Lognormal Random Variables Research Articles

A dynamical Gaussian, lognormal, and reverse lognormal Kalman filter

AbstractWe derive a generalization of the Kalman filter that allows for non‐Gaussian background and observation errors. The Gaussian assumption is replaced by considering that the errors come from a mixed distribution of Gaussian, lognormal, and reverse lognormal random variables. We detail the derivation for reverse lognormal errors and extend the results to mixed distributions, where the number of Gaussian, lognormal, and reverse lognormal state variables can change dynamically every analysis time. We test the dynamical mixed Kalman filter robustly on two different systems based on the Lorenz 1963 model, and demonstrate that non‐Gaussian techniques generally improve the analysis skill if the observations are sparse and uncertain, compared with the Gaussian Kalman filter.

Probabilistic Assessment of the Seismic Peak Ground Acceleration (PGA) at seven Colombian Cities

This paper presents a probabilistic assessment of peak ground acceleration (PGA) observations at seven Colombian cities located in regions with intermediate and high seismic activity. Frequency histograms of the annual exceedance rates were determined using historical data of the annual rate of exceedance for pre-established values of PGA. After determining distribution parameters by means of the method of moments, several probability density functions were examined in order to evaluate the performance of each function. In most sites, the maximum annual PGA appears to be best modeled by a lognormal distribution, which led to its adoption as parent distribution for the entire group of cities. Once the lognormal function is selected as parent distribution, series of PGA values for arbitrary periods of time may be generated through random simulation. Next, series of maximum values in fifty years of observation were simulated for the seven cities, assuming a Lognormal parent distribution. Analytical solutions for extreme values of different sample sizes of lognormal random variables are not known to the authors. However, since extreme values for large samples of lognormal random variables tend asymptotically to a Type I (Gumbel) probability distribution, the latter was adopted as a possible model for the 50 years PGA distributions suggested in the paper for several Colombian cities, as a useful tool for engineering design, whenever the PGA is regarded as an appropriate seismic intensity measure.

From Multi- to Univariate: A Product Random Variable with an Application to Electricity Market Transactions: Pareto and Student’s t-Distribution Case

Multivariate modelling of economics data is crucial for risk and profit analyses in companies. However, for the final conclusions, a whole set of variables is usually transformed into a single variable describing a total profit/balance of company’s cash flows. One of the possible transformations is based on the product of market variables. Thus, in this paper, we study the distribution of products of Pareto or Student’s t random variables that are ubiquitous in various risk factors analysis. We review known formulas for the probability density functions and derive their explicit forms for the products of Pareto and Gaussian or log-normal random variables. We also study how the Pareto or Student’s t random variable influences the asymptotic tail behaviour of the distribution of their product with the Gaussian or log-normal random variables and discuss how the dependency between the marginal random variables of the same type influences the probabilistic properties of the final product. The theoretical results are then applied for an analysis of the distribution of transaction values, being a product of prices and volumes, from a continuous trade on the German intraday electricity market.

Exact Statistical Distribution and Correlation of Human Height and Weight: Analysis and Experimental Confirmation

The statistical relationship between human height and weight is of especial importance to clinical medicine, epidemiology, and the biology of human development. Yet, after more than a century of anthropometric measurements and analyses, there has been no consensus on this relationship. The purpose of this article is to provide a definitive statistical distribution function from which all desired statistics (probabilities, moments, and correlation functions) can be determined. The statistical analysis reported in this article provides strong evidence that height and weight in a diverse population of healthy adults constitute correlated bivariate lognormal random variables. This conclusion is supported by a battery of independent tests comparing empirical values of 1) probability density patterns, 2) linear and higher order correlation coefficients, 3) statistical and hyperstatistics moments up to 6th order, and 4) distance correlation (dCor) values to corresponding theoretical quantities: 1) predicted by the lognormal distribution and 2) simulated by use of appropriate random number generators. Furthermore, calculation of the conditional expectation of weight, given height, yields a theoretical power law that specifies conditions under which body mass index (BMI) can be a valid proxy of obesity. The consistency of the empirical data from a large, diverse anthropometric survey partitioned by gender with the predictions of a correlated bivariate lognormal distribution was found to be so extensive and close as to suggest that this outcome is not coincidental or approximate, but may be a consequence of some underlying biophysical mechanism.

