On approximating copulas by finite mixtures

Copulas are now frequently used to approximate or estimate multivariate distributions because of their ability to take into account the multivariate dependence of the variables while controlling the approximation properties of the marginal densities. Copula based multivariate models can often also be more parsimonious than fitting a flexible multivariate model, such as a mixture of normals model, directly to the data. However, to be effective, it is imperative that the family of copula models considered is sufficiently flexible. Although finite mixtures of copulas have been used to construct flexible families of copulas, their approximation properties are not well understood and we show that natural candidates such as mixtures of elliptical copulas and mixtures of Archimedean copulas cannot approximate a general copula arbitrarily well. Our article develops fundamental tools for approximating a general copula arbitrarily well by a mixture and proposes a family of finite mixtures that can do so. We illustrate empirically on a financial data set that our approach for estimating a copula can be much more parsimonious and results in a better fit than approximating the copula by a mixture of normal copulas.Comment: 26 pages and 1 figure and 2 table

Probabilistic description of extreme events in intermittently unstable systems excited by correlated stochastic processes

In this work, we consider systems that are subjected to intermittent instabilities due to external stochastic excitation. These intermittent instabilities, though rare, have a large impact on the probabilistic response of the system and give rise to heavy-tailed probability distributions. By making appropriate assumptions on the form of these instabilities, which are valid for a broad range of systems, we formulate a method for the analytical approximation of the probability distribution function (pdf) of the system response (both the main probability mass and the heavy-tail structure). In particular, this method relies on conditioning the probability density of the response on the occurrence of an instability and the separate analysis of the two states of the system, the unstable and stable state. In the stable regime we employ steady state assumptions, which lead to the derivation of the conditional response pdf using standard methods for random dynamical systems. The unstable regime is inherently transient and in order to analyze this regime we characterize the statistics under the assumption of an exponential growth phase and a subsequent decay phase until the system is brought back to the stable attractor. The method we present allows us to capture the statistics associated with the dynamics that give rise to heavy-tails in the system response and the analytical approximations compare favorably with direct Monte Carlo simulations, which we illustrate for two prototype intermittent systems: an intermittently unstable mechanical oscillator excited by correlated multiplicative noise and a complex mode in a turbulent signal with fixed frequency, where multiplicative stochastic damping and additive noise model interactions between various modes.Comment: 29 pages, 15 figure

A sequential sampling strategy for extreme event statistics in nonlinear dynamical systems

We develop a method for the evaluation of extreme event statistics associated with nonlinear dynamical systems, using a small number of samples. From an initial dataset of design points, we formulate a sequential strategy that provides the 'next-best' data point (set of parameters) that when evaluated results in improved estimates of the probability density function (pdf) for a scalar quantity of interest. The approach utilizes Gaussian process regression to perform Bayesian inference on the parameter-to-observation map describing the quantity of interest. We then approximate the desired pdf along with uncertainty bounds utilizing the posterior distribution of the inferred map. The 'next-best' design point is sequentially determined through an optimization procedure that selects the point in parameter space that maximally reduces uncertainty between the estimated bounds of the pdf prediction. Since the optimization process utilizes only information from the inferred map it has minimal computational cost. Moreover, the special form of the metric emphasizes the tails of the pdf. The method is practical for systems where the dimensionality of the parameter space is of moderate size, i.e. order O(10). We apply the method to estimate the extreme event statistics for a very high-dimensional system with millions of degrees of freedom: an offshore platform subjected to three-dimensional irregular waves. It is demonstrated that the developed approach can accurately determine the extreme event statistics using limited number of samples

Defining Strategic Position and Busines Model of CV Energi Selaras Alam

Fuel –addictive industry is a growing industry in Indonesia. With potential market of 80 million vehicles in Indonesia where most modern car use a subsidized –low quality fuel, market opportunity of this industry is very big. Although this industry is very influenced by world oil price and government regulation about subsidized fuel. CV.Energy Selaras Alam is the new comer in this industry. Founded in 2011, this firm provides a low price and highly efficient octane booster product to the society. Along the decreased issue of subsidized fuel price, this firm difficult to maintain their sales growth and expand to broader market that leads to significant decrease on its profit. If this condition allowed for the next couple month, this will lead company to bankruptcy.To understanding company's condition better, this research use methods of observation, literature survey and interview. Resource analysis and value chain analysis are used for internal analysis. PESTEL, Porter's 5 forces, and strategic group are used for external analysis. By using interrelationship diagram, it known that the root cause of this company's issue is the un-clarity of its strategic positioning.To formulate the solution, SWOT, IFAS, EFAS and SFAS matrix used to find CV.ESA new strategic position. Alternative strategy is generated from TOWS Matrix, result 13 strategies. Every strategy is integrated and mapped back into a new business model to create more integrated result for this company. The conclusion from this proposed strategy is the new strategy of CV ESA will be based on cost focus strategic position. This new strategy proposed to diversify product and marketing channel that focus on car and motorcycle user and also local industry that located only in West Java. Action Plan is derived from business model formulation, and prepared for 3 year implementation Plan. The implementation plan of those strategies is suggested to adjust the number of marketing and sales armada. This armada will be focused on creating new alliances and partnership that will be a success foundation of strategy proposed. Strategic Implementation will be implemented in the next 3 years and will start in 2013. Hopefully, this Research result would contribute a better understanding related with the importance of Strategic position and business Model in Startup Compan

Mixed Marginal Copula Modeling

This article extends the literature on copulas with discrete or continuous marginals to the case where some of the marginals are a mixture of discrete and continuous components. We do so by carefully defining the likelihood as the density of the observations with respect to a mixed measure. The treatment is quite general, although we focus focus on mixtures of Gaussian and Archimedean copulas. The inference is Bayesian with the estimation carried out by Markov chain Monte Carlo. We illustrate the methodology and algorithms by applying them to estimate a multivariate income dynamics model.Comment: 46 pages, 8 tables and 4 figure