Tail Characteristics Research Articles

Robust multivariate L1 principal component analysis and dimensionality reduction

Further to our recent work on the robust L1 PCA we introduce a new version of robust PCA model based on the so-called multivariate Laplace distribution (called L1 distribution) proposed in Eltoft et al. [2006. On the multivariate Laplace distribution. IEEE Signal Process. Lett. 13(5), 300–303]. Due to the heavy tail and high component dependency characteristics of the multivariate L1 distribution, the proposed model is expected to be more robust against data outliers and fitting component dependency. Additionally, we demonstrate how a variational approximation scheme enables effective inference of key parameters in the probabilistic multivariate L1-PCA model. By doing so, a tractable Bayesian inference can be achieved based on the variational EM-type algorithm.

Robust L1 Principal Component Analysis and Its Bayesian Variational Inference

We introduce a robust probabilistic L1-PCA model in which the conventional gaussian distribution for the noise in the observed data was replaced by the Laplacian distribution (or L1 distribution). Due to the heavy tail characteristics of the L1 distribution, the proposed model is supposed to be more robust against data outliers. In this letter, we demonstrate how a variational approximation scheme enables effective inference of key parameters in the probabilistic L1-PCA model. As the L1 density can be expanded as a superposition of infinite number of gaussian densities, we express the L1-PCA model as a marginalized model over the superpositions. By doing so, a tractable Bayesian inference can be achieved based on the variational expectation-maximization-type algorithm.

Selection and Performance Analysis of Asia-Pacific Hedge Funds

Hedge funds that invest in Asia-Pacific markets have grown rapidly in recent years. Asia-Pacific focused hedge funds are poised to play a bigger role in investor portfolios judging by the growing number of investment companies that are based in the Asian region. This article examines the skew and kurtosis of Asia-Pacific hedge fund returns, and tests the hypothesis that they are normally distributed. The results indicate that the distribution of returns are not necessarily Gaussian, but instead exhibit fat tail characteristics. As such, standard deviation itself is insufficient to capture the full risks inherent within these funds. In order to account for fat tail distributions this article uses conditional value-at-risk (CVaR) and conditional drawdown (CDD) to assess the impact of negative tail risks of hedge funds. Further an optimal portfolio of hedge funds is constructed that is subject to constraints on CVaR and CDD. <b>TOPICS:</b>Real assets/alternative investments/private equity, emerging, VAR and use of alternative risk measures of trading risk, portfolio construction

Alloying induced degradation of the absorption edge of InAsxSb1−x

In As x Sb 1 − x alloys show a strong bowing in the energy gap, the energy gap of the alloy can be less than the gap of the two parent compounds. The authors demonstrate that a consequence of this alloying is a systematic degradation in the sharpness of the absorption edge. The alloy disorder induced band-tail (Urbach tail) characteristics are quantitatively studied for InAs0.05Sb0.95.

The Normal Inverse Gaussian Distribution for Synthetic CDO Pricing

One of the harder new derivatives problems in recent years is estimating the probability distribution of credit losses on a portfolio of risky bonds or credit default swaps. This is the essential valuation problem for securitized debt instruments, including CDOs, CLOs, and CBOs, as well as exotic derivative contracts like „nth to default” swaps. The standard Gaussian copula model is not entirely satisfactory because it constrains correlation in the tails of the distribution in such a way that it is typically very hard to calibrate it to market tranche prices. The Student t-distribution has more plausible tail behavior, but because it is not stable under convolution, it becomes computationally very unwieldy as the number of variables increases. This article proposes the Normal Inverse Gaussian (NIG) distribution as a more tractable alternative. Comparison tests on several standard CDS index portfolios show that the NIG distribution has better tail characteristics than the Normal and it is much more efficient for large scale computations than the multivariate Student t. <b>TOPICS:</b>Credit default swaps, CLOs, CDOs, and other structured credit, statistical methods

Temperature dependence of the optical properties in GaMnN

Temperature-dependent transmission measurements are carried out on ion-implanted GaMnN with different Mn-implantation doses. With a detailed procedure developed for analyzing the transmission spectra, we obtain the temperature effects in optical properties of GaMnN (including GaN), such as absorption coefficient, band gap, Urbach band tail characteristics, refractive index, and extinction coefficient. Two sets of temperature- and photon energy-dependent empirical formulas are established, which not only unify the various experimental data reported in the literature, but also provide an experimental database of optical properties in GaMnN. Furthermore, we reveal that the dependence of these optical properties on the Mn-implantation doses is closely related to the impact of Mn on the crystal structure.

