Multivariate Lognormal Distribution Research Articles

Quantile-Based Multivariate Log-Normal Distribution

We introduce a quantile-based multivariate log-normal distribution, providing a new multivariate skewed distribution with positive support. The parameters of this distribution are interpretable in terms of quantiles of marginal distributions and associations between pairs of variables, a desirable feature for statistical modeling purposes. We derive statistical properties of the quantile-based multivariate log-normal distribution involving the transformations, closed-form expressions for the mixed moments, expected value, covariance matrix, mode, Shannon entropy, and Kullback–Leibler divergence. We also present results on marginalization, conditioning, and independence. Additionally, we discuss parameter estimation and verify its performance through simulation studies. We evaluate the model fitting based on Mahalanobis-type distances. An application to children data is presented.

Spatio-Temporal Analysis and Prediction of Mass Telecommunication Base Station Failure Events

Large-scale telecommunication systems are lifeline infrastructure in modern society. A telecommunication system typically consists of a huge number of base stations with diverse geographical locations across a country, which highly complicates maintenance operations. To allocate maintenance resource properly, it is important to have a good understanding on the failure pattern of these base stations. Statistical inference of recurrent failures of these base stations is challenging because of the large number of base stations and the spatial correlation of their failure processes. Based on eight-month failure data of telecommunication base stations in Harbin, China, we propose a customized nonhomogeneous Poisson process (NHPP) model for recurrent failure data from telecommunication systems. The model consists of two layers, where the temporal layer applies an NHPP with station-specific frailty for failures of each base station, and the spatial layer uses a multivariate lognormal distribution to characterize the correlation among the frailties. The Monte Carlo EM (MCEM) algorithm is applied to estimate parameters included in the proposed model. We demonstrate the proposed model using the Harbin telecommunication system example with 7725 base stations and 4615 failure records.

Estimating Earthquake-Induced Tsunami Height Probabilities without Sampling

Given a distribution of earthquake-induced seafloor elevations, we present a method to compute the probability of the resulting tsunamis reaching a certain size on shore. Instead of sampling, the proposed method relies on optimization to compute the most likely fault slips that result in a seafloor deformation inducing a large tsunami wave. We model tsunamis induced by bathymetry change using the shallow water equations on an idealized slice through the sea. The earthquake slip model is based on a sum of multivariate log-normal distributions, and follows the Gutenberg-Richter law for seismic moment magnitudes ranging from 7 to 9. For a model problem inspired by the Tohoku-Oki 2011 earthquake and tsunami, we quantify annual probabilities of differently sized tsunami waves. Our method also identifies the most effective tsunami mechanisms. These mechanisms have smoothly varying fault slip patches that lead to an expansive but moderately large bathymetry change. The resulting tsunami waves are compressed as they approach shore and reach close-to-vertical leading wave edge close to shore.

MvLognCorrEst: an R package for sampling from multivariate lognormal distributions and estimating correlations from uncomplete correlation matrix

Background and ObjectivePharmacometrics (PMX) is a quantitative discipline which supports decision-making processes in all stages of drug development. PMX leverages Modeling and Simulations (M&S), which represents a powerful tool to characterize and predict the behavior and the effect of a drug. M&S-based methods, such as Sensitivity Analysis (SA) and Global Sensitivity Analysis (GSA), are gaining interest in PMX as they allow the evaluation of model-informed inference quality. Simulations should be correctly designed to obtain reliable results. Neglecting correlations between model parameters can significantly alter the results of simulations. However, the introduction of a correlation structure between model parameters can cause some issues. Sampling from a multivariate lognormal distribution, which is the typically distribution assumed for PMX model parameters, is not straightforward when a correlation structure is introduced. Indeed, correlations need to respect some constraints which depend by the CVs (i.e., coefficients of variation) of lognormal variables. In addition, when correlation matrices have some unspecified values, they should be properly fixed preserving the positive semi-definiteness of the correlation structure. In this paper, we present mvLognCorrEst, an R package developed to address these issues. MethodsThe proposed sampling strategy was based on reconducting the extraction from the multivariate lognormal distribution of interest to the underlying Normal distribution. However, with high lognormal CVs, a positive semi-definite Normal covariance matrix cannot be obtained due to the violation of some theoretical constraints. In these cases, the Normal covariance matrix was approximated to its nearest positive definite matrix using Frobenius norm as matrix distance. For the estimation of unknown correlations terms, the graph theory was used to represent the correlation structure as weighed undirected graph. Plausible value ranges for the unspecified correlations were derived considering the paths between variables. Then, their estimation was performed by solving a constrained optimization problem. Results: Package functions are presented and applied on a real case study, that is the GSA of a PMX model that has been recently developed to support preclinical oncological studies. ConclusionsmvLognCorrEst package is an R tool to support simulation-based analysis for which sampling from multivariate lognormal distributions with correlated variables and/or estimation of partially defined correlation matrix are required.

