The extrapolation problem and how population modeling can help

We argue that population modeling can add value to ecological risk assessment by reducing uncertainty when extrapolating from ecotoxicological observations to relevant ecological effects. We review other methods of extrapolation, ranging from application factors to species sensitivity distributions to suborganismal (biomarker and "-omics'') responses to quantitative structure activity relationships and model ecosystems, drawing attention to the limitations of each. We suggest a simple classification of population models and critically examine each model in an extrapolation context. We conclude that population models have the potential for adding value to ecological risk assessment by incorporating better understanding of the links between individual responses and population size and structure and by incorporating greater levels of ecological complexity. A number of issues, however, need to be addressed before such models are likely to become more widely used. In a science context, these involve challenges in parameterization, questions about appropriate levels of complexity, issues concerning how specific or general the models need to be, and the extent to which interactions through competition and trophic relationships can be easily incorporated.

A Contribution to the Estimation of Binary Halide and Pseudo-Halide Equilibrium Constants using a Linear Extrapolation Methodology

The molar single ion activity coefficient (y(F)) of fluoride ions was determined at 25 degrees C and ionic strengths between 0.100 and 3.00 mol L(-1) NaClO(4) using an ion-selective electrode. The activity coefficient dependency on ionic strength was determined to be Phi(F) = log y(F) = 0.2315I-0.041I(2). The function Phi(F)(I), combined with functions obtained in previous work for copper (Phi(Cu)) and hydrogen (Phi(H)), allowed us to make the estimation of the stoichiometric and thermodynamic protonation constants of some halides and pseudo-halides as well as the formation constants of some pseudo-halides and fluoride 1:1 bivalent cation complexes. The calculation procedure proposed in this paper is consistent with critically-selected experimental data. It was demonstrated that it is possible to use Phi(F)(I) for predicting the thermodynamic equilibrium parameters independently of Pearson's hardness of acids and bases.

Extrapolation of zircon fission-track annealing models

One of the purposes of this study is to give further constraints on the temperature range of the zircon partial annealing zone over a geological time scale using data from borehole zircon samples, which have experienced stable temperatures for ∼1 Ma. In this way, the extrapolation problem is explicitly addressed by fitting the zircon annealing models with geological timescale data. Several empirical model formulations have been proposed to perform these calibrations and have been compared in this work. The basic form proposed for annealing models is the Arrhenius-type model. There are other annealing models, that are based on the same general formulation. These empirical model equations have been preferred due to the great number of phenomena from track formation to chemical etching that are not well understood. However, there are two other models, which try to establish a direct correlation between their parameters and the related phenomena. To compare the response of the different annealing models, thermal indexes, such as closure temperature, total annealing temperature and the partial annealing zone, have been calculated and compared with field evidence. After comparing the different models, it was concluded that the fanning curvilinear models yield the best agreement between predicted index temperatures and field evidence. © 2012 Elsevier Ltd. All rights reserved.

Characterization of an extrapolation chamber for low-energy X-rays: Experimental and Monte Carlo preliminary results

The extrapolation chamber is a parallel-plate ionization chamber that allows variation of its air-cavity volume. In this work, an experimental study and MCNP-4C Monte Carlo code simulations of an ionization chamber designed and constructed at the Calibration Laboratory at IFEN to be used as a secondary dosimetry standard for low-energy X-rays are reported. The results obtained were within the international recommendations, and the simulations showed that the components of the extrapolation chamber may influence its response up to 11.0%. (C) 2011 Elsevier Ltd. All rights reserved.

Squared Extrapolation Methods (SQUAREM): A New Class of Simple and Efficient Numerical Schemes for Accelerating the Convergence of the EM Algorithm

We derive a new class of iterative schemes for accelerating the convergence of the EM algorithm, by exploiting the connection between fixed point iterations and extrapolation methods. First, we present a general formulation of one-step iterative schemes, which are obtained by cycling with the extrapolation methods. We, then square the one-step schemes to obtain the new class of methods, which we call SQUAREM. Squaring a one-step iterative scheme is simply applying it twice within each cycle of the extrapolation method. Here we focus on the first order or rank-one extrapolation methods for two reasons, (1) simplicity, and (2) computational efficiency. In particular, we study two first order extrapolation methods, the reduced rank extrapolation (RRE1) and minimal polynomial extrapolation (MPE1). The convergence of the new schemes, both one-step and squared, is non-monotonic with respect to the residual norm. The first order one-step and SQUAREM schemes are linearly convergent, like the EM algorithm but they have a faster rate of convergence. We demonstrate, through five different examples, the effectiveness of the first order SQUAREM schemes, SqRRE1 and SqMPE1, in accelerating the EM algorithm. The SQUAREM schemes are also shown to be vastly superior to their one-step counterparts, RRE1 and MPE1, in terms of computational efficiency. The proposed extrapolation schemes can fail due to the numerical problems of stagnation and near breakdown. We have developed a new hybrid iterative scheme that combines the RRE1 and MPE1 schemes in such a manner that it overcomes both stagnation and near breakdown. The squared first order hybrid scheme, SqHyb1, emerges as the iterative scheme of choice based on our numerical experiments. It combines the fast convergence of the SqMPE1, while avoiding near breakdowns, with the stability of SqRRE1, while avoiding stagnations. The SQUAREM methods can be incorporated very easily into an existing EM algorithm. They only require the basic EM step for their implementation and do not require any other auxiliary quantities such as the complete data log likelihood, and its gradient or hessian. They are an attractive option in problems with a very large number of parameters, and in problems where the statistical model is complex, the EM algorithm is slow and each EM step is computationally demanding.

The Birge Sponer Extrapolation

The Birge Sponer extrapolation is explained using examples.

Design of long-term animal bioassay for low-dose extrapolation using a multistage model

Conventional designs of animal bioassays allocate the same number of animals into control and dose groups to explore the spontaneous and induced tumor incidence rates, respectively. The purpose of such bioassays are (a) to determine whether or not the substance exhibits carcinogenic properties, and (b) if so, to estimate the human response at relatively low doses. In this study, it has been found that the optimal allocation to the experimental groups which, in some sense, minimize the error of the estimated response for low dose extrapolation is associated with the dose level and tumor risk. The number of dose levels has been investigated at the affordable experimental cost. The pattern of the administered dose, 1 MTD, 1/2 MTD, 1/4 MTD,....., etc. plus control, gives the most reasonable arrangement for the low dose extrapolation purpose. The arrangement of five dose groups may make the highest dose trivial. A four-dose design can circumvent this problem and has also one degree of freedom for testing the goodness-of-fit of the response model.^ An example using the data on liver tumors induced in mice in a lifetime study of feeding dieldrin (Walker et al., 1973) is implemented with the methodology. The results are compared with conclusions drawn from other studies. ^

A contribution to appendage drag extrapolation using computational tools

In this paper, the foundations of the beta method, widely used in todays ship appendage extrapolations, are explored. The present work pretends to validate the Beta Method using experimental and computational tools. The ship used is a rounded bow tugboat with two significant appendages, namely, a midship protective structure for the propulsion system and a stern keel. The experimental and computational data was obtained through Towing Tank trials and a RANSE CFD code, respectively.

Boiling Nuclear Superheater (BONUS) Power Station : supplementary study: extrapolation to large central station intergral nuclear superheat plant /

"January 28, 1960."

Method of determining the coefficient of storage from straight-line plots without extrapolation /

Mode of access: Internet.

Extrapolation, interpolation, and smoothing of stationary time series, with engineering applications.

"First published during the war as a classified report to Section D2, National Defense Research Committee."