. Exact closed-form expressions are derived for the probability density function, the cumulative distribution function, and the moments of the output signal-to-noise ratio (SNR). Important performance criteria, such as average output SNR, amount of fading, outage probability, and average bit-error probability for several modulation schemes are studied. Furthermore, for these performance criteria, novel closed-form analytical expressions are derived. The proposed analysis is complemented by various performance evaluation results, including the effects of the input SNR's unbalancing, fading severity, and fading correlation on the overall system's performance. Computer simulation results have verified the validity and accuracy of the proposed analysis.
TL;DR: Methods from order statistics are applied to the problem of satisfying regulations that specify individual criteria to be met by each of a number of outputs from a computer code simulating nuclear accidents to obtain expressions for the confidence, β, or probability that these desired extents will be covered in N runs of the code.
TL;DR: This paper compares four methods for the reliable propagation of uncertainty through calculations involving the binary operations of addition, multiplication, subtraction and division and shows that they converge to equivalent methods when they are restricted to cumulative distribution functions on the positive reals.
TL;DR: An algorithm is described to efficiently compute the cumulative distribution and probability density functions of the diffusion process with trial-to-trial variability in mean drift rate, starting point, and residual reaction time with closed-form solutions.
Abstract: An algorithm is described to efficiently compute the cumulative distribution and probability density functions of the diffusion process (Ratcliff, 1978) with trial-to-trial variability in mean drift rate, starting point, and residual reaction time. Some, but not all, of the integrals appearing in the model’s equations have closed-form solutions, and thus we can avoid computationally expensive numerical approximations. Depending on the number of quadrature nodes used for the remaining numerical integrations, the final algorithm is at least 10 times faster than a classical algorithm using only numerical integration, and the accuracy is slightly higher. Next, we discuss some special cases with an alternative distribution for the residual reaction time or with fewer than three parameters exhibiting trialto-trial variability.
TL;DR: This paper proposes an Iterative Rescaling Method (IRM) for constructing a random set with corresponding belief and plausibility measures that are a close outer approximation to the lower and upper probabilities.
TL;DR: In this paper, the authors considered the stationary and the ordinary discrete renewal risk models and derived the defective probability function of the claim causing ruin in both the discount free case and under the compound binomial model.
Abstract: This paper considers the stationary and the ordinary discrete renewal risk models. The main result is an expression of the Gerber–Shiu discounted penalty function in the stationary model in terms of the corresponding Gerber–Shiu function in the ordinary model. Subsequently, this relationship is considered in more detail in both the discount free case and under the compound binomial model. The latter case may be viewed as a discrete analog of the classical Poisson model. Simplifications of the general relationship are obtained, and a connection between the defective joint cumulative distribution functions of the surplus prior to ruin and the deficit at ruin in the stationary and the ordinary renewal risk models is established. Moreover, the defective probability function of the claim causing ruin is derived in the compound binomial case.
TL;DR: An analytical expression for the cumulative distribution function of travel time for a vehicle traversing a freeway link of arbitrary length is derived and a numerical inversion algorithm is presented to invert the transforms.
Abstract: We derive an analytical expression for the cumulative distribution function of travel time for a vehicle traversing a freeway link of arbitrary length. The vehicle's speed is assumed to be modulated by a random environment that can be modeled as a stochastic process. We first present a partial differential equation (PDE) describing the travel time distribution and obtain a solution in terms of Laplace transforms. Next, we present a numerical inversion algorithm to invert the transforms. The technique is demonstrated on two example problems. Numerical results indicate great promise for this approach to the link travel-time problem.
TL;DR: In this paper, the wind energy potential of Elazig is statistically analyzed based on hourly measured wind speed data over the five-year period from 1998 to 2002, and two probability density functions are fitted to the measured probability distribution on a yearly basis.
Abstract: In this study, the wind energy potential of Elazig is statistically analyzed based on hourly measured wind speed data over the five-year period from 1998 to 2002. The probability density distributions are derived from cumulative distribution functions. Two probability density functions are fitted to the measured probability distribution on a yearly basis. The wind energy potential of the location is studied based on the Weibull and Rayleigh distributions. It was found that the numerical values of both Weibull parameters (k and c) for Elazig vary over a wide range. The yearly values of k range from 1.653 to 1.878 with an average value of 1.819, while those of c are in the range of 2.757–2.994 m/s with an average value of 2.824 m/s. In addition, yearly mean wind speed and mean power density of Elazig is found as 2.79 m/s and 38.76 W/m2, respectively. The wind speed distributions are represented by Weibull distribution and also by Rayleigh distribution, with a special case of the Weibull distributio...
TL;DR: In this letter, an iterative algorithm is provided to compute the exact confidence interval from the cumulative distribution function, using the iterativegorithm and cubic spline interpolation.
Abstract: The magnitude-squared coherence function is widely used in many applications. The approximate confidence interval is only reliable for large data segments. In this letter, an iterative algorithm is provided to compute the exact confidence interval from the cumulative distribution function. In order to use the confidence interval conveniently in practice, some libraries are provided, using the iterative algorithm and cubic spline interpolation.