Land Plots Evaluation for Agriculture and Green Energy Projects: How to Overcome the Conflict Using Mathematics

Seventeen sustainable development goals were formulated to create a harmonious world order for the benefit of different nations and peoples. At the same time, economic practice provides a lot of examples of conflicts of an economic nature between individual sustainable development goals. One of these conflicts is the need for environmental imperatives and economic growth when a massive assessment of land used for crop production and green energy projects is needed. The present paper considers a non-traditional approach to the mass evaluation of land plots on the condition that geographic information systems provide the main source of information, such as the case of land allocation for green energy facilities and evaluation of agricultural plots. The novelty of the proposed approach firstly means the development of a comparative approach, which receives much less attention in the valuation literature than cost and income approaches, as it can give an adequate picture of the current state of the market. The model includes the study of the entire dataset, the selection of model distributions and the construction of estimates based on model distributions. The methodology of multivariate lognormal distribution of factors and prices of analogues is used. The peculiarity of the market evaluation of land plots in such cases is, as a rule, the absence of rank predictors and sufficient number of continuous predictors, which provides a base for the application of a novel approach. The method of express testing of hypotheses about joint normality of logarithms of values of pricing factors and prices is proposed. The market value is estimated as an estimate of the modal value of conditional lognormal price distribution. Secondly, the problem of market valuation is solved in case of the almost complete absence of information about price-forming factors in the areas being assessed, and thirdly, the factors are determined based on geoinformation databases (distance to the nearest large city, regional center, federal highway, large rivers, lakes, and solid waste landfills), which allow for market assessment in the absence of information on pricing factors for land plots, except for the offer price and the plot area. The research was necessitated by the claim to determine on a specific date the cadastral value of agricultural land for the purposes of taxation, corresponding to the market value, in the almost complete absence of information on pricing factors in the assessed areas. The value of land reflects a complex combination of factors, so the use of the proposed mathematical toolkit allows for building a consistent model for the evaluation of land where improvements are absent or have no value in terms of land acquisition purposes.

Analysis of an incomplete binary outcome dichotomized from an underlying continuous variable in clinical trials.

In many clinical trials, outcomes of interest are binary-valued. It is not uncommon that a binary-valued outcome is dichotomized from a continuous outcome at a threshold of clinical interest. To analyze such data, common approaches include (a) fitting a generalized linear mixed model (GLMM) to the dichotomized longitudinal binary outcome; and (b) the multiple imputation (MI) based method: imputing missing values in the continuous outcome, dichotomizing it into a binary outcome, and then fitting a generalized linear model to the "complete" data. We conducted comprehensive simulation studies to compare the performance of the GLMM versus the MI-based method for estimating the risk difference and the logarithm of odds ratio between two treatment arms at the end of study. In those simulation studies, we considered a range of multivariate distribution options for the continuous outcome (including a multivariate normal distribution, a multivariate t-distribution, a multivariate log-normal distribution, and the empirical distribution from a real clinical trial data) to evaluate the robustness of the estimators to various data-generating models. Simulation results demonstrate that both methods work well under those considered distribution options, but the MI-based method is more efficient with smaller mean squared errors compared to the GLMM. We further applied both the GLMM and the MI-based method to 29 phase 3 diabetes clinical trials, and found that the MI-based method generally led to smaller variance estimates compared to the GLMM.

New bivariate and multivariate log-normal distributions as models for insurance data

The body of most multivariate financial data sets can be well modeled by log-normal distributions. Yet not many multivariate log-normal distributions are available in the literature. In this paper, we propose many new multivariate log-normal distributions. An application to an insurance data set is given.

Mvlogncorrest: An R Package for Sampling from Multivariate Lognormal Distributions and Estimating Correlations from Uncomplete Correlation Matrix

Mvlogncorrest: An R Package for Sampling from Multivariate Lognormal Distributions and Estimating Correlations from Uncomplete Correlation Matrix

Habits in frequency of purchase models: the case of fish in France

ABSTRACT The quantity purchased in a period is the result of two decisions: the frequency of purchase and the average purchase on each occasion. We introduce habit formation into demand systems modelling each of these decisions. An econometric model is estimated by Bayesian methods. The data generating processes of the frequency and quantity decisions are assumed to follow, respectively, a multivariate Poisson log-normal distribution and a multivariate gamma log-normal distribution. We estimate the systems using French scanner data for purchases of fish. The results suggest that habits in purchase frequencies are important, while habits in average purchased quantities are less important. Furthermore, we find that price changes are important for explaining average quantities purchased but have only minor effects on purchase frequencies.

