Non-exponential Models Research Articles

Improving CPMG liver iron estimates with a -corrected proton density estimator.

CPMG spin echo acquisitions are attractive for diagnosing and monitoring liver iron concentration in iron overload disorders due to their time efficiency and potential to reveal unique information about tissue iron distribution. Clinical adoption remains low due to the insensitivity of CPMG-based estimates to liver iron concentration (LIC) when common fitting techniques are applied. In this work, we demonstrate that the inclusion of a proton density estimator (PDE) derived from the CPMG acquisition increase the sensitivity of CPMG estimates to LIC in both simulated and in-vivo human data. CPMG acquisitions from 50 clinically indicated MRI studies in patients with iron overload were analyzed with and without PDE constraints. Liver regions of interest were fit to monoexpontial and nonexponential signal decay equations. LIC by served as the reference standard. The observed calibration between CPMG values and LIC were compared to results predicted from a previously validated Monte Carlo model. The sensitivity of CPMG-derived triples when a proton density constraint is applied. When compared with -LIC estimates, both monoexponential and nonexponential models were unbiased but demonstrated broad 95% confidence intervals particularly for LIC values below 12mg/g. Absolute error did not increase with LIC. A proton density constraint can increase the sensitivity of CPMG-based models to iron. CPMG acquisitions are time-efficient and could potentially improve the dynamic range of single spin echo techniques as well as providing insight into tissue iron distribution.

An efficient algorithm for computation of information matrix in phase-type fitting

Phase-type (PH) fitting is a technique to approximate any general distribution as a PH distribution, which is a probability distribution representing an absorbing time of a Markov chain. Since the PH distribution is described as a discrete- or continuous-time Markov chain (CTMC), the PH fitting can provide approximate Markov models to any non-exponential stochastic models. Thus, the PH fitting is helpful for model-based performance evaluation. On the other hand, from the statistical point of view, the PH fitting is categorized as parameter estimation from data. Some efficient PH fitting techniques are based on the maximum-likelihood principle. Therefore, it is crucial to evaluate statistical errors, i.e., the variance and covariance of estimators. In maximum-likelihood estimation, the Fisher information matrix is a well-known method to compute the variance and covariance of estimators, and is obtained as the second derivative of the log-likelihood function (LLF). In this article, we propose an algorithm for efficiently computing the Fisher information matrix in PH fitting. By applying the uniformization technique to a CTMC, we design the algorithm for computing the second derivatives of LLF in PH fitting.

Physical and practical hydrograph recession modeling in karstic sinkholes

Abstract Karstic spring discharge is related to the hydraulic head recession through a power function with an exponent <1. In the literature, analytical solutions are available for exponential and non-exponential models based on a set of restrictive physical and mathematical assumptions. The models search for a holistic and deductive solution without basic physical and practical interpretations, simple logical inferences leading to mathematical analytical or empirical formulations. In this paper, an inductive, logical, practical, and instead of holistic modeling, physically plausible piecewise solutions are proposed with detailed inferences and interpretations. In the proposed methodology, the discharge and hydraulic head records are decomposed first into a set of verbal classes and, subsequently, physical meaning for each class is explained leading to simple general but empirical models. For this purpose, Wakula and St. Marks River (Florida) hydrograph records are used for the general solution sinkhole discharge and hydraulic head variations. The solution methodology presented in this paper does not make any distinction between relatively small or large sinkhole heads. The calibration and verification of the methodology is shown with a comparison of the available record values to partial power models. Finally, it is concluded that the proposed methodology is reliable and can be applied to hydraulic head availability with discharge records in any part of the world for karstic aquifer domains.

On measuring selection in cancer from subclonal mutation frequencies.

Recently available cancer sequencing data have revealed a complex view of the cancer genome containing a multitude of mutations, including drivers responsible for cancer progression and neutral passengers. Measuring selection in cancer and distinguishing drivers from passengers have important implications for development of novel treatment strategies. It has recently been argued that a third of cancers are evolving neutrally, as their mutational frequency spectrum follows a 1/f power law expected from neutral evolution in a particular intermediate frequency range. We study a stochastic model of cancer evolution and derive a formula for the probability distribution of the cancer cell frequency of a subclonal driver, demonstrating that driver frequency is biased towards 0 and 1. We show that it is difficult to capture a driver mutation at an intermediate frequency, and thus the calling of neutrality due to a lack of such driver will significantly overestimate the number of neutrally evolving tumors. Our approach provides quantification of the validity of the 1/f statistic across the entire range of relevant parameter values. We also show that our conclusions remain valid for non-exponential models: spatial 3d model and sigmoidal growth, relevant for early- and late stages of cancer growth.

