Lognormal Variables Research Articles

The Effective Modulus of Random Checkerboard Plates

We investigate the elastic effective modulus Eeff of two-dimensional checkerboard specimens in which square tiles are randomly assigned to one of two component phases. This is a model system for a wide class of multiphase polycrystalline materials such as granitic rocks and many ceramics. We study how the effective stiffness is affected by different characteristics of the specimen (size relative to the tiles, stiff fraction, and modulus contrast between the phases) and obtain analytical approximations to the probability distribution of Eeff as a function of these parameters. In particular, we examine the role of percolation of the soft and stiff phases, a phenomenon that is important in polycrystalline materials and composites with inclusions. In small specimens, we find that the onset of percolation causes significant discontinuities in the effective modulus, whereas in large specimens, the influence of percolation is smaller and gradual. The analysis is an extension of the elastic homogenization methodology of Dimas et al. (2015, “Random Bulk Properties of Heterogeneous Rectangular Blocks With Lognormal Young's Modulus: Effective Moduli,” ASME J. Appl. Mech., 82(1), p. 011003), which was devised for blocks with lognormal spatial variation of the modulus. Results are validated through Monte Carlo simulation. Compared with lognormal specimens with comparable first two moments, checkerboard plates have more variable effective modulus and are on average less compliant if there is prevalence of stiff tiles and more compliant if there is prevalence of soft tiles. These differences are linked to percolation.

Metallic Mineral Resources in the Twenty-First Century: II. Constraints on Future Supply

Supplying worldwide demand of metallic raw materials throughout the rest of this century may require 5–10 times the amount of metals contained in known ore deposits. This demand can be met only if mineral deposits containing the required masses of metals, in excess of present day ore reserves, exist in the Earth’s crust. It is, by definition, not known whether or not such mineral deposits exist. On the basis of the statistical distribution of metal tonnages contained in known ore deposits, however, it is possible to place constraints on the size distribution of the deposits that must be discovered in order to meet the expected demand. A nondimensional analysis of the distribution of metal tonnages in deposits of 20 metals shows that most of them follow distributions that, although not strictly lognormal, share important characteristics with a lognormal distribution. Chief among these is the observation that frequency falls off symmetrically and geometrically with deposit size, relative to a median deposit size that is approximately equal to the geometric mean deposit size. An immediate consequence of this behavior is that most of the metal endowment is concentrated in deposits that are several orders of magnitude larger than the median deposit size, and that are much rarer than the most common deposits that cluster around the median deposit size. The analysis reveals remarkable similarities among the statistical distributions of most of the metals included in this study, in particular, the fact that distribution of most metals can be fully described with essentially the same value (about 2–3) of the scale parameter, σ, which is the only parameter needed to describe the behavior of a normalized lognormal variable. This observation makes it possible to derive the following general conclusions, which are applicable to most metals—both scarce and abundant. First, it is unlikely that undiscovered mineral deposits of sizes comparable to those that contain most of the known metal endowment exist in sufficient quantities to supply the expected worldwide demand throughout the rest of this century. Second, if the expected demand is to be met, one must hope that very large deposits, perhaps up to one order of magnitude larger than the largest known deposits, exist in accessible portions of the Earth’s crust, and that these deposits are discovered.

The Lognormal Race: A Cognitive-Process Model of Choice and Latency with Desirable Psychometric Properties

We present a cognitive process model of response choice and response time performance data that has excellent psychometric properties and may be used in a wide variety of contexts. In the model there is an accumulator associated with each response option. These accumulators have bounds, and the first accumulator to reach its bound determines the response time and response choice. The times at which accumulator reaches its bound is assumed to be lognormally distributed, hence the model is race or minima process among lognormal variables. A key property of the model is that it is relatively straightforward to place a wide variety of models on the logarithm of these finishing times including linear models, structural equation models, autoregressive models, growth-curve models, etc. Consequently, the model has excellent statistical and psychometric properties and can be used in a wide range of contexts, from laboratory experiments to high-stakes testing, to assess performance. We provide a Bayesian hierarchical analysis of the model, and illustrate its flexibility with an application in testing and one in lexical decision making, a reading skill.

