Joint Multivariate Distribution Research Articles

Multivariate Profiling of Hulls for Linear Cryptanalysis

Extensions of linear cryptanalysis making use of multiple approximations, such as multiple and multidimensional linear cryptanalysis, are an important tool in symmetric-key cryptanalysis, among others being responsible for the best known attacks on ciphers such as Serpent and present. At CRYPTO 2015, Huang et al. provided a refined analysis of the key-dependent capacity leading to a refined key equivalence hypothesis, however at the cost of additional assumptions. Their analysis was extended by Blondeau and Nyberg to also cover an updated wrong key randomization hypothesis, using similar assumptions. However, a recent result by Nyberg shows the equivalence of linear dependence and statistical dependence of linear approximations, which essentially invalidates a crucial assumption on which all these multidimensional models are based. In this paper, we develop a model for linear cryptanalysis using multiple linearly independent approximations which takes key-dependence into account and complies with Nyberg’s result. Our model considers an arbitrary multivariate joint distribution of the correlations, and in particular avoids any assumptions regarding normality. The analysis of this distribution is then tailored to concrete ciphers in a practically feasible way by combining a signal/noise decomposition approach for the linear hulls with a profiling of the actual multivariate distribution of the signal correlations for a large number of keys, thereby entirely avoiding assumptions regarding the shape of this distribution. As an application of our model, we provide an attack on 26 rounds of present which is faster and requires less data than previous attacks, while using more realistic assumptions and far fewer approximations. We successfully extend the attack to present the first 27-round attack which takes key-dependence into account.

Modeling tamping recovery of track geometry using the copula-based approach

Assessing and maintaining track geometry within acceptable limits are key components of railroad infrastructure maintenance operations. Track geometry conditions have a significant influence on rider comfort and safety. To maintain the ride quality and safety of the track, maintenance activities pertaining to track geometry, such as tamping, are performed. Tamping enhances the track geometry quality but fails to return the track geometry to an as-good-as-new condition. Majority of studies have evaluated tamping recovery using deterministic techniques, which assume that tamping recovery is dependent on the track geometry quality prior to tamping. However, they fail to capture the uncertainty of the recovery values. Probabilistic approaches are increasingly being used to account for the uncertainty but fail to model the underlying dependence between the variables, which may exhibit nonlinear dependences such as tail or asymmetric dependence. To accurately model the tamping recovery phenomenon, this research employs the copula models in combining arbitrary marginal distributions to form a joint multivariate distribution with the underlying dependence. Copula models are used to estimate the tamping recovery of track geometry parameters such as surface (longitudinal level), alignment, cross level, gage, and warp.

Multivariate analysis of microscopic traffic variables using copulas in staggered car-following conditions

ABSTRACTThe current study investigates the applicability of the copula approach to capture the potential dependence among microscopic traffic variables (time headway, speed and centreline separation) and the joint probabilistic behaviour of the variables in staggered car-following conditions, using trajectory data from video recording of a four-lane divided road in Kolkata, India. As importantly, the study indicates that although Frank, Clayton and Gaussian copulas provide better fits for the two-dimensional traffic variables; considering both the positive and negative associations among the three microscopic variables, and checked by the goodness-of-fit measures based on Rosenblatt’s transformation, the Gaussian copula is found acceptable to model the multivariate joint distribution and is hence, employed to construct the joint and conditional probabilities among the three variables. Overall, the findings of the study justify the need to explicitly consider centreline separation and vehicle speed in time headway modelling for elucidating the integrated driving behaviour of non-lane-based traffic streams, a proper consideration of which will further ameliorate the realistic replication of rider’s behaviour in simulation modelling.

A Model-Free Approach to Multivariate Option Pricing

We propose a novel model-free approach to extract a joint multivariate distribution, which is consistent with options written on individual stocks as well as on various available indices. To do so, we first use the market prices of traded options to infer the risk-neutral marginal distributions for the stocks and the linear combinations given by the indices and then apply a new combinatorial algorithm to find a compatible joint distribution. Armed with the joint distribution, we can price general path-independent multivariate options.

Inverse regression approach to robust nonlinear high-to-low dimensional mapping

The goal of this paper is to address the issue of nonlinear regression with outliers, possibly in high dimension, without specifying the form of the link function and under a parametric approach. Nonlinearity is handled via an underlying mixture of affine regressions. Each regression is encoded in a joint multivariate Student distribution on the responses and covariates. This joint modeling allows the use of an inverse regression strategy to handle the high dimensionality of the data, while the heavy tail of the Student distribution limits the contamination by outlying data. The possibility to add a number of latent variables similar to factors to the model further reduces its sensitivity to noise or model misspecification. The mixture model setting has the advantage of providing a natural inference procedure using an EM algorithm. The tractability and flexibility of the algorithm are illustrated in simulations and real high-dimensional data with good performance that compares favorably with other existing methods.

