Estimating Mean Landmark Triangles

The paper deals with the statistical estimation of mean triangles of landmark data. For the model introduced by Bookstein (1986) three methods of estimating the “ideal” triangle are compared: the maximum likelihood method based on the exact distribution given in Mardia and Dryden (1989a), a moment method and an iterative algorithm yielding a mean triangle in the sense of Fréchet. These methods are compared by Monte Carlo simulation applied also to models with variances greater than those required for Bookstein's normal approximation.     

Estimating the hidden learning representations

Successful adaptation relies on the ability to learn the consequence of our actions in different environments. However, understanding the neural bases of this ability still represents one of the great challenges of system neuroscience. In fact, the neuronal plasticity changes occurring during learning cannot be fully controlled experimentally and their evolution is hidden. Our approach is to provide hypotheses about the structure and dynamics of the hidden plasticity changes using behavioral learning theory. In fact, behavioral models of animal learning provide testable predictions about the hidden learning representations by formalizing their relation with the observables of the experiment (stimuli, actions and outcomes). Thus, we can understand whether and how the predicted learning processes are represented at the neural level by estimating their evolution and correlating them with neural data. Here, we present a bayesian model approach to estimate the evolution of the internal learning representations from the observations of the experiment (state estimation), and to identify the set of models' parameters (parameter estimation) and the class of behavioral model (model selection) that are most likely to have generated a given sequence of actions and outcomes. More precisely, we use Sequential Monte Carlo methods for state estimation and the maximum likelihood principle (MLP) for model selection and parameter estimation. We show that the method recovers simulated trajectories of learning sessions on a single-trial basis and provides predictions about the activity of different categories of neurons that should participate in the learning process. By correlating the estimated evolutions of the learning variables, we will be able to test the validity of different models of instrumental learning and possibly identify the neural bases of learning.     

Boolean Models: Maximum Likelihood Estimation from Circular Clumps

This paper deals with the problem of making inferences on the maximum radius and the intensity of the Poisson point process associated to a Boolean Model of circular primary grains with uniformly distributed random radii. The only sample information used is observed radii of circular clumps (DUPAC, 1980). The behaviour of maximum likelihood estimation has been evaluated by means of Monte Carlo methods.     

Estimating genealogies from unlinked marker data: a Bayesian approach

An issue often encountered in statistical genetics is whether, or to what extent, it is possible to estimate the degree to which individuals sampled from a background population are related to each other, on the basis of the available genotype data and some information on the demography of the population. In this article, we consider this question using explicit modelling of the pedigrees and gene flows at unlinked marker loci, but then restricting ourselves to a relatively recent history of the population, that is, considering the genealogy at most some tens of generations backwards in time. As a computational tool we use a Markov chain Monte Carlo numerical integration on the state space of genealogies of the sampled individuals. As illustrations of the method, we consider the question of relatedness at the level of genes/genomes (IBD estimation), using both simulated and real data.     

Numerical non-identifiability regions of the minimal model of glucose kinetics: superiority of Bayesian estimation

The so-called minimal model (MM) of glucose kinetics is widely employed to estimate insulin sensitivity (S(I)) both in clinical and epidemiological studies. Usually, MM is numerically identified by resorting to Fisherian parameter estimation techniques, such as maximum likelihood (ML). However, unsatisfactory parameter estimates are sometimes obtained, e.g. S(I) estimates virtually zero or unrealistically high and affected by very large uncertainty, making the practical use of MM difficult. The first result of this paper concerns the mathematical demonstration that these estimation difficulties are inherent to MM structure which can expose S(I) estimation to the risk of numerical non-identifiability. The second result is based on simulation studies and shows that Bayesian parameter estimation techniques are less sensitive, in terms of both accuracy and precision, than the Fisherian ones with respect to these difficulties. In conclusion, Bayesian parameter estimation can successfully deal with difficulties of MM identification inherently due to its structure.     

Small Sample Properties of the Likelihood Estimator in Proportional Hazard Regression Models with Omitted Variables

The paper deals with the effects of incorrectly omitted regressor variables in a parametric proportional hazard regression model. By studying conditions for equality between the estimators of correct and incorrect models it is demonstrated analytically that such cases are not to be expected in practise. A small sample Monte Carlo experiment indicates severe negative effects on the retained parameters both in terms of bias and mean square error.     

A concept of a general computer program for the iterative calculation of maximum likelihood estimators with application to qualitative dose response analyses

In this article we describe the construction of a general computer program for the iterative calculation of maximum likelihood estimators. The program is general in the sense that it allows the maximization of any given likelihood function. The user only has to write a subroutine LKLHD, in which the special likelihood function and their first and second derivatives will be calculated. This subroutine is an input parameter of the optimization program. This enables the user to employ one main program for the maximization of various likelihood functions. This advantage will be shown for the evaluation of qualitative dose response relationships (quantal assays: probit-, logit-analysis).     

Estimating rate constants from single ion channel currents when the initial distribution is known

Single ion channel currents can be analysed by hidden or aggregated Markov models. A classical result from Fredkin et al. (Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer, vol I, pp 269–289, 1985) states that the maximum number of identifiable parameters is bounded by 2n_on_c, where n_o and n_c denote the number of open and closed states, respectively. We show that this bound can be overcome when the probabilities of the initial distribution are known and the data consist of several sweeps.     