Theoretical approach for uncertainty quantification in probabilistic safety assessment using sum of lognormal random variables

Probabilistic safety assessment is widely used to quantify the risks of nuclear power plants and their uncertainties. When the lognormal distribution describes the uncertainties of basic events, the uncertainty of the top event in a fault tree is approximated with the sum of lognormal random variables after minimal cutsets are obtained, and rare-event approximation is applied. As handling complicated analytic expressions for the sum of lognormal random variables is challenging, several approximation methods, especially Monte Carlo simulation, are widely used in practice for uncertainty analysis. In this study, a theoretical approach for analyzing the sum of lognormal random variables using an efficient numerical integration method is proposed for uncertainty analysis in probability safety assessments. The change of variables from correlated random variables with a complicated region of integration to independent random variables with a unit hypercube region of integration is applied to obtain an efficient numerical integration. The theoretical advantages of the proposed method over other approximation methods are shown through a benchmark problem. The proposed method provides an accurate and efficient approach to calculate the uncertainty of the top event in probabilistic safety assessment when the uncertainties of basic events are described with lognormal random variables.

On the Laplace transform of the lognormal distribution: Analytic continuation and series approximations

We study the analytical properties of the Laplace transform of the lognormal distribution. Two integral expressions for the analytic continuation of the Laplace transform of the lognormal distribution are provided, one of which takes the form of a Mellin–Barnes integral. As a corollary, we obtain an integral expression for the characteristic function. We present two approximations for the Laplace transform of the lognormal distribution, both valid in ℂ∖(−∞,0]. In the last section, we discuss how one may use our results to compute the density of a sum of independent lognormal random variables.

A note on using simulated implicit likelihood estimation to analyze summaries from a designed experiment

In this article, we use Simulated Implicit Likelihood Estimation (SimILE) to analyze summary statistics, the sample mean and variance of lognormally distributed replicates, when only these summaries are available at each run of a designed experiment. Because the likelihood for the summary statistics is unknown, by simulating lognormal random variables, SimILE estimates the likelihood and uses it in a Bayesian analysis. We first consider normally distributed replicates where the exact likelihood is known and then lognormally distributed replicates. We illustrate these cases with simulated data where we know the answers and compare the SimILE results in the cases where the exact likelihood is known.

Outlier accommodation with semiparametric density processes: A study of Antarctic snow density modelling

In many settings, data acquisition generates outliers that can obscure inference. Therefore, practitioners often either identify and remove outliers or accommodate outliers using robust models. However, identifying and removing outliers is often an ad hoc process that affects inference, and robust methods are often too simple for some applications. In our motivating application, scientists drill snow cores and measure snow density to infer densification rates that aid in estimating snow water accumulation rates and glacier mass balances. Advanced measurement techniques can measure density at high resolution over depth but are sensitive to core imperfections, making them prone to outliers. Outlier accommodation is challenging in this setting because the distribution of outliers evolves over depth and the data demonstrate natural heteroscedasticity. To address these challenges, we present a two-component mixture model using a physically motivated snow density model and an outlier model, both of which evolve over depth. The physical component of the mixture model has a mean function with normally distributed depth-dependent heteroscedastic errors. The outlier component is specified using a semiparametric prior density process constructed through a normalized process convolution of log-normal random variables. We demonstrate that this model outperforms alternatives and can be used for various inferential tasks.

