Edgeworth Series Expansion Research Articles

Approximation of Probability Density Functions for PDEs with Random Parameters Using Truncated Series Expansions

The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two stochastic finite element methods, Monte Carlo sampling and the stochastic Galerkin method with global polynomials. The random variable is a functional of the solution of the random PDE, such as the average over the physical domain. The truncated series are obtained considering a finite number of terms in the Gram-Charlier or Edgeworth series expansions. These expansions approximate the PDF of a random variable in terms of another PDF, and involve coefficients that are functions of the known cumulants of the random variable. To the best of our knowledge, their use in the framework of PDEs with random parameters has not yet been explored.

Stochastic dynamic analysis of the rotor–bearing system considering the randomness of the radial clearance

A stochastic dynamic model of the rotor–bearing system considering the randomness of the radial clearance is established in this research. Orthogonal polynomial approximation method is chosen to solve the stochastic dynamic equations. Taylor series expansion is used to approximate the nonlinear restoring force derived via the Hertz contact theory. The determination coefficient of equivalent Hertz contact is supposed to follow Bernoulli distribution. Then the probability of the contact between the jth ball and the raceways is solved. The first four order moments of the maximum displacement are obtained. And the CDF of the maximum displacement are calculated according to the Edgeworth series expansion theory. The effects of the factors such as the mean and the coefficient of variation of the radial clearances, eccentricity and the rotating speeds are discussed in time domain and frequency domain, respectively. Monte Carlo simulation verifies the effectiveness of the study.

Object-based multiscale method for SAR image change detection

This paper proposed an object-based multiscale method for synthetic aperture radar (SAR) image change detection based on the statistical model. Rather than the pixel-based analysis conducted in the traditional way, the object-based image analysis was employed to take a collection of pixels as the unit of analysis, which reduced small spurious changes and was less strict relative to registration. In addition, a multiscale concept was adopted to exhibit the inherent multiscale characteristics of the target. To achieve object-based, multiscale change detection results, the multidate segmentation was performed on two temporal SAR images and extended to a set of suitable scales. Then, the Edgeworth series expansion was employed to estimate the probability density function, and the Kullback–Leibler divergence was adopted to calculate the distance between pairs of pixel collections. Next, the divergence index maps were divided into changed and unchanged classes to obtain change detection results for each scale. Finally, the subresults were combined to obtain a more accurate detection result. The experimental results obtained using real data demonstrated the effectiveness of the proposed method.

Inhomogeneous quadratic tests in transient signal detection: Closed-form upper bounds and application in GNSS

This paper focuses on transient signal detection based on inhomogeneous quadratic tests, which involve the sum of a dependent non-central chi-square with a Gaussian random variable. These tests arise in many signal processing-related problems in biomedicine, finance or engineering, where sudden changes in the magnitude under analysis need to be promptly detected. Unfortunately, no closed-form expression is available for the density of inhomogeneous quadratic tests, which poses some concerns and limitations in their practical implementation. In particular, when trying to assess their detection performance in terms of probability of detection and probability of false alarm. In order to circumvent this limitation, two closed-form approximations based on results from Edgeworth series expansions and Extreme Value Theory (EVT) are proposed in this work. The use of these approximations is shown through a specific case of study in the context of transient detection for signal quality monitoring in Global Navigation Satellite Systems (GNSS). Numerical results are provided to assess the goodness of the proposed approximations, and to highlight their interest in real life applications.

A Multi-Region Segmentation Method for SAR Images based on the Multi-Texture Model with Level Sets.

Synthetic Aperture Radar (SAR) image segmentation is a difficult problem due to the presence of strong multiplicative noise. To attain multi-region segmentation for SAR images, this paper presents a parametric segmentation method based on the multi-texture model with level sets. Segmentation is achieved by solving level set functions obtained from minimizing the proposed energy functional. To fully utilize image information, edge feature and region information are both included in the energy functional. For the need of level set evolution, the Ratio of Exponentially Weighted Averages (ROEWA) operator is modified to obtain edge feature. Region information is obtained by the Improved Edgeworth Series Expansion (IESE), which can adaptively model a SAR image distribution with respect to various kinds of regions. The performance of the proposed method is verified by three high resolution SAR images. The experimental results demonstrate that SAR images can be segmented into multiple regions accurately without any speckle pre-processing steps by the proposed method.

Reliability-based robust design optimization using anchored ANOVA expansion and truncated Edgeworth series approximation

Based on the basic concept of reliability-based design optimization and robust design optimization, a reliability-based robust design optimization is achieved with the moment method in this work. At the first step, two methods, i.e. the anchored ANOVA expansion and truncated Edgeworth series approximation, are combined to solve the reliability constraints in the optimization procedure. The computational effort of the reliability analysis involved in the reliability-based design optimization is then reduced by the moment method. The reliability-based robust design optimization is then formulated by introducing the reliability sensitivity into the reliability-based design optimization as a sub-objective function, which is derived based on the truncated Edgeworth series expansion. The numerical examples illustrate the effectiveness and applicability of the proposed methodology and that the moment-based reliability-based robust design optimization satisfies (1) the reliability target of each constraint and (2) the minimum reliability sensitivity with respect to the mean value of the input random variables. The proposed reliability-based robust design optimization procedure could be applied to the product design under uncertain environment.