Uncertainty Quantification of Microstructural Properties due to Experimental Variations

Electron backscatter diffraction scans are an important experimental input for microstructure generation and homogenization. Multiple electron backscatter diffraction scans can be used to sample the uncertainty in orientation distribution function: both point to point within a specimen as well as across multiple specimens that originate from the same manufacturing process. However, microstructure analysis methods typically employ only the mean values of the orientation distribution function to predict properties, and the stochastic information is lost. In this work, analytical methods are developed to account for the uncertainty in the electron backscatter diffraction data during property analysis. To this end, a linear smoothing scheme is developed in the Rodrigues fundamental region to compute the orientation distribution function from the electron backscatter diffraction data. The joint multivariate probability distributions of the orientation distribution function are then modeled using a Gaussian assumption. The uncertainty in engineering properties that are obtained by homogenization are also computed. It is shown that the uncertainty in nonlinear properties can be analytically obtained using direct transformation of random variables in the homogenization approach.

Probabilistic load flow analysis of active distribution network adopting improved sequence operation methodology

In allusion to the uncertainty and correlation of distributed generation (DG) output in the active distribution network, a new method of probabilistic load flow (PLF) analysis was proposed based on improved sequence operation method. This method utilised the non-parametric kernel density estimation method and established a probabilistic model of DG daily output considering the timing characteristics. Then this method proposed the multivariate joint distribution probabilistic sequence based on the present sequence operation theory to describe the sequence transformation of the probabilistic model. Finally, an improved dependent probabilistic sequence method was proposed and combined with the traditional linear power flow calculation to achieve the probability distribution information of system power flow. Simulation tests were carried out on the modified 33-bus distribution system. Test results indicated that the proposed method can achieve more accurate results of PLF calculation and has higher computational efficiency.

Stochastic response surface method for reliability problems involving correlated multivariates with non-Gaussian dependence structure: Analysis under incomplete probability information

This paper aims to provide a stochastic response surface method (SRSM) that can consider non-Gaussian dependent random variables under incomplete probability information. The Rosenblatt transformation is adopted to map the random variables from the original space into the mutually independent standard normal space for the stochastic surrogate model development. The multivariate joint distribution is reconstructed by the pair-copula decomposition approach, in which the pair-copula parameters are retrieved from the incomplete probability information. The proposed method is illustrated in a tunnel excavation example. Three different dependence structures characterized by normal copulas, Frank copulas, and hybrid copulas are respectively investigated to demonstrate the effect of dependence structure on the reliability results. The results show that the widely used Nataf transformation is actually a special case of the proposed method if all pair-copulas are normal copulas. The effect of conditioning order is also examined. This study provides a new insight into the SRSM-based reliability analysis from the copula viewpoint and extends the application of SRSM under incomplete probability information.

Estimation of extreme Component-wise Excess design realization: a hydrological application

The classic univariate risk measure in environmental sciences is the Return Period (RP). The RP is traditionally defined as “the average time elapsing between two successive realizations of a prescribed event”. The notion of design quantile related with RP is also of great importance. The design quantile represents the “value of the variable(s) characterizing the event associated with a given RP”. Since an individual risk may strongly be affected by the degree of dependence amongst all risks, the need for the provision of multivariate design quantiles has gained ground. In contrast to the univariate case, the design quantile definition in the multivariate setting presents certain difficulties. In particular, Salvadori, G., De Michele, C. and Durante F. define in the paper called “On the return period and design in a multivariate framework” (Hydrol Earth Syst Sci 15:3293–3305, 2011) the design realization as the vector that maximizes a weight function given that the risk vector belongs to a given critical layer of its joint multivariate distribution function. In this paper, we provide the explicit expression of the aforementioned multivariate risk measure in the Archimedean copula setting. Furthermore, this measure is estimated by using Extreme Value Theory techniques and the asymptotic normality of the proposed estimator is studied. The performance of our estimator is evaluated on simulated data. We conclude with an application on a real hydrological data-set.