The closed-form approximation to price basket options under stochastic interest rate

The paper presents closed-form approximation formulas for pricing basket options. We assume that the underlying asset prices follow geometric Brownian motions and the interest rate follows the one-factor Hull-White model. Under a given forward measure, we obtain the analytical lower and upper bounds of basket options. By finding a simple random variable to replace the sum of the lognormal random variables, we combine the conditioning and the moment matching approaches, and derive approximation formulas to price basket options. Numerical results illustrate that our results fall in the sharp lower and upper bounds of basket options and are consistent with Monte Carlo simulation results.

Development of models for facing tensile forces of soil nail walls using statistical approaches

ABSTRACT The facing is a key component for the design of soil nail walls. However, current design specifications of soil nail walls in China do not specify methods for estimation of facing tensile forces, which has posed challenges for engineers to produce consistent, safe, and cost-effective facing designs. This study first constructs the facing tensile force as the product of the maximum nail load and a facing factor. Then a database containing 91 measured facing tensile force data collected from the literature is developed and used to calibrate the facing factor matching six maximum nail load models that are commonly adopted in China, namely, the simplified incremental calculation models based on the Rankine and Coulomb earth pressure theories, the China Academy of Building Research model, and the China Association for Engineering Construction Standardization model. Based on the developed database, the accuracies of the six proposed models for facing tensile forces are demonstrated to be unbiased on average and the dispersions in the prediction accuracy are ranked from low to medium. Furthermore, the influences of three wall working conditions on prediction accuracy of the developed models are investigated and shown to be practically insignificant. Last, the model biases are characterised as lognormal random variables.

A fast method to seek the mean and variance of the summation of lognormal variables

The sum of lognormal variables has been a topic of interest in several fields of research such as engineering, biology and finance, among others. For example, in the field of telecommunications, the aggregate interference of radio frequency signals is modeled as a sum of lognormal variables. To date, there is no closed expression for the probability distribution function (PDF) of this sum. Several authors have proposed approximations for this PDF, with which they calculate the mean and variance. However, each method has limitations in its range of parameters for mean, variance and number of random variables to be added. In other cases, long approximations as power series are used, which makes the analytical treatment impractical and reduces the computational performance of numerical operations. This paper shows an alternative method for calculating the mean and variance of the sum of lognormal random variables from a computational performance approach. Our method has been evaluated extensively by Monte Carlo simulations. As a result, this method is computationally efficient and yields a low approximation error computation for a wide range of mean values, variances and number of random variables.

Random finite element analysis of backward erosion piping

Backward erosion piping (BEP) is a type of internal erosion that threatens the integrity of dams and levees. BEP has been shown to be highly sensitive to spatial variations in soil properties; however, there are presently no assessment methods that permit incorporating spatial variation in soil properties into BEP analysis. The Random Finite Element Method (RFEM) is a numerical approach for incorporating spatially variable properties into finite element analysis. In this study, an RFEM approach to simulating BEP was developed to assess pipe progression through variable soils. The soil hydraulic conductivity and critical hydraulic gradient for pipe progression were treated as log-normal random variables. Analyses were conducted for a range of hydraulic conductivity and critical gradient random fields with varied spatial correlation lengths, distribution parameters, and field correlations. Results indicate that the probability of failure increases with increasing spatial correlation length. Additionally, increased variance in soil permeability was shown to increase the probability of failure for large correlation lengths and decrease the probability of failure for short correlation lengths. Regardless of correlation length, increasing values of the mean critical hydraulic gradient led to decreased failure probabilities, and increasing values of critical hydraulic gradient variance led to increased failure probabilities.

