Residential House and Zone Volumes in the United States: Empirical and Estimated Parametric Distributions

This research reports empirical distributions and estimated univariate parametric probability distributions for house volume and certain zone volumes within households for residential structures in the United States. The author derived the distributions from two separate databases. The volumes were found to be exceptionally well fit by lognormal distributions (adjusted R² >> 0.95) in almost all cases. In addition, data from one database indicates that the correlation between house volume and air changes per hour is very weak.     

Estimated Distributions for Average Daily Consumption of Total and Self-Caught Fish for Adults in Michigan Angler Households

Fish consumption rates play a critical role in the assessment of human health risks posed by the consumption of fish from chemically contaminated water bodies. Based on data from the 1989 Michigan Sport Anglers Fish Consumption Survey, we examined total fish consumption, consumption of self-caught fish, and consumption of Great Lakes fish for all adults, men, women, and certain higher risk subgroups such as anglers. We present average daily consumption rates as compound probability distributions consisting of a Bernoulli trial (to distinguish those who ate fish from those who did not) combined with a distribution (both empirical and parametric) for those who ate fish. We found that the average daily consumption rates for adults who ate fish are reasonably well fit by lognormal distributions. The compound distributions may be used as input variables for Monte Carlo simulations in public health risk assessments.     

Quantification of Variability and Uncertainty for Censored Data Sets and Application to Air Toxic Emission Factors

Many environmental data sets, such as for air toxic emission factors, contain several values reported only as below detection limit. Such data sets are referred to as "censored." Typical approaches to dealing with the censored data sets include replacing censored values with arbitrary values of zero, one-half of the detection limit, or the detection limit. Here, an approach to quantification of the variability and uncertainty of censored data sets is demonstrated. Empirical bootstrap simulation is used to simulate censored bootstrap samples from the original data. Maximum likelihood estimation (MLE) is used to fit parametric probability distributions to each bootstrap sample, thereby specifying alternative estimates of the unknown population distribution of the censored data sets. Sampling distributions for uncertainty in statistics such as the mean, median, and percentile are calculated. The robustness of the method was tested by application to different degrees of censoring, sample sizes, coefficients of variation, and numbers of detection limits. Lognormal, gamma, and Weibull distributions were evaluated. The reliability of using this method to estimate the mean is evaluated by averaging the best estimated means of 20 cases for small sample size of 20. The confidence intervals for distribution percentiles estimated with bootstrap/MLE method compared favorably to results obtained with the nonparametric Kaplan-Meier method. The bootstrap/MLE method is illustrated via an application to an empirical air toxic emission factor data set.     

Lognormal Distributions for Total Water Intake and Tap Water Intake by Pregnant and Lactating Women in the United States

Using probability plots and Maximum Likelihood Estimation (MLE), we fit lognormal distributions to data compiled by Ershow et al. for daily intake of total water and tap water by three groups of women (controls, pregnant, and lactating; all between 15–49 years of age) in the United States. We also develop bivariate lognormal distributions for the joint distribution of water ingestion and body weight for these three groups. Overall, we recommend the marginal distributions for water intake as fit by MLE for use in human health risk assessments.     

Estimating Risk Assessment Exposure Point Concentrations when the Data Are Not Normal or Lognormal

The U.S. Environmental Protection Agency (EPA) recommends the use of the one-sided 95% upper confidence limit of the arithmetic mean based on either a normal or lognormal distribution for the contaminant (or exposure point) concentration term in the Superfund risk assessment process. When the data are not normal or lognormal this recommended approach may overestimate the exposure point concentration (EPC) and may lead to unecessary cleanup at a hazardous waste site. The EPA concentration term only seems to perform like alternative EPC methods when the data are well fit by a lognormal distribution. Several alternative methods for calculating the EPC are investigated and compared using soil data collected from three hazardous waste sites in Montana, Utah, and Colorado. For data sets that are well fit by a lognormal distribution, values for the Chebychev inequality or the EPA concentration term may be appropriate EPCs. For data sets where the soil concentration data are well fit by gamma distributions, Wong's method may be used for calculating EPCs. The studentized bootstrap-t and Hall's bootstrap-t transformation are recommended for EPC calculation when all distribution fits are poor. If a data set is well fit by a distribution, parametric bootstrap may provide a suitable EPC.     

Lognormal Distributions for Water Intake by Children and Adults

We fit lognormal distributions to data collected in a national survey for both total water intake and tap water intake by children and adults for these age groups in years: 0 less than age less than 1; 1 less than or equal to age less than 11; 11 less than or equal to age less than 20; 20 less than or equal to age less than 65; 65 less than or equal to age; and all people in the survey taken as a single group. These distributions are suitable for use in public health risk assessments.     

