Copula Analysis Research Articles

Simulation of spatially correlated wind power in small geographic areas—Sampling methods and evaluation

In clusters of wind generators spread over small geographic areas, the spatial correlation of wind power production is strong. Simulation of joint power production in such cases—such as for instance for determining the available power in a microgrid—is flawed if the correlation is not properly defined.Several methods have been proposed in the literature for producing scenarios of correlated samples; mostly focused on wind speed. In this paper we analyze three popular choices: classical Monte Carlo (with correlation induced by Cholesky factorization), Latin Hypercube Sampling (with correlation induced by rank sorting), and the recent copula theory. We put together a variety of statistical tools to transform an uncorrelated multivariate sample into a correlated one; and supplement other works by introducing a detailed definition of the wind power distribution and by expanding the Archimedean copula analysis to dimensions beyond the bivariate case analyzed in some related works.We analyze a year of wind production of 210 wind site from NREL data base. We cluster them to give a view of prospective microgrids, and employ several statistical techniques to measure the adequacy of the simulated samples to the original measured data.Our results show that, for generation in small geographic areas, the higher the number of generators, the better the wind power dependence structure is described by LHS. On the contrary, copulas—Gumbel or Gaussian for two- and three-dimensional problems, and Gaussian for higher dimensions—are better suited for representing correlated wind speed. The results are different when the generators are spread over large geographic areas.Compared with LHS endowed with rank sorting for inducing correlation, copula theory is in some sense cumbersome to apply for modeling and simulating wind power data. However, simulations can be performed in prospective microgrids in small geographical areas with larger accuracy by means of LHS if wind power is analyzed rather than wind speed. This advantage is lost for large distances or when small number of generators is considered.

Nonlinear Behaviors of Tail Dependence and Cross-Correlation of Financial Time Series Model

Nonlinear behaviors of tail dependence and cross-correlation of financial time series are reproduced and investigated by stochastic voter dynamic system. The voter process is a continuous-time Markov process and is one of the interacting dynamic systems. The tail dependence of return time series for pairs of Chinese stock markets and the proposed financial models is studied by copula analysis, in an attempt to detect and illustrate the existence of relevant correlation relationships. Further, the multifractality of cross-correlations for return series is studied by multifractal detrended cross-correlation analysis, which indicates the analogous cross-correlations and some fractal characters for both actual data and simulative data and provides an intuitive evidence for market inefficiency.

Spatial and Temporal Variability of Precipitation in Haihe River Basin, China: Characterization and Management Implications

Data analysis and characterization of precipitation in the Haihe River Basin (HRB) of China are required for management practices for the purpose of flood water control and utilization. In the companion paper, we presented precipitation data in the HRB during 1951–2010 and reported its basic statistics such as temporal trend and spatial variability. In this study, spatiotemporal variability on the precipitation was further investigated comprehensively for the underlying physics and the implication to water resource management. During the summer flood season of the study area, basin-wide precipitation was negatively correlated to average NINO3.4 index. Spatially, summer precipitation was correlated with gridded sea surface temperature (SST) observed in the eastern tropic Pacific Ocean and the western tropic Indian Ocean. SST in two representative areas was identified as potential predictors for precipitation in the HRB. No spatial or temporal correlations were confirmed between precipitation and soil moisture as annual averages in the study area. Copula analysis suggested about 40% possibility in a year with a potential for cross-watershed water diversion within HRB.

Gold prices and exchange rates: a time-varying copula analysis

We investigate the dynamic dependence structure between specific currencies (the GBP, the EUR and the JPY) and gold. The primary findings are as follows. First, the lower and upper conditional dependences between the currencies and gold were weaker during the financial turmoil period than in the normal period, implying that the currencies mostly deviated from their real value during this time. Second, the conditional upper tail dependences are stronger and fluctuate more than the conditional lower dependences for the GBP/gold and the JPY/gold pairs. However, the lower tail dependence for the EUR/gold pair remains constant. Furthermore, the negative dynamic dependences during the market crash imply that gold showed its real value during the crisis. Finally, the dependence structure between the gold price and the exchange rate is asymmetric. Our results provide useful information for investors interested in portfolio diversification, risk management and international asset allocation.