On fractional modeling of viscoelastic foams

Empiric models have been introduced to describe frequency dependence of dielectric permittivity. Simple exponential models are often not satisfactory, while advanced non-exponential models (usually referred as “anomalous relaxation”) are commonly required to better explain experimental observations of complex systems. For viscoelastic materials, the so-called fractional derivatives models are powerful for both dynamic and loss moduli prediction. In this paper, the analysis of the main models used in the characterization of dielectric and viscoelastic materials such as five-parameter fractional Zener model and empiric Havriliak–Negami model are analysed. The fractional shape parameters describing the symmetric and asymmetric broadening of the complex modulus don't have the same influence in low and high frequencies. In contrast to the five-parameter Zener model, the empiric model asymmetry parameter has an influence on complex modulus at low frequencies comparing to the loss modulus peak frequency. A no resonance technique based on a forced vibrations procedure is used to investigate the frequency dependent complex shear modulus of a polyurethane foam, not influenced by its fluid phase, in the range 0.1–500 Hz. It is shown that the Havriliak–Negami model can predict the frequency dependence for a wide frequency range.

A Generic Decomposition Formula for Pricing Vanilla Options under Stochastic Volatility Models

We obtain a decomposition of the call option price for a very general stochastic volatility diffusion model, extending a previous decomposition formula for the Heston model. We realize that a new term arises when the stock price does not follow an exponential model. The techniques used for this purpose are nonanticipative. In particular, we also see that equivalent results can be obtained by using Functional Itô Calculus. Using the same generalizing ideas, we also extend to nonexponential models the alternative call option price decomposition formula written in terms of the Malliavin derivative of the volatility process. Finally, we give a general expression for the derivative of the implied volatility under both the anticipative and the nonanticipative cases.

Non-linearity and heterogeneity in modeling of population dynamics

The study of population growth reveals that the behaviors that follow the power law appear in numerous biological, demographical, ecological, physical and other contexts. Parabolic models appear to be realistic approximations of real-life replicator systems, while hyperbolic models were successfully applied to problems of global demography and appear relevant in quasispecies and hypercycle modeling. Nevertheless, it is not always clear why non-exponential growth is observed empirically and what possible origins of the non-exponential models are.In this paper the power equation is considered within the frameworks of inhomogeneous population models; it is proven that any power equation describes the total population size of a frequency-dependent model with Gamma-distributed Malthusian parameter. Additionally, any super-exponential equation describes the dynamics of inhomogeneous Malthusian density-dependent population model. All statistical characteristics of the underlying inhomogeneous models are computed explicitly. The results of this analysis show that population heterogeneity can be a reasonable explanation for power law accurately describing total population growth.

Modeling computer virus with the BSDE approach

The aim of this paper is to contribute to the connection between computer viruses spreading and epidemiological models. To this end, the block-structured state-dependent event (BSDE) approach is used to study the number of cases of infection in a computer network. The goal of the BSDE approach is the possibility of dealing with non-exponential models with correlated flows, but keeping tractable the dimensionality of the underlying Markov chain. The obtained results are illustrated by numerical experiments which show how the BSDE approach is helpful to strengthen the computer security when it is used in connection with a warning mechanism. An application to the propagation of the CodeRed-II virus is also included.

Resonances of light atoms in electric and magnetic fields

Empiric models have been introduced to describe frequency dependence of dielectric permittivity. Simple exponential models are often not satisfactory, while advanced non-exponential models (usually referred as “anomalous relaxation”) are commonly required to better explain experimental observations of complex systems. For viscoelastic materials, the so-called fractional derivatives models are powerful for both dynamic and loss moduli prediction. In this paper, the analysis of the main models used in the characterization of dielectric and viscoelastic materials such as five-parameter fractional Zener model and empiric Havriliak–Negami model are analysed. The fractional shape parameters describing the symmetric and asymmetric broadening of the complex modulus don't have the same influence in low and high frequencies. In contrast to the five-parameter Zener model, the empiric model asymmetry parameter has an influence on complex modulus at low frequencies comparing to the loss modulus peak frequency. A no resonance technique based on a forced vibrations procedure is used to investigate the frequency dependent complex shear modulus of a polyurethane foam, not influenced by its fluid phase, in the range 0.1–500 Hz. It is shown that the Havriliak–Negami model can predict the frequency dependence for a wide frequency range.