Joint Modeling of Received Power, Mean Delay, and Delay Spread for Wideband Radio Channels

We propose a multivariate log-normal distribution to jointly model received power, mean delay, and root mean square (rms) delay spread of wideband radio channels, referred to as the standardized temporal moments. The model is validated using experimental data collected from five different measurement campaigns (four indoor scenarios and one outdoor scenario). We observe that the received power, the mean delay, and the rms delay spread are correlated random variables, and therefore, should be simulated jointly. Joint models are able to capture the structure of the underlying process, unlike the independent models considered in the literature. The proposed model of the multivariate log-normal distribution is found to be a good fit for a large number of wideband data sets.

A Multilevel Empirical Bayesian Approach to Estimating the Unknown Redshifts of 1366 BATSE Catalog Long-duration Gamma-Ray Bursts

Abstract We present a catalog of probabilistic redshift estimates for 1366 individual Long-duration Gamma-ray Bursts (LGRBs) detected by the Burst And Transient Source Experiment (BATSE). This result is based on a careful selection and modeling of the population distribution of 1366 BATSE LGRBs in the five-dimensional space of redshift and the four intrinsic prompt gamma-ray emission properties: the isotropic 1024 ms peak luminosity ( ), the total isotropic emission ( ), the spectral peak energy ( ), as well as the intrinsic duration ( ), while carefully taking into account the effects of sample incompleteness and the LGRB-detection mechanism of BATSE. Two fundamental plausible assumptions underlie our purely probabilistic approach: (1) LGRBs trace, either exactly or closely, the cosmic star formation rate, with a possibility of the excess rates of LGRBs in the nearby universe, and (2) the joint four-dimensional distribution of the aforementioned prompt gamma-ray emission properties is well described by a multivariate log-normal distribution. Our modeling approach enables us to constrain the redshifts of individual BATSE LGRBs to within 0.36 and 0.96 average uncertainty ranges at 50% and 90% confidence levels, respectively. Our redshift predictions are completely at odds with the previous redshift estimates of BATSE LGRBs that were computed via the proposed phenomenological high-energy relations, specifically, the apparently strong correlation of LGRBs’ peak luminosity with the spectral peak energy, lightcurve variability, and spectral lag. The observed discrepancies between our predictions and the previous works can be explained by the strong influence of detector threshold and sample incompleteness in shaping these phenomenologically proposed high-energy correlations in the literature. Finally, we also discuss the potential effects of an excess cosmic rate of LGRBs at low redshifts and the possibility of a luminosity evolution of LGRBs on our results.

Market and cadastral data comparison for the real estate market value forecasting

The aim of the research is to develop theoretical and methodological approaches to market value forecasting in the real estate market. The relevance of the research is determined by the system-forming place that the real estate market occupies in the economy of the country and regions, affecting the interests of owners of various forms of ownership, construction and development companies, insurance companies, banks. Another aspect that determines the actuality of the study is the discrepancy between well-structured cadastral databases and market data dispersed between different owners of information resources, and the unstructured nature of market data, which in most cases is focused on advertising, rather than on analytical market research.Materials and methods. The study uses a model of a multidimensional logarithmically normal distribution law of the ensemble of prices for residential real estate at equidistant points of time and cadastral value, the ARIMA model for predicting market value, taking into account the features of the logarithmically normal distribution of prices, as a distribution with positive asymmetry. As a statistical material, we used market data on residential real estate published in the periodical press in the period from the end of 2012 to 2018. The volume of samples of weekly publications is 15000-20000 objects; data for 21 quarters (more than five years) was used. As a comparison base, we used data from cadastral registration of real estate objects in Saint Petersburg for 2018. The total volume of the cadastral database of residential real estate in Saint Petersburg (individual apartments) is 2 226734 objects with a fairly complete (and well-structured) set of price-forming factors. The authors propose a method for estimating the most likely movement of the market value for a pre-selected real estate object that has passed cadastral registration and has a cadastral value entered in the register and predicting the market value in the future period.Results. The theoretical significance of the work is the proposed algorithm for estimating the most probable trajectory of the market value of the investigated object, based on the conditional multivariate log-normal distribution for a given value of the cadastral value. A well-developed and studied ARIMA time series forecasting model is applied to the logarithms of the obtained time series, the return from logarithmic prices to real prices is carried out taking into account the peculiarities of the logarithmically normal distribution. Results are compared with median scores and estimates, obtained by average values.Conclusion. The paper shows that the introduction of cadastral value in the Russian Federation opens up new opportunities for analyzing and forecasting market prices, since cadastral databases contain the most complete lists of real estate objects, including the cadastral value, which now, in accordance with the law, must be updated at least once every three years and, as of 2015 and 2018, was determined as the market value, therefore, until the next cadastral assessment, can serve as a basis for constant comparison with market data, which are constantly changing, primarily in the composition of objects.