Estimation of Dietary Exposure to Chemicals: A Case Study Illustrating Methods of Distributional Analyses for Food Consumption Data

There are a number of sources of variability in food consumption patterns and residue levels of a particular chemical (e.g., pesticide, food additive) in commodities that lead to an expected high level of variability in dietary exposures across a population. This paper focuses on examples of consumption pattern survey data for specific commodities, namely that for wine and grape juice, and demonstrates how such data might be analyzed in preparation for performing stochastic analyses of dietary exposure. Data from the NIAAA/NHIS wine consumption survey were subset for gender and age group and, with matched body weight data from the survey database, were used to define empirically-based percentile estimates for wine intake (μl wine/kg body weight) for the strata of interest. The data for these two subpopulations were analyzed to estimate 14-day consumption distributional statistics and distributions for only those days on which wine was consumed. Data subsets for all wine-consuming adults and wine-consuming females ages 18 through 45, were determined to fit a lognormal distribution ( R ²= 0.99 for both datasets). Market share data were incorporated into estimation of chronic exposures to hypothetical chemical residues in imported table wine. As a separate example, treatment of grape juice consumption data for females, ages 18–40, as a simple lognormal distribution resulted in a significant underestimation of intake, and thus exposure, because the actual distribution is a mixture (i.e., multiple subpopulations of grape juice consumers exist in the parent distribution). Thus, deriving dietary intake statistics from food consumption survey data requires careful analysis of the underlying empirical distributions.     

Lognormal Distributions for Fish Consumption by the General U.S. Population

The rate of fish consumption is a critical variable in the assessment of human health risk from water bodies affected by chemical contamination and in the establishment of federal and state Ambient Water Quality Criteria (AWQC). For 1973 and 1974, the National Marine Fisheries Service (NMFS) analyzed data on the consumption of salt-water finfish, shellfish, and freshwater finfish from all sources in 10 regions of the United States for three age groups in the general population: children (ages 1 through 11 years), teenagers (ages 12 through 18 years), and adults (ages 19 through 98 years). Even though the NMFS data reported in Ref. 14 are 20 years old, they remain the most complete data on the overall consumption of all fish by the general U.S. population and they have been widely used to select point values for consumption. Using three methods, we fit lognormal distributions to the results of the survey as analyzed and published in Ref. 14. Strong lognormal fits were obtained for most of the 90 separate data sets. These results cannot necessarily be used to model the consumption of fish by sport or subsistence anglers from specific sites or from single water bodies.     

Correlated Inputs in Quantitative Risk Assessment: The Effects of Distributional Shape

Application of Monte Carlo simulation methods to quantitative risk assessment are becoming increasingly popular. With this methodology, investigators have become concerned about correlations among input variables which might affect the resulting distribution of risk. We show that the choice of input distributions in these simulations likely has a larger effect on the resultant risk distribution than does the inclusion or exclusion of correlations. Previous investigators have studied the effect of correlated input variables for the addition of variables with any underlying distribution and for the product of lognormally distributed variables. The effects in the main part of the distribution are small unless the correlation and variances are large. We extend this work by considering addition, multiplication and division of two variables with assumed normal, lognormal, uniform and triangular distributions. For all possible pairwise combinations, we find that the effects of correlated input variables are similar to those observed for lognormal distributions, and thus relatively small overall. The effect of using different distributions, however, can be large.     

Modeling and Forecasting Realized Volatility

We provide a framework for integration of high–frequency intraday data into the measurement, modeling, and forecasting of daily and lower frequency return volatilities and return distributions. Building on the theory of continuous–time arbitrage–free price processes and the theory of quadratic variation, we develop formal links between realized volatility and the conditional covariance matrix. Next, using continuously recorded observations for the Deutschemark/Dollar and Yen/Dollar spot exchange rates, we find that forecasts from a simple long–memory Gaussian vector autoregression for the logarithmic daily realized volatilities perform admirably. Moreover, the vector autoregressive volatility forecast, coupled with a parametric lognormal–normal mixture distribution produces well–calibrated density forecasts of future returns, and correspondingly accurate quantile predictions. Our results hold promise for practical modeling and forecasting of the large covariance matrices relevant in asset pricing, asset allocation, and financial risk management applications.     