Sensitivity analysis of a stochastic discrete event simulation model of harvest operations in a static rose cultivation system

Greenhouse crop system design for maximum efficiency and quality of labour is an optimisation problem that benefits from model-based design evaluation. This study focussed on the harvest process of roses in a static system as a step in this direction. The objective was to identify parameters with strong influence on labour performance as well as the effect of uncertainty in input parameters on key performance indicators. Differential sensitivity was analysed and results were tested for model linearity and superposability and verified using the robust Monte Carlo analysis method since in the literature, performance and applicability of differential sensitivity analysis has been questioned for models with internal stochastic behaviour. Greenhouse section length and width, single rose cut time, and yield influence labour performance most, but greenhouse section dimensions and yield also affect the number of harvested stems directly. Throughput, i.e. harvested stems per second, being the preferred metric for labour performance, is most affected by single rose cut time, yield, number of harvest cycles per day, greenhouse length and operator transport velocity. The model is insensitive for σ of lognormal distributed stochastic variables describing the duration of low frequent operations in the harvest process, like loading and unloading rose nets. In uncertainty analysis, the coefficient of variation for the most important outputs, labour time and throughput, is around 5%. Total sensitivity as determined using differential sensitivity analysis and Monte Carlo analysis essentially agreed. The combination of both methods gives full insight into both individual and total sensitivity of key performance indicators.

Optimising Reliability: Portfolio Modeling of Contract Types for Retail Water Providers

This paper considers the retail water provider’s purchasing decision of a portfolio of permanent contracts from wholesalers with multiple volatile water sources. We consider the reliability of two contract types: (1) fixed annual quantities, and (2) an inflow harvest function with storage. Our four-reservoir case in Sydney (Australia) has cross-correlated inflow data. To accommodate multi-reservoir cross-correlation we adapt Portfolio Theory from finance to lognormal reservoir inflows, re-framing traditional storage theory from the wholesaler’s optimal operating policy to the retailer’s optimal purchasing policy. We find that Reliability improves with access to a source pool (cf. fixed quantities from separate sources), demonstrating the ‘insurance effect’, and the portfolio that minimises lognormal variance also minimises harvest (and thus environmental impact). Reform direction in Australian (and other international) water markets is towards multi-provider vertical disintegration, which may reduce pool opportunities and negate the insurance effect. We consider diminishing reliability returns as reservoir harvesting increases, and conclude a retail portfolio of permanent contracts from reservoirs, plus short-term contracts from alternative sources (either independently or negatively cross-correlated) efficiently secures high reliability. The challenge in incomplete water markets remains in encouraging and sustaining supply diversification that may only be needed aperiodically during extreme droughts.

On Group Choice Procedures for Problems of Classification and Reliability in the Case of Lognormal Variance*

This paper is the first to obtain results by applying group analysis to the problems of statistical estimation, statistical pattern recognition, and robustness in the case of populations with lognormal distributions. Maximum likelihood unbiased Bayesian estimators are constructed, a variety of decision rules for group recognition is developed, and their statistical properties and qualitative characteristics are examined. Analytical expressions are found for the probability of the linear stochastic inequality as well as various statistical estimates for this probability. The relations between recognition problems and the probability of a linear inequality are analyzed.

The Sum of Two Correlated Lognormal Random Variables — WKB Approximation

In this paper we apply the idea of the WKB method to derive an effective single lognormal approximation for the probability distribution of the sum of two correlated lognormal variables. An approximate probability distribution of the sum is determined in closed form, and illustrative numerical examples are presented to demonstrate the validity and accuracy of the approximate distribution. Our analysis shows that the proposed method is able to provide a simple, efficient and accurate approximation to the probability distribution of the sum of two correlated lognormal variables. We also discuss how this new approach can be straightforwardly extended to study the sum of N lognormals.

WKB approximation for the sum of two correlated lognormal random variables

In this paper we apply the idea of the WKB method to derive an eective single lognormal approximation for the probability distribution of the sum of two correlated lognormal variables. An approximate probability distribution of the sum is determined in closed form, and illustrative numerical examples are presented to demonstrate the validity and accuracy of the approximate distribution. Our analysis shows that the proposed method is able to provide a simple, ecient and accurate approximation to this probability distribution of the sum of two correlated lognormal variables. We also discuss how this new approach can be straightforwardly extended to study the sum of N lognormals.

Probabilistic Characteristics of Residual Displacements of SDOF Systems

Probabilistic residual displacement analysis plays an important role in determination of technical or economical feasibility of repairing the damaged emgineering structures after an earthquake. Probabilistic characteristics of residual displacements of SDOF systems were quantificationally investigated in this study through a comprehensive statistical analysis. The influences of the P-Δ effect, period of vibration, normalized yield strength and post-yield stiffness ratio on probabilistic characteristics of residual displacements were investigated based on 69 selected earthquake records. In particular, the probability models for the normalized residual displacements were proposed and tested. Results show that the P-Δ effect and post-yield stiffness ratio have significant impacts on the residual displacements; the period of vibration also obviously influences the residual displacements for rigid systems; the residual displacement can be described as the Lognormal distribution variable.