Generation of virtual patient data for in-silico cardiomyopathies drug development using tree ensembles: a comparative study.

In-silico clinical platforms have been recently used as a new revolutionary path for virtual patients (VP) generation and further analysis, such as, drug development. Advanced individualized models have been developed to enhance flexibility and reliability of the virtual patient cohorts. This study focuses on the implementation and comparison of three different methodologies for generating virtual data for in-silico clinical trials. Towards this direction, three computational methods, namely: (i) the multivariate log-normal distribution (log- MVND), (ii) the supervised tree ensembles, and (iii) the unsupervised tree ensembles are deployed and evaluated against their performance towards the generation of high-quality virtual data using the goodness of fit (gof) and the dataset correlation matrix as performance evaluation measures. Our results reveal the dominance of the tree ensembles towards the generation of virtual data with similar distributions (gof values less than 0.2) and correlation patterns (average difference less than 0.03).

The basic reproductive number for disease systems with multiple coupled heterogeneities

In mathematical epidemiology, a well-known formula describes the impact of heterogeneity on the basic reproductive number, R0, for situations in which transmission is separable and for which there is one source of variation in susceptibility and one source of variation in infectiousness. This formula is written in terms of the magnitudes of the heterogeneities, as quantified by their coefficients of variation, and the correlation between them. A natural question to ask is whether analogous results apply when there are multiple sources of variation in susceptibility and/or infectiousness. In this paper we demonstrate that with three or more coupled heterogeneities, R0 under separable transmission depends on details of the distribution of the heterogeneities in a way that is not seen in the well-known simpler situation. We provide explicit formulae for the cases of multivariate normal and multivariate log-normal distributions, showing that R0 can again be expressed in terms of the magnitudes of the heterogeneities and the pairwise correlations between them. The formulae, however, differ between the two multivariate distributions, demonstrating that no formula of this type applies generally when there are three or more coupled heterogeneities. We see that the results of the formulae are approximately equal when heterogeneities are relatively small and show that an earlier result in the literature (Koella, 1991) should be viewed in this light. We provide numerical illustrations of our results and discuss a setting in which coupled heterogeneities are likely to have a major impact on the value of R0. We also describe a rather surprising result: in a system with three heterogeneities, R0 can exhibit non-monotonic behavior with increasing levels of heterogeneity, in marked contrast to the familiar two heterogeneity setting in which R0 either increases or decreases with increasing heterogeneity.

Global alcohol exposure between 1990 and 2017 and forecasts until 2030: a modelling study

Alcohol use is a leading risk factor for global disease burden, and data on alcohol exposure are crucial to evaluate progress in achieving global non-communicable disease goals. We present estimates on the main indicators of alcohol exposure for 189 countries from 1990-2017, with forecasts up to 2030. Adult alcohol per-capita consumption (the consumption in L of pure alcohol per adult [≥15 years]) in a given year was based on country-validated data up to 2016. Forecasts up to 2030 were obtained from multivariate log-normal mixture Poisson distribution models. Using survey data from 149 countries, prevalence of lifetime abstinence and current drinking was obtained from Dirichlet regressions. The prevalence of heavy episodic drinking (30-day prevalence of at least one occasion of 60 g of pure alcohol intake among current drinkers) was estimated with fractional response regressions using survey data from 118 countries. Between 1990 and 2017, global adult per-capita consumption increased from 5·9 L (95% CI 5·8-6·1) to 6·5 L (6·0-6·9), and is forecasted to reach 7·6 L (6·5-10·2) by 2030. Globally, the prevalence of lifetime abstinence decreased from 46% (42-49) in 1990 to 43% (40-46) in 2017, albeit this was not a significant reduction, while the prevalence of current drinking increased from 45% (41-48) in 1990 to 47% (44-50) in 2017. We forecast both trends to continue, with abstinence decreasing to 40% (37-44) by 2030 (annualised 0·2% decrease) and the proportion of current drinkers increasing to 50% (46-53) by 2030 (annualised 0·2% increase). In 2017, 20% (17-24) of adults were heavy episodic drinkers (compared with 1990 when it was estimated at 18·5% [15·3-21·6%], and this prevalence is expected to increase to 23% (19-27) in 2030. Based on these data, global goals for reducing the harmful use of alcohol are unlikely to be achieved, and known effective and cost-effective policy measures should be implemented to reduce alcohol exposure. Centre for Addiction and Mental Health and the WHO Collaborating Center for Addiction and Mental Health at the Centre for Addiction and Mental Health.