A Probabilistic Framework for the Reference Dose (Probabilistic RfD)

Determining the probabilistic limits for the uncertainty factors used in the derivation of the Reference Dose (RfD) is an important step toward the goal of characterizing the risk of noncarcinogenic effects from exposure to environmental pollutants. If uncertainty factors are seen, individually, as "upper bounds" on the dose-scaling factor for sources of uncertainty, then determining comparable upper bounds for combinations of uncertainty factors can be accomplished by treating uncertainty factors as distributions, which can be combined by probabilistic techniques. This paper presents a conceptual approach to probabilistic uncertainty factors based on the definition and use of RfDs by the US. EPA. The approach does not attempt to distinguish one uncertainty factor from another based on empirical data or biological mechanisms but rather uses a simple displaced lognormal distribution as a generic representation of all uncertainty factors. Monte Carlo analyses show that the upper bounds for combinations of this distribution can vary by factors of two to four when compared to the fixed-value uncertainty factor approach. The probabilistic approach is demonstrated in the comparison of Hazard Quotients based on RfDs with differing number of uncertainty factors.     

The Expected Value of Information and the Probability of Surprise

Risk assessors attempting to use probabilistic approaches to describe uncertainty often find themselves in a data-sparse situation: available data are only partially relevant to the parameter of interest, so one needs to adjust empirical distributions, use explicit judgmental distributions, or collect new data. In determining whether or not to collect additional data, whether by measurement or by elicitation of experts, it is useful to consider the expected value of the additional information. The expected value of information depends on the prior distribution used to represent current information; if the prior distribution is too narrow, in many risk-analytic cases the calculated expected value of information will be biased downward. The well-documented tendency toward overconfidence, including the neglect of potential surprise, suggests this bias may be substantial. We examine the expected value of information, including the role of surprise, test for bias in estimating the expected value of information, and suggest procedures to guard against overconfidence and underestimation of the expected value of information when developing prior distributions and when combining distributions obtained from multiple experts. The methods are illustrated with applications to potential carcinogens in food, commercial energy demand, and global climate change.     

基于二次分解策略和模糊时间序列模型的航空客运需求预测研究

提高航空客运需求预测的准确性对于航空公司以及整个航空运输系统的发展都具有重要的现实意义。以往研究普遍采用单一分解策略去处理航空客运需求时序中存在的复杂特征,以此提升组合模型的预测性能。然而传统的分解策略存在着特征提取不完全、分解方法带有固有缺陷等问题,导致组合模型预测效果不能得到充分的提升。为此,本文提出一种基于二次分解策略和模糊时间序列模型的航空客运需求预测方法。该方法首先利用季节调整模型（X12-ARIMA）将原始时序分解成季节成分序列与季节调整后序列,继而利用改进的自适应噪声集成经验模态分解方法（ICEEMDAN）将季节调整后序列分解成一系列不同时间尺度的本征模态函数（IMF）和残差序列（Residue）。然后使用基于模糊C均值算法（FCM）划分论域区间的FTS模型对季节成分序列、各IMF分量以及残差序列分别进行预测。最后将各分量序列的预测结果进行集成,重构出航空客运需求的预测值。实证结果表明,本文所提出的二次分解策略表现显著优于传统的分解策略,并且本文所提出模型对于航空客运需求预测有着较高的准确性。     

A comparison of several approximations to the lead time demand distribution

Several distributions have been used for approximating the lead time demand distribution in inventory systems. We compare five distributions, the normal, the logistic, the lognormal, the gamma and the Weibull for obtaining the expected number of back orders, the reorder levels to have a given protection and the optimal order quantity, reorder levels in continuous review models of (Q, r) type. The normal and the logistic distributions are inadequate to represent the situations where the coefficient of variation (the ratio of the standard deviation to the mean) of the lead time demand distribution is large. The lognormal, the gamma and the Weibull distributions are versatile and adequate; however the lognormal seems to be a viable candidate because of its computational simplicity.     