Dependence structure among international stock markets: a GARCH–copula analysis

In this article, we investigate the dependence structure among international stock markets, with particular emphasis on developed and emerging stock markets, as proxied for by major country-level exchanges. Specifically, we adopt the copula model for the presented analysis and find that an asymmetric dependence relationship only exists between developed and emerging markets. In particular, emerging markets are sensitive to outside negative news (downside risk) from developed markets. We also compare the dependence structure of the analysed stock markets in the pre- and post-2007 financial crisis periods and draw three broad conclusions. First, the correlations among these markets increase during the crisis period because of the contagion effect. Second, even though the dependence for both markets is weaker during the pre-crisis period, this tendency is more obvious for emerging markets. Finally, the dependence structure changes considerably across these sub-periods, mainly because each country implements ...

Financial dependence analysis: applications of vine copulas

This paper features the application of a novel and recently developed method of statistical and mathematical analysis to the assessment of financial risk, namely regular vine copulas. Dependence modelling using copulas is a popular tool in financial applications but is usually applied to pairs of securities. Vine copulas offer greater flexibility and permit the modelling of complex dependence patterns using the rich variety of bivariate copulas that can be arranged and analysed in a tree structure to facilitate the analysis of multiple dependencies. We apply regular vine copula analysis to a sample of stocks comprising the Dow Jones index to assess their interdependencies and to assess how their correlations change in different economic circumstances using three different sample periods around Global Financial Crisis (GFC).: pre‐GFC (January 2005 to July 2007), GFC (July 2007 to September 2009) and post‐GFC periods (September 2009 to December 2011). The empirical results suggest that the dependencies change in a complex manner, and there is evidence of greater reliance on the Student‐t copula in the copula choice within the tree structures for the GFC period, which is consistent with the existence of larger tails in the distributions of returns for this period. One of the attractions of this approach to risk modelling is the flexibility in the choice of distributions used to model co‐dependencies. The practical application of regular vine metrics is demonstrated via an example of the calculation of the Value at Risk of a portfolio of stocks.

Characterisation of Drought Properties with Bivariate Copula Analysis

Drought severity and duration are usually modelled independently. However, these two characteristics are known to be related. To model this relationship, a joint distribution of drought severity and duration using a bivariate copula model is proposed and applied to daily rainfall data (1976–2007) of 30 rain gauge stations in Peninsular Malaysia. The drought characteristics are classified using the standardized precipitation index (SPI) and their univariate marginal distributions are further identified by fitting exponential, gamma, generalized extreme value, generalized gamma, generalized logistics, generalized pareto, gumbel max, gumbel min, log-logistic, log-pearson3, log-normal, normal, pearson 5, pearson 6 and weibull distributions. The three-parameter log-normal distribution is identified as the best fitting distribution for drought severity while the generalized pareto distribution is determined as the most appropriate distribution for drought duration with respect to the application of the Anderson-Darling procedure. The dependency among the drought properties is analysed using Kendall’s τ method. The maximum likelihood estimation of the univariate marginal distributions and the maximisation of the bivariate likelihood are employed to compute the Akaike Information Criterion (AIC) values in verifying the best fitting copula distribution. The Galambos distribution is recognised as the most appropriate copula distribution for describing the relationship between drought severity and duration. The conditional drought probability and drought return period are further described to explain the drought properties comprehensively. The probabilities of drought occurrences under certain circumstances with a specific seriousness or duration can be determined in order to verify the possibility of drought episodes. The return period of a recurrent drought has also been investigated to identify the time-interval for repeated drought occurrences under similar situation.

Wind Turbine Condition Assessment Through Power Curve Copula Modeling

Power curves constructed from wind speed and active power output measurements provide an established method of analyzing wind turbine performance. In this paper, it is proposed that operational data from wind turbines are used to estimate bivariate probability distribution functions representing the power curve of existing turbines so that deviations from expected behavior can be detected. Owing to the complex form of dependency between active power and wind speed, which no classical parameterized distribution can approximate, the application of empirical copulas is proposed; the statistical theory of copulas allows the distribution form of marginal distributions of wind speed and power to be expressed separately from information about the dependency between them. Copula analysis is discussed in terms of its likely usefulness in wind turbine condition monitoring, particularly in early recognition of incipient faults such as blade degradation, yaw, and pitch errors.