Correction for bias of models with lognormal distributed variables in absence of original data

The logarithmic transformation of the dependent variables for models developed using regression analysis induces bias that should be corrected, regardless its magnitude. The simplest correction for bias was proposed by Sprugel (1983), which basically multiplies the back-transformed estimates with the constant value of exponential of half the variance of the errors of the logarithmically transformed variable. While this correction is fast and easy to implement does not supplies estimates of the variability existing in the original data. Consequently, a procedure based on generated data was developed to provide unbiased estimates for both attribute of interest and variability existing along the model. The procedure reveals that valid estimates can be obtained if large number of values is generated (e.g., 5000 values/x). The procedures supplies accurate estimates for the attribute of interest and its variability, but encounters significant data processing difficulties for models with more than one predictor variable. Nevertheless, irrespective the number of predictor of variables and magnitude of the correction factor computed by Sprugel, the estimates determined using logarithmic transformations should be corrected for bias, to avoid cumulated errors or chaotic effects associated with nonlinear models.

A Quasi‐Analytical Pricing Model for Arithmetic Asian Options

We develop a quasi‐analytical pricing method for discretely sampled arithmetic Asian options. We derive an asymptotic approximation of the arithmetic average with the geometric average of lognormal variables. Numerical experiments show that the asymptotic approximation is accurate and the absolute error converges very quickly as the number of observations increases. The absolute error is of the order of 10−5 to 10−6 for daily average. We then derive quasi‐analytical formulas for arithmetic Asian options under the Black–Scholes framework, in which the probability density of the geometric average is used. Extensive experiments are conducted to compare the proposed method with the various existing semianalytical methods. The overall accuracy of the proposed method is better than any other methods tested. The proposed method performs much better than the second best one for at‐the‐money Asian options under high volatility. The mean pricing error of the proposed method for a daily average Asian option is 37.5% less than the second best one. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:1143–1166, 2013

The Sum and Difference of Two Lognormal Random Variables

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of N lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

Quality measures for soil surveys by lognormal kriging

If we know the variogram of a random variable then we can compute the prediction error variances (kriging variances) for kriged estimates of the variable at unsampled sites from sampling grids of different design and density. In this way the kriging variance is a useful pre-survey measure of the quality of statistical predictions, which can be used to design sampling schemes to achieve target quality requirements at minimal cost. However, many soil properties are lognormally distributed, and must be transformed to logarithms before geostatistical analysis. The predicted values on the log scale are then back-transformed. It is possible to compute the prediction error variance for a prediction by this lognormal kriging procedure. However, it does not depend only on the variogram of the variable and the sampling configuration, but also on the conditional mean of the prediction. We therefore cannot use the kriging variance directly as a pre-survey measure of quality for geostatistical surveys of lognormal variables. In this paper we present an alternative. First we show how the limits of a prediction interval for a variable predicted by lognormal kriging can be expressed as dimensionless quantities, proportions of the unknown median of the conditional distribution. This scaled prediction interval can be used as a presurvey quality measure since it depends only on the sampling configuration and the variogram of the log-transformed variable. Second, we show how a similar scaled prediction interval can be computed for the median value of a lognormal variable across a block, in the case of block kriging. This approach is then illustrated using variograms of lognormally distributed data on concentration of elements in the soils of a part of eastern England.

The Sum and Difference of Two Lognormal Random Variables

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie‐Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of N lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean‐reversion.

Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.

Variable offspring size as an adaptation to environmental heterogeneity in a clonal plant species: integrating experimental and modelling approaches

Summary1. The production of variably sized offspring has been hypothesized to be adaptive to temporal variability in environmental conditions.2. This is difficult to verify empirically, and theoretical models are typically generic and not parameterized with data from real populations; studies integrating theoretical and empirical approaches to this problem are rare.3. Here, we present experimental data on the growth of Scirpus maritimus, a clonal aquatic macrophyte that grows through vegetative extensions involving tubers.4. The experiments show that offspring fitness (biomass productivity) is dependent on environmental conditions (water depth).5. Experimental data indicate that variation in offspring (tuber) sizes can be approximated by a lognormal distribution.6. We use these data to develop a different equation model of S. maritimus growth to test whether producing variably sized offspring is adaptive. The model compares fitness under the lognormal strategy to several hypothetical strategies with qualitatively different variance in offspring size.7. The model results suggest that lognormal variation in S. maritimus tuber size may be adaptive to the temporal variation in water levels that characterize its natural Mediterranean environment.8. We illustrate how the underlying principles that lend adaptive value to offspring size variation may apply to other species experiencing similar environmental conditions.9. Synthesis. Close integration of data and theoretical models creates a unique tool for investigating the adaptive value of life‐history traits in clonal aquatic plants.

