Multivariate Probability Distribution Function Research Articles

Charging management of electric vehicles with the presence of renewable resources

Considering the increasing use of electric vehicles, the establishment of charging stations to exchange power between the grid and electric devices, and the integration of charging stations with solar power generation sources, the optimal use of electric vehicle charging stations in the power system. The purpose of cost reduction in the presence of the intelligent environment is a challenge that must be investigated so that this platform is suitable for predicting the behaviour of vehicles and, as a result, optimizing their presence in the power network. This research presents a relatively complete radial distribution network development planning model in two scenarios. In the first scenario, the effects of electric vehicles are not considered, and only the effects of distributed production (renewable and dispatchable) are considered. Studies have been done on a sample 54-bus network, a common system in most Distribution expansion planning (DEP) articles for distribution networks. In addition, the real data of American highways have been used to create raw input data. Also, due to the distance limit, the information on vehicles under 100 miles has been received as electric vehicle information. The clustering method and Capiola multivariate probability distribution functions have created suitable vehicle scenarios during different planning years. Capiola’s method increases the accuracy of vehicle load forecasting according to a predetermined growth rate. The DEP problem in this research is modeled as an optimization problem based on scenario, dynamic, and in 5 one-year time frames (5-year time horizon and one-year accuracy). The results indicate that, in the presence of electric vehicles and distributed production sources, the technical characteristics of the network are improved.Similarly, the use of DGs, in addition to reducing the cost of equipment, has reduced undistributed energy in the system. But 10,000 vehicles, which have been applied to the network as an uncontrolled load, have caused an increase in undistributed energy. The cost of equipment required for the network development is almost as much as 5%.

Interactive Visualization and Computation of 2D and 3D Probability Distributions.

Mathematical modeling, probability estimation, and statistical inference represent core elements of modern artificial intelligence (AI) approaches for data-driven prediction, forecasting, classification, risk-estimation, and prognosis. Currently there are many tools that help calculate and visualize univariate probability distributions, however, very few resources venture beyond into multivariate distributions, which are commonly used in advanced statistical inference and AI decision-making. This article presents a new web-calculator that enables some calculation and visualization of bivariate and trivariate probability distributions. Several methods are explored to compute the joint bivariate and trivariate probability densities, including the optimal multivariate modeling using Gaussian copula. We developed an interactive webapp to visually illustrate the parallels between the mathematical formulation, computational implementation, and graphical depiction of multivariate probability density and cumulative distribution functions. To ensure the interface and functionality are hardware platform independent, scalable, and functional, the app and its component widgets are implemented using HTML5 and JavaScript. We validated the webapp by testing the multivariate copula models under different experimental conditions and inspecting the performance in terms of accuracy and reliability of the estimated multivariate probability densities and distribution function values. This article demonstrates the construction, implementation, and utilization of multivariate probability calculators. The proposed webapp implementation is freely available online (https://socr.umich.edu/HTML5/BivariateNormal/BVN2/) and can be used to assist with education and research of a diverse array of data scientists, STEM instructors, and AI learners.

Risk probabilistic assessment of ultrahigh arch dams through regression panel modeling on deformation behavior

The deformations of ultrahigh arch dams can comprehensively indicate the dynamic variations of their structural behavior to judge the normal or not for timely discovering anomalies. First, the panel data features on the deformation behavior are extracted to effectively indicate the overall structural evolution of the Jinping I arch dam combing with the time series and the cross-section series. Afterwards, a regression panel model (RPM) on the multi-dimensional variables is proposed to model the deformation panel data consisted of multi-monitoring points synchronously. Subsequently, an innovative functional relationship between the measured values and the real-time risk probabilistic function is established due to the RPM estimation accuracy. In order to estimate the risk probability of the whole arch dam, the Copula function is used to build a multivariate joint probability distribution function to indicate the correlation among the random variables. The proposed methods are validated by an application on the Jinping I arch dam to evaluate its risk probability, which explores a novel approach for the arch dam safety assessment.