COMPUTATIONAL STUDIES OF DISCRETE BREATHERS — FROM BASICS TO COMPETING LENGTH SCALES

This work provides a description of the main computational tools for the study of discrete breathers. It starts with the observation of breathers through simple numerical runs, the study uses targeted initial conditions, and discrete breather impact on transient processes and thermal equilibrium. We briefly describe a set of numerical methods to obtain breathers up to machine precision. In the final part of this work we apply the discussed methods to study the competing length scales for breathers with purely anharmonic interactions — favoring superexponential localization — and long range interactions, which favor algebraic decay in space. As a result, we observe and explain the presence of three different spatial tail characteristics of the considered localized excitations.

Study of thermodynamic parameters in amphiphilic systems by lattice Monte Carlo: effect of tails and heads

Results from three-dimensional lattice Monte Carlo simulations of amphiphile–solvent mixtures are presented. The chemical potential is derived from the monomer distribution in different clusters rather than using a Widom particle insertion approach. The effect of tail and head characteristics on the non-ideality of these systems, aggregation number, and premicellar phenomena is considered. The aggregation number and CMC behavior of the simulated amphiphilic systems are compared with existing experimental results for non-ionic amphiphiles. Two kinds of polydispersity changing with total concentration of surfactants are observed which are related to phase transition phenomena. Shape variations in clusters are studied by calculating the eigenvalues of the gyration matrix; it is shown that large clusters are non-spherical. With the Maclaurin’s expansion of activity coefficient into volume fraction, the distribution of excess chemical potential with changing aggregation number is considered. Study of the degree of non-ideality of these amphiphiles reveals that asymmetric amphiphiles are characterized by greater non-ideality than symmetric amphiphiles. Goldstein’s parameters are calculated taking non-ideality into consideration. The difference between the phenomenological model and the simulation data is investigated.

Preferential attachment with information filtering—node degree probability distribution properties

A network growth mechanism based on a two-step preferential rule is investigated as a model of network growth in which no global knowledge of the network is required. In the first filtering step a subset of fixed size m of existing nodes is randomly chosen. In the second step the preferential rule of attachment is applied to the chosen subset. The characteristics of thus formed networks are explored using two approaches: computer simulations of network growth and a theoretical description based on a master equation. The results of the two approaches are in excellent agreement. Special emphasis is put on the investigation of the node degree probability distribution. It is found that the tail of the distribution has the exponential form given by exp ( - k / m ) . Implications of the node degree distribution with such tail characteristics are briefly discussed.

Dating Breaks in the Extremal Behavior of Financial Time Series

This paper studies the extremal behavior of emerging market bonds in relation to equities and bond returns of developed markets. Returns on emerging markets fixed income investments are known to behave like equity returns, yet many studies have used techniques that do not account for their thick tailed behavior. Since emerging markets are characterized by highly turbulent periods, we develop in this paper a test statistic for testing and dating breaks in the tail behavior of a series. Our test for extremal breaks is valid for GARCH data and for multiple breaks where the number of breaks can be known or unknown. Our finding is that while extremal movements of emerging market debt is characterized by three regimes with breakdates corresponding to the Asian/Russian crisis of 1997/1998 and the Argentinian crisis of 2001, extremal movements of equities and bond returns of developed markets are characterized by only a single break in 1997/1998. These findings and the methods introduced here have implications for value at risk methodologies that assume time invariant tail characteristics.