Gender and Race in Beliefs about Outdoor Air Pollution

Universal need for, or reactions to, risk communications should not be assumed; potential differences across demographic groups in environmental risk beliefs, attitudes, and behaviors could affect risk levels or opportunities for risk reduction. This article reports relevant findings from a survey experiment involving 1,100 potential jurors in Philadelphia concerning public responses to outdoor air pollution and air quality information. Flynn et al. (1994) and Finucane et al. (2000) found significant differences in risk ratings for multiple hazards, and in generic risk beliefs, between white men (or a subset) and all others (white women, nonwhite men, and nonwhite women). This study examined whether white men had significantly different responses to air pollution and air pollution information. An opportunity sample of volunteers from those awaiting potential jury duty in city courts (matching census estimates for white versus nonwhite proportions, but more female than the city's adult population and more likely to have children) filled out questionnaires distributed quasi-randomly. On most measures there were no statistically significant differences among white men (N = 192), white women (N = 269), nonwhite men (N = 165), and nonwhite women (N = 272). Nonwhites overall (particularly women) reported more concern about and sensitivity to air pollution than whites, and were more concerned by (even overly sensitive to) air pollution information provided as part of the experiment. Nonwhites also were more likely (within-gender comparisons) to report being active outdoors for at least four hours a day, a measure of potential exposure to air pollution, and to report intentions to reduce such outdoor activity after reading air pollution information. Differences between men and women were less frequent than between whites and nonwhites; the most distinctive group was nonwhite women, followed by white men. Flynn et al. (1994) and Finucane et al. (2000) found a far larger proportion of significant differences, with white men as most distinctive, probably due to use of different measures, study design, and population samples. However, all three studies broadly confirm the existence of gender and race interactions in risk beliefs and attitudes (particularly for white men and nonwhite women) that deserve more attention from researchers.     

Fitting the Lognormal Distribution to Surgical Procedure Times*

Minimum surgical times are positive and often large. The lognormal distribution has been proposed for modeling surgical data, and the three‐parameter form of the lognormal, which includes a location parameter, should be appropriate for surgical data. We studied the goodness‐of‐fit performance, as measured by the Shapiro‐Wilk p‐value, of three estimators of the location parameter for the lognormal distribution, using a large data set of surgical times. Alternative models considered included the normal distribution and the two‐parameter lognormal model, which sets the location parameter to zero. At least for samples with n > 30, data adequately fit by the normal had significantly smaller skewness than data not well fit by the normal, and data with larger relative minima (smallest order statistic divided by the mean) were better fit by a lognormal model. The rule “If the skewness of the data is greater than 0.35, use the three‐parameter lognormal with the location parameter estimate proposed by Muralidhar & Zanakis (1992), otherwise, use the two‐parameter model” works almost as well at specifying the lognormal model as more complex guidelines formulated by linear discriminant analysis and by tree induction.     

A Combined Monte Carlo and Possibilistic Approach to Uncertainty Propagation in Event Tree Analysis

In risk analysis, the treatment of the epistemic uncertainty associated to the probability of occurrence of an event is fundamental. Traditionally, probabilistic distributions have been used to characterize the epistemic uncertainty due to imprecise knowledge of the parameters in risk models. On the other hand, it has been argued that in certain instances such uncertainty may be best accounted for by fuzzy or possibilistic distributions. This seems the case in particular for parameters for which the information available is scarce and of qualitative nature. In practice, it is to be expected that a risk model contains some parameters affected by uncertainties that may be best represented by probability distributions and some other parameters that may be more properly described in terms of fuzzy or possibilistic distributions. In this article, a hybrid method that jointly propagates probabilistic and possibilistic uncertainties is considered and compared with pure probabilistic and pure fuzzy methods for uncertainty propagation. The analyses are carried out on a case study concerning the uncertainties in the probabilities of occurrence of accident sequences in an event tree analysis of a nuclear power plant.     

Production planning problem in manufacturing systems with general failure and repair time distributions

We study the optimal flow control for a manufacturing system subject to random failures and repairs. In most previous work, it has been proved that, for constant demand rates and exponential failure and repair times distributions of machines, the hedging point policy is optimal. The aim of this study is to extend the hedging point policy to non-exponential failure and repair times distributions and random demand rates models. The performance measure is the cost related to the inventory and back order penalties. We find that the structure of the hedging point policy can be parametrized by a single factor representing the critical stock level or threshold. With the corresponding hedging point policy, simulation experiments are used to construct input-output data from which an estimation of the incurred cost function is obtained through a regression analysis. The best parameter value of the related hedging point policy is derived from a minimum search of the obtained cost function. The extended hedging point policy is validated and shown to be quite effective. We find that the hedging point policy is also applicable to a wide variety of complex problems (i.e. non-exponential failure and repair times distributions and random demand rates), where analytical solutions may not be easily obtained.     

Lognormal Distributions for Body Weight as a Function of Age for Males and Females in the United States, 1976–1980

Based on results reported from the NHANES II Survey (the National Health and Nutrition Examination Survey II) for people living in the United States during 1976–1980, we use exploratory data analysis, probability plots, and the method of maximum likelihood to fit lognormal distributions to percentiles of body weight for males and females as a function of age from 6 months through 74 years. The results are immediately useful in probabilistic (and deterministic) risk assessments.