A machine learning approach to galaxy properties: joint redshift–stellar mass probability distributions with Random Forest

ABSTRACTWe demonstrate that highly accurate joint redshift–stellar mass probability distribution functions (PDFs) can be obtained using the Random Forest (RF) machine learning (ML) algorithm, even with few photometric bands available. As an example, we use the Dark Energy Survey (DES), combined with the COSMOS2015 catalogue for redshifts and stellar masses. We build two ML models: one containing deep photometry in the griz bands, and the second reflecting the photometric scatter present in the main DES survey, with carefully constructed representative training data in each case. We validate our joint PDFs for 10 699 test galaxies by utilizing the copula probability integral transform and the Kendall distribution function, and their univariate counterparts to validate the marginals. Benchmarked against a basic set-up of the template-fitting code bagpipes, our ML-based method outperforms template fitting on all of our predefined performance metrics. In addition to accuracy, the RF is extremely fast, able to compute joint PDFs for a million galaxies in just under 6 min with consumer computer hardware. Such speed enables PDFs to be derived in real time within analysis codes, solving potential storage issues. As part of this work we have developed galpro1, a highly intuitive and efficient python package to rapidly generate multivariate PDFs on-the-fly. galpro is documented and available for researchers to use in their cosmology and galaxy evolution studies.

A fault detection and isolation technique using nonlinear support vectors dichotomizing multi-class parity space residuals

Parity equation based residual generation with a statistical decision-making scheme has been proved to be effective for fault detection and isolation (FDI) in process systems. However, to implement a statistical decision-making scheme on residuals generated through parity equations, a multivariate probability distribution function of the residuals must be identified for each fault class. Often, the statistical properties of these residuals depend on the noise characteristics associated with the measurement systems, in addition to modeling errors. However, identification of the correct probability distribution function for the noise is a nontrivial task. The issue is further complicated due to the effects of the noise propagating through the transformation designed for minimizing the effects of the normal operating envelope and disturbances. To overcome such problems, a support vector machine (SVM) classification algorithm can be used in lieu of a statistical decision-making scheme. Optimal fault classifications can be achieved under the SVM framework using properly designed hyper-planes. To process the residuals using SVMs, several additional issues have to be dealt with, such as the non-linear classification problem, over-fitting issues, and multi-class fault classification. The objective of this paper is to describe a framework within which SVMs can be effectively integrated with parity equation based residual generation schemes. The developed framework has been evaluated on a physical process control platform under various fault scenarios. It has been shown that the scheme is feasible and particularly suitable for situations where it is difficult to obtain probability distribution functions for all potential fault classes.

Predicting sprinkler spray dispersion in FireFOAM

Accurate representation of fire sprinkler spray enables quantitative engineering analysis of fire suppression performance. Increasingly, fire sprinkler systems are evaluated using computer fire models in which sprinkler spray is simulated with Lagrangian particles. However, limited guidance exists as to how to predict the behavior of complex, spatio-stochastic fire sprinkler spray or how to accurately represent the dispersion of spray in terms of Lagrangian particles. The current work predicts the dispersion of the fire sprinkler spray generated by a canonical axisymmetric sprinkler using a Lagrangian particle tracking model within FireFOAM, an open source computational fluid dynamics fire model. In this work, the initial fire sprinkler spray is described with a new spray injection model characterized by a multivariate probability distribution function related to spatially resolved breakup radius, volume flux, drop size, and drop velocity. This function is stochastically sampled to generate Lagrangian particles representative of the near-field spray and the dispersion of these Lagrangian particles is in turn simulated in FireFOAM to predict the far-field spray. Our new model is compared to a baseline FireFOAM spray injection model which incorporated fewer spray details. Modeled results are compared to highly resolved far-field measurements of axisymmetric sprinkler sprays generated by the Spatially-Resolved Spray Scanning System (4S).

Random taste heterogeneity in discrete choice models: Flexible nonparametric finite mixture distributions

This study proposes a mixed logit model with multivariate nonparametric finite mixture distributions. The support of the distribution is specified as a high-dimensional grid over the coefficient space, with equal or unequal intervals between successive points along the same dimension; the location of each point on the grid and the probability mass at that point are model parameters that need to be estimated. The framework does not require the analyst to specify the shape of the distribution prior to model estimation, but can approximate any multivariate probability distribution function to any arbitrary degree of accuracy. The grid with unequal intervals, in particular, offers greater flexibility than existing multivariate nonparametric specifications, while requiring the estimation of a small number of additional parameters. An expectation maximization algorithm is developed for the estimation of these models. Multiple synthetic datasets and a case study on travel mode choice behavior are used to demonstrate the value of the model framework and estimation algorithm. Compared to extant models that incorporate random taste heterogeneity through continuous mixture distributions, the proposed model provides better out-of-sample predictive ability. Findings reveal significant differences in willingness to pay measures between the proposed model and extant specifications. The case study further demonstrates the ability of the proposed model to endogenously recover patterns of attribute non-attendance and choice set formation.