Hybrid reliability analysis of structural fatigue life: Based on Taylor expansion method

A new method for computing the failure probability of the fatigue life is proposed, dealing with uncertain problems with both random and interval variables. Using a Taylor expansion and the concept of statistical moment, the first four central moments of the structural fatigue life performance function are obtained. Then, using a second Taylor expansion, the first four central moments are expanded at the midpoint of the interval variable, and the intervals of the statistical moments of the performance function are calculated. The obtained moment information is applied into an Edgeworth series expansion expression, giving the cumulative distribution function of the structural fatigue life performance function and getting its interval of the failure probability. Two numerical examples of growing complexity are employed to demonstrate the feasibility of the proposed approach.

Pricing and hedging of arithmetic Asian options via the Edgeworth series expansion approach

In this paper, we derive a pricing formula for arithmetic Asian options by using the Edgeworth series expansion. Our pricing formula consists of a Black-Scholes-Merton type formula and a finite sum with the estimation of the remainder term. Moreover, we present explicitly a method to compute each term in our pricing formula. The hedging formulas (greek letters) for the arithmetic Asian options are obtained as well. Our formulas for the long lasting question on pricing and hedging arithmetic Asian options are easy to implement with enough accuracy. Our numerical illustration shows that the arithmetic Asian options worths less than the European options under the standard Black-Scholes assumptions, verifies theoretically that the volatility of the arithmetic average is less than the one of the underlying assets, and also discovers an interesting phenomena that the arithmetic Asian option for large fixed strikes such as stocks has higher volatility (elasticity) than the plain European option. However, the elasticity of the arithmetic Asian options for small fixed strikes as trading in currencies and commodity products is much less than the elasticity of the plain European option. These findings are consistent with the ones from the hedgings with respect to the time to expiration, the strike, the present underlying asset price, the interest rate and the volatility.

SAR image segmentation based on the advanced level set

Image segmentation takes an important role in SAR image processing. In this paper, a SAR image segmentation method based on level set evolution combining edge feature and statistic information is proposed. In order to enhance the impact of edge on image segmentation, all edge values are homogenized according to the calculated ROA operator. Different from traditional method where the SAR distribution is often specified based on human experiences, the Edgeworth algorithm, an approximation method for statistical distribution model, gives any SAR image distribution a statistical expression. Considering the practicability of ROA operator and the adaptivity of Edgeworth series expansion at fitting statistical distribution, an energy function based on edge and region properties is defined. To implement image division, partial differential equation (PDE) of curve evolution is obtained by minimizing the function. The proposed approach uses more information from SAR images and is appropriate for any SAR images without the need for human-specified distribution pattern. Finally, the experimental results which are obtained from the SAR images of some typical regions such as rivers and buildings show the applicability of the proposed method.

Stochastic optimal power flow by multi-variate Edgeworth expansions

Stochastic optimal power flow can provide the system operator with adequate strategies for controlling the power flow to maintain secure operation under stochastic parameter variations. One limitation of stochastic optimal power flow has been that only steady-state variable limits have been used as security constraints. In many systems voltage stability and small-signal stability also play an important role in constraining the operation.Recently an extension of the stochastic optimal power flow formulation that included constraints for voltage stability as well as small-signal stability was proposed. This was done by approximating the voltage stability and small-signal stability constraint boundaries with second order approximations in parameter space. In this article an alternative solution method to this problem will be proposed. The new improved solution method, which is based on Edgeworth series expansions, is both more efficient and accurate.We also give details on convexity of the problem and discuss some computational issues.

Closed Form Option Pricing Under Generalized Hermite Expansions

In this article, we generalize the classical Edgeworth series expansion used in the option pricing literature. We obtain a closed-form pricing formula for European options by employing a generalized Hermite expansion for the risk neutral density. The main advantage of the generalized expansion is that it can be applied to heavy-tailed return distributions, a case for which the standard Edgeworth expansions are not suitable. We also show how the expansion coefficients can be inferred directly from market option prices.

A New Model-Independent Method for Change Detection in Multitemporal SAR Images Based on Radon Transform and Jeffrey Divergence

This letter presents a new approach for change detection in multitemporal synthetic aperture radar images. Considering about the existence of speckle noise, the local statistics in a sliding window are compared instead of pixel-by-pixel comparison. Edgeworth series expansion is applied to estimate the probability density function (pdf), which is on the assumption that the pdf is not too far from normal distribution. To transcend such a limitation, in each analysis window, the image is projected onto two vectors in two independent dimensions; thus, the pdf of each projection is closer to a Gaussian density. In order to measure the distance between the two pairs of projections, the proposed algorithm uses a modified Kullback–Leibler (KL) divergence, called Jeffrey divergence, which turns out to be more numerically stable than KL divergence. Experiments on the real data show that the proposed detector outperforms all the others when a high detection rate is demanded.