Estimating the life-span of oligodendrocytes from clonal data on their development in cell culture

This paper presents a new method to analyze clonal data on oligodendrocyte development in cell culture. The process of oligodendrocyte generation from precursor cells is modelled as a multi-type Bellman-Harris branching process as suggested in an earlier paper [K. Boucher, A. Zorin, A.Y. Yakovlev, M. Mayer-Proschel, M. Noble, An alternative stochastic model of generation of oligodendrocytes in cell culture, J. Math. Biol. 43 (2001) 22]. This model has been extended to allow for death of oligodendrocytes as well as a dissimilar distribution of the first mitotic cycle duration as compared to the subsequent cycles of precursor cells, which lengths are assumed to be independent and identically distributed random variables. Since the time-span of oligodendrocytes is not directly observable in clonal data, plausible parametric assumptions are invoked to make estimation problems tractable. In particular, the time to cell death follows a two-parameter gamma distribution, while the lapse of time between the event of cell death and the event of cell disintegration is assumed to be exponentially distributed. A simulated pseudo maximum likelihood method for estimation of model parameters has been developed using simulation-based approximations of the expected numbers and variance-covariance matrices for different types of cells. Finite sample properties of the estimation procedure are studied by computer simulations. The proposed method is illustrated with an analysis of the clonal development of O-2A progenitor cells isolated from the rat optic nerve and the corpus callosum.     

Estimating the kernel parameters of premises-based stochastic models of farmed animal infectious disease epidemics using limited, incomplete, or ongoing data

Three different estimators are presented for the types of parameters present in mathematical models of animal epidemics. The estimators make use of the data collected during an epidemic, which may be limited, incomplete, or under collection on an ongoing basis. When data are being collected on an ongoing basis, the estimated parameters can be used to evaluate putative control strategies. These estimators were tested using simulated epidemics based on a spatial, discrete-time, gravity-type, stochastic mathematical model containing two parameters. Target epidemics were simulated with the model and the three estimators were implemented using various combinations of collected data to independently determine the two parameters.     

Comparative Studies on the Estimation of Genetic Variances by Several Variance Components Estimation Method

Using a quantitative genetic model, this paper compares four different methods for estimating genetic variance components. Given various genetic parameters, data were generated and estimates computed. The number of negative estimates, the sample mean, the sample variance, and the sample mean squared error were computed for each method. It is shown that, if the genetic values are not very small, the traditional MATHER -JINKS method is at least as good as any other method. The ML method might be preferable only if the genetic values are very small and the number of loci large.     

Estimation of the mean function with panel count data using monotone polynomial splines

We study nonparametric likelihood-based estimators of the meanfunction of counting processes with panel count data using monotonepolynomial splines. The generalized Rosen algorithm, proposedby Zhang & Jamshidian (2004), is used to compute the estimators.We show that the proposed spline likelihood-based estimatorsare consistent and that their rate of convergence can be fasterthan n^1/3. Simulation studies with moderate samples show thatthe estimators have smaller variances and mean squared errorsthan their alternatives proposed by Wellner & Zhang (2000).A real example from a bladder tumour clinical trial is usedto illustrate this method.     

Test of the Inverse Monte Carlo Method for the Calculation of Interatomic Potential Energies in Atomic Liquids

Abstract

The principle purpose of this paper is to demonstrate the use of the Inverse Monte Carlo technique for calculating pair interaction energies in monoatomic liquids from a given equilibrium property. This method is based on the mathematical relation between transition probability and pair potential given by the fundamental equation of the “importance sampling” Monte Carlo method. In order to have well defined conditions for the test of the Inverse Monte Carlo method a Metropolis Monte Carlo simulation of a Lennard Jones liquid is carried out to give the equilibrium pair correlation function determined by the assumed potential. Because an equilibrium configuration is prerequisite for an Inverse Monte Carlo simulation a model system is generated reproducing the pair correlation function, which has been calculated by the Metropolis Monte Carlo simulation and therefore representing the system in thermal equilibrium. This configuration is used to simulate virtual atom displacements. The resulting changes in atom distribution for each single simulation step are inserted in a set of non-linear equations defining the transition probability for the virtual change of configuration. The solution of the set of equations for pair interaction energies yields the Lennard Jones potential by which the equilibrium configuration has been determined.     

Estimating the fraction of invariable codons with a capture-recapture method

Summary A codon-based approach to estimating the number of variable sites in a protein is presented. When first and second positions of codons are assumed to be replacement positions, a capture-recapture model can be used to estimate the number of variable codons from every pair of homologous and aligned sequences. The capture-recapture estimate is compared to a maximum likelihood estimate of the number of variable codons and to previous approaches that estimate the number of variable sites (not codons) in a sequence. Computer simulations are presented that show under which circumstances the capture-recapture estimate can be used to correct biases in distance matrices. Analysis of published sequences of two genes, calmodulin and serum albumin, shows that distance corrections that employ a capture-recapture estimate of the number of variable sites may be considerably different from corrections that assume that the number of variable sites is equal to the total number of positions in the sequence. Offprint requests to: A. Sidow