Discussion: Prediction of long-term urban stormwater loads at single sites

Contribution by Fahimeh Sarmadi, Mohammad Azmi and Mana Gharun May and Sivakumar (2013) have predicted the long-term urban stormwater loads at single sites, using five different measures of central tendency. They have stated that their motivation in using measures of central tendency is one in which, in contrast to regression and process-based methods, there is no need for correlating urban stormwater loads with other variables (e.g. discharge), as well as less complexity. Overall, the flow-weighted mean concentration has produced the most accurate predictions of long-term loads. Disciplines Engineering | Science and Technology Studies Publication Details May, D., Sivakumar, M., Sarmadi, F., Azmi, M. & Gharun, M. (2014). Discussion: prediction of long-term urban stormwater loads at single sites. Proceedings of the Institution of Civil Engineers: Water Management, 167 (8), 482-484. This journal article is available at Research Online: http://ro.uow.edu.au/eispapers/3362 Proceedings of the Institution of Civil Engineers Water Management 167 September 2014 Issue WM8 Pages 482–484 http://dx.doi.org/10.1680/wama.13.00108 Paper 1300108 Published online 20/12/2013 ICE Publishing: All rights reserved Water Management Volume 167 Issue WM8 Discussion May, Sivakumar, Sarmadi, Azmi and Gharun Discussion: Prediction of long-term urban stormwater loads at single sites Daniel May BE, PhD Sustainable Water and Energy Research Group, School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong, Australia Muttucumaru Sivakumar ME, PhD, MIEAust, CPEng Associate Professor, Sustainable Water and Energy Research Group, School of Civil, Mining and Environmental Engineering, University of Wollongong, Wollongong, Australia Fahimeh Sarmadi MsC Water Engineering Department, Imam Khomeini International University, Qazvin, Iran Mohammad Azmi PhD Candidate, Civil Engineering Department, Faculty of Engineering, Monash University, Melbourne, Australia Mana Gharun PhD Candidate, Faculty of Agriculture and Environment, University of Sydney, Sydney, Australia Contribution by Fahimeh Sarmadi, Mohammad Azmi and Mana Gharun May and Sivakumar (2013) have predicted the long-term urban stormwater loads at single sites, using five different measures of central tendency. They have stated that their motivation in using measures of central tendency is one in which, in contrast to regression and process-based methods, there is no need for correlating urban stormwater loads with other variables (e.g. discharge), as well as less complexity. Overall, the flow-weighted mean concentration has produced the most accurate predictions of long-term loads. However, the following points are worth discussing: The definite superiority of the flow-weighted mean method obviously shows that there is sufficient correlation between loads and runoff, and therefore it suggests that regression methods could derive more accurate results. Interestingly, the authors claim in a different study (May and Sivakumar, 2009a) that the comparison between different statistical models shows that regression models are more applicable than simple estimates of mean concentration in urban areas. Therefore, here, May and Sivakumar should compare different mean functions at least with a simple regression model to support their new claim. From an environmental point of view, evaluating the trends, extreme events, dispersion tendency characteristics and bias are usually more effective than using a single value (e.g. long-term mean) for analysing/mitigating natural or human-induced side effects. For instance, Colin (1995) reports that urban stormwater loads have bias, which means the mode and arithmetic mean of samples are not related to the same load value; hence, pollution management for an area can be adjusted depending on the side of bias. In a different example, Vogel et al. (2005) state that the natural characteristics of long-term stormwater loads, at least in their case studies, are more sophisticated than using simple statistical methods. Therefore, Vogel et al. (2005) introduce a multivariate probability distribution function to model load– concentration–discharge relationships in order to predict short and long-term stormwater loads. Furthermore, they derive benefit from using L-moment statistical analysis to discover the complexities of this phenomenon. Overall, the current study presents valuable points; however, the title and concepts should clarify that the aim of the research is simply to compare the performance of different measures of central tendency in deriving site mean concentration, and not to predict long-term urban stormwater loads.

Flipping of multivariate aggregation functions

We consider flipping transformations of multivariate aggregation functions and we investigate the closure of these transformations with respect to the class of aggregation functions with annihilator element equal to 0. Moreover, the consecutive application of flipping transformations is also discussed. The results present interesting connections both with quasi-copulas and with suitable modifications of the n-increasing property of a multivariate probability distribution function.

Multi-variate joint PDF for non-Gaussianities: exact formulation and generic approximations

We provide an exact expression for the multi-variate joint probability distribution function of non-Gaussian fields primordially arising from local transformations of a Gaussian field. This kind of non-Gaussianity is generated in many models of inflation. We apply our expression to the non-Gaussianity estimation from Cosmic Microwave Background maps and the halo mass function where we obtain analytical expressions. We also provide analytic approximations and their range of validity. For the Cosmic Microwave Background we give a fast way to compute the PDF which is valid up to more than 7σ for fNL values (both true and sampled) not ruled out by current observations, which consists of expressing the PDF as a combination of bispectrum and trispectrum of the temperature maps. The resulting expression is valid for any kind of non-Gaussianity and is not limited to the local type. The above results may serve as the basis for a fully Bayesian analysis of the non-Gaussianity parameter.