Reliability-Based Robust Design for Structural System with Multiple Failure Modes

A reliability-based robust design method for structural system with multiple failure modes is proposed. For arbitrarily distributed random variables, the perturbation method and the Edgeworth series expansion are employed to approximate the cumulative distribution function of each failure mode. On the relevancy of failure modes, the probability network evaluation technique is used to assess the reliability of the system. Reliability sensitivities of random variables are derived with the gradient method. A reliability-based optimization model is built through minimum reliability sensitivity of design variables. Numerical examples illustrate the method proposed facilitates the robust design of structural system.

The Refined Positive Definite and Unimodal Regions for the Gram-Charlier and Edgeworth Series Expansion

Gram-Charlier and Edgeworth Series Expansions are used in the field of statistics to approximate probability density functions. The expansions have proven useful but have experienced limitations due to the values of the moments that admit a proper probability density function. An alternative approach in developing the boundary conditions for the boundary of the positive region for both series expansions is investigated using Sturm's theorem. The result provides a more accurate representation of the positive region developed by others.

Robust Misspecification Tests for the Heckman's Two-Step Estimator

This article constructs and evaluates Lagrange multiplier (LM) and Neyman's C(α) tests based on bivariate Edgeworth series expansions for the consistency of the Heckman's two-step estimator in sample selection models, that is, for marginal normality and linearity of the conditional expectation of the error terms. The proposed tests are robust to local misspecification in nuisance distributional parameters. Monte Carlo results show that testing marginal normality and linearity of the conditional expectations separately have a better size performance than testing bivariate normality. Moreover, the robust variants of the tests have better empirical size than nonrobust tests, which determines that these tests can be successfully applied to detect specific departures from the null model of bivariate normality. Finally, the tests are applied to women's labor supply data.

Analytical approximation method of option pricing under geometric mean-reverting process

This paper is concerned with analytical approximation for option pricing while the underlying asset price follows a geometric mean-reverting diffusion process, which is often used in real and commodity options. Two methods are developed, one employing the idea of Edgeworth series expansion and another based on Taylor series expansion. Numerical tests show that the second method significantly outperforms the first one, particularly for long-maturity options. In addition, we compare the analytical approximation prices with that obtained using an alternative log-normal model for commodities and find out for many cases the discrepancy in option prices is remarkable.

Some statistical applications of Faa di Bruno

The formula of Faa di Bruno is used to calculate higher order derivatives of a composition of functions. In this paper, we first review the multivariate version due to Constantine and Savits [A multivariate Faa di Bruno formula with applications, Trans. AMS 348 (1996) 503–520]. We next derive some useful recursion formulas. These results are then applied to obtain both explicit expressions and recursive formulas for the multivariate Hermite polynomials and moments associated with a multivariate normal distribution. Finally, an explicit expression is derived for the formal Edgeworth series expansion of the distribution of a normalized sum of iid random variables.

Inferring option-implied investors' risk preferences

Risk preference functions across the wealth domain are estimated from option prices and asset realized returns using: (a) a semiparametric probability model, the Edgeworth Series Expansion model, and (b) a new data set consisting of eurodollar and WTI oil markets’ data. The empirical preference functions are examined and found consistent with the market conditions of the period under study. The risk aversion estimates are also similar to these found by alternative methodologies.

Implied PDFs: Estimation, Testing and Applications in the Eurodollar Market

This paper develops a method for estimating implied PDFs for futures prices from American options. The restricting assumption of log-normally distributed returns is relaxed with the use of the more flexible distributional form of an Edgeworth Series Expansion (ESE) of a log-normal distribution. The method is applied to Eurodollar futures options market data and a plethora of tests is employed to explore its capacity. The proposed model is found to be able to estimate PDFs that can capture the general market sentiment and also incorporate isolated events causing a significant impact on the market. In addition to exploring the 'economic' sensibility of the recovered densities, their statistical properties are examined against the properties of PDFs estimated with the use of alternative methodologies. A superior behaviour is present in most of the cases addressed. Based on the proposed model, a new setting that allows to explicitly quantify the information content of option prices is also developed.

Valuing Exotic Derivatives with Jump Diffusions: The Case of Basket Options

Exotic options are complicated derivatives instruments whose structure does not allow, in general, for closed-form analytic solutions, thus, making their pricing and hedging a difficult task. To overcome additional complexity such products are, as a rule, priced within a Black-Scholes framework, assuming that the underlying asset follows a Geometric Brownian Motion (GBM) stochastic process. This paper develops a more realistic framework for the pricing of exotic derivatives; and derives closed-form analytic solutions for the pricing and hedging of basket options. We relax the simplistic assumption of the GBM, by introducing the Bernoulli Jump Diffusion process (BJD) and approximate the terminal distribution of the underlying asset with a log-normal distribution. Potential extension of the model with the use of the Edgeworth Series Expansion (ESE) is also discussed. Monte Carlo simulation confirms the validity of the proposed BJD model.