The Relationship between Carbon, Commodity and Financial Markets: A Copula Analysis*

This article applies different copulas to investigate the complex dependence structure between EU emission allowance (EUA) futures returns and those of other commodities, equity and energy indices. The analysis yields important insights into the relationship between carbon, commodities and financial markets. Firstly, we find a significant relationship between EUA returns and those of the other considered variables that is most appropriately modelled by a Gaussian and Student‐t copula. These results contradict some earlier studies that report no statistically significant or even negative correlations between returns of emission allowances and other financial variables. Secondly, considering time‐varying copulas shows that the estimated copula parameters are not constant over time. We find in particular that the dependence is stronger during the period of the financial crisis. In a Value‐at‐Risk (VaR) analysis, finally, we further illustrate the advantages of copula methods. In particular, the Student‐t copula provides an appropriate quantification of VaR at different confidence levels while other models fail to specify the risk correctly. This analysis shows that ignoring the actual nature of dependence might lead to an underestimation of the risk for portfolios combining EUAs with commodities or equity investments.

Copula analysis of mixture models

Contemporary computers collect databases that can be too large for classical methods to handle. The present work takes data whose observations are distribution functions (rather than the single numerical point value of classical data) and presents a computational statistical approach of a new methodology to group the distributions into classes. The clustering method links the searched partition to the decomposition of mixture densities, through the notions of a function of distributions and of multi-dimensional copulas. The new clustering technique is illustrated by ascertaining distinct temperature and humidity regions for a global climate dataset and shows that the results compare favorably with those obtained from the standard EM algorithm method.

The Dependence Structure between Carbon Emission Allowances and Financial Markets - A Copula Analysis

This paper applies different copulas in order to investigate the complex dependence structure between EU emission allowance (EUA) futures returns and those of other commodities, equity and energy indices. The analysis yields important insights into the relationship between carbon, commodities and financial markets. First of all, we find a significant relationship between EUA returns and those of the other considered variables that is most appropriately modeled by a Gaussian and Student-t copula. These results contradict some earlier studies that report no statistically significant or even negative correlations between returns of emission allowances and other financial variables. Secondly, considering time-varying copulas shows that the estimated copula parameters are not constant over time. We find in particular that the dependence is stronger during the period of the financial crisis. In a Value-at-Risk (VaR) analysis, finally, we further illustrate the advantages of copula methods. In particular the Student-t copula provides an appropriate quantification of VaR at different confidence levels while other models fail to specify the risk correctly. This analysis shows that ignoring the actual nature of dependence might lead to an underestimation of the risk for portfolios combining EUAs with commodities or equity investments.

An empirical central limit theorem with applications to copulas under weak dependence

We state a multidimensional Functional Central Limit Theorem for weakly dependent random vectors. We apply this result to copulas. We get the weak convergence of the empirical copula process and of its smoothed version. The finite dimensional convergence of smoothed copula densities is also proved. A new definition and the theoretical analysis of conditional copulas and their empirical counterparts are provided.

Statistical Mechanics of Neocortical Interactions: Testing Theories with Multiple Imaging Data

To test theories of neocortical processing of macroscopic phenomena, e.g., attention and consciousness, based on relatively microscopic neocortical interactions, e.g., synaptic and neuronal interactions, both theoretical and experimental designs must account for the nonlinear nonequilibrium multivariate stochastic nature of these phenomena. Statistical mechanics of neocortical interactions (SMNI) is one theoretical description that has been tested on short-term memory and electroencephalographic data, and is proposed as a framework to test other theories and mechanisms of neocortical activity. Copula analysis of multivariate stochastic systems, previously developed for financial markets, is an important approach to developing a portfolio of physiological indicators to create cost/objective functions to fit experimental data, e.g., multiple synchronous imaging data.