Probabilistic projections of transient climate change

This paper describes a Bayesian methodology for prediction of multivariate probability distribution functions (PDFs) for transient regional climate change. The approach is based upon PDFs for the equilibrium response to doubled carbon dioxide, derived from a comprehensive sampling of uncertainties in modelling of surface and atmospheric processes, and constrained by multiannual mean observations of recent climate. These PDFs are sampled and scaled by global mean temperature predicted by a Simple Climate Model (SCM), in order to emulate corresponding transient responses. The sampled projections are then reweighted, based upon the likelihood that they correctly replicate observed historical changes in surface temperature, and combined to provide PDFs for 20 year averages of regional temperature and precipitation changes to the end of the twenty-first century, for the A1B emissions scenario. The PDFs also account for modelling uncertainties associated with aerosol forcing, ocean heat uptake and the terrestrial carbon cycle, sampled using SCM configurations calibrated to the response of perturbed physics ensembles generated using the Hadley Centre climate model HadCM3, and other international climate model simulations. Weighting the projections using observational metrics of recent mean climate is found to be as effective at constraining the future transient response as metrics based on historical trends. The spread in global temperature response due to modelling uncertainty in the carbon cycle feedbacks is determined to be about 65–80 % of the spread arising from uncertainties in modelling atmospheric, oceanic and aerosol processes of the climate system. Early twenty-first century aerosol forcing is found to be extremely unlikely to be less than −1.7 W m−2. Our technique provides a rigorous and formal method of combining several lines of evidence used in the previous IPCC expert assessment of the Transient Climate Response. The 10th, 50th and 90th percentiles of our observationally constrained PDF for the Transient Climate Response are 1.6, 2.0 and 2.4 °C respectively, compared with the 10–90 % range of 1.0–3.0 °C assessed by the IPCC.

Modeling parameters of structured clays as a multivariate normal distribution

This study explores the possibility of modeling liquidity index, undrained shear strength, remolded undrained shear strength, preconsolidation stress, and vertical effective stress of structured clays (sensitive or quick clays) as a multivariate normal distribution. The literature is replete with correlation equations between two soil parameters. Consistent synthesis of more than two soil parameters through construction of a multivariate probability distribution function is rare, despite obvious practical usefulness of such an approach. This study compiles a large database of structured clays to construct the multivariate probability distribution among the aforementioned five soil parameters. This multivariate distribution is then used to simulate the correlations between soil parameters of interest and to derive useful equations for Bayesian inference. This constructed multivariate distribution and equations are further validated by another independent database of structured clays as well as by empirical equations proposed in the literature.

The entropy in finite N-unit nonextensive systems: The normal average and q-average

We discuss the Tsallis entropy in finite N-unit nonextensive systems by using the multivariate q-Gaussian probability distribution functions (PDFs) derived by the maximum entropy methods with the normal average and the q-average (q: the entropic index). The Tsallis entropy obtained by the q-average has an exponential N dependence: Sq(N)/N≃e(1−q)NS1(1) for large N (⪢1/(1−q)>0). In contrast, the Tsallis entropy obtained by the normal average is given by Sq(N)/N≃[1/(q−1)N] for large N (⪢1/(q−1)>0). N dependences of the Tsallis entropy obtained by the q- and normal averages are generally quite different, although both results are in fairly good agreement for |q−1|⪡1.0. The validity of the factorization approximation (FA) to PDFs, which has been commonly adopted in the literature, has been examined. We have calculated correlations defined by Cm=⟨(δxiδxj)m⟩−⟨(δxi)m⟩⟨(δxj)m⟩ for i≠j where δxi=xi−⟨xi⟩, and the bracket ⟨⋅⟩ stands for the normal and q-averages. The first-order correlation (m=1) expresses the intrinsic correlation and higher-order correlations with m≥2 include nonextensivity-induced correlation, whose physical origin is elucidated in the superstatistics.

