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- ItemA model for phenotype change in a stochastic framework(American Institute of Mathematical Sciences, 2008) Wake GC
Show more In some species, an inducible secondary phenotype will develop some time after the environmental change that evokes it. Nishimura (2006) [4] showed how an individual organism should optimize the time it takes to respond to an environmental change ("waiting time''). If the optimal waiting time is considered to act over the population, there are implications for the expected value of the mean fitness in that population. A stochastic predator-prey model is proposed in which the prey have a fixed initial energy budget. Fitness is the product of survival probability and the energy remaining for non-defensive purposes. The model is placed in the stochastic domain by assuming that the waiting time in the population is a normally distributed random variable because of biological variance inherent in mounting the response. It is found that the value of the mean waiting time that maximises fitness depends linearly on the variance of the waiting time.Show more - ItemLinear models with perturbed and truncated Laplace response functions: The asymptotic theory of MLE with application to epigenetics(Hindawi Publishing Corporation, 20/11/2012) Hassell-Sweatman CZW; Wake GC; Pleasants T; McLean CA; Sheppard AM
Show more The statistical application considered here arose in epigenomics, linking the DNA methylation proportions measured at specific genomic sites to characteristics such as phenotype or birth order. It was found that the distribution of errors in the proportions of chemical modification (methylation) on DNA, measured at CpG sites, may be successfully modelled by a Laplace distribution which is perturbed by a Hermite polynomial. We use a linear model with such a response function. Hence, the response function is known, or assumed well estimated, but fails to be differentiable in the classical sense due to the modulus function. Our problem was to estimate coefficients for the linear model and the corresponding covariance matrix and to compare models with varying numbers of coefficients. The linear model coefficients may be found using the (derivative-free) simplex method, as in quantile regression. However, this theory does not yield a simple expression for the covariance matrix of the coefficients of the linear model. Assuming response functions which are 2 except where the modulus function attains zero, we derive simple formulae for the covariance matrix and a log-likelihood ratio statistic, using generalized calculus. These original formulae enable a generalized analysis of variance and further model comparisons.Show more - ItemThe evolution of a truncated Gaussian probability density through time: Modelling animal liveweights after selection(Elsevier, 2003) Wake GC; Soboleva TK; Pleasants AB
Show more The form of the probability density derived from the evolution in time of a previously truncated frequency distribution of animal Liveweights is of interest in animal husbandry. Truncated frequency distributions arise when the heavier animals are sold for slaughter and the lighter animals retained. The demands of modern quality assurance schemes require that, given information on animal growth, the farmer is able to estimate the number of animals that would meet the specifications at some time in the future after truncation. Assuming that animal growth can be described by a linear stochastic differential equation, we derive an explicit expression for the probability density of animal Liveweights at any time after the truncation of an initial Gaussian density. It is shown that this probability density converges rapidly to a Gaussian density, so that after about 20 days of typical growth rates for lambs, the resulting density is practically indistinguishable from Gaussian.Show more