ANALYSIS OF RANDOM AND SYSTEMATIC ERROR EFFECTS ON UNCERTAINTY PROPAGATION IN PROCESS DESIGN AND SIMULATION USING DISTRIBUTION TAIL CHARACTERIZATION

An exponential and heavy tail analysis method is presented to study the effect of systematic and random errors present in thermodynamic data on chemical process design and simulation. The true distribution tail characteristics (important for high levels of quality assurance) can be far from the estimates obtained with typical Gaussian distribution analysis. Pareto (heavy) or exponential distributions may represent the tail behavior better under many circumstances. Heavy tails diminish at an algebraic rate rather than at an exponential rate. Different error types such as random and systematic error can potentially cause different effects in the behavior of probability distributions, particularly in the tails. In this work, we use the tail behavior of cumulative frequency distributions produced from uncertainty analyses to characterize the error propagation in process design and simulation. The diminishing rate of the tail of a given distribution can be related to the error types involved in the process and also can be used to determine which error exhibits stochastic dominance. Case studies of process performance evaluations for liquid-liquid extraction operations are presented to illustrate the approach. It is observed that random and systematic errors (coupled with typical nonlinear chemical engineering models) can cause the tails of the uncertainty probability distributions to be exponential.

Investigation of the Microstructure of Micelles Formed by Hard-Sphere Chains Interacting via Size Nonadditivity by Discontinuous Molecular Dynamics Simulation

Micelle formation by short nonadditive hard surfactant chains was investigated at different size ratios, reduced densities, and nonadditivity parameters using molecular dynamics simulation. It was found that spherical, cylindrical, lamellar, and reverse micelles can form in systems with different head, tail, and solvent characteristics. Hard-core surfactant chains composed of a head segment and three tail segments were simulated in a solvent of hard spheres. The formation of micelles was found to be a strong function of the packing fraction and nonadditivity parameter. Micelles were more stable at higher densities and larger nonadditivity parameters. At lower densities, micelles tended to break into small, dynamic globules.

Monte Carlo Simulations of Polyion−Macroion Complexes. 2. Polyion Length and Charge Density Dependence

The complexation between a polyion and an oppositely charged spherical macroion in the framework of the primitive model has been studied by the use of Monte Carlo simulations. The polyion length, linear charge density, and bare persistence length are varied systematically, while the properties of the macroion are kept constant. The polyion charge to macroion charge ratio is varied between 1/4 and 4. The structure of the complex is investigated by direct visualization; polyion bead complexation probability; loop, tail, and train characteristics; degree of overcharging; and tail joint probability functions. The strongest complexes are observed for flexible chains, where the polyion is folded around the macroion. In the case of fully flexible chains, a transition from a collapsed state to a fluctuating two-tail state and eventually to a one-tail state are observed as the chain length is increased. As the stiffness is increased, several complex structures, such as multiloop, single-loop, and solenoid arrangements, and finally a structure involving only a single contact between the polyion and the macroion occur. In particular, for long and highly charged polyions, a transition from the one-tail state to a two-tail state appears as the chain stiffness is increased. A discussion with recent theories and other simulation studies is also provided.

Diagnostics for Dependence within Time Series Extremes

SummaryThe analysis of extreme values within a stationary time series entails various assumptions concerning its long- and short-range dependence. We present a range of new diagnostic tools for assessing whether these assumptions are appropriate and for identifying structure within extreme events. These tools are based on tail characteristics of joint survivor functions but can be implemented by using existing estimation methods for extremes of univariate independent and identically distributed variables. Our diagnostic aids are illustrated through theoretical examples, simulation studies and by application to rainfall and exchange rate data. On the basis of these diagnostics we can explain characteristics that are found in the observed extreme events of these series and also gain insight into the properties of events that are more extreme than those observed.