Quantification of Optimal Choices of Parameters in Lognormal Variational Data Assimilation and Their Chaotic Behavior

An important property of variational-based data assimilation is the ability to define a functional formulation such that the minimum of that functional can be any state that is desired. Thus, it is possible to define cost functions such that the minimum of the background error component is the mean, median or the mode of a multivariate lognormal distribution, where, unlike the multivariate Gaussian distributions, these statistics are not equivalent. Therefore, for lognormal distributions it is shown here that there are regions where each one of these three statistics are optimal at minimizing the errors, given estimates of an a priori state. Also, as part of this work, a chaotic signal was detected with respect to the first guess to the Newton–Raphson solver that affect the accuracy of the solution to several decimal places.

Evaluation of the path integral for flow through random porous media.

We present a path integral formulation of Darcy's equation in one dimension with random permeability described by a correlated multivariate lognormal distribution. This path integral is evaluated with the Markov chain Monte Carlo method to obtain pressure distributions, which are shown to agree with the solutions of the corresponding stochastic differential equation for Dirichlet and Neumann boundary conditions. The extension of our approach to flow through random media in two and three dimensions is discussed.

State space modeling of autocorrelated multivariate Poisson counts

Although many applications involve autocorrelated multivariate counts, there is a scarcity of research on their statistical modeling. To fill this research gap, this article proposes a state space model to describe autocorrelated multivariate counts. The model builds upon the multivariate log-normal mixture Poisson distribution and allows for serial correlations by considering the Poisson mean vector as a latent process driven by a nonlinear autoregressive model. In this way, the model allows for flexible cross-correlation and autocorrelation structures of count data and can also capture overdispersion. The Monte Carlo Expectation Maximization algorithm, together with particle filtering and smoothing methods, provides satisfactory estimators for the model parameters and the latent process variables. Numerical studies show that, compared with other state-of-the-art models, the proposed model has superiority and more generality with respect to describing count data generated from different mechanisms of the process of counts. Finally, we use this model to analyze counts of different types of damage collected from a power utility system as a case study. Supplementary materials are available for this article. Go to the publisher’s online edition of IISE Transactions for additional tables and figures.

Statistical dependency analysis of multiple competing failure causes of fuel cell engines

Given the current strict regulations on fuel emissions, hydrogen fuel cell vehicles have been considered as ideal non-polluting vehicles. As the kernel subsystem of hydrogen fuel cell vehicles, the fuel cell engine, which is a complex system, may fail because of multiple competing failure causes. In fuel cell engine reliability engineering, discriminating and measuring dependency among multiple failure causes is an urgent problem that should be addressed. This study proposes a statistical dependency analysis model under competing risks based on reliability field data of fuel cell engines. The multivariate lognormal distribution is employed to model joint failure distribution. Then, using simulated annealing algorithm to estimate the parameters of the conditional probability likelihood function. Moreover, p-value hypothesis test procedures are developed to determine the significance of the dependence degree among multiple competing failure causes. In this article, the failure mechanisms of fuel cell engine are comprehensively discussed to prove the feasibility and effectiveness of the proposed method. Results can provide technical support for studies on optimal maintenance strategies.

The logarithmic vector multiplicative error model: an application to high frequency NYSE stock data

We develop a general form logarithmic vector multiplicative error model (log-vMEM). The log-vMEM improves on existing models in two ways. First, it is a more general form model as it allows the error terms to be cross-dependent and relaxes weak exogeneity restrictions. Second, the log-vMEM specification guarantees that the conditional means are non-negative without any restrictions imposed on the parameters. We further propose a multivariate lognormal distribution and a joint maximum likelihood estimation strategy. The model is applied to high frequency data associated with a number of NYSE-listed stocks. The results reveal empirical support for full interdependence of trading duration, volume and volatility, with the log-vMEM providing a better fit to the data than a competing model. Moreover, we find that unexpected duration and volume dominate observed duration and volume in terms of information content, and that volatility and volatility shocks affect duration in different directions. These results are interpreted with reference to extant microstructure theory.