Computing the partition function of the Sherrington–Kirkpatrick model is hard on average

We establish the average-case hardness of the algorithmic problem of exact computation of the partition function associated with the Sherrington–Kirkpatrick model of spin glasses with Gaussian couplings and random external field. In particular, we establish that unless P=#P, there does not exist a polynomial-time algorithm to exactly compute the partition function on average. This is done by showing that if there exists a polynomial time algorithm, which exactly computes the partition function for inverse polynomial fraction (1/nO(1)) of all inputs, then there is a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding P=#P. The computational model that we adopt is finite-precision arithmetic, where the algorithmic inputs are truncated first to a certain level N of digital precision. The ingredients of our proof include the random and downward self-reducibility of the partition function with random external field; an argument of Cai et al. (In STACS 99 (Trier) (1999) 90–99 Springer) for establishing the average-case hardness of computing the permanent of a matrix; a list-decoding algorithm of Sudan (In 37th Annual Symposium on Foundations of Computer Science (Burlington, VT, 1996) (1996) 164–172 IEEE Comput. Soc. Press), for reconstructing polynomials intersecting a given list of numbers at sufficiently many points; and near-uniformity of the log-normal distribution, modulo a large prime p. To the best of our knowledge, our result is the first one establishing a provable hardness of a model arising in the field of spin glasses. Furthermore, we extend our result to the same problem under a different real-valued computational model, for example, using a Blum–Shub–Smale machine (In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science (1988) 387–397 IEEE) operating over real-valued inputs. We establish that, if there exists a polynomial time algorithm which exactly computes the partition function for 34+1nO(1) fraction of all inputs, then there exists a polynomial time algorithm, which exactly computes the partition function for all inputs, with high probability, yielding P=#P. Our proof uses the random self-reducibility of the partition function, together with a control over the total variation distance for log-normal random variables in presence of a convex perturbation, and the Berlekamp–Welch algorithm.

Solving the multi-objective stochastic interval-valued linear fractional integer programming problem

In this paper, we consider a Multi-Objective Stochastic Interval-Valued Linear Fractional Integer Programming problem (MOSIVLFIP). We especially deal with a multi-objective stochastic fractional problem involving an inequality type of constraints, where all quantities on the right side are log-normal random variables, and the objective functions coefficients are fractional intervals. The proposed solving procedure is divided in three steps. In the first one, the probabilistic constraints are converted into deterministic ones by using the chance constrained programming technique. Then, the second step consists of transforming the studied problem objectives on an optimization problem with an interval-valued objective functions. Finally, by introducing the concept of weighted sum method, the equivalent converted problem obtained from the two first steps is transformed into a single objective deterministic fractional problem. The effectiveness of the proposed procedure is illustrated through a numerical example.

Modeling the complete kinetics of coxsackievirus B3 reveals human determinants of host-cell feedback.

Complete kinetic models are pervasive in chemistry but lacking in biological systems. We encoded the complete kinetics of infection for coxsackievirus B3 (CVB3), a compact and fast-acting RNA virus. The model consists of separable, detailed modules describing viral binding-delivery, translation-replication, and encapsidation. Specific module activities are dampened by the type I interferon response to viral double-stranded RNAs (dsRNAs), which is itself disrupted by viral proteinases. The experimentally validated kinetics uncovered that cleavability of the dsRNA transducer mitochondrial antiviral signaling protein (MAVS) becomes a stronger determinant of viral outcomes when cells receive supplemental interferon after infection. Cleavability is naturally altered in humans by a common MAVS polymorphism, which removes a proteinase-targeted site but paradoxically elevates CVB3 infectivity. These observations are reconciled with a simple nonlinear model of MAVS regulation. Modeling complete kinetics is an attainable goal for small, rapidly infecting viruses and perhaps viral pathogens more broadly. A record of this paper's transparent peer review process is included in the Supplemental information.

Moment-matching approximations for stochastic sums in non-Gaussian Ornstein–Uhlenbeck models

In this paper, we recall actuarial and financial applications of sums of dependent random variables that follow a non-Gaussian mean-reverting process and contemplate distribution approximations. Our work complements previous related studies restricted to lognormal random variables; we revisit previous approximations and suggest new ones. We then derive moment-based distribution approximations for random sums attuned to Asian option pricing and computation of risk measures of random annuities. Various numerical experiments highlight the speed–accuracy benefits of the proposed methods.