FlowFP: A Bioconductor Package for Fingerprinting Flow Cytometric Data

A new software package called flowFP for the analysis of flow cytometry data is introduced. The package, which is tightly integrated with other Bioconductor software for analysis of flow cytometry, provides tools to transform raw flow cytometry data into a form suitable for direct input into conventional statistical analysis and empirical modeling software tools. The approach of flowFP is to generate a description of the multivariate probability distribution function of flow cytometry data in the form of a “fingerprint.” As such, it is independent of a presumptive functional form for the distribution, in contrast with model-based methods such as Gaussian Mixture Modeling. FlowFP is computationally efficient and able to handle extremely large flow cytometry data sets of arbitrary dimensionality. Algorithms and software implementation of the package are described. Use of the software is exemplified with applications to data quality control and to the automated classification of Acute Myeloid Leukemia.

A focus on high‐content cytometry

A focus on high‐content cytometry

Cytometric fingerprinting: Quantitative characterization of multivariate distributions

Recent technological advances in flow cytometry instrumentation provide the basis for high-dimensionality and high-throughput biological experimentation in a heterogeneous cellular context. Concomitant advances in scalable computational algorithms are necessary to better utilize the information that is contained in these high-complexity experiments. The development of such tools has the potential to expand the utility of flow cytometric analysis from a predominantly hypothesis-driven mode to one of discovery, or hypothesis-generating research. A new method of analysis of flow cytometric data called Cytometric Fingerprinting (CF) has been developed. CF captures the set of multivariate probability distribution functions corresponding to list-mode data and then "flattens" them into a computationally efficient fingerprint representation that facilitates quantitative comparisons of samples. An experimental and synthetic data were generated to act as reference sets for evaluating CF. Without the introduction of prior knowledge, CF was able to "discover" the location and concentration of spiked cells in ungated analyses over a concentration range covering four orders of magnitude, to a lower limit on the order of 10 spiked events in a background of 100,000 events. We describe a new method for quantitative analysis of list-mode cytometric data. CF includes a novel algorithm for space subdivision that improves estimation of the probability density function by dividing space into nonrectangular polytopes. Additionally it renders a multidimensional distribution in the form of a one-dimensional multiresolution hierarchical fingerprint that creates a computationally efficient representation of high dimensionality distribution functions. CF supports both the generation and testing of hypotheses, eliminates sources of operator bias, and provides an increased level of automation of data analysis.

Monge Property and Bounding Multivariate Probability Distribution Functions with Given Marginals and Covariances

Multivariate probability distributions with given marginals are considered, along with linear functionals, to be minimized or maximized, acting on them. The functionals are supposed to satisfy the Monge or inverse Monge or some higher order convexity property, and they may be only partially known. Existing results in connection with Monge arrays are reformulated and extended in terms of linear programming dual feasible bases. Lower and upper bounds are given for the optimum value as well as for unknown coefficients of the objective function, based on the knowledge of some dual feasible bases and corresponding objective function coefficients. In the two- and three-dimensional cases dual feasible bases are obtained for the problem, where not only the univariate marginals but also the covariances of the pairs of random variables are known.

Application of large deviation methods to the pricing of index options in finance

We develop an asymptotic formula for calculating the implied volatility of European index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for evaluating the multivariate probability distribution function for stock prices, which is based on large-deviation estimates of diffusion processes densities by Varadhan (Comm. Pure Appl. Math. 20 (1967)). A detailed version of these results can be found in (RISK 15 (10) (2002)). To cite this article: M. Avellaneda et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Reconstruction of Volatility: Pricing Index Options by the Steepest Descent Approximation

We propose a formula for calculating the implied volatility of index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for the multivariate probability distribution function of forward prices. A simple financial justification is provided. We apply the formula to compute the implied volatilities of liquidly-traded options on exchange-traded funds (ETF) across different strikes. Our theoretical results were found to be in good agreement with contemporaneous quotes on the Chicago Board of Options Exchange (CBOE) and the American Stock Exchange (AMEX).

Probability bounds given by hypercherry trees

In this article new lower and upper bounds are given for the probability of the union of events. For this purpose the new concept of hypercherry trees has been introduced. Earlier the concept of cherry tree and its application for bounding the probability of union of events was introduced by Bukszár and Prékopa. This, based on the cherry tree bound, is always an upper bound, and it can be regarded as a generalisation of the upper bound introduced by Hunter by means of maximum weight spanning trees. Later the Hunter bound was generalised by Tomescu. He used the concept of hypertrees in the framework of uniform hypergraphs and on the basis of these new hypergraph structures it became possible to define not only upper but also lower bounds on the probability of union of events. The new bounds of the paper are generalisations of Tomescu's bounds in the same sense as the upper bound by Bukszár and Prékopa was a generalisation of the Hunter bound. The efficiency of the new bounds is illustrated on some test problems according to multivariate normal probability distribution function calculations.