TL;DR: In this paper, the authors proposed a new method for computing the probability distribution of reliability indices, where the random sums introduced by the randomness of the number of fault occurrences in the time interval of analysis are handled by using a characteristic functions-based approach.
Abstract: In reliability analysis of distribution systems, random events like the occurrence of a fault or the time to restore the service after a fault are represented by using random variables (RVs), so that the reliability indices built on the basis of these RVs also become RVs. Existing techniques for the evaluation of the probability distributions of reliability indices are typically based on Monte Carlo and analytical simulations. This paper presents a new method for computing the probability distribution of reliability indices. The random sums introduced by the randomness of the number of fault occurrences in the time interval of analysis are handled by using a characteristic functions-based approach. The direct convolution of the probability density functions is avoided by resorting to the properties of the compound Poisson process. In addition, the direct and inverse discrete Fourier transforms are used to allow for handling any type of probability distribution. The proposed method is an effective alternative to the existing methods, providing a fast and simple computation of probability distributions and moments for local and global reliability indices. Results obtained for large real urban distribution systems are presented.
TL;DR: In this article, the authors used a conceptual approach to fit the model parameters (friction coefficients and the volume of snow involved in the avalanches) to the field data and adjusted appropriate statistical distributions.
Abstract: Investigating snow avalanches using a purely statistical approach raises several issues. First, even in the heavily populated areas of the Alps, there are few data on avalanche motion or extension. Second, most of the field data are related to the point of furthest reach in the avalanche path (run-out distance or altitude). As data of this kind are tightly dependent on the avalanche path profile, it is a priori not permissible to extrapolate the cumulative distribution function fitted to these data without severe restrictions or further assumptions. Using deterministic models is also problematic, as these are not really physically based models. For instance, they do not include all the phenomena occurring in the avalanche movement, and the rheological behaviour of the snow is not known. Consequently, it is not easy to predetermine extreme-event extensions. Here, in order to overcome this problem, we propose to use a conceptual approach. First, using an avalanche-dynamics numerical model, we fitted the model parameters (friction coefficients and the volume of snow involved in the avalanches) to the field data. Then, using these parameters as random variables, we adjusted appropriate statistical distributions. The last steps involved simulating a large number of (fictitious) avalanches using the Monte Carlo approach. Thus, the cumulative distribution function of the run-out distance can be computed over a much broader range than was initially possible with the historical data. In this paper, we develop the proposed method through a complete case study, comparing two different models.
IBM1
TL;DR: A non-linear feature space transformation which forces the individual dimensions of the acoustic data for every speaker to be Gaussian distributed and achieves minimum divergence between the density function of the transformed adaptation data and the normal density with zero mean and unit variance is proposed.
Abstract: We propose a non-linear feature space transformation for speaker/environment adaptation which forces the individual dimensions of the acoustic data for every speaker to be Gaussian distributed. The transformation is given by the preimage under the Gaussian cumulative distribution function (CDF) of the empirical CDF on a per dimension basis. We show that, for a given dimension, this transformation achieves minimum divergence between the density function of the transformed adaptation data and the normal density with zero mean and unit variance. Experimental results on both small and large vocabulary tasks show consistent improvements over the application of linear adaptation transforms only.
TL;DR: The cumulative incidence estimators obtained are considerably more efficient than the usual nonparametric estimator, particularly with regard to interpolation of cumulative incidence at early or intermediate time points within the range of data used to fit the function.
Abstract: In analyses of time-to-failure data with competing risks, cumulative incidence functions may be used to estimate the time-dependent cumulative probability of failure due to specific causes. These functions are commonly estimated using nonparametric methods, but in cases where events due to the cause of primary interest are infrequent relative to other modes of failure, nonparametric methods may result in rather imprecise estimates for the corresponding subdistribution. In such cases, it may be possible to model the cause-specific hazard of primary interest parametrically, while accounting for the other modes of failure using nonparametric estimators. The cumulative incidence estimators so obtained are simple to compute and are considerably more efficient than the usual nonparametric estimator, particularly with regard to interpolation of cumulative incidence at early or intermediate time points within the range of data used to fit the function. More surprisingly, they are often nearly as efficient as fully parametric estimators. We illustrate the utility of this approach in the analysis of patients treated for early stage breast cancer.
Patent•
06 Apr 2004TL;DR: In this article, a probability density function is derived based on the number of tuples in the sample that satisfy the query expression, and the cumulative distribution for the probability density functions is solved for the given threshold to determine a selectivity estimate at the given confidence value.
Abstract: Selectivity estimates are produced that meet a desired confidence threshold. To determine the confidence level of a given selectivity estimate for a query expression, the query expression is evaluated on a sample tuples. A probability density function is derived based on the number of tuples in the sample that satisfy the query expression. The cumulative distribution for the probability density function is solved for the given threshold to determine a selectivity estimate at the given confidence value.
01 Jan 2004
TL;DR: This paper develops conservative bounding methods both for space-based augmentation systems (SBAS) and for ground-based augmentation systems (GBAS) by using excess-mass functions, of which the first focuses on probability density functions (EMP overbounding) and the second on cumulative distribution functions (EMC overbounded).
