Exponential Mixture Research Articles

Multistate Traffic Flow Headway Fitting

Influenced by a variety of factors, such as driving skill, traffic composition, signal control, traffic management, and traffic environment and so on, the traffic flow characteristic parameter statistics embodies multi-state characteristic which is incapable to describe precisely by using a single distribution model. In order to fit the headway characteristic parameter of multi-state traffic flow, based on the denseness of mixture Gamma distribution, the data could be fitted more accurately with the method of expectation maximization algorithm and by means of controlling the number of branches under the condition of precision beyond 95%. Take actual survey data as the example, this paper compares the fitting errors of negative exponential distribution, Erlang distribution and mixture Gamma distribution and the result shows that the fitting result based on mixture Gamma distribution is closer to actual case.

A recursive approach to mortality-linked derivative pricing

In this paper, we develop a recursive method to derive an exact numerical and nearly analytical representation of the Laplace transform of the transition density function with respect to the time variable for time-homogeneous diffusion processes. We further apply this recursion algorithm to the pricing of mortality-linked derivatives. Given an arbitrary stochastic future lifetime T , the probability distribution function of the present value of a cash flow depending on T can be approximated by a mixture of exponentials, based on Jacobi polynomial expansions. In case of mortality-linked derivative pricing, the required Laplace inversion can be avoided by introducing this mixture of exponentials as an approximation of the distribution of the survival time T in the recursion scheme. This approximation significantly improves the efficiency of the algorithm.

Ant colony optimization for wavelet-based image interpolation using a three-component exponential mixture model

Wavelet-based image interpolation typically treats the input image as the low frequency subbands of an unknown wavelet-transformed high-resolution image, and then produces the unknown high-resolution image by estimating the wavelet coefficients of the high frequency subbands. For that, a new approach is proposed in this paper, the contribution of which are twofold. First, unlike that the conventional Gaussian mixture (GM) model only exploits the magnitude information of the wavelet coefficients, a three-component exponential mixture (TCEM) model is proposed in this paper to investigate both the magnitude information and the sign information of the wavelet coefficients. The proposed TCEM model consists of a Gaussian component, a positive exponential component and a negative exponential component. Second, to address the parameter estimation challenge of the proposed TCEM model, the ant colony optimization (ACO) technique is exploited in this paper to classify the wavelet coefficients into one of three components of the proposed TCEM model for estimating their parameters. Experiments are conducted to demonstrate that the proposed approach outperform a number of approaches developed in the literature.

On a Sparre Andersen Risk Model with Time-Dependent Claim Sizes and Jump-Diffusion Perturbation

In this paper, we consider a Sparre Andersen risk model where the interclaim time and claim size follow some bivariate distribution. Assuming that the risk model is also perturbed by a jump-diffusion process, we study the Gerber–Shiu functions when ruin is due to a claim or the jump-diffusion process. By using a q-potential measure, we obtain some integral equations for the Gerber–Shiu functions, from which we derive the Laplace transforms and defective renewal equations. When the joint density of the interclaim time and claim size is a finite mixture of bivariate exponentials, we obtain the explicit expressions for the Gerber–Shiu functions.

On moments based Padé approximations of ruin probabilities

In this paper, we investigate the quality of the moments based Padé approximation of ultimate ruin probabilities by exponential mixtures. We present several numerical examples illustrating the quick convergence of the method in the case of Gamma processes. While this is not surprising in the completely monotone case (which holds when the shape parameter is less than 1), it is more so in the opposite case, for which we improve even further the performance by a fix-up which may be of special importance due to its potential use in the four moments Gamma approximation. We also review the connection of the exponential mixtures approximation to Padé approximation, orthogonal polynomials, and Gaussian quadrature. These connections may turn out useful for providing rates of convergence.

Iterative algorithms for learning a linear Gaussian observation model with an exponential power scale mixture prior

The authors study an iterative algorithm for learning a linear Gaussian observation model with an exponential power scale mixture prior (EPSM). This is a generalisation of previous study based on the Gaussian scale mixture prior. The authors use the principle of majorisation minimisation to derive the general iterative algorithm which is related to a reweighted lp-minimisation algorithm. The authors then show that the Gaussian and Laplacian scale mixtures are two special cases of the EPSM and the corresponding learning algorithms are related to the reweighted l2-and l1-minimisation algorithms, respectively. The authors also study a particular case of the EPSM which is a Pareto distribution and discuss Bayesian methods for parameter estimation.

Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

Using multiple chemical indicators to characterize and determine the age of groundwater from selected vents of the Silver Springs Group, central Florida, USA

The Silver Springs Group, Florida (USA), forms the headwaters of the Silver River and supports a diverse ecosystem. The 30 headwater springs divide into five subgroups based on chemistry. Five selected spring vents were sampled in 2007 to better understand the contaminant sources and groundwater flow system. Elevated nitrate-N concentrations (>0.8 mg/L) in the five spring vents likely originate from inorganic (fertilizers) and organic sources, based on nitrogen and oxygen isotope ratios of nitrate. Evidence for denitrification in the Lost River Boil spring includes enriched δ15N and δ18O, excess N2 gas, and low dissolved O2 concentrations (<0.5 mg/L). Multiple age-tracer data (SF6, 3H, tritiogenic 3He) for the two uppermost springs (Mammoth East and Mammoth West) indicate a binary mixture dominated by recent recharge water (mean age 6–7 years, and 87–97% young water). Tracer data for the three downstream spring vents (Lost River Boil, Catfish Hotel-1, and Catfish Conventional Hall-1) indicate exponential mixtures with mean ages of 26–35 years. Contamination from non-atmospheric sources of CFCs and SF5CF3 precluded their use as age tracers here. Variations in chemistry were consistent with mean groundwater age, as nitrate-N and dissolved O2 concentrations were higher in younger waters, and the Ca/Mg ratio decreased with increasing mean age.

A Corrected Likelihood Approach for the Nonlinear Transformation Model with Application to Fluorescence Lifetime Measurements Using Exponential Mixtures

A fast and efficient estimation method is proposed that compensates the distortion in nonlinear transformation models. A likelihood-based estimator is developed that can be computed by an EM-type algorithm. The consistency of the estimator is shown and its limit distribution is provided. The new estimator is particularly well suited for fluorescence lifetime measurements, where only the shortest arrival time of a random number of emitted fluorescence photons can be detected and where arrival times are often modeled by a mixture of exponential distributions. The method is evaluated on real and synthetic data. Compared to currently used methods in fluorescence, the new estimator should allow a reduction of the acquisition time of an order of magnitude.

Data-driven approach to decomposing complex enzyme kinetics with surrogate models

The temporal autocorrelation (AC) function associated with monitoring order parameters characterizing conformational fluctuations of an enzyme is analyzed using a collection of surrogate models. The surrogates considered are phenomenological stochastic differential equation (SDE) models. It is demonstrated how an ensemble of such surrogate models, each surrogate being calibrated from a single trajectory, indirectly contains information about unresolved conformational degrees of freedom. This ensemble can be used to construct complex temporal ACs associated with a "non-Markovian" process. The ensemble of surrogates approach allows researchers to consider models more flexible than a mixture of exponentials to describe relaxation times and at the same time gain physical information about the system. The relevance of this type of analysis to matching single-molecule experiments to computer simulations and how more complex stochastic processes can emerge from a mixture of simpler processes is also discussed. The ideas are illustrated on a toy SDE model and on molecular-dynamics simulations of the enzyme dihydrofolate reductase.

TVaR-based capital allocation with copulas

Because of regulation projects from control organisations such as the European solvency II reform and recent economic events, insurance companies need to consolidate their capital reserve with coherent amounts allocated to the whole company and to each line of business. The present study considers an insurance portfolio consisting of several lines of risk which are linked by a copula and aims to evaluate not only the capital allocation for the overall portfolio but also the contribution of each risk over their aggregation. We use the tail value at risk (TVaR) as risk measure. The handy form of the FGM copula permits an exact expression for the TVaR of the sum of the risks and for the TVaR-based allocations when claim amounts are exponentially distributed and distributed as a mixture of exponentials. We first examine the bivariate model and then the multivariate case. We also show how to approximate the TVaR of the aggregate risk and the contribution of each risk when using any copula.

A mixture of exponentials distribution for a simple and precise assessment of the volcanic hazard

Abstract. The assessment of volcanic hazard is the first step for disaster mitigation. The distribution of repose periods between eruptions provides important information about the probability of new eruptions occurring within given time intervals. The quality of the probability estimate, i.e., of the hazard assessment, depends on the capacity of the chosen statistical model to describe the actual distribution of the repose times. In this work, we use a mixture of exponentials distribution, namely the sum of exponential distributions characterized by the different eruption occurrence rates that may be recognized inspecting the cumulative number of eruptions with time in specific VEI (Volcanic Explosivity Index) categories. The most striking property of an exponential mixture density is that the shape of the density function is flexible in a way similar to the frequently used Weibull distribution, matching long-tailed distributions and allowing clustering and time dependence of the eruption sequence, with distribution parameters that can be readily obtained from the observed occurrence rates. Thus, the mixture of exponentials turns out to be more precise and much easier to apply than the Weibull distribution. We recommended the use of a mixture of exponentials distribution when regimes with well-defined eruption rates can be identified in the cumulative series of events. As an example, we apply the mixture of exponential distributions to the repose-time sequences between explosive eruptions of the Colima and Popocatépetl volcanoes, México, and compare the results obtained with the Weibull and other distributions.