Change analysis of a dynamic copula for measuring dependence in multivariate financial data

This paper proposes a new approach to measure dependencies in multivariate financial data. Data in finance and insurance often cover a long time period. Therefore, the economic factors may induce some changes within the dependence structure. Recently, two methods have been proposed using copulas to analyse such changes. The first approach investigates changes within the parameters of the copula. The second determines the sequence of copulas using moving windows. In this paper we take into account the non-stationarity of the data and analyse the impact of (1) time-varying parameters for a copula family, and (2) the sequence of copulas, on the computations of the VaR and ES measures. We propose tests based on conditional copulas and the goodness-of-fit to decide the type of change, and further give the corresponding change analysis. We illustrate our approach using the Standard & Poor 500 and Nasdaq indices in order to compute risk measures using the two previous methods.

Ideas by Statistical Mechanics (Ism)

Ideas by Statistical Mechanics (ISM) is a generic program to model evolution and propagation of ideas/patterns throughout populations subjected to endogenous and exogenous interactions. The program is based on the author's work in Statistical Mechanics of Neocortical Interactions (SMNI), and uses the author's Adaptive Simulated Annealing (ASA) code for optimizations of training sets, as well as for importance-sampling to apply the author's copula financial risk-management codes, Trading in Risk Dimensions (TRD), for assessments of risk and uncertainty. This product can be used for decision support for projects ranging from diplomatic, information, military, and economic (DIME) factors of propagation/evolution of ideas, to commercial sales, trading indicators across sectors of financial markets, advertising and political campaigns, etc. It seems appropriate to base an approach for propagation of ideas on the only system so far demonstrated to develop and nurture ideas, i.e., the neocortical brain. A statistical mechanical model of neocortical interactions, developed by the author and tested successfully in describing short-term memory and EEG indicators, is the proposed model. ISM develops subsets of macrocolumnar activity of multivariate stochastic descriptions of defined populations, with macrocolumns defined by their local parameters within specific regions and with parameterized endogenous inter-regional and exogenous external connectivities. Parameters with a given subset of macrocolumns will be fit using ASA to patterns representing ideas. Parameters of external and inter-regional interactions will be determined that promote or inhibit the spread of these ideas. Tools of financial risk management, developed by the author to process correlated multivariate systems with differing non-Gaussian distributions using modern copula analysis, importance-sampled using ASA, will enable bona fide correlations and uncertainties of success and failure to be calculated. Marginal distributions will be evolved to determine their expected duration and stability using algorithms developed by the author, i.e., PATHTREE and PATHINT codes.

Copy constructions and their interaction with the copula in Korean

We argue here for a lexicalist analysis of the Korean copula (following Kim, Sells and Wescoat (2004)), on the basis of different properties of sequences of noun-plus-copula, which shows word-like behavior, in contrast to noun and negative copula, which are independent syntactic units. The interactions of these items with various copy constructions brings out their clear differences. The analysis is formalized in HPSG using Lexical Sharing, from Wescoat (2002).

The Meta-elliptical Distributions with Given Marginals

Based on an analysis of copulas of elliptically contoured distributions, joint densities of continuous variables with given strictly increasing marginal distributions are constructed. A method utilized for this procedure is to embed the spherical distribution quantile transformation of each variable into an elliptically contoured distribution. The new class of distributions is then called meta-elliptical distributions. The corresponding analytic forms of the density, conditional distribution functions, and dependence properties are derived. This new class of distributions has the same Kendall's rank correlation coefficient as meta-Gaussian distributions. As an extension of elliptically contoured distributions, some new classes of distributions are also obtained.

Defining the envelope of linguistic variation: The case of “don't count” forms in the copula analysis of African American Vernacular English

ABSTRACTEver since Labov, Cohen, Robbins, and Lewis's (1968) pioneering study, it has been commonplace to set aside certain tokens in analyzing variability in the English copula as “don't count” (DC) forms. These cases are most often occurrences of the copula that exhibit categorical behavior (as with the full copula in clause-final position), as well as those that are ambiguous or indeterminate. In this article, I propose a set of copula forms that should be set aside from variable analysis as instances of DC forms to allow for systematic comparisons among studies. I review the major alternative descriptions of DC copula cases in the literature and analyze the behavior of the traditional DC categories. New data are presented to support the exclusion of particular DC cases from analyses of copula variability. Among the conclusions are that [was], [thas], and [is] should be excluded from quantitative analyses of variation in the copula because of their invariant status, and that a number of tokens commonly included (e.g., questions) should be excluded on various grounds.