Interval Reliability Based Condition Assessment of Bridges

Recent research indicates that small changes in input parameters of reliability models significantly affect outcomes of reliability computations. This is a considerable concern especially in reliability based condition assessments of old bridges which have lower reliabilities. To overcome such errors of input parameters, structural reliability can be combined with interval analysis (i.e., interval reliability). This paper presents an interval reliability based condition assessment procedure of bridges. Interval reliability of a component is introduced when resistance and load behave as: (1) normal variables; (2) log normal variables. Similarly, interval reliabilities of other components can be estimated. Finally, system interval reliability is estimated. Bridge condition is assessed by comparing with the acceptable reliability of the bridge. A case study, a steel girder bridge, is selected, and its interval reliability indices are estimated for possible errors of statistical parameters. The obtained results are then compared with an existing method without considering interval reliability. Results indicate that there are significant variations in reliability when interval reliability is considered.

(Weighted) Sum of 'n' Correlated Lognormals: Probability Density Function – Numerical calculation

Probability density function (pdf) for (weighted) sum of n correlated lognormal variables is deducted. The method uses joint pdf of multivariate correlated lognormal variables and an extended method of convolution. The formula contains (n-1)-fold integral, which can be evaluated by numerical integration or an alternative direct discrete computing method. As an example a 4 dimensional weighted case is provided, the deducted theoretical pdf is checked by Monte Carlo simulation.

Evaluation des propriétés hydrauliques des aquifères fracturés des formations cristalline et métamorphique dans la région des Lacs (centre de la Côte d’Ivoire)

Groundwater is the main source of water supply for rural populations in the Region des Lacs area in central Cote d’Ivoire (West Africa). The area is underlain by the metamorphic and crystalline fractured hard rock aquifers. This paper focuses on the assessment of their hydraulic properties. To this end, a data base comprising pumping tests data and the technical reports were gathered. 105 values of transmissivity (T ) and specific capacity (Q/s) have been deduced after pumping tests interpretation by the Jacob recovery method. Statistical analyses of all these data have been done. Depth of wells range from 49.50 to 99 m with an average of 69 m. The thickness of the weathered zone has averaged 16.52 m and lies between 1.90 and 63.10 m. Wells average rate is 2.32 m3/s. The depths of open fractures are significantly in the first 60 meters of hard rock drilling and averaged 42.94 m. The transmissivity and specific capacity of each aquifer span over several orders of magnitude revealing the strong heterogeneity of the aquifer. Both variables are lognormal variables A significant empirical relationship between T and Q/s was found T = 0,937(Q/s)^1.118 with a coefficient of determination (R^2 = 0.82). This relationship enabled the transmissivity data to be supplemented with the 95% prediction interval in order to assess the uncertainty associated with the estimates of transmissivity in the area of the study. These results are significant and can be used as an input in forthcoming modeling these aquifers and facilitate groundwater management policy.

Numerical Monte Carlo analysis of the influence of pore‐scale dispersion on macrodispersion in 2‐D heterogeneous porous media

We investigate the influences of pore‐scale dispersion and of larger‐scale permeability heterogeneities on the macrodispersion without the molecular diffusion. Permeability follows a lognormal exponentially correlated distribution characterized by its correlation length λ and its lognormal variance σ2. Macrodispersion is evaluated numerically by using parallel simulations on grids of characteristic size ranging from 200λ to 1600λ. We note αL and αT the pore‐scale longitudinal and transversal dispersivities. For αL/λ < 10−2 and αT/λ < 10−3, the influence of pore‐scale dispersion on the macrodispersion is smaller than 5% of the macrodispersion due only to permeability heterogeneities. Larger dispersivities (αL/λ ≥ 10−2 or αT/λ ≥ 10−3) induce larger effects than those obtained by the semianalytical expression of Salandin and Fiorotto (1998) for σ2 > 1. The effects of local dispersion on the longitudinal macrodispersion remain limited to 25% at most of the macrodispersion due only to permeability heterogeneities. For σ2 > 1, isotropic local dispersion induces a reduction of the longitudinal macrodispersion, whereas anisotropic local dispersion lets it increase. The longitudinal and transverse local dispersions induce opposite effects on the longitudinal macrodispersion, which are respectively an increase and a reduction. The transverse macrodispersion null without local dispersion or molecular diffusion becomes strictly positive with local dispersion. Because of the velocity field heterogeneities, it is amplified by a factor of 2 to 50 from the grid scale to the macro scale. The transverse dispersion is triggered by both longitudinal and transverse local dispersions. A reduction of a factor of 2 of the transverse local dispersion at fixed longitudinal local dispersion yields only a reduction of a factor of 4 at most of the transverse macrodispersion for σ2 ≥ 2.25.