The expectation-maximization algorithm for autoregressive models with normal inverse Gaussian innovations

The normal inverse Gaussian (NIG) distribution has previously been shown to be a versatile tool to model heavy-tailed processes. A cumulant-matching estimator of the NIG parameters was introduced in Oigard and Hansen (2001). Here, we analyze the performance of this estimator. Next, we study whether NIG models are appropriate models for multi-user interference. Finally, we empirically test the goodness of fit of squared-NIG models to tele-traffic data, which have been fitted with alpha-stable models.

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

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTNumerical estimation of adsorption energy distributions from adsorption isotherm data with the expectation-maximization methodBrett J. Stanley and Georges GuiochonCite this: J. Phys. Chem. 1993, 97, 30, 8098–8104Publication Date (Print):July 1, 1993Publication History Published online1 May 2002Published inissue 1 July 1993https://doi.org/10.1021/j100132a046RIGHTS & PERMISSIONSArticle Views352Altmetric-Citations89LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (716 KB) Get e-Alerts

Many diseases show dichotomous phenotypic variation but do not follow a simple Mendelian pattern of inheritance. Variances of these binary diseases are presumably controlled by multiple loci and environmental variants. A least-squares method has been developed for mapping such complex disease loci by treating the binary phenotypes (0 and 1) as if they were continuous. However, the least-squares method is not recommended because of its ad hoc nature. Maximum Likelihood (ML) and Bayesian methods have also been developed for binary disease mapping by incorporating the discrete nature of the phenotypic distribution. In the ML analysis, the likelihood function is usually maximized using some complicated maximization algorithms (e.g. the Newton-Raphson or the simplex algorithm). Under the threshold model of binary disease, we develop an Expectation Maximization (EM) algorithm to solve for the maximum likelihood estimates (MLEs). The new EM algorithm is developed by treating both the unobserved genotype and the disease liability as missing values. As a result, the EM iteration equations have the same form as the normal equation system in linear regression. The EM algorithm is further modified to take into account sexual dimorphism in the linkage maps. Applying the EM-implemented ML method to a four-way-cross mouse family, we detected two regions on the fourth chromosome that have evidence of QTLs controlling the segregation of fibrosarcoma, a form of connective tissue cancer. The two QTLs explain 50-60% of the variance in the disease liability. We also applied a Bayesian method previously developed (modified to take into account sex-specific maps) to this data set and detected one additional QTL on chromosome 13 that explains another 26% of the variance of the disease liability. All the QTLs detected primarily show dominance effects.

Finite mixtures of multivariate skew t (MST) distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour. Recently, they have been exploited as an effective tool for modelling flow cytometric data. A number of algorithms for the computation of the maximum likelihood (ML) estimates for the model parameters of mixtures of MST distributions have been put forward in recent years. These implementations use various characterizations of the MST distribution, which are similar but not identical. While exact implementation of the expectation-maximization (EM) algorithm can be achieved for `restricted' characterizations of the component skew t-distributions, Monte Carlo (MC) methods have been used to fit the `unrestricted' models. In this paper, we review several recent fitting algorithms for finite mixtures of multivariate skew t-distributions, at the same time clarifying some of the connections between the various existing proposals. In particular, recent results have shown that the EM algorithm can be implemented exactly for faster computation of ML estimates for mixtures with unrestricted MST components. The gain in computational time is effected by noting that the semi-infinite integrals on the E-step of the EM algorithm can be put in the form of moments of the truncated multivariate non-central t-distribution, similar to the restricted case, which subsequently can be expressed in terms of the non-truncated form of the central t-distribution function for which fast algorithms are available. We present comparisons to illustrate the relative performance of the restricted and unrestricted models, and demonstrate the usefulness of the recently proposed methodology for the unrestricted MST mixture, by some applications to three real datasets.