Analytical approach to the recovery of short fluorescence lifetimes from fluorescence decay curves

Considerable effort in instrument development has made possible detection of picosecond fluorescence lifetimes by time-correlated single-photon counting. In particular, efforts have been made to narrow markedly the instrument response function (IRF). Less attention has been paid to analytical methods, especially to problem of discretization of the convolution integral, on which the detection and quantification of short lifetimes critically depends. We show that better discretization methods can yield acceptable results for short lifetimes even with an IRF several times wider than necessary for the standard discretization based on linear approximation (LA). A general approach to discretization, also suitable for nonexponential models, is developed. The zero-time shift is explicitly included. Using simulations, we compared LA, quadratic, and cubic approximations. The latter two proved much better for detection of short lifetimes and, in that respect, they do not differ except when the zero-time shift exceeds two channels, when one can benefit from using the cubic approximation. We showed that for LA in some cases narrowing the IRF beyond FWHM = 150 ps is actually counterproductive. This is not so for quadratic and cubic approximations, which we recommend for general use.

Probing dynamical disorder kinetics with chemical relaxation techniques

A number of theoretical models describing non-exponential kinetic behavior in complex media exist. It is often difficult to discriminate between these models on the basis of their ability to provide good statistical fits to experimental data. In this work the use of chemical relaxation techniques to discrminate non-exponential models is explored. A class of simple dynamical disorder models is considered and it is shown that relaxation techniques can be used to distinguish these models from structural disorder models. In these dynamical disorder models the activation energy of the chemical process is assumed to be linearly related to a fluctuating thermodynamic variable of the system. These fluctuations alter the rate constant for the chemical process and, therefore, the integrated rate expression must be averaged to obtain the macroscopic law. Highly accurate approximations can be derived for the model using a cumulant expansion. For irreversible chemical processes, the systems' response to an external periodic perturbation is derived. This response can be used to separate parameters related to the internal fluctuations from those related to the chemical process. For chemical relaxations about equilibrium (reversible dynamical processes), it is shown that double pulse perturbations can be used to distinguish general classes of structural disorder models from dynamical disorder models. Experimental design strategies for discriminating non-exponential models are also discussed.

Monotonic and Insensitive Optimal Policies for Control of Queues with Undiscounted Costs

We consider the problem of controlling the service and/or arrival rates in queues, with the objective of minimizing the total expected cost to reach state zero. We present a unified, simple method for proving that an optimal policy is monotonic in the number of customers in the system. Applications to average-cost minimization over an infinite horizon are given. Both exponential and nonexponential models are considered; the essential characteristic is a left-skip-free transition structure and a nondecreasing (not necessarily convex) holding-cost function. Some of our results are insensitive to service-time distributions.

Decomposing the reaction time distribution: Pure insertion and selective influence revisited

This paper investigates the consequences of extending the assumptions of pure insertion and selective influence (popular in RT theorizing) to the level of the distribution. In the case of pure insertion and under the additional assumption that the additive random variable is exponentially distributed, a solution is obtained which not only allows estimation of the exponential-rate parameter but also provides a test of the assumptions. The result is shown to be applicable not only when processing is serial but also for certain parallel models. In addition, discrimination between self-terminating and exhaustive search strategies is provided, and in the case of either, both parameter estimation and tests of the model are possible. Extensions to nonexponential models are investigated and a general method of moments solution is outlined. In the case of selective influence a general nonparametric alternative to Sternberg's additive factor method is developed. The problem of empirical estimation and application is then considered. Simulations which place bounds on the type I and II error are reported. Finally the first theorem is provided an illustrative application with data from a memory scanning experiment. The results provide some support for the double assumption of pure insertion and that the additive random variable is distributed exponentially.

A Semi-Markov Model of a Three-State Generating Unit

This paper shows how the theory of semi-Markov processes can be used to obtain some important figures of merit from non-exponential failure models of generating units. The approach is used to obtain the time dependent state probabilities, the distribution of the time to transit from one state to another and the distribution of the number of visits to a given state in a fixed amount of time for a three-state model in which the occupancy times of two of the states is not exponential.

Numerical evaluation of Chapman's grazing incidence integral ch (X, χ)

A formula that is both numerically convenient to use and accurate to within 2% for all χ when X > 50 may be used to approximate ch (X, χ). The approximation may be applied to optical depth calculations in nonexponential models of the terrestrial atmosphere, the resulting errors still remaining smaller than 4% for most ionospheric applications.