Dual Stochastic Dominance and Quantile Risk Measures

Following the seminal work by Markowitz, the portfolio selection problem is usually modeled as a bicriteria optimization problem where a reasonable trade–off between expected rate of return and risk is sought. In the classical Markowitz model, the risk is measured with variance. Several other risk measures have been later considered thus creating the entire family of mean–risk (Markowitz type) models. In this paper, we analyze mean–risk models using quantiles and tail characteristics of the distribution. Value at risk (VAR), defined as the maximum loss at a specified confidence level, is a widely used quantile risk measure. The corresponding second order quantile measure, called the worst conditional expectation or Tail VAR, represents the mean shortfall at a specified confidence level. It has more attractive theoretical properties and it leads to LP solvable portfolio optimization models in the case of discrete random variables, i.e., in the case of returns defined by their realizations under the specified scenarios. We show that the mean–risk models using the worst conditional expectation or some of its extensions are in harmony with the stochastic dominance order. For this purpose, we exploit duality relations of convex analysis to develop the quantile model of stochastic dominance for general distributions.

Monte Carlo Simulations of Polyion−Macroion Complexes. 1. Equal Absolute Polyion and Macroion Charges

Intermolecular structures of complexes formed between a charged polymer and a spherical and oppositely charged macroion have been studied by employing the primitive model solved by Monte Carlo simulations. The strong-complex case involving a polyion and a macroion with equal absolute charges and without small ions was considered. The influence of the polyion flexibility with a bare persistence length ranging from 7 to 1000 A for four different systems characterized by different polyion linear charge densities and macroion sizes has been examined. Radial distributions, polyion bead complexation probabilities, loop, tail, and train characteristics, and energetic analysis have been performed. The strongest and most compact complex, involving a collapsed polyion wrapping the macroion, was formed for a semiflexible chain. As the stiffness was increased, this state was transformed into a range of different structures comprising “tennis ball seam”-like, solenoid, multiloop (“rosette”), and single-loop arrangements as well as structures involving only a single polyion-macroion contact region.

Dual Stochastic Dominance and Related Mean-Risk Models

We consider the problem of constructing mean-risk models which are consistent with the second degree stochastic dominance relation. By exploiting duality relations of convex analysis we develop the quantile model of stochastic dominance for general distributions. This allows us to show that several models using quantiles and tail characteristics of the distribution are in harmony with the stochastic dominance relation. We also provide stochastic linear programming formulations of these models.

Imerys North American Pigments and Additives Group chooses ‘IFS Applications’ software

To solve the problem that existing grey prediction models are prone to be affected by outliers and show poor stability, a novel grey prediction model based on quantile regression technology is proposed (abbreviated as the QGM(1,1) model). The improved simplex algorithm is utilized to solve the programming problem in the model so that the structural parameters are estimated at different quantiles. Compared with the parameters estimation of traditional grey prediction models, the QGM(1,1) model describes the influence of the independent variable on the range of the dependent variable and the shape of the conditional distribution accurately, and captures the tail characteristics of the distribution. Moreover, the results of interval prediction are given according to the estimated values of parameters at different quantiles by setting a corresponding error criterion. The effectiveness of the novel model is verified by three analytical examples. The results show that the QGM(1,1) model not only can conclude the predicted value of the distribution center but also can predict the dynamic trends of upper and lower limits of the distribution. The prediction accuracy is significantly improved and the robustness is greatly enhanced.