水库月径流概率预报研究

运用奇异谱分析(SSA)方法对资料系列进行降噪处理，采用人工神经网络(ANN)和支持向量机(SVM)建立确定性预报模型，构建基于Copula函数的联合分布，推求概率预报区间，并获得中值预报结果。以丹江口水库的月径流系列资料为研究对象，分析比较概率预报与确定性预报的结果。概率预报不仅可提高精度，同时还给出了预报区间，有利于水库调度决策人员定量考虑预报的不确定性，为决策支持提供技术支撑。 The Singular Spectrum Analysis (SSA) was applied to preprocess the original flow series, and Artificial Neural Network (ANN) and Support Vector Machine (SVM) were used to simulate and predict the reconstructed data series. The multivariate joint distribution based on copula function and probability forecast model were proposed. With a case study of Danjiangkou reservoir monthly inflow series, the probability forecast results were compared with that of deterministic model. It is shown that the probability forecast model not only can improve the accuracy of middle value prediction to a certain extent but also give probability interval, which helps reservoir managers to consider uncertainty quantitatively and provides technical support for decision-making.

Uncertainty quantification of microstructural properties due to variability in measured pole figures

Experimental pole figures are an important input for microstructure homogenization models. In this paper, we derive an exact analytical formulation to quantify the uncertainties in homogenized properties due to uncertainty in the experimentally measured pole figures. The pole figures are acquired from a set of Ti-7Al alloy samples. These samples were obtained from the same process: by compressing a beta forged ingot at room temperature followed by annealing. The samples were taken from different regions of the original ingot, and this created variability in the resulting pole figures. The joint multivariate probability distributions of the computed orientation distribution function (ODF) is then found using the method of characteristic functions based on a Gaussian model of the pole figures. Engineering properties such as elastic modulus can be obtained by volume averaging over the ODF. We also show that uncertainty in elastic properties can be analytically obtained using direct transformation of random variables in the homogenization approach.

Hidden Markov Model-based population synthesis

Micro-simulation travel demand and land use models require a synthetic population, which consists of a set of agents characterized by demographic and socio-economic attributes. Two main families of population synthesis techniques can be distinguished: (a) fitting methods (iterative proportional fitting, updating) and (b) combinatorial optimization methods. During the last few years, a third outperforming family of population synthesis procedures has emerged, i.e., Markov process-based methods such as Monte Carlo Markov Chain (MCMC) simulations. In this paper, an extended Hidden Markov Model (HMM)-based approach is presented, which can serve as a better alternative than the existing methods. The approach is characterized by a great flexibility and efficiency in terms of data preparation and model training. The HMM is able to reproduce the structural configuration of a given population from an unlimited number of micro-samples and a marginal distribution. Only one marginal distribution of the considered population can be used as a boundary condition to “guide” the synthesis of the whole population. Model training and testing are performed using the Survey on the Workforce of 2013 and the Belgian National Household Travel Survey of 2010. Results indicate that the HMM method captures the complete heterogeneity of the micro-data contrary to standard fitting approaches. The method provides accurate results as it is able to reproduce the marginal distributions and their corresponding multivariate joint distributions with an acceptable error rate (i.e., SRSME=0.54 for 6 synthesized attributes). Furthermore, the HMM outperforms IPF for small sample sizes, even though the amount of input data is less than that for IPF. Finally, simulations show that the HMM can merge information provided by multiple data sources to allow good population estimates.

Generic Identifiability of Linear Structural Equation Models by Ancestor Decomposition

AbstractLinear structural equation models, which relate random variables via linear interdependencies and Gaussian noise, are a popular tool for modelling multivariate joint distributions. The models correspond to mixed graphs that include both directed and bidirected edges representing the linear relationships and correlations between noise terms, respectively. A question of interest for these models is that of parameter identifiability, whether or not it is possible to recover edge coefficients from the joint covariance matrix of the random variables. For the problem of determining generic parameter identifiability, we present an algorithm building upon the half‐trek criterion. Underlying our new algorithm is the idea that ancestral subsets of vertices in the graph can be used to extend the applicability of a decomposition technique.

Study of vertical breakwater reliability based on copulas

The reliability of a vertical breakwater is calculated using direct integration methods based on joint density functions. The horizontal and uplifting wave forces on the vertical breakwater can be well fitted by the lognormal and the Gumbel distributions, respectively. The joint distribution of the horizontal and uplifting wave forces is analyzed using different probabilistic distributions, including the bivariate logistic Gumbel distribution, the bivariate lognormal distribution, and three bivariate Archimedean copulas functions constructed with different marginal distributions simultaneously. We use the fully nested copulas to construct multivariate distributions taking into account related variables. Different goodness fitting tests are carried out to determine the best bivariate copula model for wave forces on a vertical breakwater. We show that a bivariate model constructed by Frank copula gives the best reliability analysis, using marginal distributions of Gumbel and lognormal to account for uplifting pressure and horizontal wave force on a vertical breakwater, respectively. The results show that failure probability of the vertical breakwater calculated by multivariate density function is comparable to those by the Joint Committee on Structural Safety methods. As copulas are suitable for constructing a bivariate or multivariate joint distribution, they have great potential in reliability analysis for other coastal structures.