Generalized Linear Models for Site-Specific Density Forecasting of U.K. Daily Rainfall

Abstract Site-specific probability density rainfall forecasts are needed to price insurance premiums, contracts, and other financial products based on precipitation. The spatiotemporal correlations in U.K. daily rainfall amounts over the Thames Valley are investigated and statistical Markov chain generalized linear models (Markov GLM) of rainfall are constructed. The authors compare point and density forecasts of total rainfall amounts, and forecasts of probability of occurrence of rain from these models and from other proposed density models, including persistence, statistical climatology, Markov chain, unconditional gamma and exponential mixture models, and density forecasts from GLM regression postprocessed NCEP numerical ensembles, at up to 45-day forecast horizons. The Markov GLMs and GLM processed ensembles produced skillful 1-day-ahead and short-term point forecasts. Diagnostic checks show all models are well calibrated, but GLMs perform best under the continuous-ranked probability score. For lead times of greater than 1 day, no models were better than the GLM processed ensembles at forecasting occurrence probability. Of all models, the ensembles are best able to account for the serial correlations in rainfall amounts. In conclusion, GLMs for future site-specific density forecasting are recommended. Investigations explain this conclusion in terms of the interaction between the autocorrelation properties of the data and the structure of the models tested.

Some properties of the generalized TTT transform

Recently Li and Shaked [2007. A general family of univariate stochastic orders. J. Statist. Plann. Inference 137, 3601–3610] introduced the generalized total time on test (GTTT) transform with respect to a given function ϕ . In this paper we study some properties of it which are related with stochastic orderings. A concept of Lehmann and Rojo [1992. Invariant directional orderings. Ann. Statist. 20, 2100–2110] is applied to a new setting and the GTTT transform is used to define invariance properties and distances of some stochastic orders. Iterations of the GTTT transforms are also studied and their relations with exponential mixtures of gamma distributions are established.

CramÉr–Rao Bounds for Multiple Poles and Coefficients of Quasi-Polynomials in Colored Noise

In this paper, we provide analytical expressions of the Cramer-Rao bounds for the frequencies, damping factors, amplitudes, and phases of complex exponentials in colored noise. These expressions show the explicit dependence of the bounds of each distinct parameter with respect to the amplitudes and phases, leading to readily interpretable formulae, which are then simplified in an asymptotic context. The results are presented in the general framework of the polynomial amplitude complex exponentials (PACE) model, also referred to as the quasi-polynomial model in the literature, which accounts for systems involving multiple poles and represents a signal as a mixture of complex exponentials modulated by polynomials. This work looks further and generalizes the studies previously undertaken on the exponential and the quasi-polynomial models.

Convergence of stochastic approximation algorithms under irregular conditions

We consider a class of stochastic approximation (SA) algorithms for solving a system of estimating equations. The standard condition for the convergence of the SA algorithms is that the estimating functions are locally Lipschitz continuous. Here, we show that this condition can be relaxed to the extent that the estimating functions are bounded and continuous almost everywhere. As a consequence, the use of the SA algorithm can be extended to some problems with irregular estimating functions. Our theoretical results are illustrated by solving an estimation problem for exponential power mixture models.

Bregman bubble clustering

In classical clustering, each data point is assigned to at least one cluster. However, in many applications only a small subset of the available data is relevant for the problem and the rest needs to be ignored in order to obtain good clusters. Certain nonparametric density-based clustering methods find the most relevant data as multiple dense regions, but such methods are generally limited to low-dimensional data and do not scale well to large, high-dimensional datasets. Also, they use a specific notion of “distance”, typically Euclidean or Mahalanobis distance, which further limits their applicability. On the other hand, the recent One Class Information Bottleneck (OC-IB) method is fast and works on a large class of distortion measures known as Bregman Divergences, but can only find a single dense region. This article presents a broad framework for finding k dense clusters while ignoring the rest of the data. It includes a seeding algorithm that can automatically determine a suitable value for k . When k is forced to 1, our method gives rise to an improved version of OC-IB with optimality guarantees. We provide a generative model that yields the proposed iterative algorithm for finding k dense regions as a special case. Our analysis reveals an interesting and novel connection between the problem of finding dense regions and exponential mixture models; a hard model corresponding to k exponential mixtures with a uniform background results in a set of k dense clusters. The proposed method describes a highly scalable algorithm for finding multiple dense regions that works with any Bregman Divergence, thus extending density based clustering to a variety of non-Euclidean problems not addressable by earlier methods. We present empirical results on three artificial, two microarray and one text dataset to show the relevance and effectiveness of our methods.

Block–Savits Characterization and Star Ordering of Exponential Mixtures

Block and Savits (1980) established a characterization of life distributions using the Laplace transform. In this article, we remark that one of the necessary conditions to be IFRA distribution is equivalent to the star ordering of exponential mixtures. It leads to the definition of two new classes of life distributions, called LIFR and LIFRA, and their dual classes: LDFR and LDFRA. It occurs that these classes have many useful aging properties and preserve known reliability operations. Properties of the classes are studied and relations with known classes are established.

Infinite divisibility of the spacings of a Kotz–Kozubowski–Podgórski generalized Laplace model

The infinite divisibility of the Laplace distribution and its applicability as a statistical model were the motivation for the study of some properties of the spacings of a Kotz–Kozubowski–Podgórski generalized Laplace model. This model is an extension of the classical symmetric Laplace model for the case of asymmetric tails. In this note we shall show that the spacings are generalized exponential mixtures or gamma mixtures and, hence, preserve the infinite divisibility of the parent model.