This paper presents a nonsubsampled shearlet transform (NSST) based remote sensing (RS) image retrieval technique where the NSST subband coefficients are modeled with symmetric normal inverse Gaussian (SNIG) distribution. The NSST is popular for its very good directional selectivity and shift invariance. The SNIG is a four parameter distribution which can describe a wide range of heavy tailed distributions. Through Kolmogorov-Smirnov (KS) goodness of fit test, it is shown that the SNIG best approximates the shape of distribution of detail NSST coefficients. For Maximum Likelihood (ML) estimation of SNIG parameters, an Expectation-Maximization (EM) type of algorithm is used. The RS images are first decomposed with NSST and the feature vector is formed with the estimated SNIG parameters from detail NSST coefficients. Here, the UC Merced land use (UCM) and the High-resolution satellite scene (HRS) datasets are used for the experiments conducted. The results of the proposed method reveal superior performance, when compared with some well known techniques.

Abstract. An expectation maximization (EM) algorithm is proposed to find fibre length distributions in standing trees. The available data come from cylindric wood samples (increment cores). The sample contains uncut fibres as well as fibres cut once or twice. The sample contains not only fibres, but also other cells, the so‐called ‘fines’. The lengths are measured by an automatic fibre‐analyser, which is not able to distinguish fines from fibres and cannot tell if a cell has been cut. The data thus come from a censored version of a mixture of the fine and fibre length distributions in the tree. The parameters of the length distributions are estimated by a stochastic version of the EM algorithm, and an estimate of the corresponding covariance matrix is derived. The method is applied to data from northern Sweden. A simulation study is also presented. The method works well for sample sizes commonly obtained from increment cores.

Objective: Data being missed is an unavoidable problem in cohort studies. This paper compares the imputation effect of eight common missing data imputation methods involved in cutting longitudinal data through simulation study to provide a valuable reference for the treatment of missing data in longitudinal studies. Methods: The simulation study is based on R language software and generates missing longitudinal data by the Monte Carlo method. By comparing the average absolute deviation, average relative deviation, and TypeⅠerror from the regression analysis of different imputation methods, the imputation effect of varying imputation methods on missing longitudinal data and the influence on subsequent multivariate analysis are evaluated. Results: The mean imputation, k nearest neighbor (KNN), regression imputation, and random forest all have a similar imputation effect, which is also steady. However, the hot deck is inferior to the above imputation methods. K-means clustering and expectation maximization (EM) algorithm are among the worst and unstable. Mean imputation, EM algorithm, random forest, KNN, and regression imputation can control TypeⅠerror. Still, multiple imputations, hot deck, and K-means clustering cannot effectively manage the TypeⅠerror. Conclusions: For missing data in longitudinal studies, mean imputation, KNN, regression imputation, and random forest can be used as better imputation methods under the mechanism of missing at random. When the missing ratio is not too large, multiple imputations and hot deck can also perform well, but K-means clustering and EM algorithm are not recommended.

Joint modeling of survival and longitudinal data has been studied extensively in the recent literature. The likelihood approach is one of the most popular estimation methods employed within the joint modeling framework. Typically, the parameters are estimated using maximum likelihood, with computation performed by the expectation maximization (EM) algorithm. However, one drawback of this approach is that standard error (SE) estimates are not automatically produced when using the EM algorithm. Many different procedures have been proposed to obtain the asymptotic covariance matrix for the parameters when the number of parameters is typically small. In the joint modeling context, however, there may be an infinite-dimensional parameter, the baseline hazard function, which greatly complicates the problem, so that the existing methods cannot be readily applied. The profile likelihood and the bootstrap methods overcome the difficulty to some extent; however, they can be computationally intensive. In this paper, we propose two new methods for SE estimation using the EM algorithm that allow for more efficient computation of the SE of a subset of parametric components in a semiparametric or high-dimensional parametric model. The precision and computation time are evaluated through a thorough simulation study. We conclude with an application of our SE estimation method to analyze an HIV clinical trial dataset.

Computation: Expectation-Maximization Algorithm

The EM algorithm is a widely used technique for finding maximum likelihood (ML) estimates when the data are not fully observed. Despite its popularity for computing ML estimates in unrestricted problems and the need for simplified computations for problems with equality and inequality restrictions, there have been few applications of the algorithm to restricted ML estimation. The EM algorithm is presented for restricted ML estimation and provides its applications to the probit model under equality and inequality restrictions using two small data sets.