Number of appearances of events in random sequences: A new approach to non-overlapping runs

ABSTRACTDistributions of non-overlapping runs in multistate systems are derived using a new generating function (GF) approach, which has been applied previously mainly to exact length runs. By utilizing this method, univariate and multivariate joint distributions of non-overlapping runs are derived in a unified way. The GF approach also leads naturally to the asymptotic Gaussian distributions, with the mean, variance, and covariance linear in the size of the system.

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

In this paper, we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses. For a Fréchet class of discrete distributions with overlapping marginals, we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming. To guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information, we make use of a graph theoretic property known as the running intersection property. Building on this property, we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst-case conditional value-at-risk measure. Lastly, we use a data-driven approach with financial return data to identify the Fréchet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions. Numerical results in two different data sets show that the distributionally robust portfolio optimization model improves on the sample-based approach.

Loss scenario analysis and loss aggregation for process facilities

This study presents an overall loss modelling methodology for process facilities. The methodology comprises loss scenario identification and aggregation of losses due to process deviations. The identification of loss scenarios and determination of the time periods at which a process experiences each scenario are described first. Then, the application of the copula functions and their integration with the Monte Carlo (MC) approach are proposed to address the existing challenge of loss aggregation for multiple-loss scenarios. The proposed loss aggregation provides a flexible and realistic approach to construct joint multivariate distribution of the losses by considering their interdependence. The sensitivity of the model to the choice of correlation parameters is investigated. The results serve as a reminder to risk analysts about the significance of choosing an appropriate loss aggregation model for risk analysis purposes. The application of the methodology is demonstrated using a distillation column case study.

A New Time Series Model for Transaction Data and Marked Point Processes

We propose a new time series model for the bivariate/multivariate transaction data, where model their joint transition dynamics by directly modeling the conditional distribution of the current observation to be a mixture of conditional distributions given the past p-lags information. Each joint multivariate conditional distribution in the mixture is constructed via a certain class of Copula. Such model can successfully capture some distinguished nonlinear non-Gaussian time series features, such as the irregular bursts or jumps that are so widely observed in the transaction data set and other more general marked point processes. We illustrate the new time series model and related methods by modeling three pieces of IBM stock transaction data, and show its advantages over some benchmark models.

DPSynthesizer: Differentially Private Data Synthesizer for Privacy Preserving Data Sharing.

Differential privacy has recently emerged in private statistical data release as one of the strongest privacy guarantees. Releasing synthetic data that mimic original data with Differential privacy provides a promising way for privacy preserving data sharing and analytics while providing a rigorous privacy guarantee. However, to this date there is no open-source tools that allow users to generate differentially private synthetic data, in particular, for high dimensional and large domain data. Most of the existing techniques that generate differentially private histograms or synthetic data only work well for single dimensional or low-dimensional histograms. They become problematic for high dimensional and large domain data due to increased perturbation error and computation complexity. We propose DPSynthesizer, a toolkit for differentially private data synthesization. The core of DPSynthesizer is DPCopula designed for high-dimensional and large-domain data. DPCopula computes a differentially private copula function from which synthetic data can be sampled. Copula functions are used to describe the dependence between multivariate random vectors and allow us to build the multivariate joint distribution using one-dimensional marginal distributions. DPSynthesizer also implements a set of state-of-the-art methods for building differentially private histograms, suitable for low-dimensional data, from which synthetic data can be generated. We will demonstrate the system using DPCopula as well as other methods with various data sets and show the feasibility, utility, and efficiency of various methods.

An Approximation Solution to Refinery Crude Oil Scheduling Problem with Demand Uncertainty Using Joint Constrained Programming

This paper is devoted to develop an approximation method for scheduling refinery crude oil operations by taking into consideration the demand uncertainty. In the stochastic model the demand uncertainty is modeled as random variables which follow a joint multivariate distribution with a specific correlation structure. Compared to deterministic models in existing works, the stochastic model can be more practical for optimizing crude oil operations. Using joint chance constraints, the demand uncertainty is treated by specifying proximity level on the satisfaction of product demands. However, the joint chance constraints usually hold strong nonlinearity and consequently, it is still hard to handle it directly. In this paper, an approximation method combines a relax-and-tight technique to approximately transform the joint chance constraints to a serial of parameterized linear constraints so that the complicated problem can be attacked iteratively. The basic idea behind this approach is to approximate, as much as possible, nonlinear constraints by a lot of easily handled linear constraints which will lead to a well balance between the problem complexity and tractability. Case studies are conducted to demonstrate the proposed methods. Results show that the operation cost can be reduced effectively compared with the case without considering the demand correlation.