Abstract: Safety-of-life GNSS augmentation systems must provide bounds on the probability with which hazardous navigation errors occur. This paper develops conservative bounding methods both for space-based augmentation systems (SBAS) and for ground-based augmentation systems (GBAS) by using excess-mass functions. The excess-mass concept, which employs conservative bounding functions with integrated density greater than unity, is applied to develop two new bounding strategies, of which the first focuses on probability density functions (EMP overbounding) and the second on cumulative distribution functions (EMC overbounding). These strategies can bound arbitrary error distributions, even those that are asymmetric, multimodal, or non-zero mean. To compare the two strategies to each other, and to existing methods such as paired-CDF overbounding and moment overbounding, a set of metrics are introduced to evaluate overbound performance given anomalies in the actual error distribution. These performance metrics provide a basis for application-specific trade studies that would balance the availability benefits of various overbounding methods against required modifications to the broadcast signal and protection limits. In the generic case, assuming identical error sources for all satellites and neglecting broadcast-message bandwidth constraints, the performance metrics favour the EMC approach, which tightly bounds unknown biases and heavy-tailed errors. The major drawback of the EMC approach is its sensitivity to outliers in sampled error distributions.
TL;DR: It is shown how information about the correlation between values of given random variables can lead to better envelopes around the cumulative distribution of a function of their values.
Abstract: Given two random variables whose dependency relationship is unknown, if a new random variable is defined whose samples are some function of samples of the given random variables, the distribution of this function is not fully determined. However, envelopes can be computed that bound the space through which its cumulative distribution function must pass. If those envelopes could be made to bound a smaller space, the cumulative distribution, while still not fully determined, would at least be more constrained. We show how information about the correlation between values of given random variables can lead to better envelopes around the cumulative distribution of a function of their values.
TL;DR: In this article, a method for predicting rain attenuation in terrestrial links is developed using data from measurements carried out in tropical regions, together with a larger data set from temperate climates, previously available.
Abstract: A method for prediction of rain attenuation in terrestrial links is developed using data from measurements carried out in tropical regions, together with a larger data set from temperate climates, previously available. The proposed method uses the complete rainfall rate cumulative probability distribution as input data and shows significant improvement in prediction accuracy.
TL;DR: The present study underlines the need to perform uncertainty analyses instead of either using a set of simple rules or just looking at certain parameters on the basis of the uncertainty of this regional model of EUSES.
Abstract: The European Union System for the Evaluation of Substances (EUSES) is a computerized model system to facilitate and harmonize health and environmental risk assessment of previously notified and new substances. For calculation of regional background exposure, a multimedia distribution model is used. In the present study, the uncertainty of this regional model is analyzed. Environmental parameters were collected for North Rhine Westphalia (Germany), which resembles the standard region of EUSES. Probability distribution functions of various types (uniform, triangular, normal, log normal) depending on data availability were derived for environmental input parameters, including geometric parameters. Generic log-normal distribution functions with fixed standard deviations were chosen for solubility in air, water, and n-octanol as well as for degradation half-lives. Monte Carlo simulations were carried out for 10 reference substances having different properties. Contribution of environmental parameter uncertainty to total output uncertainties is higher than that of substance parameters. Range of output uncertainty, defined as the ratio of the logarithms of the 90th and 10th percentiles of the cumulative probability distribution function, shows an increase from air and water to soil. The highest-occurring range is 1.4 orders of magnitude, which means that total uncertainty of the regional model is relatively low and, usually, is lower than the range of measured values. The median of output probability distributions lies above the point estimate. Influence of input parameters was estimated as their rank correlation coefficients to output uncertainty. Substance and environmental parameters contribute differently to output variance depending on individual substance properties and environmental compartment. Hence, the present study underlines the need to perform uncertainty analyses instead of either using a set of simple rules or just looking at certain parameters.
TL;DR: This work considers the problem of approximating the moment generating function (MGF) of a truncated random variable in terms of the MGF of the underlying (i.e., untruncated) random variable, and derives two types of representation which are referred to as an exponential convolution representation and an exponential tilting representation.
Abstract: We consider the problem of approximating the moment generating function (MGF) of a truncated random variable in terms of the MGF of the underlying (i.e., untruncated) random variable. The purpose of approximating the MGF is to enable the application of saddlepoint approximations to certain distributions determined by truncated random variables. Two important statistical applications are the following: the approximation of certain multivariate cumulative distribution functions; and the approximation of passage time distributions in ion channel models which incorporate time interval omission. We derive two types of representation for the MGF of a truncated random variable. One of these representations is obtained by exponential tilting. The second type of representation, which has two versions, is referred to as an exponential convolution representation. Each representation motivates a different approximation. It turns out that each of the three approximations is extremely accurate in those cases “to which it is suited.” Moreover, there is a simple rule of thumb for deciding which approximation to use in a given case, and if this rule is followed, then our numerical and theoretical results indicate that the resulting approximation will be extremely accurate.