In this paper, a computationally efficient algorithm is presented for joint maximum likelihood (ML) timing and carrier phase estimation of OFDM systems employing M-PSK modulation scheme with additive Gaussian noise, based on the expectation-maximization (EM) algorithm. A nondata-aided (NDA) scheme is considered for the joint timing and phase synchronizer which maximizes the low SNR limit of the likelihood function averaged over the M-PSK signal constellation. For this, an EM algorithm is derived which estimates the timing offset and the phase rotations of each subcarrier iteratively and which converges to the true ML estimation of the unknown timing and phase. It is shown that the algorithm becomes independent of the signal-to-noise ratio for both low and high SNR cases. The algorithm is applied to the QPSK modulated OFDM systems and it is concluded that for SNR values greater than 10 dB the convergence is achieved in first iteration and for SNR values less than 10 dB, at most in three iterations. It is also concluded that the convergence is independent of the initial starting points.

The Weibull regression model is a Weibull distribution that is directly influenced by covariates. The Weibull regression model discussed in this study was the Weibull survival and the Weibull hazard regression model. The Weibull regression model in this study was applied to the hospitalization time data of COVID-19 patients from May to September 2021 at the RSUD Abdul Wahab Sjahranie Samarinda. The event of the study is recovery of patient. This study aims to obtain Weibull survival and hazard regression model to the hospitalization time data of Covid-19 patients, to obtain the factors that affect the chance of not recovering (survive) and the recovery rate of Covid-19 patients, and also to interpret Weibull survival and hazard regression models based on the obtained model. In this study, the Maximum Likelihood Estimation (MLE) was used as the parameter estimation method. The closed form of the Maximum Likelihood (ML) estimator cannot be found analytically, and the approximation of ML estimator was found using Newton-Raphson iterative method. Based on the test results, the factors that influence the chance of not recovering and the recovery rate of COVID-19 patients were comorbidities history. The chance of not not recovering (survive) for patients who have a history of comorbidities is greater than the chance of not recovering (survive) for patients who have no history of comorbidities. The recovery rate for COVID-19 patients who have a history of comorbidities is 0,5358 times the recovery rate for patients without a history of comorbidities.

Helioseismic data are often interrupted by gaps, which diminish the quality of the data. In the frequency domain, these gaps lead to systematical effects with misleading interpretation of the power spectra. We propose a gap filling method that is based on modeling solar oscillation data with a statistical process, i.e., the stochastic nature of a single oscillation is taken into account by regarding it as realization of an autoregressive (AR) processes of second order. From the whole oscillation time series given as the superposition of the realization of many excited modes, the process parameters are estimated via the expectation maximization (EM) algorithm. Then the estimated model is used to predict the further course of the oscillatory process during occurring gaps. We demonstrate the applicability of this procedure on the basis of both simulations and data obtained with the DIFOS satellite experiment suffering from gaps of 30 min duration occurring regularly every 90 min due to the orbit around the Earth.

This paper proposes a method for obtaining nonlinear models of distribution grid based on available measurements from the power grid. We formulate a maximum likelihood estimation (MLE) problem that estimates unknown line parameters—specifically, the impedance between nodes—using measured voltage magnitudes and net active and reactive power injections at each node. The nonlinear model for the distribution grid uses a nonlinear approximation of the DistFlow model, which includes line losses and is parameterized by the unknown line impedances. We solve the resulting MLE problem using an expectation maximization (EM) algorithm, tailored for the nonlinear model, and provide a numerically robust implementation. The proposed method is demonstrated on the IEEE 37-node test network, and we compare it with the state-of-the-art methods. The proposed method achieves a 70% reduction in voltage error and an error for state variables that is more than 10000 times smaller. A final comparison uses data from a real network, and the proposed method achieves parameter estimates with errors 100 times smaller than competing methods.