Statistical calibration of federal highway administration simplified models for facing tensile forces of soil nail walls

This study first develops a database containing 56 measured facing tensile forces from instrumented soil nail walls during or at completion of construction. Based on the compiled database, the accuracies of both default and modified federal highway administration (FHWA) simplified models for estimation of short-term facing tensile forces are evaluated. Here, accuracy is defined by the model bias computed as the ratio of measured to predicted facing tensile force. The analysis results show that predictions are highly conservative and highly dispersive using the default model, and moderately conservative and medium dispersive using the modified model. Moreover, the prediction accuracy is statistically correlated with magnitudes of the computed facing tensile forces and several model input parameters. An out-of-sample approach is used to develop and validate a recalibrated FHWA model which is then demonstrated to have least empirical constants but best accuracy compared to the default and modified models. The biases for the three models are characterized as lognormal random variables. An example of reliability-based analysis for facing flexure limit state is illustrated to both elaborate the application and highlight the benefit of using the recalibrated model for design practice from the perspective of cost-effectiveness.

Mechanical‐probabilistic formulation of the soil–structure interaction, accounting for the average shear wave velocity

SummaryThis work proposes a mechanical‐probabilistic formulation of the soil–structure interaction, taking into account the variability of the mean velocity of shear wave seismic load. This work aims at studying the sensitivity of the seismic response of a reinforced concrete frame structure by accounting for different seismic scenarios. For this purpose, the simplified model of a two‐story structure system has been developed. This model is based both on a macro‐element approach and on a stochastic collocation method. The sensitivity of the maximum displacement of the structure under different average shear wave velocities is studied. This aim in view, Gaussian or lognormal random variables are arbitrarily chosen to model this velocity. Analysis of the seismic vulnerability of the soil–structure system leads to cumulative distribution functions, which sometimes show significant differences for the same soil class in the sense of the European standard for the design of structures for earthquake resistance.

Non-intrusive reliability analysis of unsaturated embankment slopes accounting for spatial variabilities of soil hydraulic and shear strength parameters

Existing reliability analyses of embankment slopes usually considered the spatial variabilities of shear strength parameters and saturated hydraulic conductivity separately. Additionally, the variability in the soil–water characteristic curve (SWCC) was ignored. A non-intrusive approach is proposed for reliability analyses of unsaturated embankment slopes under four different cases on a steady-state seepage analysis. The spatial variabilities of hydraulic parameters (including fitting parameters, a and n, of SWCC and saturated hydraulic conductivity) and shear strength parameters are accounted for simultaneously. To illustrate the proposed approach, a hypothetical embankment under unsaturated seepage is investigated. The soil parameters of the embankment and foundation are modeled as lognormal random fields and lognormal random variables, respectively, to take into account the inherent uncertainties. Parametric sensitivity studies are then performed to account explicitly for the influence of the spatial variation of hydraulic parameters on the reliability of embankment slope stability. The proposed approach can act as a practical and rigorous tool for estimating the reliability of embankment slopes with small levels of probability of failure (i.e., 10−4–10−3). An interesting finding that the probability of failure of the unsaturated embankment slope decreases with the variability of the fitting parameter n of SWCC is observed and explained.

Artificial increasing returns to scale and the problem of sampling from lognormals

We show how increasing returns to scale in urban scaling can artificially emerge, systematically and predictably, without any sorting or positive externalities. We employ a model where individual productivities are independent and identically distributed lognormal random variables across all cities. We use extreme value theory to demonstrate analytically the paradoxical emergence of increasing returns to scale when the variance of log-productivity is larger than twice the log-size of the population size of the smallest city in a cross-sectional regression. Our contributions are to derive an analytical prediction for the artificial scaling exponent arising from this mechanism and to develop a simple statistical test to try to tell whether a given estimate is real or an artifact. Our analytical results are validated analyzing simulations and real microdata of wages across municipalities in Colombia. We show how an artificial scaling exponent emerges in the Colombian data when the sizes of random samples of workers per municipality are 1% or less of their total size.