Regional Growth and Convergence: Evidence from the United States

TSIONAS E. G. (2000) Regional growth and convergence: evidence from the United States, Reg. Studies 34 , 231‐238. The paper considers regional convergence in the US. Both g - and σ -convergence tests indicate that in the period 1977‐96 regional incomes do not converge, yet kernel density estimates of the distribution show some variation in the tail characteristics of the distribution. Using finite normal mixtures to examine time invariance of the distribution in a more formal way, the paper concludes that the distribution of the logarithms of regional income has indeed shown very little time variation. The distribution of income percentages shows some mobility, there is a structural break in mixing probabilities, mixing means and variances, but the mobility is not strong enough to imply a radical change in the co-movement patterns of regional means. TSIONAS E. G. (2000) La croissance et la convergence régionales: des preuves provenant des Etats-Unis, Reg. Studies 34 , 231‐238. Cet article cherche à étudier la convergence aux Etats-Unis. Et le test de convergence g - et le test de convergence σ - laissent voir que les revenus régionaux ne convergent pas entre 1977 et 1996. Toujours est-il que les estimations fondamentales de densité de la distribution laisse voir une certaine variation des caractéristiques de queue de la distribution. Employant des mélanges normaux finis pour examiner de façon plus formelle l'invariance temps de la distribution, l'article conclut que la distribution des logarithmes du revenu régional a en effet varié très peu sur le temps. La distribution des pourcentages de revenu montre une certaine mobilité. Il y a une rupture structurelle des probabilités de mélange des moyennes de mélanges et des variances. Cependant, la mobilité ne s'avère pas suffisamment forte pour laisser supposer un changement radical des distributions de mouvement simultané des moyennes régionales. TSIONAS E. G. (2000) Regionales Wachstum und Konvergenz: Beweise aus den Vereinigten Staaten, Reg. Studies 34 , 231‐238. Dieser Aufsatz beschäftigt sich mit regionaler Konvergenz in den Vereinigten Staaten. Sowohl g als auch σ Konvergenzprüfungen zeigen, daß regionale Einkommen sich im Zeitraum 1977‐1996 nicht aneinander annähern, doch Kerndichteschätzungen der Verteilung weisen bei den zuletzt aufgeführten Merkmalen einige Abweichungen in der Verteilung auf. Der Aufsatz benutzt begrenzte Normalmischungen zur Untersuchung der Zeitinvarianz der Verteilung auf formalere Art, und folgert, daß die Verteilung der Logarithmen eines regionalen einkommens tatsächlich nur sehr geringe zeitliche Abweichungen aufgewiesen hat. Die Verteilung der Einkommensprozentsätze erweist sich als etwas beweglicher, es gibt eine strukturelle Unterbrechung bei der Mischung von Wahrscheinlichkeiten, der Mischung von Mitteln und Abweichungen, doch ist die Beweglichkeit nicht stark genug, eine radikale Wende in den simultanen Bewegungsmustern regionaler Mittel anzuzeigen.

Random Walk with a Heavy-Tailed Jump Distribution

The classical random walk of which the one-step displacement variable u has a first finite negative moment is considered. The R.W. possesses an unique stationary distribution; x is a random variable with this distribution. It is assumed that the right-hand and/or the left-hand tail of the distribution of u are heavy-tailed. For the type of heavy-tailed distribution considered in this study a contraction factor Δ(a) exists with Δ(a) ↓ 0 for a ↑ 1, and a↑1 is equivalent with E{u} ↑ 0. It is shown that Δ(a)x converges in distribution for a↑1. It is the analysis of the tail of this limiting distribution of Δ(a)x which is the main purpose of the present study in particular when u is a mix of stochastic variables ui, ie1,…,N, each ui having its own heavy tail characteristics for its right- and left-hand tails. For an important case it is shown that for the tail of the distribution of Δ(a)x an asymptotic expression in the variables Δ(a) and t for Δ(a) ↓ 0 and t→∞ can be derived. For this asymptotic relation the dominating term is completely determined by the heavier tail of the 2N tails of the ui; the other terms of the asymptotic relation show the influence of less heavy tails and, depending on t, the terms may have a contribution which is not always negligible. The study starts with the derivation of a functional equation for the L.S.-transform of the distribution of x and that of the excess distribution of the stationary idle time distribution. For several important cases this functional equation could be solved and thus has led to the above mentioned asymptotic result. The derivation of it required quite some preparation, because it needed an effective description of the heavy-tailed jump sequence u. It was obtained by prescribing the heavy-tailed distributions of [u]+emax (0,u) and [u]−emin (0,u). It is shown that the random walk can serve as a model for the actual waiting process of a GI/G/1 queueing model; in that case the distribution of u is that of the difference of the service time and the interarrival time. The analysis of the present study then describes the heavy-traffic theory for the case with heavy-tailed service- and/or interarrival time distribution.